{"moduleDocstrings":{"FormalConjectures.Arxiv.«0911.2077».Conjecture6_3":"# Central Binomial Tail Bounds, Conjecture 6.3\n\n*Reference:* [arxiv/0911.2077](https://arxiv.org/abs/0911.2077)\n**Central Binomial Tail Bounds**\nby *Matus Telgarsky*\n","FormalConjectures.Arxiv.«0912.2382».CurlingNumberConjecture":"# The Curling Number Conjecture\n\n*Reference:* [arxiv/0912.2382](https://arxiv.org/abs/0912.2382)\n**The Curling Number Conjecture**\nby *Benjamin Chaffin and N. J. A. Sloane*\n","FormalConjectures.Arxiv.«1102.4662».AtiyahSutcliffe":"# The first Atiyah--Sutcliffe conjecture\n\nAtiyah and Sutcliffe associate a homogeneous binary polynomial to each point\nin a configuration of distinct points in Euclidean three-space. Their first\nconjecture says that these polynomials are always linearly independent.\n\n*References:*\n- M. F. Atiyah and P. M. Sutcliffe,\n  [The Geometry of Point Particles](https://doi.org/10.1098/rspa.2001.0913)\n- Marcin Mazur and Bogdan V. Petrenko,\n  [On the conjectures of Atiyah and Sutcliffe](https://arxiv.org/abs/1102.4662)\n","FormalConjectures.Arxiv.«1104.1579».CunninghamChain":"# Cunningham chains — Jones's conjecture\n\nA Cunningham chain is a sequence of primes satisfying either $p_{i+1}=2p_i+1$\n(first kind) or $p_{i+1}=2p_i-1$ (second kind). It is conjectured that there\nare infinitely many chains of every positive exact length, of both kinds.\n\nA chain has **exact length k** when its first $k$ terms are prime and the\n$(k+1)$-th generated term is composite.\n\nLenny Jones conjectures that for every positive integer $k$, infinitely many\nprimes start a chain of exact length $k$, for each of the two kinds.\n\n*References:*\n- Lenny Jones, [Polynomial Cunningham Chains](https://arxiv.org/abs/1104.1579)\n- [OEIS A181697](https://oeis.org/A181697), first-kind chain lengths\n- [OEIS A181715](https://oeis.org/A181715), second-kind chain lengths\n","FormalConjectures.Arxiv.«1308.0994».BoxdotConjecture":"# Boxdot Conjecture\n\nThe Boxdot Conjecture was originally formulated by French and Humberstone and\nhas been studied in several works. In particular, see:\n\n*References:*\n- [arxiv/1308.0994](https://arxiv.org/abs/1308.0994)\n  **Cluster Expansion and the Boxdot Conjecture** by *Emil Jeřábek*\n- [The Boxdot Conjecture and the Generalized McKinsey Axiom](https://ojs.victoria.ac.nz/ajl/article/view/4891)\n  by *Christopher Steinsvold*, Australasian Journal of Logic\n\n","FormalConjectures.Arxiv.«1601.03081».UniqueCrystalComponents":"# Unique Crystal Components\n\n*Reference:* [arxiv/1601.03081](https://arxiv.org/abs/1601.03081)\n**The Biharmonic mean**\nby *Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru*\n","FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples":"# The length of an $s$-increasing sequence of $r$-tuples\n\nThis file contains the formalisation of [GoLo21] up to and\nincluding Conjecture 1.8.\n\n*References:*\n- [arxiv/1609.08688](https://arxiv.org/abs/1609.08688)\n  **The length of an $s$-increasing sequence of $r$-tuples** by *W. T. Gowers, J. Long*\n- [GoLo21](https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/length-of-an-sincreasing-sequence-of-rtuples/7301418D47DB1ECD6BE71C20E8A98D0A)\n  **The length of an $s$-increasing sequence of $r$-tuples**\n  by *W. T. Gowers, J. Long*, Combinatorics, Probability and Computing (2021), 686-721\n","FormalConjectures.Arxiv.«2001.02665».RingelConjecture":"# Ringel's Conjecture and Kotzig's Conjecture for large $n$\n\n*Reference:* [arxiv/2001.02665](https://arxiv.org/abs/2001.02665)\n**A proof of Ringel's Conjecture**\nby *Richard Montgomery, Alexey Pokrovskiy, Benny Sudakov*\n\nThe original conjecture of Ringel (1963), which remains open, is stated in\n`Paper/RingelConjecture.lean`. The referenced paper proves it for all sufficiently large $n$.\n\nMontgomery–Pokrovskiy–Sudakov actually prove the strictly stronger statement conjectured by\nKotzig: the decomposition can always be realized by cyclic shifts of a single copy of $T$.\nKotzig's conjecture (all $n$) remains open; see `Paper/KotzigConjecture.lean`.\n","FormalConjectures.Arxiv.«2104.00502».BarkerSequence":"# Barker sequences\n\nA Barker sequence is a finite sequence of $\\pm 1$ values whose nontrivial aperiodic\nautocorrelations all have magnitude at most one. The Barker conjecture says that no such sequence\nhas length greater than $13$.\n\n*References:*\n* J. Willms, [A note on Barker sequences of even length](https://arxiv.org/abs/2104.00502)\n* [Barker code](https://en.wikipedia.org/wiki/Barker_code)\n* [OEIS A091704](https://oeis.org/A091704)\n","FormalConjectures.Arxiv.«2107.00295».IndependentDomination":"# Independent Domination of Regular Graphs, Conjecture 1.6\n\n*Reference:* [arxiv/2107.00295](https://arxiv.org/abs/2107.00295)\n**On independent domination of regular graphs**\nby *Eun-Kyung Cho, Ilkyoo Choi, Boram Park*\n","FormalConjectures.Arxiv.«2107.12475».CollatzLike":"# Digit $2$ in base $3$ representation of $2^n$\n\n*References:*\n- [Some Unconventional Problems in Number Theory](https://doi.org/10.2307/2689842)\n  by *Paul Erdös*, Mathematics Magazine 52, no. 2, p. 67, 1979\n- [arxiv/2107.12475](https://arxiv.org/abs/2107.12475)\n  **Hardness of busy beaver value BB(15)** by *Tristan Stérin, Damien Woods*\n- [Hardness of Busy Beaver Value BB(15)](https://doi.org/10.1007/978-3-031-72621-7_9)\n  by *Tristan Stérin, Damien Woods*, Reachability Problems, Lecture Notes in Computer Science\n  15050, Springer, Cham (2024)\n","FormalConjectures.Arxiv.«2208.14736».ZariskiCancellation":"# Zariski Cancellation\n\n*Reference:* [arxiv/2208.14736](https://arxiv.org/abs/2208.14736)\n**The Zariski Cancellation Problem and related problems in Affine Algebraic Geometry**\nby *Neena Gupta*\n","FormalConjectures.Arxiv.«2209.04540».SpectralSetsAndWeakTiling":"# Spectral sets and weak tiling\n\nThis file formalizes Problems 7.1 and 7.2 from Kolountzakis, Lev, and Matolcsi.\n\n*References:*\n- [KLM2023] Mihail N. Kolountzakis, Nir Lev, and Máté Matolcsi,\n  [Spectral sets and weak tiling](https://arxiv.org/abs/2209.04540).\n- [GL16] Rachel Greenfeld and Nir Lev, Spectrality and tiling by cylindric domains,\n  *Journal of Functional Analysis* 271 (2016), 2808–2821.\n- [GL20] Rachel Greenfeld and Nir Lev, Spectrality of product domains and Fuglede's conjecture\n  for convex polytopes, *Journal d'Analyse Mathématique* 140 (2020), 409–441.\n","FormalConjectures.Arxiv.«2303.01089».FurstenbergTimesPTimesQ":"# Furstenberg's `times p, times q` conjectures\n\n*Reference:* [arxiv/2303.01089](https://arxiv.org/abs/2303.01089)\n**Around Furstenberg's times $p$, times $q$ conjecture: times $p$-invariant measures\nwith some large Fourier coefficients**\nby *Catalin Badea, Sophie Grivaux*\n","FormalConjectures.Arxiv.«2402.13202».CirculantHadamard":"# The circulant Hadamard conjecture\n\nA circulant matrix is generated by cyclically shifting one row. The circulant Hadamard conjecture,\nalso attributed to Ryser, says that no circulant Hadamard matrix has order greater than $4$.\n\n*Reference:*\n* S. Steinerberger,\n  [A note on approximate Hadamard matrices](https://arxiv.org/abs/2402.13202)\n","FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS":"# An Arithmetic Sum Associated with the Classical Theta Function\n\n*Reference:* [arxiv/2501.03234](https://arxiv.org/abs/2501.03234)\n**An Arithmetic Sum Associated with the Classical Theta Function**\nby *Bruce C. Berndt, Raghavendra N. Bhat, Jeffrey L. Meyer, Likun Xie, Alexandru Zaharescu*\n","FormalConjectures.Arxiv.«2504.17644».Margulis":"# A conjecture by Margulis on matrix groups\n\n*Reference:* [arxiv/2504.17644v3](https://arxiv.org/abs/2504.17644v3)\n**Bounded diagonal orbits in homogeneous spaces over function fields**\nby *Qianlin Huang, Ronggang Shi*\n","FormalConjectures.Arxiv.«2602.05192».FirstProof4":"# First Proof, Theorem 4\n\n*Reference:* [arxiv/2602.05192v2](https://arxiv.org/abs/2602.05192v2)\n**First Proof**\nby *Mohammed Abouzaid, Andrew J. Blumberg, Martin Hairer, Joe Kileel, Tamara G. Kolda, Paul D. Nelson, Daniel Spielman, Nikhil Srivastava, Rachel Ward, Shmuel Weinberger, Lauren Williams*\n","FormalConjectures.Arxiv.«2602.05192».FirstProof6":"# First Proof, Theorem 6\n\n*Reference:* [arxiv/2602.05192v2](https://arxiv.org/abs/2602.05192v2)\n**First Proof**\nby *Mohammed Abouzaid, Andrew J. Blumberg, Martin Hairer, Joe Kileel, Tamara G. Kolda, Paul D. Nelson, Daniel Spielman, Nikhil Srivastava, Rachel Ward, Shmuel Weinberger, Lauren Williams*\n","FormalConjectures.Arxiv.«2604.08040».Conjecture5_5":"# Group structure via subgroup counts\n\n*Reference:* [arXiv:2604.08040v1](https://arxiv.org/abs/2604.08040v1)\n**Group Structure via Subgroup Counts**\nby *Angsuman Das, Hiranya Kishore Dey, Khyati Sharma* (2026)\n\nFor a finite group $G$, let $\\mathrm{cyc}(G)$ denote the number of cyclic subgroups of $G$,\nand let $t = \\pi(G)$ denote the number of distinct prime divisors of $|G|$.\n\nThe paper establishes several structural results of the form\n\"if $\\mathrm{cyc}(G)$ or $\\mathrm{sub}(G)$ is small relative to $2^t$, then $G$\nhas a strong structural property\":\n\n* $\\mathrm{cyc}(G) < 5 \\cdot 2^{t-2} \\implies G$ is nilpotent (Theorem 3.1)\n* $\\mathrm{cyc}(G) < 2^{t+1} \\implies G$ is supersolvable (Theorem 4.2)\n* $\\mathrm{sub}(G) < 59 \\cdot 2^{t-3} \\implies G$ is solvable (Theorem 5.3)\n\n**Conjecture 5.5** proposes the analogue for solvability via cyclic subgroup count:\nif $\\mathrm{cyc}(G) < 2^{t+2}$, then $G$ is solvable.\n\n* **In-Paper Location:** Conjecture 5.5, Section 5 \"Solvability of a group from $\\mathrm{sub}(G)$\"\n  ([PDF page 15](https://arxiv.org/pdf/2604.08040v1#page=15))\n* **OpenConjecture ID:** 1512\n","FormalConjectures.Arxiv.«2605.02731».DeanCycles":"# Dean's conjecture on cycles of length divisible by `k`\n\n*References:*\n- [arxiv/2605.02731](https://arxiv.org/abs/2605.02731)\n  **Existence of cycles of length divisible by 3 or 4**\n  by *Ilkyoo Choi, Hojin Chu, Ringi Kim, Boram Park*, where this is Conjecture 1.1.\n- [De88] Dean, N., Open problem, in Cycles and Rays. (1988).\n- [DeLeSa93] Dean, N. and Lesniak, L. and Saito, A., Cycles of length 0 modulo 4 in graphs.\n  Discrete Math. (1993), 133--139.\n- [ChSa94] Chen, G. and Saito, A., Graphs with a cycle of length divisible by three.\n  J. Combin. Theory Ser. B (1994), 277--292.\n","FormalConjectures.Arxiv.«2605.12342».Conjecture1":"# Fernandes' conjecture on the 2-generation of even direct product permutation groups\n\n*Reference:* [arxiv/2605.12342](https://arxiv.org/abs/2605.12342)\n**Groups of permutations that are even on maximal proper subsets, and related monoids**\nby *Vítor H. Fernandes*\n\nFor positive integers $m, n \\ge 2$, let $\\mathrm{S}_m \\times \\mathrm{S}_n$ be the direct product of\nsymmetric groups on $[m] = \\{1, \\dots, m\\}$ and $[n'] = \\{1', \\dots, n'\\}$. Define\n$$\n\\Gamma_{m \\oplus n} = \\{(\\sigma_1, \\sigma_2) \\in \\mathrm{S}_m \\times \\mathrm{S}_n :\n  \\mathrm{sgn}(\\sigma_1) = \\mathrm{sgn}(\\sigma_2)\\}\n$$\nto be the index-$2$ subgroup of pairs of permutations with equal parity, i.e., those whose\nproduct action on $[m] \\cup [n']$ is an even permutation.\n\nIt is known that $\\mathrm{rank}(\\Gamma_{2 \\oplus 2}) = 1$, $\\mathrm{rank}(\\Gamma_{3 \\oplus 3}) = 3$,\n$\\mathrm{rank}(\\Gamma_{4 \\oplus 3}) = 3$, and $\\mathrm{rank}(\\Gamma_{4 \\oplus 4}) = 3$.\n\n**Conjecture 1 (Fernandes, 2026):** For all integers $m \\ge n \\ge 2$ such that\n$(m, n) \\notin \\{(2,2), (3,3), (4,3), (4,4)\\}$, the group $\\Gamma_{m \\oplus n}$ has\nrank $2$ (i.e., is $2$-generated).\n","FormalConjectures.Arxiv.«2606.03696».BondyLongestCycles":"# Bondy's conjecture on longest cycles in highly connected graphs\n\n*References:*\n- [arxiv/2606.03696](https://arxiv.org/abs/2606.03696)\n  **Longest cycles and Dirac-type results in highly connected graphs**\n  by *Jie Ma, Bo Ning, Ziyuan Zhao*, where this is Conjecture 1.\n- [Bo80] Bondy, J. A., Longest paths and cycles in graphs of high degree. (1980).\n\nTake a `k`-connected graph with a large minimum degree. Bondy says that the graph outside any\nlongest cycle holds no long path.\n\nThe case `k = 1` is Dirac's theorem and the case `k = 2` is the theorem of Nash-Williams. The\ncase `k = 3` is proved. The cases `k ≥ 4` are open.\n","FormalConjectures.Arxiv.«2607.03582».LpRogersShephard":"# Planar $L_p$-Rogers-Shephard, the equality case\n\n*Reference:* [arxiv/2607.03582](https://arxiv.org/abs/2607.03582)\n**$L_p$-Rogers-Shephard type inequalities for $L_p$-zonoids and symmetric bodies**\nby *Matthieu Fradelizi, Auttawich Manui, Mark Meyer, Cheikh Saliou Ndiaye*\n\nCorollary 29 bounds $|K \\oplus_p -K|$ against $|K|$ for planar convex bodies with a centre of\nsymmetry containing the origin, and notes that parallelograms with a vertex at the origin\nattain it. Conjecture 5 asks whether they are the only bodies that do.\n","FormalConjectures.Arxiv.«2607.05349».MicroscopicWeighting":"# The microscopic weighting on a metric space\n\n*Reference:* [arxiv/2607.05349](https://arxiv.org/abs/2607.05349)\n**The microscopic weighting on a metric space**\nby *Emily Roff, Simon Willerton*\n","FormalConjectures.Arxiv.«2607.05739».TanArctanSum":"# Integer values of $\\tan(\\arctan 1 + \\arctan 2 + \\cdots + \\arctan n)$\n\n*References:*\n- [arxiv/2607.05739](https://arxiv.org/abs/2607.05739)\n  **Integer values of $\\tan(\\arctan 1+\\arctan 2+\\cdots+\\arctan n)$ are rare** by *Ken Ono*\n- [AMM08] T. Amdeberhan, L. A. Medina, and V. H. Moll, *Arithmetical properties of a sequence\n  arising from an arctangent sum*, J. Number Theory 128 (2008), no. 6, 1807-1846.\n- [TanArctan](https://github.com/AxiomMath/TanArctan), a Lean formalisation of the three\n  results of [Ono26], MIT licensed. Its `P`, `A`, `B` and `x` are the definitions used\n  here.\n","FormalConjectures.Arxiv.«2607.06396».AlonTarsi":"# The Alon-Tarsi short cycle cover conjecture\n\n*References:*\n- [AlTa85] Alon, N. and Tarsi, M., Covering multigraphs by simple circuits.\n  SIAM J. Algebraic Discrete Methods (1985), 345--350.\n- [arxiv/2607.06396](https://arxiv.org/abs/2607.06396)\n  **Some new results on Sylvester colorings of cubic graphs**\n  by *Luca Ferrarini, Vahan Mkrtchyan*, where this is Conjecture 4.\n\nEvery bridgeless graph has a list of cycles covering every edge whose lengths sum to at most\n$\\frac{7}{5}|E|$.\n","FormalConjectures.Arxiv.«2607.08366».MinModulus":"# Minimum modulus for the unique multiset-sum problem\n\n*References:*\n- [arxiv/2607.08366](https://arxiv.org/abs/2607.08366)\n  **Minimum modulus for the unique multiset-sum problem**\n  by *José A. R. Fonollosa*\n- [jarfo/min-modulus](https://github.com/jarfo/min-modulus), the author's Lean development of\n  the paper's Main Theorem. Section 7 of the paper describes it.\n\nThe paper's Main Theorem fixes the super-increasing set $\\{2^k - 1\\}$ and pins the least modulus\nat which *it* is valid. Conjecture 1 says no other set of $n$ residues does better, and is open.\n","FormalConjectures.Arxiv.«math.0110202».BanachMazurRotation":"# Banach-Mazur Rotation Problem\n\n*References:*\n- [arxiv/math.0110202](https://arxiv.org/abs/math/0110202)\n  **A note on Banach--Mazur problem** by *Beata Randrianantoanina*\n- [mathoverflow/41211](https://mathoverflow.net/questions/41211/easy-proof-of-the-fact-that-isotropic-spaces-are-euclidean)\n  **Easy proof of the fact that isotropic spaces are Euclidean**\n","FormalConjectures.Arxiv.«math.0608009».PoissonConjecture":"# The Poisson Conjecture\n\n*References:*\n- [AvdE07] [arxiv/math.0608009](https://arxiv.org/abs/math/0608009)\n  **On the equivalence of the Jacobian, Dixmier and Poisson Conjectures in any characteristic**\n  by *Kossivi Adjamagbo, Arno van den Essen*. Published as *A proof of the equivalence of the\n  Dixmier, Jacobian and Poisson conjectures*, Acta Math. Vietnam. 32 (2007), 205–214.\n- [BCW82] [The Jacobian conjecture: reduction of degree and formal expansion of the inverse](https://doi.org/10.1090/S0273-0979-1982-15032-7)\n  by *Hyman Bass, Edwin H. Connell, David Wright*, Bull. Amer. Math. Soc. 7 (1982), 287–330.\n- [Alp26] [Counterexample to the Jacobian conjecture](https://x.com/__alpoge__/status/2079028340955197566)\n  by *Levent Alpöge* (2026), disproving the Jacobian conjecture in dimension $3$\n\nThe Poisson Conjecture $PC_n$ ([AvdE07], Notations 5) asserts that, over a field of\ncharacteristic zero, every endomorphism of the `n`-th canonical Poisson algebra\n$P_n(K)$ — the polynomial algebra $K[X_1, \\dots, X_{2n}]$ equipped with the canonical\nPoisson bracket — is an automorphism.\n\nBy [AvdE07], Theorem 7 (the \"United Conjectures Theorem\"), for every `n` there is a chain\nof implications\n$$JC_{2n} \\Longrightarrow PC_n \\Longrightarrow DC_n \\Longrightarrow JC_n,$$\nwhere $JC_n$ is the Jacobian Conjecture in dimension `n` and $DC_n$ the Dixmier Conjecture\nfor the `n`-th Weyl algebra (the implication $DC_n \\Longrightarrow JC_n$ being classical [BCW82]).\nSince the Jacobian conjecture was disproved in dimension $3$ [Alp26] — and hence in every\ndimension $n ≥ 3$ by padding with identity coordinates — $PC_n$ is false for all $n ≥ 3$:\ncomposing the counterexample's failure of $JC_3$ through $PC_3 \\Longrightarrow DC_3 \\Longrightarrow JC_3$ refutes\n$PC_3$, and directly, the cotangent (symplectic) lift of the counterexample map\n([AvdE07], Theorem 1 in reverse) is a non-invertible Poisson endomorphism of $P_3(K)$.\nThe cases $n = 1$ and $n = 2$ remain open.\n\nIndexing note: we index the $2n$ variables of $P_n(K)$ by `Fin n ⊕ Fin n`, with\n`Sum.inl i` playing the role of $X_i$ and `Sum.inr i` the role of $X_{i+n}$ of [AvdE07],\nso that the canonical bracket reads $\\{X_i, X_{i+n}\\} = 1$; see\n`MvPolynomial.poissonBracket`.\n","FormalConjectures.Books.BorweinSineSeries":"# Convergence of the Borwein Series with Sinusoidal Coefficient\n\n*References:*\n- [MathWorld, Harmonic Series](https://mathworld.wolfram.com/HarmonicSeries.html)\n- Borwein, J.; Bailey, D.; Girgensohn, R. *Experimentation in Mathematics: Computational Paths\n  to Discovery*, A K Peters, 2004, p. 56.\n","FormalConjectures.Books.BugeaudDistributionModuloOne.IntDistanceDistribution":"# Bugeaud Collection of Conjectures and Open Questions: Fractional Parts of Powers\n\nChapter 10 of the book collects open questions. This file formalizes Problems 10.1,\n10.2, 10.3 and the unnumbered conjecture by Waldschmidt.\n\n*References:*\n  - [Bug12] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\n    Vol. 193. Cambridge University Press, 2012. Chapter 10.\n  - [Har19] Hardy, Gr H. \"A problem of Diophantine approximation.\"\n    J. Indian Math. Soc 11 (1919): 162-166.\n  - [Kok45] Koksma, J. F. \"Sur la théorie métrique des approximations diophantiques.\"\n    Indag. Math 7 (1945): 54-70.\n  - [Mah53] Mahler, Kurt. \"On the approximation of logarithms of algebraic numbers.\"\n    Philosophical Transactions of the Royal Society of London. Series A,\n    Mathematical and Physical Sciences 245.898 (1953): 371-398.\n  - [Wal03](http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Cetraro.pdf)\n    Waldschmidt, Michel. \"Linear independence measures for logarithms of algebraic numbers.\"\n    Diophantine Approximation: Lectures given at the CIME Summer School held in Cetraro, Italy,\n    June 28–July 6, 2000. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. 249-344.\n","FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_4":"# Bugeaud Collection of Conjectures and Open Questions: Spectrum of Sequence\n*References:*\n  - [Bug12] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\n    Vol. 193. Cambridge University Press, 2012. Chapter 10.\n  - [Men73] Mendès France, Michel. \"Les ensembles de Bésineau.\"\n    Séminaire Delange-Pisot-Poitou 15.1 (1973): 1-6.\n","FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_5":"# Bugeaud Collection of Conjectures and Open Questions: Lacunary Sequences in Real Number Fields\n\nThe following problems were proposed and discussed by Dubickas as Conjecture 2 in [Dub09].\n\n*References:*\n  - [Bug12] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\n    Vol. 193. Cambridge University Press, 2012. Chapter 10.\n  - [Dub09] Dubickas, Artūras. \"An approximation property of lacunary sequences.\"\n    Israel Journal of Mathematics 170.1 (2009): 95-111.\n","FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_6":"# Bugeaud Collection of Conjectures and Open Questions: Rapidly Increasing Sequences Dense Modulo One\n\n*References:*\n  - [Bos94] Boshernitzan, Michael D. \"Density modulo 1 of dilations of sublacunary sequences.\"\n    Advances in Mathematics 108.1 (1994): 104-117.\n  - [Bug12] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\n    Vol. 193. Cambridge University Press, 2012. Chapter 10.\n  - [Fur67] Furstenberg, H. \"Disjointness in ergodic theory, minimal sets, and a problem\n    in diophantine approximation\". Math. Systems Theory 1, 1–49 (1967).\n  - [Mat80] de Mathan, Bernard. \"Numbers contravening a condition in density modulo 1.\"\n    Acta Mathematica Hungarica 36.3-4 (1980): 237-241.\n  - [Pol79] Pollington, Andrew Douglas. \"On the density of sequence $\\{n_ {k}\\xi\\} $.\"\n    Illinois Journal of Mathematics 23.4 (1979): 511-515.\n","FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_7":"# Bugeaud Collection of Conjectures and Open Questions: Confined Powers of Non-Pisot Numbers\n\n*References:*\n  - [Bug12a] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\n    Vol. 193. Cambridge University Press, 2012. Chapter 10.\n  - [Bug12b] Bugeaud, Yann, and Nikolay Moshchevitin. \"On fractional parts of powers\n    of real numbers close to 1.\" Mathematische Zeitschrift 271.3 (2012): 627-637.\n","FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_8":"# Bugeaud Collection of Conjectures and Open Questions: $p$-adic Littlewood Conjecture\n\nThis is the $p$-adic analogue of the Littlewood conjecture, posed by de Mathan and\nTeulié. A liminf-based formulation also appears in the file\n`FormalConjectures/Wikipedia/LittlewoodConjecture.lean` as `padic_littlewood_conjecture`.\n\n*References:*\n  - [Bug12] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\n    Vol. 193. Cambridge University Press, 2012. Chapter 10.\n  - [dMT04] de Mathan, Bernard, and Olivier Teulié. \"Problèmes diophantiens simultanés.\"\n    Monatshefte für Mathematik 143.3 (2004): 229-245.\n  - [EK07] Einsiedler, Manfred, and Dmitry Kleinbock. \"Measure rigidity and $p$-adic\n    Littlewood-type problems.\" Compositio Mathematica 143.3 (2007): 689-702.\n","FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_9":"# Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers\n\nSee also FormalConjectures/Wikipedia/Mahler32.lean.\n\n*References:*\n  - [Bug12] Bugeaud, Yann. \"Distribution modulo one and Diophantine approximation.\"\n    Vol. 193. Cambridge University Press, 2012. Chapter 10.\n  - [Mah68] Mahler, Kurt. \"An unsolved problem on the powers of 3/2.\"\n    Journal of the Australian Mathematical Society 8.2 (1968): 313-321.\n  - [FLP95] Flatto, Leopold, Jeffrey C. Lagarias, and Andrew D. Pollington.\n    \"On the range of fractional parts $\\{\\xi(p/q)^n\\}$.\"\n    Acta Arithmetica 70.2 (1995): 125-147.\n","FormalConjectures.Books.UniformDistributionOfSequences.Equidistribution":"# Equidistributed Sequences\n\nCorollary 4.2 of Chapter 1 states that the sequence $(x^n), n = 1, 2, ... ,$ is equidistributed modulo 1 for\nalmost all x > 1. And a little bit further down:\n\"one does not know whether sequences such as $(e^n)$, $(π^n)$, or even $((\\frac 3 2)^n)$\"\nare equidistributed modulo 1 or not.\n\n*References:*\n  - [Uniform Distribution of Sequences](https://store.doverpublications.com/products/9780486149998)\nby *L. Kuipers* and *H. Niederreiter*, 1974\n  - [Wikipedia](https://en.wikipedia.org/wiki/Equidistributed_sequence)\n","FormalConjectures.ErdosProblems.«1000»":"# Erdős Problem 1000\n\n*References:*\n- [erdosproblems.com/1000](https://www.erdosproblems.com/1000)\n- [Ca50b] Cassels, J. W. S., *Some metrical theorems in Diophantine approximation. I*. Proc.\n  Cambridge Philos. Soc. (1950), 209-218.\n- [Er64b] Erdős, P., *Problems and results on diophantine approximations*. Compositio Math. (1964),\n  52-65.\n- [Ha] Haight, J. A., *Metric Diophantine approximation and related topics*. PhD thesis.\n","FormalConjectures.ErdosProblems.«1002»":"# Erdős Problem 1002\n\n*References:*\n- [erdosproblems.com/1002](https://www.erdosproblems.com/1002)\n- [Ke60] Kesten, Harry, Uniform distribution {${\\rm mod}\\,1$}. Ann. of Math. (2) (1960), 445--471.\n","FormalConjectures.ErdosProblems.«1003»":"# Erdős Problem 1003\n\n*Reference:* [erdosproblems.com/1003](https://www.erdosproblems.com/1003)\n","FormalConjectures.ErdosProblems.«1004»":"# Erdős Problem 1004\n\n*Reference:* [erdosproblems.com/1004](https://www.erdosproblems.com/1004)\n","FormalConjectures.ErdosProblems.«1007»":"# Erdős Problem 1007\n\n*References:*\n- [erdosproblems.com/1007](https://www.erdosproblems.com/1007)\n- [ChNo16] Chaffee, Joe and Noble, Matt, *Dimension 4 and dimension 5 graphs with minimum edge\n  set*. Australas. J. Combin. (2016), 327-333.\n- [Ho13] House, Roger F., *A 4-dimensional graph has at least 9 edges*. Discrete Math. (2013),\n  1783-1789.\n","FormalConjectures.ErdosProblems.«1008»":"# Erdős Problem 1008\n\n*References:*\n- [erdosproblems.com/1008](https://www.erdosproblems.com/1008)\n- [CFS14b] Conlon, D. and Fox, J. and Sudakov, B., *Large subgraphs without complete bipartite\n  graphs*. arXiv:1401.6711 (2014).\n- [Er71] Erdős, P., *Some unsolved problems in graph theory and combinatorial analysis*.\n  Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.\n","FormalConjectures.ErdosProblems.«100»":"# Erdős Problem 100\n*References:*\n* [erdosproblems.com/100](https://www.erdosproblems.com/100)\n* [Kanold](No references found)\n* [GuKa15](Guth, Larry and Katz, Nets Hawk, On the Erd\\H{o}s distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.)\n* [Piepmeyer](No references found)\n","FormalConjectures.ErdosProblems.«1014»":"# Erdős Problem 1014\n\n*References:*\n- [erdosproblems.com/1014](https://www.erdosproblems.com/1014)\n- [Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial\n  Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.\n- [OpenAI26] *On the ratio of $R(k,\\ell)$ and $R(k,\\ell+1)$*, proof due to an internal model at\n  OpenAI (2026).\n  https://cdn.openai.com/pdf/6dc7175d-d9e7-4b8d-96b8-48fe5798cd5b/Ramsey.pdf\n","FormalConjectures.ErdosProblems.«101»":"# Erdős Problem 101\n\n*Reference:* [erdosproblems.com/101](https://www.erdosproblems.com/101)\n","FormalConjectures.ErdosProblems.«1020»":"# Erdős Problem 1020\n\n*References:*\n- [erdosproblems.com/1020](https://www.erdosproblems.com/1020)\n- [BDE76] Bollobás, B. and Daykin, D. E. and Erdős, P., *Sets of independent edges of a hypergraph*.\n  Quart. J. Math. Oxford Ser. (2) (1976), 25--32.\n- [Er65d] Erdős, P., *A problem on independent {$r$}-tuples*. Ann. Univ. Sci. Budapest. Eötvös Sect.\n  Math. (1965), 93--95.\n- [ErGa59] Erdős, P. and Gallai, T., *On maximal paths and circuits of graphs*. Acta Math. Acad.\n  Sci. Hungar. (1959), 337-356 (unbound insert).\n- [FLM12] Frankl, Peter and Łuczak, Tomasz and Mieczkowska, Katarzyna, *On matchings in\n  hypergraphs*. Electron. J. Combin. (2012), Paper 42, 5.\n- [FRR12] Frankl, Peter and Rödl, Vojtech and Ruciński, Andrzej, *On the maximum number of edges in\n  a triple system not containing a disjoint family of a given size*. Combin. Probab. Comput. (2012),\n  141--148.\n- [Fr17] Frankl, Peter, *Proof of the {E}rdős matching conjecture in a new range*. Israel J. Math.\n  (2017), 421--430.\n- [Fr87] Frankl, Peter, *The shifting technique in extremal set theory*. (1987), 81--110.\n- [HLS12] Huang, Hao and Loh, Po-Shen and Sudakov, Benny, *The size of a hypergraph and its matching\n  number*. Combin. Probab. Comput. (2012), 442--450.\n- [Kl68] Kleitman, Daniel J., *Maximal number of subsets of a finite set no {$k$} of which are\n  pairwise disjoint*. J. Combinatorial Theory (1968), 157--163.\n- [KoKu23] Kolupaev, Dmitriy and Kupavskii, Andrey, *Erdős matching conjecture for almost perfect\n  matchings*. Discrete Math. (2023), Paper No. 113304, 9.\n- [LuMi14] Łuczak, Tomasz and Mieczkowska, Katarzyna, *On {E}rdős' extremal problem on matchings in\n  hypergraphs*. J. Combin. Theory Ser. A (2014), 178--194.\n","FormalConjectures.ErdosProblems.«1022»":"# Erdős Problem 1022\n\n*References:*\n- [erdosproblems.com/1022](https://www.erdosproblems.com/1022)\n- [Er71] Erdős, P., *Some unsolved problems in graph theory and combinatorial analysis*.\n  Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.\n- [Lo68] Lovász, L., *On covering of graphs*. Theory of Graphs (Proc. Colloq., Tihany, 1966)\n  (1968), 231-236.\n- [Wo13b] Wood, D. R., *Hypergraph colouring and degeneracy*. arXiv:1310.2972 (2013).\n","FormalConjectures.ErdosProblems.«1023»":"# Erdős Problem 1023\n\n*References:*\n- [erdosproblems.com/1023](https://www.erdosproblems.com/1023)\n- [Er71, p.105] Erdős, P., *Some unsolved problems in graph theory and combinatorial analysis*.\n  Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.\n","FormalConjectures.ErdosProblems.«1026»":"# Erdős Problem 1026\n\n*References:*\n- [erdosproblems.com/1026](https://www.erdosproblems.com/1026)\n- [Er71] Erdős, P., *Some unsolved problems in graph theory and combinatorial analysis*.\n  Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.\n- [Ha57] Hanani, Haim, *On the number of monotonic subsequences*. Bull. Res. Council Israel\n  Sect. F (1957/58), 11-13.\n- [St95] Steele, J. Michael, *Variations on the monotone subsequence theme of Erdős and\n  Szekeres*. (1995), 111-131.\n- [TWY16] Tidor, J. and Wang, V. and Yang, B., *$1$-color avoiding paths, special tournaments,\n  and incidence geometry*. arXiv:1608.04153 (2016).\n- [Wa17] Wagner, Adam Zsolt, *Large subgraphs in rainbow-triangle free colorings*. J. Graph\n  Theory (2017), 141-148.\n","FormalConjectures.ErdosProblems.«1028»":"# Erdős Problem 1028\n\n*References:*\n- [erdosproblems.com/1028](https://www.erdosproblems.com/1028)\n- [Er63d] Erdős, Pál, *On combinatorial questions connected with a theorem of Ramsey and van der\n  Waerden*. Mat. Lapok (1963), 29-37.\n- [Er71] Erdős, P., *Some unsolved problems in graph theory and combinatorial analysis*.\n  Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.\n- [ErSp71] Erdős, P. and Spencer, J., *Imbalances in $k$-colorations*. Networks (1971/72),\n  379-385.\n","FormalConjectures.ErdosProblems.«1029»":"# Erdős Problem 1029\n\n*References:*\n- [erdosproblems.com/1029](https://www.erdosproblems.com/1029)\n- [Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory.\n  Quaestiones Math. (1993), 333-350.\n- [ErSz35] Erdős, P. and Szekeres, G., A combinatorial problem in geometry. Compos. Math. (1935),\n  463-470.\n- [Sp75] Spencer, J., Ramsey's theorem - a new lower bound. J. Combin. Theory Ser. A (1975),\n  108-115.\n","FormalConjectures.ErdosProblems.«1030»":"# Erdős Problem 1030\n\n*References:*\n- [erdosproblems.com/1030](https://www.erdosproblems.com/1030)\n- [BEFS89] Burr, S. and Erdős, P. and Faudree, R. J. and Schelp, R. H., On the difference between\n  consecutive Ramsey numbers. Utilitas Math. (1989), 115-118.\n","FormalConjectures.ErdosProblems.«1034»":"# Erdős Problem 1034\n\n*References:*\n- [erdosproblems.com/1034](https://www.erdosproblems.com/1034)\n- [Er93] Erdős, Paul, *Some of my favorite solved and unsolved problems in graph theory*.\n  Quaestiones Math. (1993), 333-350.\n- [MaTa25] Ma, Jie and Tang, Quanyu, *On Erdős problem #1034*.\n  [staff.ustc.edu.cn/~jiema/Erdos-1034.pdf](http://staff.ustc.edu.cn/~jiema/Erdos-1034.pdf)\n","FormalConjectures.ErdosProblems.«1035»":"# Erdős Problem 1035\n\n*References:*\n- [erdosproblems.com/1035](https://www.erdosproblems.com/1035)\n- [Er93] Erdős, Paul, *Some of my favorite solved and unsolved problems in graph theory*.\n  Quaestiones Math. (1993), 333-350.\n","FormalConjectures.ErdosProblems.«1036»":"# Erdős Problem 1036\n\n*References:*\n- [erdosproblems.com/1036](https://www.erdosproblems.com/1036)\n- [AlHa91] Alon, N. and Hajnal, A., *Ramsey graphs contain many distinct induced subgraphs*.\n  Graphs Combin. (1991), 1-6.\n- [ErHa89b] Erdős, P. and Hajnal, A., *On the number of distinct induced subgraphs of a graph*.\n  Discrete Math. (1989), 145-154.\n- [Sh98] Shelah, Saharon, *Erdős and Rényi conjecture*. J. Combin. Theory Ser. A (1998), 179-185.\n","FormalConjectures.ErdosProblems.«1037»":"# Erdős Problem 1037\n\n*References:*\n- [erdosproblems.com/1037](https://www.erdosproblems.com/1037)\n- [Er93] Erdős, Paul, *Some of my favorite solved and unsolved problems in graph theory*.\n  Quaestiones Math. (1993), 333-350.\n","FormalConjectures.ErdosProblems.«1038»":"# Erdős Problem 1038\n\n*Reference:*\n - [erdosproblems.com/1038](https://www.erdosproblems.com/1038)\n - [Tao25] Tao, Terence. Sublevel Sets of Logarithmic Potentials. Terry Tao’s Blog, Dec. 2025\n  (https://terrytao.wordpress.com/wp-content/uploads/2025/12/erdos-1038-1.pdf)\n","FormalConjectures.ErdosProblems.«1041»":"# Erdős Problem 1041\n\n*Reference:* [erdosproblems.com/1041](https://www.erdosproblems.com/1041)\n","FormalConjectures.ErdosProblems.«1043»":"# Erdős Problem 1043\n\n*References:*\n- [erdosproblems.com/1043](https://www.erdosproblems.com/1043)\n- [EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J.\n  Analyse Math. (1958), 125-148.\n- [Po59] Pommerenke, Ch., On some problems by Erdős, Herzog and Piranian. Michigan Math. J.\n  (1959), 221-225.\n- [Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961),\n  97-115.\n","FormalConjectures.ErdosProblems.«1044»":"# Erdős Problem 1044\n\n*References:*\n- [erdosproblems.com/1044](https://www.erdosproblems.com/1044)\n- [EHP58] Erdős, P. and Herzog, F. and Piranian, G., *Metric properties of polynomials*.\n  J. Analyse Math. (1958), 125-148.\n- [Ta26] Tang, Quanyu, *On Erdős Problem 1044* (2026),\n  [github.com/QuanyuTang/erdos-problem-1044](https://github.com/QuanyuTang/erdos-problem-1044).\n","FormalConjectures.ErdosProblems.«1047»":"# Erdős Problem 1047\n\n*References:*\n- [erdosproblems.com/1047](https://www.erdosproblems.com/1047)\n- [EHP58] Erdős, P. and Herzog, F. and Piranian, G., *Metric properties of polynomials*.\n  J. Analyse Math. (1958), 125-148.\n- [Go66] Goodman, A. W., *On the convexity of the level curves of a polynomial*.\n  Proc. Amer. Math. Soc. (1966), 358-361.\n- [Po61] Pommerenke, Ch., *On metric properties of complex polynomials*. Michigan Math. J.\n  (1961), 97-115.\n","FormalConjectures.ErdosProblems.«1048»":"# Erdős Problem 1048\n\n*References:*\n- [erdosproblems.com/1048](https://www.erdosproblems.com/1048)\n- [EHP58] Erdős, P. and Herzog, F. and Piranian, G., *Metric properties of polynomials*.\n  J. Analyse Math. (1958), 125-148.\n- [Po61] Pommerenke, Ch., *On metric properties of complex polynomials*. Michigan Math. J.\n  (1961), 97-115.\n","FormalConjectures.ErdosProblems.«1049»":"# Erdős Problem 1049\n\n*References:*\n- [erdosproblems.com/1049](https://www.erdosproblems.com/1049)\n- [Er48] Erdős, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.)\n  (1948), 63-66.\n","FormalConjectures.ErdosProblems.«104»":"# Erdős Problem 104\n\n*References:*\n- [erdosproblems.com/104](https://www.erdosproblems.com/104)\n- [El84] Elekes, G., *{$n$} points in the plane can determine $n^{3/2}$ unit circles*. Combinatorica\n  (1984), 131.\n- [Er75h] Erdős, P., *Some problems on elementary geometry*. Austral. Math. Soc. Gaz. (1975), 2-3.\n- [Er81d] Erdős, P., *Some applications of graph theory and combinatorial methods to number theory\n  and geometry*. Algebraic methods in graph theory, Vol. I, II (Szeged, 1978) (1981), 137-148.\n- [Er92e] Erdős, Pál, *Some Unsolved problems in Geometry, Number Theory and Combinatorics*. Eureka\n  (1992), 44-48.\n- [HaMe86] Harborth, Heiko and Mengersen, Ingrid, *Point sets with many unit circles*. Discrete\n  Math. (1986), 193--197.\n","FormalConjectures.ErdosProblems.«1050»":"# Erdős Problem 1050\n\n*References:* \n- [erdosproblems.com/1050](https://www.erdosproblems.com/1050)\n- [Bo91] Borwein, Peter B., On the irrationality of {$\\sum(1/(q^n+r))$}. J. Number Theory (1991), 253--259.\n- [Bo92] Borwein, Peter B., On the irrationality of certain series. Math. Proc. Camb. Phil. Soc. (1992), 141--146.\n- [Er48] Erdős, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.\n- [Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\n\nIs $\\sum_{n=1}^\\infty \\frac{1}{2^n - 3}$ irrational?\n\nThe answer is **yes**, proved by P. B. Borwein [Bo91] (with a cleaner self-contained proof in [Bo92]),\nspecialized to $q = 2$, $r = -3$.\n\nA formal Lean proof is given in an external repository,\n[`gotrevor/lean-gallery`](https://github.com/gotrevor/lean-gallery), formalized by Trevor Morris with\nClaude Code and Harmonic's Aristotle.\n","FormalConjectures.ErdosProblems.«1051»":"# Erdős Problem 1051\n\n*References:*\n- [erdosproblems.com/1051](https://www.erdosproblems.com/1051)\n- [BKKKZ26] K. Barreto, J. Kang, S.-H. Kim, V. Kovač, and S. Zhang, Irrationality of rapidly\n  converging series: a problem of Erdős and Graham. arXiv:2601.21442 (2026).\n- [Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in\n  transcendence theory (Durham, 1986) (1988), 102-109.\n- [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\n  theory. Monographies de L'Enseignement Mathematique (1980).\n- [Fe26] T. Feng et al, Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős\n  Problems. arXiv:2601.22401 (2026).\n","FormalConjectures.ErdosProblems.«1052»":"# Erdős Problem 1052\n\n*Reference:* [erdosproblems.com/1052](https://www.erdosproblems.com/1052)\n","FormalConjectures.ErdosProblems.«1054»":"# Erdős Problem 1054\n\n*Reference:* [erdosproblems.com/1054](https://www.erdosproblems.com/1054)\n","FormalConjectures.ErdosProblems.«1055»":"# Erdős Problem 1055\n\n*Reference:* [erdosproblems.com/1055](https://www.erdosproblems.com/1055)\n","FormalConjectures.ErdosProblems.«1056»":"# Erdős Problem 1056\n\n*Reference:* [erdosproblems.com/1056](https://www.erdosproblems.com/1056)\n","FormalConjectures.ErdosProblems.«1057»":"# Erdős Problem 1057\n\n*References:*\n- [erdosproblems.com/1057](https://www.erdosproblems.com/1057)\n- [AGP94] Alford, W. R. and Granville, Andrew and Pomerance, Carl, There are infinitely many\n  Carmichael numbers. Ann. of Math. (2) (1994), 703--722.\n- [Er56c] Erdős, P., On pseudoprimes and Carmichael numbers. Publ. Math. Debrecen (1956),\n  201--206.\n- [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n- [Ha08] Harman, Glyn, Watt's mean value theorem and Carmichael numbers. Int. J. Number Theory\n  (2008), 241--248.\n- [Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes\n  without large prime factors. arXiv:2211.09641 (2022).\n- [Po89] Pomerance, Carl, Two methods in elementary analytic number theory. (1989), 135--161.\n","FormalConjectures.ErdosProblems.«1059»":"# Erdős Problem 1059\n\n*Reference:* [erdosproblems.com/1059](https://www.erdosproblems.com/1059)\n","FormalConjectures.ErdosProblems.«105»":"# Erdős Problem 105\n\n*References:*\n- [erdosproblems.com/105](https://www.erdosproblems.com/105)\n- [Be83] Beck, József, *On the lattice property of the plane and some problems of Dirac, Motzkin\n  and Erdős in combinatorial geometry*. Combinatorica (1983), 281-297.\n- [ErPu95] Erdős, P. and Purdy, G., *Two combinatorial problems in the plane*. Discrete Comput.\n  Geom. (1995), 441-443.\n- [SzTr83] Szemerédi, Endre and Trotter, Jr., William T., *Extremal problems in discrete\n  geometry*. Combinatorica (1983), 381-392.\n","FormalConjectures.ErdosProblems.«1060»":"# Erdős Problem 1060\n\n*Reference:* [erdosproblems.com/1060](https://www.erdosproblems.com/1060)\n","FormalConjectures.ErdosProblems.«1061»":"# Erdős Problem 1061\n\n*References:*\n - [erdosproblems.com/1061](https://www.erdosproblems.com/1061)\n - [Gu04] Guy, Richard K., _Unsolved problems in number theory_. (2004), Problem B15.\n","FormalConjectures.ErdosProblems.«1062»":"# Erdős Problem 1062\n\n*Reference:* [erdosproblems.com/1062](https://www.erdosproblems.com/1062)\n","FormalConjectures.ErdosProblems.«1063»":"# Erdős Problem 1063\n\n*References:*\n * [erdosproblems.com/1063](https://www.erdosproblems.com/1063)\n * [ErSe83] Erdos, P. and Selfridge, J. L., Problem 6447. Amer. Math. Monthly (1983), 710.\n * [Gu04] Guy, Richard K., _Unsolved problems in number theory_. (2004), Problem B31.\n * [Mo85] Monier, Jean-Marie, _Problems and Solutions: Solutions of Advanced Problems: 6447_.\n   Amer. Math. Monthly **92** (1985), 435-436.\n","FormalConjectures.ErdosProblems.«1064»":"# Erdős Problem 1064\n\n*Reference:* [erdosproblems.com/1064](https://www.erdosproblems.com/1064)\n","FormalConjectures.ErdosProblems.«1065»":"# Erdős Problem 1065\n\n*Reference:* [erdosproblems.com/1065](https://www.erdosproblems.com/1065)\n","FormalConjectures.ErdosProblems.«1067»":"# Erdős Problem 1067\n\n*References:*\n- [erdosproblems.com/1067](https://www.erdosproblems.com/1067)\n- [BoPi24] N. Bowler and M. Pitz, A note on uncountably chromatic graphs. arXiv:2402.05984 (2024).\n- [ErHa66] Erdős, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Math. Acad.\n  Sci. Hungar. (1966), 61-99.\n- [Ko13] Komjáth, Péter, A note on chromatic number and connectivity of infinite graphs. Israel\n  J. Math. (2013), 499--506.\n- [So15] Soukup, Dániel T., Trees, ladders and graphs. J. Combin. Theory Ser. B (2015), 96--116.\n- [Th17] Thomassen, Carsten, Infinitely connected subgraphs in graphs of uncountable chromatic\n  number. Combinatorica (2017), 785--793.\n","FormalConjectures.ErdosProblems.«1068»":"# Erdős Problem 1068\n\n*Reference:* [erdosproblems.com/1068](https://www.erdosproblems.com/1068)\n","FormalConjectures.ErdosProblems.«1071»":"# Erdős Problem 1071\n\n*References:*\n* [erdosproblems.com/1071](https://www.erdosproblems.com/1071)\n* [Da85] Danzer, L., _Some combinatorial and metric problems in geometry_.\n  Intuitive geometry (Siófok, 1985), 167-177.\n","FormalConjectures.ErdosProblems.«1072»":"# Erdős Problem 1072\n\n*Reference:* [erdosproblems.com/1072](https://www.erdosproblems.com/1072)\n","FormalConjectures.ErdosProblems.«1073»":"# Erdős Problem 1073\n\n*Reference:* [erdosproblems.com/1073](https://www.erdosproblems.com/1073)\n","FormalConjectures.ErdosProblems.«1074»":"# Erdős Problem 1074\n\n*Reference:* [erdosproblems.com/1074](https://www.erdosproblems.com/1074)\n","FormalConjectures.ErdosProblems.«1077»":"# Erdős Problem 1077\n\n*Reference:* [erdosproblems.com/1077](https://www.erdosproblems.com/1077)\n","FormalConjectures.ErdosProblems.«107»":"# Erdős Problem 107\n\n*References:*\n- [erdosproblems.com/107](https://www.erdosproblems.com/107)\n- [Wikipedia](https://en.wikipedia.org/wiki/Happy_ending_problem)\n","FormalConjectures.ErdosProblems.«1080»":"# Erdős Problem 1080\n\n*References:*\n- [erdosproblems.com/1080](https://www.erdosproblems.com/1080)\n- [DeSz92] de Caen, D. and Székely, L. A., The maximum size of {$4$}- and {$6$}-cycle free bipartite\n  graphs on {$m,n$} vertices. (1992), 135--142.\n- [Er75] Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975),\n  3-14.\n- [LUW94] Lazebnik, F. and Ustimenko, V. A. and Woldar, A. J., New constructions of bipartite graphs\n  on {$m,n$} vertices with many edges and without small cycles. J. Combin. Theory Ser. B (1994),\n  111--117.\n","FormalConjectures.ErdosProblems.«1082»":"# Erdős Problem 1082\n\n*Reference:* [erdosproblems.com/1082](https://www.erdosproblems.com/1082)\n","FormalConjectures.ErdosProblems.«1083»":"# Erdős Problem 1083\n\n*References:*\n- [erdosproblems.com/1083](https://www.erdosproblems.com/1083)\n- [APST04] Aronov, Boris and Pach, János and Sharir, Micha and Tardos, Gábor, *Distinct distances in\n  three and higher dimensions*. Combin. Probab. Comput. (2004), 283--293.\n- [CEGSW90] Clarkson, Kenneth L. and Edelsbrunner, Herbert and Guibas, Leonidas J. and Sharir, Micha\n  and Welzl, Emo, *Combinatorial complexity bounds for arrangements of curves and spheres*. Discrete\n  Comput. Geom. (1990), 99--160.\n- [Er46b] Erdős, P., *On sets of distances of {$n$} points*. Amer. Math. Monthly (1946), 248--250.\n- [SoVu08] Solymosi, József and Vu, Van H., *Near optimal bounds for the {E}rdős distinct distances\n  problem in high dimensions*. Combinatorica (2008), 113--125.\n","FormalConjectures.ErdosProblems.«1084»":"# Erdős Problem 1084\n\n*Reference:* [erdosproblems.com/1084](https://www.erdosproblems.com/1084)\n\nLet `f_2(n)` be the maximum number of pairs of points at distance exactly `1`\namong any set of `n` points in `ℝ²`, under the condition that all pairwise\ndistances are at least `1`.\n\nEstimate the growth of `f_2(n)`.\n\nStatus: open.\n","FormalConjectures.ErdosProblems.«1085»":"# Erdős Problem 1085\n\nLet f_d(n) be minimal such that, in any set of n points in ℝ^d, there exist at most f_d(n) pairs\nof points which are distance 1 apart. Estimate f_d(n).\n\n*Reference:* [erdosproblems.com/1085](https://www.erdosproblems.com/1085)\n","FormalConjectures.ErdosProblems.«1088»":"# Erdős Problem 1088\n\n*Reference:* [erdosproblems.com/1088](https://www.erdosproblems.com/1088)\n","FormalConjectures.ErdosProblems.«108»":"# Erdős Problem 108\n\n*Reference:* [erdosproblems.com/108](https://www.erdosproblems.com/108)\n","FormalConjectures.ErdosProblems.«1090»":"# Erdős Problem 1090\n\n*References:*\n- [erdosproblems.com/1090](https://www.erdosproblems.com/1090)\n- [Er75f] Erdős, Paul, *On some problems of elementary and combinatorial geometry*. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n","FormalConjectures.ErdosProblems.«1092»":"# Erdős Problem 1092\n\n*References:*\n- [Erdős Problem 1092](https://www.erdosproblems.com/1092)\n- [Ro82] V. Rödl, *On the chromatic number of subgraphs of a given graph*, Proc. Amer. Math. Soc. **85** (1982), 382–386\n","FormalConjectures.ErdosProblems.«1093»":"# Erdős Problem 1093\n\n*Reference:* [erdosproblems.com/1093](https://www.erdosproblems.com/1093)\n","FormalConjectures.ErdosProblems.«1094»":"# Erdős Problem 1094\n\n*Reference:* [erdosproblems.com/1094](https://www.erdosproblems.com/1094)\n","FormalConjectures.ErdosProblems.«1095»":"# Erdős Problem 1095\n\n*References:*\n- [erdosproblems.com/1095](https://www.erdosproblems.com/1095)\n- [EES74] Ecklund, Jr., E. F. and Erd\\H{o}s, P. and Selfridge, J. L., A new function associated with\n  the prime factors of {$(\\sp{n}\\sb{k})$}. Math. Comp. (1974), 647--649.\n- [ELS93] Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor\n  of a binomial coefficient. Math. Comp. (1993), 215--224.\n- [GrRa96] Granville, Andrew and Ramaré, Olivier, Explicit bounds on exponential sums and the\n  scarcity of squarefree binomial coefficients. Mathematika (1996), 73--107.\n- [Ko99b] Konyagin, S. V., Estimates of the least prime factor of a binomial coefficient.\n  Mathematika (1999), 41--55.\n- [SSW20] Sorenson, Brianna and Sorenson, Jonathan and Webster, Jonathan, An algorithm and estimates\n  for the {E}rdős-{S}elfridge function. (2020), 371--385.\n","FormalConjectures.ErdosProblems.«1096»":"# Erdős Problem 1096\n\n*References:*\n- [erdosproblems.com/1096](https://www.erdosproblems.com/1096)\n- [ErKo98] Erdős, P. and Komornik, V., Developments in non-integer bases.\n  Acta Math. Hungar. (1998), 57--83.\n- [Fe16] Feng, D.-J., On the topology of polynomials with bounded integer coefficients.\n  J. Eur. Math. Soc. (2016), 181--193.\n","FormalConjectures.ErdosProblems.«1097»":"# Erdős Problem 1097\n\n*References:*\n- [erdosproblems.com/1097](https://www.erdosproblems.com/1097)\n- [Bo99] Bourgain, J., On the dimension of {K}akeya sets and related maximal\ninequalities. Geom. Funct. Anal. (1999), 256--282\n- [KaTa99] Katz, Nets Hawk and Tao, Terence, Bounds on arithmetic projections, and applications to the\n{K}akeya conjecture. Math. Res. Lett. (1999), 625--630.\n- [Le15] Lemm, Marius, New counterexamples for sums-differences. Proc. Amer. Math. Soc. (2015), 3863--3868.\n- [GGTW25] B. Georgiev, J. Gómez-Serrano, T. Tao, and A. Wagner, Mathematical exploration and discovery at scale. arXiv:2511.02864 (2025).\n","FormalConjectures.ErdosProblems.«1098»":"# Erdős Problem 1098\n\n*References:*\n- [erdosproblems.com/1098](https://www.erdosproblems.com/1098)\n- [Ne76] Neumann, B. H., *A problem of Paul Erdős on groups*. J. Austral. Math. Soc. Ser. A (1976),\n  467-472.\n","FormalConjectures.ErdosProblems.«109»":"# Erdős Problem 109\n\n*References:*\n- [erdosproblems.com/109](https://www.erdosproblems.com/109)\n- [MRR19] J. Moreira, F.K. Richter, and D. Robertson, A proof of a sumset conjecture of Erdős,\n  Annals of Math. 189 (2019), 605-652.\n","FormalConjectures.ErdosProblems.«10»":"# Erdős Problem 10\n\n*Reference:* [erdosproblems.com/10](https://www.erdosproblems.com/10)\n","FormalConjectures.ErdosProblems.«1101»":"# Erdős Problem 1101\n\n*Reference:* [erdosproblems.com/1101](https://www.erdosproblems.com/1101)\n","FormalConjectures.ErdosProblems.«1102»":"# Erdős Problem 1102\n\n*Reference:* [erdosproblems.com/1102](https://www.erdosproblems.com/1102)\n","FormalConjectures.ErdosProblems.«1104»":"# Erdős Problem 1104\n\n*Reference:* https://www.erdosproblems.com/1104\n","FormalConjectures.ErdosProblems.«1105»":"# Erdős Problem 1105\n\n*References:*\n- [erdosproblems.com/1105](https://www.erdosproblems.com/1105)\n- [ESS75] Erdős, P. and Simonovits, M. and Sós, V. T., Anti-{R}amsey theorems. (1975), 633--643.\n- [MoNe05] Montellano-Ballesteros, J. J. and Neumann-Lara, V., An anti-{R}amsey theorem on cycles.\n  Graphs Combin. (2005), 343--354.\n- [SiSo84] Simonovits, Miklós and Sós, Vera T., On restricted colourings of {$K_n$}. Combinatorica\n  (1984), 101--110.\n- [Yu21] L.-T. Yuan, The anti-Ramsey number for paths. arXiv:2102.00807 (2021).\n","FormalConjectures.ErdosProblems.«1106»":"# Erdős Problem 1106\n\n*Reference:* [erdosproblems.com/1064](https://www.erdosproblems.com/1106)\n","FormalConjectures.ErdosProblems.«1107»":"# Erdős Problem 1107\n\n*References:*\n- [erdosproblems.com/1107](https://www.erdosproblems.com/1107)\n- [He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988)\n","FormalConjectures.ErdosProblems.«1108»":"# Erdős Problem 1108\n\n*Reference:* [erdosproblems.com/1108](https://www.erdosproblems.com/1108)\n","FormalConjectures.ErdosProblems.«1109»":"# Erdős Problem 1109\n\n*References:*\n- [erdosproblems.com/1109](https://www.erdosproblems.com/1109)\n- [ErSa87] P. Erdős and A. Sárközy, *On divisibility properties of integers of the form\n  $a+a'$*, Acta Math. Hungar. (1987), 117--122.\n- [Gy01] Katalin Gyarmati, *On divisibility properties of integers of the form $ab+1$*,\n  Period. Math. Hungar. (2001), 71--79.\n- [Ko04] S. V. Konyagin, *Problems of the set of square-free numbers*,\n  Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63--90.\n- [Sa92c] G. N. Sárközy, *On a problem of P. Erdős*, Acta Math. Hungar.\n  (1992), 271--282.\n","FormalConjectures.ErdosProblems.«1110»":"# Erdős Problem 1110\n\n*Reference:* [Erdős Problem 1110](https://www.erdosproblems.com/1110)\n","FormalConjectures.ErdosProblems.«1113»":"# Erdős Problem 1113\n\n*References:*\n- [erdosproblems.com/1113](https://www.erdosproblems.com/1113)\n- [ErGr80] Erdős, P. and Graham, R. L., Old and New Problems and Results in Combinatorial\n  Number Theory. Monographie de l'Enseignement Mathématique, No. 28 (1980).\n- [Si60] Sierpiński, W., Elementary Theory of Numbers. Państwowe Wydawnictwo Naukowe,\n  Warsaw (1960).\n- [FFK08] Filaseta, M., Finch, C., and Kozek, M., On powers associated with Sierpiński numbers,\n  Riesel numbers and Polignac's conjecture. Journal of Number Theory 128 (2008), 1916–1940.\n\nA positive odd integer $k$ is a *Sierpiński number* if $k \\cdot 2^n + 1$ is composite for all\n$n \\geq 0$. A *covering set* for $k$ is a finite set of primes $P$ such that every number of\nthe form $k \\cdot 2^n + 1$ is divisible by at least one prime in $P$.\n\nSierpiński (1960) proved that infinitely many Sierpiński numbers exist using covering systems.\nThe smallest known Sierpiński number is 78557 (Selfridge). Erdős and Graham conjectured that\nthere exist Sierpiński numbers with no finite covering set. A negative answer would imply\ninfinitely many Fermat primes.\n\nNote: The notion of a covering set for a Sierpiński number is closely related to a\n`CoveringSystem` of $\\mathbb{Z}$ (see\n`FormalConjecturesForMathlib.NumberTheory.CoveringSystem`): a finite covering set of primes\nfor $k$ works because the exponents $n$ for which each prime divides $k \\cdot 2^n + 1$ form\nresidue classes whose union covers all of $\\mathbb{Z}$, i.e. a covering system.\n\nSee also Erdős Problems [203](https://www.erdosproblems.com/203) and\n[276](https://www.erdosproblems.com/276).\n","FormalConjectures.ErdosProblems.«1119»":"# Erdős Problem 1119\n\n*References:*\n- [erdosproblems.com/1119](https://www.erdosproblems.com/1119)\n- [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n- [Er64g] Erdős, P., An interpolation problem associated with the continuum hypothesis.\n  Michigan Math. J. (1964), 9--10.\n- [KuSh17] Kumar, Ashutosh and Shelah, Saharon, On a question about families of entire\n  functions. Fund. Math. (2017), 279--288.\n- [ScWe24] Schilhan, Jonathan and Weinert, Thilo, Wetzel families and the continuum.\n  J. Lond. Math. Soc. (2) (2024), Paper No. e12918, 27.\n","FormalConjectures.ErdosProblems.«1121»":"# Erdős Problem 1121\n\n*References:*\n- [erdosproblems.com/1121](https://www.erdosproblems.com/1121)\n- [BeLi16] Bezdek, Károly and Litvak, Alexander E., *Packing convex bodies by cylinders*.\n  Discrete Comput. Geom. (2016), 725-738.\n- [GoGo45] Goodman, A. W. and Goodman, R. E., *A circle covering theorem*. Amer. Math. Monthly\n  (1945), 494-498.\n- [Ha47] Hadwiger, H., *Nonseparable convex systems*. Amer. Math. Monthly (1947), 583-585.\n","FormalConjectures.ErdosProblems.«1125»":"# Erdős Problem 1125\n\n*References:*\n- [erdosproblems.com/1125](https://www.erdosproblems.com/1125)\n- [Er81b] Erdős, P., *My Scottish Book 'Problems'*. The Scottish Book (1981), 27-35.\n- [Ke69] Kemperman, J. H. B., *On the regularity of generalized convex functions*. Trans. Amer. Math. Soc. (1969), 69-93.\n- [La84] Laczkovich, M., *On Kemperman's inequality $2f(x)\\leq f(x+h)+f(x+2h)$*. Colloq. Math. (1984), 109-115.\n","FormalConjectures.ErdosProblems.«1126»":"# Erdős Problem 1126\n\n*References:*\n- [erdosproblems.com/1126](https://www.erdosproblems.com/1126)\n- [Er60c] Erdős, P., *Problem 310*. Colloq. Math., 311.\n- [dB66] de Bruijn, N. G., *On almost additive functions*. Colloq. Math. (1966), 59-63.\n- [Ju65] Jurkat, Wolfgang B., *On Cauchy's functional equation*. Proc. Amer. Math. Soc. (1965), 683-686.\n","FormalConjectures.ErdosProblems.«1128»":"# Erdős Problem 1128\n\n*Reference:* [erdosproblems.com/1128](https://www.erdosproblems.com/1128)\n","FormalConjectures.ErdosProblems.«1133»":"# Erdős Problem 1133\n\n*References:*\n- [erdosproblems.com/1133](https://www.erdosproblems.com/1133)\n- [Er67] Erdős, P., Problems and results on the convergence and divergence properties of the\n  Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73.\n","FormalConjectures.ErdosProblems.«1135»":"# Erdős Problem 1135\n\nThe Collatz conjecture states that for any positive integer $n$, there exists a natural\nnumber $m$ such that the $m$-th term of the sequence is 1.\n\n*References:*\n- [erdosproblems.com/1135](https://www.erdosproblems.com/1135)\n- [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n- [La10] Lagarias, Jeffrey C., The {$3x+1$} problem: an overview. (2010), 3--29.\n- [La16] Lagarias, Jeffrey C., Erdős, Klarner, and the {$3x+1$} problem. Amer. Math. Monthly\n  (2016), 753--776.\n- [La85] Lagarias, Jeffrey C., The {$3x+1$} problem and its generalizations. Amer. Math. Monthly\n  (1985), 3--23.\n\nThis file points to the canonical formalization in\n`FormalConjectures.Wikipedia.CollatzConjecture`.\n","FormalConjectures.ErdosProblems.«1136»":"# Erdős Problem 1136\n\n*References:*\n- [erdosproblems.com/1136](https://www.erdosproblems.com/1136)\n- [Mu11] Müller, Helmut, *Über ein additiv-zahlentheoretisches Problem von P. Erdős*.\n  Mitt. Math. Ges. Hamburg (2011), 75-78.\n","FormalConjectures.ErdosProblems.«1137»":"# Erdős Problem 1137\n\n*Reference:* [erdosproblems.com/1137](https://www.erdosproblems.com/1137)\n","FormalConjectures.ErdosProblems.«1138»":"# Erdős Problem 1138\n\n*References:*\n- [erdosproblems.com/1138](https://www.erdosproblems.com/1138)\n- [Va99] Vardi, I., Prime census. (1999).\n- [Kum26] Kumrawat, S., [Disproof of Erdős Problem 1138](https://sourish-kumrawat.github.io/papers/Erdos_1138.pdf).\n\nNote that the conjecture has a claimed disproof found at:\nhttps://sourish-kumrawat.github.io/papers/Erdos_1138.pdf,\nsee the discussion section on the Erdos problems website for more information.\n","FormalConjectures.ErdosProblems.«1139»":"# Erdős Problem 1139\n\n*Reference:* [erdosproblems.com/1139](https://www.erdosproblems.com/1139)\n","FormalConjectures.ErdosProblems.«1141»":"# Erdős Problem 1141\n\n*References:*\n- [erdosproblems.com/1141](https://www.erdosproblems.com/1141)\n- [A214583](https://oeis.org/A214583)\n- [APSSV26b] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant,\n  [Short proofs in combinatorics, probability and number theory II](https://arxiv.org/abs/2604.06609).\n  arXiv:2604.06609 (2026).\n- [Or26] Y. Oriike, [Lean formalisation of Erdős problem 1141](https://github.com/yuta0x89/ErdosProblems/blob/a1319f732cdee5140faf47d984e2c451c1184803/Erdos1141.lean) (2026)\n- [Po17] P. Pollack, Bounds for the first several prime character nonresidues. Proc. Amer. Math. Soc.\n  (2017), 2815--2826.\n- [Me1874] F. Mertens, Ein Beitrag zur analytischen Zahlentheorie. J. Reine Angew. Math. (1874),\n  46--62.\n- [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős\n  and his mathematics\", Budapest, July 1999 (1999).\n","FormalConjectures.ErdosProblems.«1142»":"# Erdős Problem 1142\n\n*References:*\n- [erdosproblems.com/1142](https://www.erdosproblems.com/1142)\n- [A039669](https://oeis.org/A039669)\n- [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős\n  and his mathematics\", Budapest, July 1999 (1999).\n- [MiWe69] Mientka, W. E. and Weitzenkamp, R. C., On f-plentiful numbers, Journal of\n  Combinatorial Theory, Volume 7, Issue 4, December 1969, pages 374-377.\n\n","FormalConjectures.ErdosProblems.«1145»":"# Erdős Problem 1145\n\n*References:*\n- [erdosproblems.com/28](https://www.erdosproblems.com/28)\n- [erdosproblems.com/1145](https://www.erdosproblems.com/1145)\n","FormalConjectures.ErdosProblems.«1146»":"# Erdős Problem 1146\n\n*References:*\n- [erdosproblems.com/1146](https://www.erdosproblems.com/1146)\n- [Ru99] Ruzsa, I., Erdős and the Integers. Journal of Number Theory (1999), 115-163.\n","FormalConjectures.ErdosProblems.«1148»":"# Erdős Problem 1148\n\n*References:*\n- [erdosproblems.com/1148](https://www.erdosproblems.com/1148)\n- [Ch26] P. Chojecki, [Bounded Representations by $x^2 + y^2 - z^2$](https://www.ulam.ai/research/erdos1148-full.pdf) (2026)\n- [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős\n  and his mathematics\", Budapest, July 1999 (1999).\n","FormalConjectures.ErdosProblems.«1150»":"# Erdős Problem 1150\n\n*Reference:* [erdosproblems.com/1150](https://www.erdosproblems.com/1150)\n","FormalConjectures.ErdosProblems.«1159»":"# Erdős Problem 1159\n\n*References:*\n- [erdosproblems.com/1159](https://www.erdosproblems.com/1159)\n- [ESS83] Erdős, P. and Silverman, R. and Stein, A., *Intersection properties of families containing\n  sets of nearly the same size*. Ars Combin. (1983), 247--259.\n- [Er81] Erdős, P., *On the combinatorial problems which I would most like to see solved*.\n  Combinatorica (1981), 25-42.\n","FormalConjectures.ErdosProblems.«115»":"# Erdős Problem 115\n\n*References:*\n- [erdosproblems.com/115](https://www.erdosproblems.com/115)\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [Er90] Erdős, Paul, *Some of my favourite unsolved problems*. A tribute to Paul Erdős (1990),\n  467-478.\n- [ErLe94] Erëmenko, A. and Lempert, L., *An extremal problem for polynomials*. Proc. Amer. Math.\n  Soc. (1994), 191-193.\n- [Po59a] Pommerenke, Ch., *On the derivative of a polynomial*. Michigan Math. J. (1959), 373-375.\n","FormalConjectures.ErdosProblems.«1167»":"# Erdős Problem 1167\n\n*Reference:* [erdosproblems.com/1167](https://www.erdosproblems.com/1167)\n\nThe original Erdős–Hajnal problem list gives the additional conditions $\\gamma \\geq 2$, $r < \\omega$,\nand $\\kappa_\\alpha > r$. Without $\\gamma \\geq 2$, the statement is false: taking $\\gamma = 1$ and\n$\\kappa_0 = \\aleph_1$ with $\\lambda = \\aleph_0$ gives a counterexample, since the partition relation\nwith one color degenerates to a cardinality comparison (see `erdos_1167.unrestricted_is_false`).\n","FormalConjectures.ErdosProblems.«1175»":"# Erdős Problem 1175\n\n*Reference:* [erdosproblems.com/1175](https://www.erdosproblems.com/1175)\n\n## Formalization notes\n\n- **Chromatic cardinal**: `SimpleGraph.chromaticCardinal` is the cardinal-valued chromatic number\n  defined in `FormalConjecturesForMathlib`. It extends the finite `chromaticNumber` (which takes\n  values in `ℕ∞`) to a `Cardinal`, and is therefore able to distinguish between different infinite\n  chromatic numbers.\n- **Triangle-free subgraph**: a subgraph `H : G.Subgraph` is triangle-free when `H.coe.CliqueFree 3`.\n  This is the standard Mathlib formulation: `CliqueFree 3` means the graph has no `K₃` as a clique.\n- **Subgraph**: we use `G.Subgraph` (a spanning subgraph record) rather than an induced subgraph\n  since the problem asks for any subgraph, not just induced ones.\n","FormalConjectures.ErdosProblems.«1176»":"# Erdős Problem 1176\n\n*Reference:* [erdosproblems.com/1176](https://www.erdosproblems.com/1176)\n","FormalConjectures.ErdosProblems.«1188»":"# Erdős Problem 1188\n\n*Reference:* [erdosproblems.com/1188](https://www.erdosproblems.com/1188)\n","FormalConjectures.ErdosProblems.«1190»":"# Erdős Problem 1190\n\n*References:*\n- [erdosproblems.com/1190](https://www.erdosproblems.com/1190)\n- [BFV13] de la Bretèche, Régis and Ford, Kevin and Vandehey, Joseph, *On non-intersecting\n  arithmetic progressions*. Acta Arith. (2013), 381-392.\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*. Ann. Discrete Math.\n  (1980), 89-115.\n","FormalConjectures.ErdosProblems.«1192»":"# Erdős Problem 1192\n\n*References:*\n- [erdosproblems.com/1192](https://www.erdosproblems.com/1192)\n- [Ru90] Ruzsa, Imre Z., A just basis. Monatsh. Math. (1990), 145--151.\n- [Er80] Erdős, Paul, A survey of problems in combinatorial number theory.\n  Ann. Discrete Math. 6 (1980), 89--115.\n","FormalConjectures.ErdosProblems.«1193»":"# Erdős Problem 1193\n\n*References:*\n- [erdosproblems.com/1193](https://www.erdosproblems.com/1193)\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*. Ann. Discrete Math.\n  (1980), 89-115.\n","FormalConjectures.ErdosProblems.«1196»":"# Erdős Problem 1196\n\n*Reference:* [erdosproblems.com/1196](https://www.erdosproblems.com/1196)\n","FormalConjectures.ErdosProblems.«1199»":"# Erdős Problem 1199\n\n*References:*\n- [erdosproblems.com/1199](https://www.erdosproblems.com/1199)\n- [Hi79] Hindman, Neil, Partitions and sums of integers with repetition.\n  J. Combin. Theory Ser. A (1979), 19--32.\n- [Ow74] J. Owings, E2494. Amer. Math. Monthly (1974), 902.\n","FormalConjectures.ErdosProblems.«119»":"# Erdős Problem 119\n\n*References:*\n- [erdosproblems.com/119](https://www.erdosproblems.com/119)\n- [Be91] Beck, J., The modulus of polynomials with zeros on the unit circle: A problem of Erdős.\n  Annals of Math. (1991), 609-651.\n- [Er57] Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300.\n- [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [Er64b] Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964),\n  52-65.\n- [Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.\n- [Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990),\n  467-478.\n- [Er97f] Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge,\n  1993) (1997), 1-10.\n- [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n- [Li77] Linden, C. N., The modulus of polynomials with zeros on the unit circle. Bull. London Math.\n  Soc. (1977), 65--69.\n- [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős\n  and his mathematics\", Budapest, July 1999 (1999).\n- [Wa80] Wagner, Gerold, On a problem of Erdős in {D}iophantine approximation. Bull. London Math.\n  Soc. (1980), 81--88.\n","FormalConjectures.ErdosProblems.«11»":"# Erdős Problem 11\n\n*Reference:* [erdosproblems.com/11](https://www.erdosproblems.com/11)\n","FormalConjectures.ErdosProblems.«1201»":"# Erdős Problem 1201\n\n*Reference:* [erdosproblems.com/1201](https://www.erdosproblems.com/1201)\n","FormalConjectures.ErdosProblems.«1203»":"# Erdős Problem 1203\n\n*References:*\n- [erdosproblems.com/1203](https://www.erdosproblems.com/1203)\n","FormalConjectures.ErdosProblems.«1206»":"# Erdős Problem 1206\n\n*References:*\n- [erdosproblems.com/1206](https://www.erdosproblems.com/1206)\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*. Ann. Discrete Math.\n  (1980), 89-115.\n- [GGK26] M. Garaev, F. Garayev, and S. Konyagin, *On Sidon sets with squares, cubes, and quartics\n  in short intervals*. arXiv:2602.08807 (2026).\n- [GaKo24] Gabdullin, M. R. and Konyagin, S. V., *Trigonometric polynomials with frequencies in the\n  set of cubes*. Math. Notes (2024), 336--340.\n","FormalConjectures.ErdosProblems.«1207»":"# Erdős Problem 1207\n\n*References:*\n- [erdosproblems.com/1207](https://www.erdosproblems.com/1207)\n- [BMP05] Brass, Peter and Moser, William and Pach, János, *Research problems in discrete geometry*.\n  (2005), xii+499.\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*. Ann. Discrete Math.\n  (1980), 89-115.\n- [PaTa02] Pach, János and Tardos, Gábor, *Isosceles triangles determined by a planar point set*.\n  Graphs Combin. (2002), 769--779.\n","FormalConjectures.ErdosProblems.«1209»":"# Erdős Problem 1209\n\n*References:*\n- [erdosproblems.com/429](https://www.erdosproblems.com/429)\n- [erdosproblems.com/1102](https://www.erdosproblems.com/1102)\n- [erdosproblems.com/1209](https://www.erdosproblems.com/1209)\n- [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math.\n  (1980), 89-115.\n","FormalConjectures.ErdosProblems.«120»":"# Erdős Problem 120\n\n*Reference:*\n- [erdosproblems.com/120](https://www.erdosproblems.com/120)\n- [St20](http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1111.pdf) Steinhaus, Hugo, Sur les distances des points dans les ensembles de measure positive. Fund. Math. (1920), 93-104.\n","FormalConjectures.ErdosProblems.«1210»":"# Erdős Problem 1210\n\n*References:*\n- [erdosproblems.com/1210](https://www.erdosproblems.com/1210)\n- [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day\n  (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.\n- [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math.\n  (1980), 89-115.\n","FormalConjectures.ErdosProblems.«1212»":"# Erdős Problem 1212\n\n*References:*\n- [erdosproblems.com/1212](https://www.erdosproblems.com/1212)\n- [Er80] Erdős, P., _Some notes on problems and results in number theory_ (1980), p. 114.\n\nThis file also records machine-checked cores of verified partial results (2026):\na composite-anchor sufficient reduction and an impossibility theorem for periodic\ncertificates; see the corresponding lemmas below.\n","FormalConjectures.ErdosProblems.«1214»":"# Erdős Problem 1214\n\n*References:*\n- [erdosproblems.com/1214](https://www.erdosproblems.com/1214)\n- [CoSc97] Corrales-Rodrigáñez, Capi and Schoof, René, The support problem and its\n  elliptic analogue. J. Number Theory (1997) [Volume 64, Issue 2], 276--290.\n","FormalConjectures.ErdosProblems.«123»":"# Erdős Problem 123\n\n*References:*\n- [erdosproblems.com/123](https://www.erdosproblems.com/123)\n- [ChYu23b] Chen, Yong-Gao and Yu, Wang-Xing, On {$d$}-complete sequences of integers, {II}. Acta\n  Arith. (2023), 161--181.\n- [Er92b] Erdős, Paul, Some of my favourite problems in various branches of combinatorics.\n  Matematiche (Catania) (1992), 231-240.\n- [Er97] Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.\n- [Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n- [ErLe96] Erdős, P. and Lewin, Mordechai, $d$-complete sequences of integers. Math. Comp. (1996),\n  837-840.\n- [MaCh16] Ma, Mi-Mi and Chen, Yong-Gao, On {$d$}-complete sequences of integers. J. Number Theory\n  (2016), 1--12.\n","FormalConjectures.ErdosProblems.«124»":"# Erdős Problem 124\n\n*References:*\n- [erdosproblems.com/124](https://www.erdosproblems.com/124)\n- [BEGL96] Burr, S. A. and Erdős, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.\n","FormalConjectures.ErdosProblems.«125»":"# Erdős Problem 125\n\n*Reference:* [erdosproblems.com/125](https://www.erdosproblems.com/125)\n\nThere are four possibilities for the density of $A+B$:\n1. $A+B$ has zero upper and lower density (and hence also zero density).\n2. $A+B$ has zero lower density, but positive upper density (and hence no density).\n3. $A+B$ has positive upper and lower density that are equal (and hence positive density).\n4. $A+B$ has positive upper and lower density that are unequal (and hence no density).\n","FormalConjectures.ErdosProblems.«126»":"# Erdős Problem 126\n\n*Reference:* [erdosproblems.com/126](https://www.erdosproblems.com/126)\n","FormalConjectures.ErdosProblems.«128»":"# Erdős Problem 128\n\n*Reference:* [erdosproblems.com/128](https://www.erdosproblems.com/128)\n","FormalConjectures.ErdosProblems.«12»":"# Erdős Problem 12\n\n*Reference:* [erdosproblems.com/12](https://www.erdosproblems.com/12)\n","FormalConjectures.ErdosProblems.«130»":"# Erdős Problem 130\n\n*Reference:* [erdosproblems.com/130](https://www.erdosproblems.com/130)\n","FormalConjectures.ErdosProblems.«134»":"# Erdős Problem 134\n\n*References:*\n- [erdosproblems.com/134](https://www.erdosproblems.com/134)\n- [Er97b] Erdős, Paul, *Some old and new problems in various branches of combinatorics*. Discrete Math. (1997), 227-231.\n","FormalConjectures.ErdosProblems.«137»":"# Erdős Problem 137\n\n*References:*\n- [erdosproblems.com/137](https://www.erdosproblems.com/137)\n","FormalConjectures.ErdosProblems.«138»":"# Erdős Problem 138\n\n*References:*\n- [erdosproblems.com/138](https://www.erdosproblems.com/138)\n- [Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.\n- [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.\n- [Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n- [Go01] Gowers, W. T., A new proof of Szemerédi's theorem. Geom. Funct. Anal. (2001), 465-588.\n","FormalConjectures.ErdosProblems.«139»":"# Erdős Problem 139\n\n*Reference:* [erdosproblems.com/139](https://www.erdosproblems.com/139)\n","FormalConjectures.ErdosProblems.«13»":"# Erdős Problem 13\n\n*Reference:* [erdosproblems.com/13](https://www.erdosproblems.com/13)\n","FormalConjectures.ErdosProblems.«141»":"# Erdős Problem 141\n\n*References:*\n- [erdosproblems.com/141](https://www.erdosproblems.com/141)\n- [Wikipedia](https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression#Consecutive_primes_in_arithmetic_progression)\n","FormalConjectures.ErdosProblems.«142»":"# Erdős Problem 142\n\n*Reference:* [erdosproblems.com/142](https://www.erdosproblems.com/142)\n","FormalConjectures.ErdosProblems.«143»":"# Erdős Problem 143\n\n*Reference:* [erdosproblems.com/143](https://www.erdosproblems.com/143)\n","FormalConjectures.ErdosProblems.«145»":"# Erdős Problem 145\n\n*Reference:* [erdosproblems.com/145](https://www.erdosproblems.com/145)\n","FormalConjectures.ErdosProblems.«146»":"# Erdős Problem 146\n\n*References:*\n- [erdosproblems.com/146](https://www.erdosproblems.com/146)\n- [ErSi84] Erdős, P. and Simonovits, M., *Cube-supersaturated graphs and related problems*.\n  Progress in graph theory (1984), 203-218.\n- [OpenAI26] OpenAI, *Ten advances in mathematics and theoretical computer science*. (2026).\n","FormalConjectures.ErdosProblems.«14»":"# Erdős Problem 14\n\n*Reference:* [erdosproblems.com/14](https://www.erdosproblems.com/14)\n","FormalConjectures.ErdosProblems.«150»":"# Erdős Problem 150\n\n*References:*\n- [erdosproblems.com/150](https://www.erdosproblems.com/150)\n- [Br24] Bradač, D., *On a question of Erdős and Nešetřil about minimal cuts in a graph*.\n  arXiv:2409.02974 (2024).\n- [Er88] Erdős, P., *Problems and results in combinatorial analysis and graph theory*.\n  Discrete Math. (1988), 81-92.\n- [FKTV08] Fomin, Fedor V. and Kratsch, Dieter and Todinca, Ioan and Villanger, Yngve, *Exact\n  algorithms for treewidth and minimum fill-in*. SIAM J. Comput. (2008), 1058-1079.\n- [FoVi12] Fomin, Fedor V. and Villanger, Yngve, *Treewidth computation and extremal\n  combinatorics*. Combinatorica (2012), 289-308.\n- [GaMa18] Gaspers, Serge and Mackenzie, Simon, *On the number of minimal separators in graphs*.\n  J. Graph Theory (2018), 653-659.\n","FormalConjectures.ErdosProblems.«152»":"# Erdős Problem 152\n\n#TODO: Formalize the corresponding conjecture for infinite Sidon sets.\n\n*References:*\n - [erdosproblems.com/152](https://www.erdosproblems.com/152)\n - [DM26a] DeepMind prover agent, [formal proof of Erdős problem 152](https://github.com/mo271/formal-conjectures/blob/29c60aa79729701905cf9e92517af23f588971f2/FormalConjectures/ErdosProblems/152.lean#L485) (2026)\n - [DM26b] DeepMind prover agent, [formal proof of the quadratic variant of Erdős problem 152](https://github.com/mo271/formal-conjectures/blob/ff58c933d53bb807bf85d98a47402703f9f14ed3/FormalConjectures/ErdosProblems/152.lean#L496) (2026)\n - [ESS94] Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number\n    Theory (1994), 329-347.\n","FormalConjectures.ErdosProblems.«153»":"# Erdős Problem 153\n\n#TODO: Formalize the corresponding conjecture for infinite Sidon sets.\n\n*References:*\n - [erdosproblems.com/153](https://www.erdosproblems.com/153)\n - [ESS94] Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number\n    Theory (1994), 329-347.\n","FormalConjectures.ErdosProblems.«154»":"# Erdős Problem 154\n\n*References:*\n- [erdosproblems.com/154](https://www.erdosproblems.com/154)\n- [Li98] Lindström, Bernt, *Well distribution of Sidon sets in residue classes*.\n  J. Number Theory (1998), 197-200.\n- [Ko99] Kolountzakis, Mihail N., *On the uniform distribution in residue classes of dense sets\n  of integers with distinct sums*. J. Number Theory (1999), 147-153.\n- [ESS94] Erdős, P. and Sárközy, A. and Sós, T., *On Sum Sets of Sidon Sets, I*.\n  Journal of Number Theory (1994), 329-347.\n","FormalConjectures.ErdosProblems.«155»":"# Erdős Problem 155\n\n*Reference:* [erdosproblems.com/155](https://www.erdosproblems.com/155)\n","FormalConjectures.ErdosProblems.«156»":"# Erdős Problem 156\n\n*References:*\n- [erdosproblems.com/156](https://www.erdosproblems.com/156)\n- [ESS94] Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number\n  Theory (1994), 329-347.\n- [Ru98b] Ruzsa, Imre Z., A small maximal Sidon set. Ramanujan J. (1998), 55-58.\n","FormalConjectures.ErdosProblems.«158»":"# Erdős Problem 158\n\n*References:*\n - [erdosproblems.com/158](https://www.erdosproblems.com/158)\n - [ESS94] Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number\n    Theory (1994), 329-347.\n","FormalConjectures.ErdosProblems.«159»":"# Erdős Problem 159\n\n*References:*\n- [erdosproblems.com/159](https://www.erdosproblems.com/159)\n- [Er78] Erdős, P., Problems and results in combinatorial analysis and combinatorial number\n  theory. Proc. Ninth Southeastern Conf. Combinatorics, Graph Theory and Computing (1978), 29-40.\n- [Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.\n","FormalConjectures.ErdosProblems.«15»":"# Erdős Problem 15: Convergence of Series with Primes\n\n*Reference:* [erdosproblems.com/15](https://www.erdosproblems.com/15)\n","FormalConjectures.ErdosProblems.«160»":"# Erdős Problem 160\n\n*Reference:* [erdosproblems.com/160](https://www.erdosproblems.com/160)\n","FormalConjectures.ErdosProblems.«163»":"# Erdős Problem 163\n\n*References:*\n- [erdosproblems.com/163](https://www.erdosproblems.com/163)\n- [BuEr75] Burr, S. A. and Erdős, P., On the Ramsey number of graphs with small degree-ratio.\n  Colloq. Math. Soc. János Bolyai (1975).\n- [Le17] Lee, C., Ramsey numbers of degenerate graphs. Ann. of Math. (2) 185 (2017), 791-829.\n","FormalConjectures.ErdosProblems.«164»":"# Erdős Problem 164\n\n*References:*\n- [erdosproblems.com/164](https://www.erdosproblems.com/164)\n- [Er76g] Erdős, P., *Problems and results on combinatorial number theory. II*. J. Indian Math. Soc.\n  (N.S.) (1976), 285-298.\n- [Er86] Erdős, P., *Problémes et résultats en théorie des nombres*. (1986).\n- [Va99] Various, *Some of Paul's favorite problems*. Booklet produced for the conference \"Paul\n  Erdős and his mathematics\", Budapest, July 1999 (1999).\n- [ABLLPSTT26] B. Alexeev, K. Barreto, Y. Li, J. D. Lichtman, L. Price, J. I. Shah, Q. Tang, and\n  T. Tao, *Primitive sets and Von Mangoldt Chains: Erdős problem #1196 and beyond*.\n  arXiv:2605.00301 (2026).\n- [Er35] Erdős, Paul, *Note on Sequences of Integers No One of Which is Divisible By Any Other*.\n  J. London Math. Soc. (1935), 126-128.\n- [Li23] Lichtman, J. D., *A proof of the Erdős primitive set conjecture*. arXiv:2202.02384 (2023).\n","FormalConjectures.ErdosProblems.«166»":"# Erdős Problem 166\n\n*References:*\n- [erdosproblems.com/166](https://www.erdosproblems.com/166)\n- [Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.\n- [AKS80] Ajtai, M., Komlós, J. and Szemerédi, E., A note on Ramsey numbers. J. Combin. Theory\n  Ser. A (1980), 354-360.\n- [MaVe23] Mattheus, S. and Verstraëte, J., The asymptotics of $r(4,t)$. Ann. of Math. (2024),\n  941-965.\n","FormalConjectures.ErdosProblems.«168»":"# Erdős Problem 168\n\n*Reference:* [erdosproblems.com/168](https://www.erdosproblems.com/168)\n","FormalConjectures.ErdosProblems.«16»":"# Erdős Problem 16\n\n*References:*\n- [erdosproblems.com/16](https://www.erdosproblems.com/16)\n- [Ch23] Chen, Yong-Gao, A conjecture of Erdős on $p+2^k$. arXiv:2312.04120 (2023).\n- [Er50] Erdős, P., On integers of the form $2^k+p$ and some related problems. Summa Brasil. Math.\n  (1950), 113-123.\n- [Ro34] Romanoff, N. P., Über einige Sätze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.\n","FormalConjectures.ErdosProblems.«170»":"# Erdős Problem 170\n\n*Reference:* [erdosproblems.com/170](https://www.erdosproblems.com/170)\n","FormalConjectures.ErdosProblems.«172»":"# Erdős Problem 172\n\n*Reference:* [erdosproblems.com/172](https://www.erdosproblems.com/172)\n","FormalConjectures.ErdosProblems.«175»":"# Erdős Problem 175\n\n*References:*\n- [erdosproblems.com/175](https://www.erdosproblems.com/175)\n- [Sa85] A. Sárközy, *On divisors of binomial coefficients, I*, J. Number Theory 20\n  (1985), 70–80.\n- [GrRa96] A. Granville and O. Ramaré, *Explicit bounds on exponential sums and the scarcity\n  of squarefree binomial coefficients*, Mathematika 43 (1996), 73–107.\n- [Ve95] G. Velammal, *Is the binomial coefficient $\\binom{2n}{n}$ square free?*,\n  Hardy-Ramanujan Journal 18 (1995), 23–45.\n\nSárközy proved the assertion for all sufficiently large `n`; Granville--Ramaré and Velammal\nindependently proved the full range `n ≥ 5`.\n","FormalConjectures.ErdosProblems.«178»":"# Erdős Problem 178\n\n*References:*\n- [erdosproblems.com/178](https://www.erdosproblems.com/178)\n- [ErGr79] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics*. Enseign. Math. (1979), 325-344.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number theory*. Monographies de L'Enseignement Mathématique (1980).\n- [Be81] Beck, József, *Balancing families of integer sequences*. Combinatorica (1981), 209-216.\n- [Be17] Beck, József, *A discrepancy problem: balancing infinite dimensional vectors*. Number theory—Diophantine problems, uniform distribution and applications (2017), 61-82.\n","FormalConjectures.ErdosProblems.«17»":"# Erdős Problem 17\n*Reference:* [erdosproblems.com/17](https://www.erdosproblems.com/17)\n","FormalConjectures.ErdosProblems.«180»":"# Erdős Problem 180\n\n*References:*\n- [erdosproblems.com/180](https://www.erdosproblems.com/180)\n- [ErSi82] Erdős, P. and Simonovits, M., *Compactness results in extremal graph theory*.\n  Combinatorica (1982), 275-288.\n- [OpenAI26] OpenAI, *Ten advances in mathematics and theoretical computer science*. (2026).\n","FormalConjectures.ErdosProblems.«181»":"# Erdős Problem 181\n\n*References:*\n- [erdosproblems.com/181](https://www.erdosproblems.com/181)\n- [Er93] Erdős, Paul, *Some of my favorite solved and unsolved problems in graph theory*.\n  Quaestiones Math. (1993), 333-350.\n- [Ti22] Tikhomirov, K., *A remark on the Ramsey number of the hypercube*. arXiv:2208.14568 (2022).\n","FormalConjectures.ErdosProblems.«183»":"# Erdős Problem 183\n\n*References:*\n- [erdosproblems.com/183](https://www.erdosproblems.com/183)\n- [Er61] Erdős, P., *Graph theory and probability. II*. Canad. J. Math. (1961), 346-352.\n- [OpenAI26] OpenAI, *Ten advances in mathematics and theoretical computer science*. (2026).\n","FormalConjectures.ErdosProblems.«184»":"# Erdős Problem 184\n\n*References:*\n- [erdosproblems.com/184](https://www.erdosproblems.com/184)\n- [BM22] Bucić, M. and Montgomery, R., Towards the Erdős-Gallai Cycle Decomposition Conjecture.\n  arXiv:2211.07689 (2022).\n- [CFS14] Conlon, David and Fox, Jacob and Sudakov, Benny, Cycle packing. Random Structures\n  Algorithms (2014), 608-626.\n- [EGP66] Erdős, Paul and Goodman, A. W. and Pósa, Lajos, The representation of a graph by set\n  intersections. Canadian J. Math. (1966), 106-112.\n- [Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial\n  Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.\n","FormalConjectures.ErdosProblems.«188»":"# Erdős Problem 188\n\n*References:*\n- [erdosproblems.com/188](https://www.erdosproblems.com/188)\n- [EGMRSS75] Erdős, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J.\n  and Straus, E. G., Euclidean {R}amsey theorems. {II}. (1975), 529--557.\n- [Ts17] Tsaturian, Sergei, A {E}uclidean {R}amsey result in the plane. Electron. J. Combin. (2017),\n  Paper No. 4.35, 9.\n","FormalConjectures.ErdosProblems.«189»":"# Erdős Problem 189\n\n*Reference:* [erdosproblems.com/189](https://www.erdosproblems.com/189)\n","FormalConjectures.ErdosProblems.«18»":"# Erdős Problem 18\n\n*Reference:*\n* [erdosproblems.com/18](https://www.erdosproblems.com/18)\n* [ErGr80] Erdős, P. and Graham, R. L. (1980). Old and New Problems and Results in Combinatorial Number\nTheory. Monographies de L'Enseignement Mathématique, 28. Université de Genève. (See the\nsections on Egyptian fractions or practical numbers).\n* [Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.\n","FormalConjectures.ErdosProblems.«193»":"# Erdős Problem 193\n\nReferences:\n- [erdosproblems.com/193](https://www.erdosproblems.com/193)\n- [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\n  theory. Monographies de L'Enseignement Mathematique (1980).\n- [GeRa79] Gerver, Joseph L. and Ramsey, L. Thomas, \"On certain sequences of lattice points.\"\n  Pacific J. Math. (1979), 357-363.\n","FormalConjectures.ErdosProblems.«194»":"# Erdős Problem 194\n\n*References:*\n- [erdosproblems.com/194](https://www.erdosproblems.com/194)\n- [ABJ11] Ardal, H. and Brown, T. and Jungić, V., Chaotic orderings of the rationals and reals. Amer. Math. Monthly (2011), 921-925.\n","FormalConjectures.ErdosProblems.«195»":"# Erdős Problem 195\n\n*References:*\n- [erdosproblems.com/195](https://www.erdosproblems.com/195)\n- [Ad22] Adenwalla, S., Avoiding Monotone Arithmetic Progressions in Permutations of Integers.\n  arXiv:2211.04451 (2022).\n- [Ge19] Geneson, Jesse, Forbidden arithmetic progressions in permutations of subsets of the\n  integers. Discrete Math. (2019), 1489-1491.\n","FormalConjectures.ErdosProblems.«196»":"# Erdős Problem 196\n\n*Reference:* [erdosproblems.com/196](https://www.erdosproblems.com/196)\n","FormalConjectures.ErdosProblems.«197»":"# Erdős Problem 197\n\n*Reference:* [erdosproblems.com/197](https://www.erdosproblems.com/197)\n","FormalConjectures.ErdosProblems.«198»":"# Erdős Problem 198\n\n*References:*\n- [erdosproblems.com/198](https://www.erdosproblems.com/198)\n- [Ba75] Baumgartner, James E., Partitioning vector spaces. J. Combinatorial Theory Ser. A (1975),\n  231-233.\n","FormalConjectures.ErdosProblems.«199»":"# Erdős Problem 199\n\n*References:*\n- [erdosproblems.com/199](https://www.erdosproblems.com/199)\n- [Ba75] Baumgartner, James E., *Partitioning vector spaces*. J. Combinatorial Theory\n  Ser. A (1975), 231-233.\n","FormalConjectures.ErdosProblems.«1»":"# Erdős Problem 1\n\n*Reference:* [erdosproblems.com/1](https://www.erdosproblems.com/1)\n","FormalConjectures.ErdosProblems.«200»":"# Erdős Problem 200\n\n*Reference:* [erdosproblems.com/200](https://www.erdosproblems.com/200)\n","FormalConjectures.ErdosProblems.«202»":"# Erdős Problem 202\n\n*References:*\n- [erdosproblems.com/202](https://www.erdosproblems.com/202)\n- [BFV13] de la Bretèche, Régis and Ford, Kevin and Vandehey, Joseph, *On non-intersecting\n  arithmetic progressions*. Acta Arith. (2013), 381-392.\n- [Ch05] Chen, Yong-Gao, *On disjoint arithmetic progressions*. Acta Arith. (2005), 143-148.\n- [Cr03b] Croot, III, Ernest S., *On non-intersecting arithmetic progressions*. Acta Arith.\n  (2003), 233-238.\n- [ErSz68] Erdős, P. and Szemerédi, E., *On a problem of P. Erdős and S. Stein*. Acta Arith.\n  (1968), 85-90.\n- [PaPh24] Park, Jinyoung and Pham, Huy Tuan, *A proof of the Kahn-Kalai conjecture*. J. Amer.\n  Math. Soc. (2024), 235-243.\n","FormalConjectures.ErdosProblems.«203»":"# Erdős Problem 203\n\n*Reference:* [erdosproblems.com/203](https://www.erdosproblems.com/203)\n","FormalConjectures.ErdosProblems.«204»":"# Erdős Problem 204\n\n*References:*\n- [erdosproblems.com/204](https://www.erdosproblems.com/204)\n- [Ad25] S. Adenwalla, A Question of Erdős and Graham on Covering Systems. arXiv:2501.15170 (2025).\n","FormalConjectures.ErdosProblems.«205»":"# Erdős Problem 205\n\n*References:*\n- [erdosproblems.com/205](https://www.erdosproblems.com/205)\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*. Ann. Discrete Math.\n  (1980), 89-115.\n- [Ro34] Romanoff, N. P., *Über einige Sätze der additiven Zahlentheorie*. Math. Ann. (1934),\n  668-678.\n","FormalConjectures.ErdosProblems.«206»":"# Erdős Problem 206\n\n*References:*\n- [erdosproblems.com/206](https://www.erdosproblems.com/206)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial\n  number theory*. Monographies de L'Enseignement Mathématique (1980).\n- [Cu22] Curtiss, D. R., *On Kellogg's Diophantine Problem*. Amer. Math. Monthly (1922),\n  380-387.\n- [Er50b] Erdős, Pál, *On a Diophantine equation*. Mat. Lapok (1950), 192-210.\n- [Na23] Nathanson, M., *Underapproximation by Egyptian fractions*. J. Number Theory (2023),\n  208-234.\n- [Ch23b] Chu, H. V., *A threshold for the best two-term underapproximation by Egyptian\n  fractions*. arXiv:2306.12564 (2023).\n- [Ko24b] Kovač, V., *On eventually greedy best underapproximations by Egyptian fractions*.\n  arXiv:2406.07218 (2024).\n","FormalConjectures.ErdosProblems.«208»":"# Erdős Problem 208\n*Reference:* [erdosproblems.com/208](https://www.erdosproblems.com/208)\n","FormalConjectures.ErdosProblems.«209»":"# Erdős Problem 209\n\n*References:*\n- [erdosproblems.com/209](https://www.erdosproblems.com/209)\n- [Er84] Erdős, P., *Research problems*. Period. Math. Hungar. (1984), 101-103.\n- [ErPu95b] Erdős, Paul and Purdy, George, *Extremal problems in combinatorial geometry*.\n  Handbook of combinatorics, Vol. 1, 2 (1995), 809-874.\n- [FuPa84] Füredi, Z. and Palásti, I., *Arrangements of lines with a large number of triangles*.\n  Proc. Amer. Math. Soc. (1984), 561-566.\n- [Es16] Escudero, Juan García, *Gallai triangles in configurations of lines in the projective\n  plane*. C. R. Math. Acad. Sci. Paris (2016), 551-554.\n","FormalConjectures.ErdosProblems.«20»":"# Erdős Problem 20\n\n*References:*\n* [erdosproblems.com/20](https://www.erdosproblems.com/20)\n* [Wikipedia](https://en.wikipedia.org/wiki/Sunflower_(mathematics))\n* [ErRa60] Erdős, Paul and Rado, Richard. Intersection theorems for systems of sets.\n  J. London Math. Soc. 35 (1960), 85--90.\n\n","FormalConjectures.ErdosProblems.«212»":"# Erdős Problem 212\n\n*Reference:* [erdosproblems.com/212](https://www.erdosproblems.com/212)\n","FormalConjectures.ErdosProblems.«213»":"# Erdős Problem 213\n\n*Reference:* [erdosproblems.com/213](https://www.erdosproblems.com/213)\n","FormalConjectures.ErdosProblems.«214»":"# Erdős Problem 214\n\n*References:*\n- [erdosproblems.com/214](https://www.erdosproblems.com/214)\n- [CsTo94] Csizmadia, György and Tóth, Géza, *Note on a Ramsey-type problem in geometry*.\n  J. Combin. Theory Ser. A (1994), 302-306.\n- [EGMRSS75] Erdős, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J.\n  and Straus, E. G., *Euclidean Ramsey theorems. II*. (1975), 529-557.\n- [Ju79] Juhász, Rozália, *Ramsey type theorems in the plane*. J. Combin. Theory Ser. A (1979),\n  152-160.\n","FormalConjectures.ErdosProblems.«218»":"# Erdős Problem 218\n\n*Reference:* [erdosproblems.com/218](https://www.erdosproblems.com/218)\n","FormalConjectures.ErdosProblems.«219»":"# Erdős Problem 219\n\n*Reference:* [erdosproblems.com/219](https://www.erdosproblems.com/219)\n","FormalConjectures.ErdosProblems.«221»":"# Erdős Problem 221\n\n*References:*\n- [erdosproblems.com/221](https://www.erdosproblems.com/221)\n- [Lo54] Lorentz, G. G., *On a problem of additive number theory*. Proc. Amer. Math. Soc. (1954), 838-841.\n- [Ru72] Ruzsa, Jr., I., *On a problem of P. Erdős*. Canad. Math. Bull. (1972), 309-310.\n","FormalConjectures.ErdosProblems.«224»":"# Erdős Problem 224\n\n*References:*\n- [erdosproblems.com/224](https://www.erdosproblems.com/224)\n- [DaGr62] Danzer, L. and Gr\\\"{u}nbaum, B., *\\\"{U}ber zwei Probleme\n  bez\\\"{u}glich konvexer K\\\"{o}rper von P. Erd\\H{o}s und von V. L. Klee*.\n  Math. Z. (1962), 95-99.\n","FormalConjectures.ErdosProblems.«226»":"# Erdős Problem 226\n\n*References:*\n- [erdosproblems.com/226](https://www.erdosproblems.com/226)\n- [BaSc70] Barth, K. F. and Schneider, W. J., *Entire functions mapping\n  countable dense subsets of the reals onto each other monotonically*.\n  J. London Math. Soc. (2) (1970), 620--626.\n- [BaSc71] Barth, K. F. and Schneider, W. J., *Entire functions mapping\n  arbitrary countable dense sets and their complements onto each other*.\n  J. London Math. Soc. (2) (1971/72), 482--488.\n- [Ha74] Hayman, W. K., *Research problems in function theory: new problems*.\n  (1974), 155--180.\n","FormalConjectures.ErdosProblems.«228»":"# Erdős Problem 228\n\n*Reference:* [erdosproblems.com/228](https://www.erdosproblems.com/228)\n","FormalConjectures.ErdosProblems.«229»":"# Erdős Problem 229\n\n*References:*\n- [erdosproblems.com/229](https://www.erdosproblems.com/229)\n- [BaSc72] Barth, K. F. and Schneider, W. J., On a problem of Erd\\H{o}s concerning the zeros of the\n  derivatives of an entire function. Proc. Amer. Math. Soc. (1972), 229--232.\n- [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n","FormalConjectures.ErdosProblems.«22»":"# Erdős Problem 22\n\nThe central problem of Ramsey–Turán theory: writing $\\mathrm{rt}(n; k, \\ell)$ for the maximum\nnumber of edges of a $K_k$-free graph on $n$ vertices whose largest independent set has size\nless than $\\ell$, is it true that for every $\\epsilon > 0$ and all sufficiently large $n$\n$$\\mathrm{rt}(n; 4, \\epsilon n) \\geq n^2/8?$$\n\nConjectured by Bollobás and Erdős [BoEr76], who constructed such a graph with\n$(1/8 + o(1))n^2$ edges. Together with the matching upper bound\n$\\mathrm{rt}(n; 4, \\epsilon n) \\leq (1/8 + o(1))n^2$ of Szemerédi [Sz72], this determines the\nRamsey–Turán density of $K_4$ to be $1/8$. The conjecture as stated was proved by Fox, Loh,\nand Zhao [FLZ15].\n\n*References:*\n* [erdosproblems.com/22](https://www.erdosproblems.com/22)\n* [BoEr76] Bollobás, B. and Erdős, P., *On a Ramsey-Turán type problem*. J. Combin. Theory\n  Ser. B 21 (1976), 166--168.\n* [Sz72] Szemerédi, E., *On graphs containing no complete subgraph with 4 vertices*\n  (Hungarian). Mat. Lapok 23 (1972), 113--116.\n* [FLZ15] Fox, J., Loh, P.-S., and Zhao, Y., *The critical window for the classical\n  Ramsey-Turán problem*. Combinatorica 35 (2015), 435--476.\n","FormalConjectures.ErdosProblems.«233»":"# Erdős Problem 233\n\n*References:*\n  - [erdosproblems.com/233](https://www.erdosproblems.com/233)\n  - [A74741](https://oeis.org/A74741)\n  - [Wikipedia](https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture)\n","FormalConjectures.ErdosProblems.«234»":"# Erdős Problem 234\n\n*Reference:* [erdosproblems.com/234](https://www.erdosproblems.com/234)\n","FormalConjectures.ErdosProblems.«236»":"# Erdős Problem 236\n\n*Reference:* [erdosproblems.com/236](https://www.erdosproblems.com/236)\n","FormalConjectures.ErdosProblems.«238»":"# Erdős Problem 238\n\n*Reference:* [erdosproblems.com/238](https://www.erdosproblems.com/238)\n","FormalConjectures.ErdosProblems.«239»":"# Erdős Problem 239\n\n*References:*\n- [erdosproblems.com/239](https://www.erdosproblems.com/239)\n- [Ha68] Halász, G., Über die Mittelwerte multiplikativer zahlentheoretischer\n  Funktionen. Acta Math. Acad. Sci. Hungar. (1968), 365-403.\n- [Wi67] Wirsing, E., Das asymptotische Verhalten von Summen über multiplikative Funk­tionen.\n  Acta Math. Acad. Sei. Hung. (1967), 411-467.\n","FormalConjectures.ErdosProblems.«23»":"# Erdős Problem 23\n\n*References:*\n* [erdosproblems.com/23](https://www.erdosproblems.com/23)\n* [OEIS A389646](https://oeis.org/A389646)\n* [Balogh-Clemen-Lidicky, Max Cuts in Triangle-free Graphs](https://arxiv.org/abs/2103.14179)\n* [McKay, Extremal graphs for bipartization of triangle-free graphs](https://users.cecs.anu.edu.au/~bdm/data/graphs.html)\n","FormalConjectures.ErdosProblems.«241»":"# Erdős Problem 241\n\n*References:*\n- [erdosproblems.com/30](https://www.erdosproblems.com/30)\n- [erdosproblems.com/241](https://www.erdosproblems.com/241)\n- [BoCh62] Bose, R. C. and Chowla, S., Theorems in the additive theory of numbers. Comment. Math.\n  Helv. (1962/63), 141-147.\n- [Gr01] Green, Ben, The number of squares and {$B_h[g]$} sets. Acta Arith. (2001), 365-390.\n- [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n","FormalConjectures.ErdosProblems.«242»":"# Erdős Problem 242\n\n*References:*\n- [erdosproblems.com/242](https://www.erdosproblems.com/242)\n- [Si56] Sierpiński, W., Sur les décompositions de nombres rationnels en fractions primaires.\n  Mathesis (1956), 16--32.\n","FormalConjectures.ErdosProblems.«243»":"# Erdős Problem 243\n\n*Reference:* [erdosproblems.com/243](https://www.erdosproblems.com/243)\n","FormalConjectures.ErdosProblems.«244»":"# Erdős Problem 244\n\n*Reference:* [erdosproblems.com/244](https://www.erdosproblems.com/244)\n","FormalConjectures.ErdosProblems.«245»":"# Erdős Problem 245\n\n*Reference:* [erdosproblems.com/245](https://www.erdosproblems.com/245)\n","FormalConjectures.ErdosProblems.«246»":"# Erdős Problem 246\n\n*References:*\n- [erdosproblems.com/246](https://www.erdosproblems.com/246)\n- [Bi59] Birch, B. J., *Note on a problem of Erd\\H{o}s*. Proc. Cambridge\n  Philos. Soc. (1959), 370-373.\n- [Ca60] Cassels, J. W. S., *On the representation of integers as the sums of\n  distinct summands taken from a fixed set*. Acta Sci. Math. (Szeged) (1960),\n  111-124.\n- [FaCh17] Fang, Jin-Hui and Chen, Yong-Gao, *A quantitative form of the\n  {E}rd\\H{o}s-{B}irch theorem*. Acta Arith. (2017), 301--311.\n- [He00b] Hegyv\\'{a}ri, N., *On the completeness of an exponential type\n  sequence*. Acta Math. Hungar. (2000), 127--135.\n- [Yu24] Yu, Wang-Xing, *On the representation of an exponential type sequence*.\n  Publ. Math. Debrecen (2024), 253--261.\n","FormalConjectures.ErdosProblems.«247»":"# Erdős Problem 247\n\n*Reference:* [erdosproblems.com/247](https://www.erdosproblems.com/247)\n","FormalConjectures.ErdosProblems.«248»":"# Erdős Problem 248\n\n*References:*\n- [erdosproblems.com/248](https://www.erdosproblems.com/248)\n- [TaTe25] T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers. arXiv:2512.01739 (2025).\n","FormalConjectures.ErdosProblems.«249»":"# Erdős Problem 249\n\n*Reference:* [erdosproblems.com/249](https://www.erdosproblems.com/249)\n","FormalConjectures.ErdosProblems.«24»":"# Erdős Problem 24\n\n*References:*\n- [erdosproblems.com/24](https://www.erdosproblems.com/24)\n- [Er90] Erdős, Paul, *Some of my favourite unsolved problems*. A tribute to Paul Erdős (1990),\n  467-478.\n- [Er97b] Erdős, Paul, *Some old and new problems in various branches of combinatorics*. Discrete\n  Math. (1997), 227-231.\n- [Er92b] Erdős, Paul, *Some of my favourite problems in various branches of combinatorics*.\n  Matematiche (Catania) (1992), 231-240.\n- [Er97f] Erdős, Paul, *Some unsolved problems*. Combinatorics, geometry and probability\n  (Cambridge, 1993) (1997), 1-10.\n- [Gr12] Grzesik, Andrzej, *On the maximum number of five-cycles in a triangle-free graph*.\n  J. Combin. Theory Ser. B (2012), 1061-1066.\n- [HHKNR13] Hatami, Hamed and Hladký, Jan and Kráľ, Daniel and Norine, Serguei and Razborov,\n  Alexander, *On the number of pentagons in triangle-free graphs*. J. Combin. Theory Ser. A\n  (2013), 722-732.\n","FormalConjectures.ErdosProblems.«250»":"# Erdős Problem 250\n\n*Reference:* [erdosproblems.com/250](https://www.erdosproblems.com/250)\n","FormalConjectures.ErdosProblems.«251»":"# Erdős Problem 251\n\n*Reference:* [erdosproblems.com/251](https://www.erdosproblems.com/251)\n","FormalConjectures.ErdosProblems.«252»":"# Erdős Problem 252\n\n*References:*\n - [erdosproblems.com/252](https://www.erdosproblems.com/252)\n - [ErSt71] Erdös, P., and E. G. Straus. \"Some number theoretic results.\" Pacific J. Math 36 (1971):\n    635-646.\n - [ErSt74] Erdős, Paul, and Ernst Straus. \"On the irrationality of certain series.\" Pacific journal\n    of mathematics 55.1 (1974): 85-92.\n - [ErKa54] P. Erdős, M. Kac, Amer. Math. Monthly 61 (1954), Problem 4518.\n - [ScPu06] Schlage-Puchta, J. C., The irrationality of a number theoretical series. Ramanujan J.\n    (2006), 455-460.\n - [FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor\n    function series. Integers (2007).\n - [Pr22] Pratt, K., The irrationality of a divisor function series of Erdős and Kac.\n    arXiv:2209.11124 (2022).\n","FormalConjectures.ErdosProblems.«253»":"# Erdős Problem 253\n\n*Reference:* [erdosproblems.com/253](https://www.erdosproblems.com/253)\n","FormalConjectures.ErdosProblems.«254»":"# Erdős Problem 254\n\n*References:*\n- [erdosproblems.com/254](https://www.erdosproblems.com/254)\n- [Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken\n  from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.\n","FormalConjectures.ErdosProblems.«257»":"# Erdős Problem 257\n\n*Reference:* [erdosproblems.com/257](https://www.erdosproblems.com/257)\n","FormalConjectures.ErdosProblems.«258»":"# Erdős Problem 258\n\n*References:*\n- [erdosproblems.com/258](https://www.erdosproblems.com/258)\n- [Ch26] P. Chojecki and GPT-5.4 Pro, [Erdős problem 258](https://www.ulam.ai/research/erdos258.pdf) (2026)\n- [St26] ster-oc, [Lean formalisation of Erdős problem 258](https://live.lean-lang.org/#project=mathlib-v4.28.0&url=https://gist.githubusercontent.com/ster-oc/2b7adcf9d753cf6e29d782f7374cc57e/raw/689a8483895cbe147634dfbf2d7b1db93a3b5b5f/Erdos258.lean) (2026)\n","FormalConjectures.ErdosProblems.«259»":"# Erdős Problem 259\n\n*Reference:* [erdosproblems.com/259](https://www.erdosproblems.com/259)\n","FormalConjectures.ErdosProblems.«25»":"# Erdős Problem 25: Logarithmic density of size-dependent congruences\n\n*Reference:* [erdosproblems.com/25](https://www.erdosproblems.com/25)\n","FormalConjectures.ErdosProblems.«260»":"# Erdős Problem 260\n\n*Reference:* [erdosproblems.com/260](https://www.erdosproblems.com/260)\n","FormalConjectures.ErdosProblems.«261»":"# Erdős Problem 261\n\n*References:*\n - [erdosproblems.com/261](https://www.erdosproblems.com/261)\n - [BoLo90] Borwein, Peter and Loring, Terry A., Some questions of Erdős and Graham on numbers\n    of the form $\\sum g_n/2^{g_n}$. Math. Comp. (1990), 377--394.\n - [Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances\n    in transcendence theory (Durham, 1986) (1988), 102--109.\n - [TUZ20] Tengely, Szabolcs and Ulas, Maciej and Zygadlo, Jakub, On a Diophantine equation of\n    Erdős and Graham. J. Number Theory (2020), 445--459.\n","FormalConjectures.ErdosProblems.«263»":"# Erdős Problem 263\n\n*Reference:* [erdosproblems.com/263](https://www.erdosproblems.com/263)\n","FormalConjectures.ErdosProblems.«264»":"# Erdős Problem 264\n\n*Reference:* [erdosproblems.com/264](https://www.erdosproblems.com/264)\n","FormalConjectures.ErdosProblems.«266»":"# Erdős Problem 266\n\n*Reference:* [erdosproblems.com/266](https://www.erdosproblems.com/266)\n","FormalConjectures.ErdosProblems.«267»":"# Erdős Problem 267\n\n*Reference:* [erdosproblems.com/267](https://www.erdosproblems.com/267)\n","FormalConjectures.ErdosProblems.«268»":"# Erdős Problem 268\n\n*Reference:*\n - [erdosproblems.com/268](https://www.erdosproblems.com/268)\n - [KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series.\n","FormalConjectures.ErdosProblems.«269»":"# Erdős Problem 269\n\n*Reference:* [erdosproblems.com/269](https://www.erdosproblems.com/269)\n","FormalConjectures.ErdosProblems.«26»":"# Erdős Problem 26\n\n*References:*\n- [erdosproblems.com/26](https://www.erdosproblems.com/26)\n- [Te19](https://arxiv.org/pdf/1908.00488) G. Tenenbaum,\n  _Some of Erdős' unconventional problems in number theory, thirty-four years later_,\n  arXiv:1908.00488 [math.NT] (2019)\n","FormalConjectures.ErdosProblems.«272»":"# Erdős Problem 272\n\n*Reference:* [erdosproblems.com/272](https://www.erdosproblems.com/272)\n","FormalConjectures.ErdosProblems.«273»":"# Erdős Problem 273\n*Reference:* [erdosproblems.com/273](https://www.erdosproblems.com/273)\n","FormalConjectures.ErdosProblems.«274»":"# Erdős Problem 274\n\n*References:*\n* [erdosproblems.com/274](https://www.erdosproblems.com/274)\n* [Wikipedia](https://en.wikipedia.org/wiki/Herzog%E2%80%93Sch%C3%B6nheim_conjecture)\n* [arXiv:1803.08301](https://arxiv.org/abs/1803.08301)\n* [arXiv:1803.03569](https://arxiv.org/abs/1803.03569)\n* [PMC7247885](https://pmc.ncbi.nlm.nih.gov/articles/PMC7247885/)\n* [arXiv:1804.11103](https://arxiv.org/abs/1804.11103)\n","FormalConjectures.ErdosProblems.«275»":"# Erdős Problem 275\n\n*References:*\n- [erdosproblems.com/275](https://www.erdosproblems.com/275)\n- [CrVE70] R.B. Crittenden and C.L. Vanden Eynden, *Any n arithmetic progressions covering the first\n  2^n integers cover all integers*, Proc. Amer. Math. Soc. 24 (1970), 475-481.\n","FormalConjectures.ErdosProblems.«276»":"# Erdős Problem 276\n\n*References:*\n[erdosproblems.com/276](https://www.erdosproblems.com/276)\n","FormalConjectures.ErdosProblems.«277»":"# Erdős Problem 277\n\n*References:*\n- [erdosproblems.com/277](https://www.erdosproblems.com/277)\n- [Ha79] Haight, J. A., Covering systems of congruences, a negative result. Mathematika (1979),\n  53--61.\n","FormalConjectures.ErdosProblems.«279»":"# Erdős Problem 279\n*Reference:* [erdosproblems.com/279](https://www.erdosproblems.com/279)\n","FormalConjectures.ErdosProblems.«280»":"# Erdős Problem 280\n\n*References:*\n- [erdosproblems.com/280](https://www.erdosproblems.com/280)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial\n  number theory*. Monographies de L'Enseignement Mathematique (1980).\n","FormalConjectures.ErdosProblems.«281»":"# Erdős Problem 281\n\n*References:*\n- [erdosproblems.com/281](https://www.erdosproblems.com/281)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [DaEr36] Davenport, H. and Erdős, P., *On sequences of positive integers*. Acta\n  Arithmetica (1936), 147-151.\n- [HaRo66] Halberstam, H. and Roth, K. F., *Sequences. Vol. I*. (1966), xx+291.\n","FormalConjectures.ErdosProblems.«282»":"# Erdős Problem 282\n\n*Reference:* [erdosproblems.com/282](https://www.erdosproblems.com/282)\n","FormalConjectures.ErdosProblems.«283»":"# Erdős Problem 283\n\n*References:*\n- [erdosproblems.com/283](https://www.erdosproblems.com/283)\n- [Al19] Alekseyev, Max A., On partitions into squares of distinct integers whose\nreciprocals sum to 1. (2019), 213--221.\n- [Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.\n- [Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.\n- [vD25] W. van Doorn, Partitions with prescribed sum of rationals: asymptotic bounds. arXiv:2502.02200 (2025).\n","FormalConjectures.ErdosProblems.«285»":"# Erdős Problem 285\n\n*Reference:* [erdosproblems.com/285](https://www.erdosproblems.com/285)\n","FormalConjectures.ErdosProblems.«287»":"# Erdős Problem 287\n\n*Reference:* [erdosproblems.com/287](https://www.erdosproblems.com/287)\n","FormalConjectures.ErdosProblems.«288»":"# Erdős Problem 288\n\n*Reference:* [erdosproblems.com/288](https://www.erdosproblems.com/288)\n","FormalConjectures.ErdosProblems.«289»":"# Erdős Problem 289\n*Reference:* [erdosproblems.com/289](https://www.erdosproblems.com/289)\n","FormalConjectures.ErdosProblems.«28»":"# Erdős Problem 28\n\n*Reference:* [erdosproblems.com/28](https://www.erdosproblems.com/28)\n","FormalConjectures.ErdosProblems.«290»":"# Erdős Problem 290\n\n*References:*\n- [erdosproblems.com/290](https://www.erdosproblems.com/290)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathématique (1980), p.34.\n- [vD24] van Doorn, W., *On the non-monotonicity of the denominator of generalized harmonic sums*.\n  arXiv:2411.03073 (2024).\n","FormalConjectures.ErdosProblems.«291»":"# Erdős Problem 291\n\n*References:*\n- [erdosproblems.com/291](https://www.erdosproblems.com/291)\n- [ErGr80, p.34] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\n  theory. Monographies de L'Enseignement Mathematique (1980).\n- [Sh16] P. Shiu, The denominators of harmonic numbers. arXiv:1607.02863 (2016).\n- [WuYa22] Wu, Bing-Ling and Yan, Xiao-Hui, On the denominators of harmonic numbers. {IV}. C. R.\n  Math. Acad. Sci. Paris (2022), 53--57.\n","FormalConjectures.ErdosProblems.«295»":"# Erdős Problem 295\n\n*Reference:* [erdosproblems.com/295](https://www.erdosproblems.com/295)\n","FormalConjectures.ErdosProblems.«296»":"# Erdős Problem 296\n\n*References:*\n- [erdosproblems.com/296](https://www.erdosproblems.com/296)\n- [Bl21] Bloom, T. F., *On a density conjecture about unit fractions*.\n  arXiv:2112.03726 (2021).\n","FormalConjectures.ErdosProblems.«298»":"# Erdős Problem 298\n\n*References:*\n- [erdosproblems.com/298](https://www.erdosproblems.com/298)\n- [Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).\n","FormalConjectures.ErdosProblems.«299»":"# Erdős Problem 299\n\n*References:*\n- [erdosproblems.com/298](https://www.erdosproblems.com/298)\n- [erdosproblems.com/299](https://www.erdosproblems.com/299)\n- [Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).\n","FormalConjectures.ErdosProblems.«2»":"# Erdős Problem 2\n\n*Reference:* [erdosproblems.com/2](https://www.erdosproblems.com/2)\n\nErdős asked whether the smallest modulus in a distinct covering system can be arbitrarily large.\nHough proved that the answer is no, and Balister, Bollobás, Morris, Sahasrabudhe, and Tiba later\ngave a simpler proof with an improved explicit upper bound.\n","FormalConjectures.ErdosProblems.«302»":"# Erdős Problem 302\n\n*References:*\n- [erdosproblems.com/302](https://www.erdosproblems.com/302)\n- [BrRo91] Brown, Tom C. and Rödl, Voijtech, Monochromatic solutions to equations with unit\n  fractions. Bull. Austral. Math. Soc. (1991), 387-392.\n- [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\n  theory. Monographies de L'Enseignement Mathematique (1980).\n- [va25](https://github.com/Woett/Mathematical-shorts/blob/main/Two-colouring%20and%20density%20lead%20to%20solutions%20to%20an%20equation%20in%20unit%20fractions.pdf)\n","FormalConjectures.ErdosProblems.«303»":"# Erdős Problem 303\n\n*References:*\n- [erdosproblems.com/303](https://www.erdosproblems.com/303)\n- [BrRo91] Brown, Tom C. and Rödl, Voijtech, Monochromatic solutions to equations with unit\n  fractions. Bull. Austral. Math. Soc. (1991), 387-392.\n","FormalConjectures.ErdosProblems.«304»":"# Erdős Problem 304\n*Reference:* [erdosproblems.com/304](https://www.erdosproblems.com/304)\n","FormalConjectures.ErdosProblems.«306»":"# Erdős Problem 306\n\n*Reference:* [erdosproblems.com/306](https://www.erdosproblems.com/306)\n","FormalConjectures.ErdosProblems.«307»":"# Erdős Problem 307\n\n*Reference:* [erdosproblems.com/307](https://www.erdosproblems.com/307)\n","FormalConjectures.ErdosProblems.«30»":"# Erdős Problem 30\n\n*Reference:* [erdosproblems.com/30](https://www.erdosproblems.com/30)\n","FormalConjectures.ErdosProblems.«312»":"# Erdős Problem 312\n\n*Reference:* [erdosproblems.com/312](https://www.erdosproblems.com/312)\n","FormalConjectures.ErdosProblems.«313»":"# Erdős Problem 313\n\n*References:*\n- [erdosproblems.com/313](https://www.erdosproblems.com/313)\n- [A54377](https://oeis.org/A54377) (Primary pseudoperfect numbers)\n","FormalConjectures.ErdosProblems.«314»":"# Erdős Problem 314\n\n*References:*\n- [erdosproblems.com/314](https://www.erdosproblems.com/314)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [LiSt24] J. Lim and Steinerberger, S., *On differences of two harmonic numbers*.\n  arXiv:2405.11354 (2024).\n","FormalConjectures.ErdosProblems.«315»":"# Erdős Problem 315\n\n*References:*\n- [erdosproblems.com/315](https://www.erdosproblems.com/315)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [Ka25] Y. Kamio, *Asymptotic analysis of infinite decompositions of a unit fraction into unit\n  fractions*. arXiv:2503.02317 (2025).\n- [LiTa25] Z. Li and Q. Tang, *On a conjecture of Erdős and Graham about the Sylvester's\n  sequence*. arXiv:2503.12277 (2025).\n","FormalConjectures.ErdosProblems.«316»":"# Erdős Problem 316\n\n*References:*\n- [erdosproblems.com/316](https://www.erdosproblems.com/316)\n- [Sa97] Sándor, Csaba, On a problem of Erdős. J. Number Theory (1997), 203-210.\n","FormalConjectures.ErdosProblems.«317»":"# Erdős Problem 317\n\n*Reference:* [erdosproblems.com/317](https://www.erdosproblems.com/317)\n","FormalConjectures.ErdosProblems.«318»":"# Erdős Problem 318\n\n*References:*\n  - [erdosproblems.com/318](https://www.erdosproblems.com/318)\n  - [ErSt75] Erdős, P. and Straus, E. G., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 183.\n  - [Sa75] Sattler, R., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 184-189.\n  - [Sa82b] Sattler, R., On Erdős property P₁ for the arithmetical sequence. Nederl. Akad. Wetensch.\n    Indag. Math. (1982), 347--352.\n  - [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\n    theory. Monographies de L'Enseignement Mathematique (1980).\n  - [La26] D. Larsen, [Erdős problem 318](https://github.com/Larsen-Daniel/Erdos-318/blob/main/318.pdf) (2026)\n","FormalConjectures.ErdosProblems.«319»":"# Erdős Problem 319\n\n*Reference:* [erdosproblems.com/319](https://www.erdosproblems.com/319)\n","FormalConjectures.ErdosProblems.«31»":"# Erdős Problem 31\n\n*References:*\n- [erdosproblems.com/31](https://www.erdosproblems.com/31)\n- [Er56] Erdős, P., *Problems and results in additive number theory*. Colloque sur la Théorie des\n  Nombres, Bruxelles, 1955 (1956), 127-137.\n- [Er59] Erdős, P., *Über einige Probleme der additiven Zahlentheorie*. Sammelband zu Ehren des 250.\n  Geburtstages Leonhard Eulers (1959), 116-119.\n- [Er65b] Erdős, Paul, *Some recent advances and current problems in number theory*. Lectures on\n  Modern Mathematics, Vol. III (1965), 196-244.\n- [Er73] Erdős, P., *Problems and results on combinatorial number theory*. A survey of combinatorial\n  theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n- [Lo54] Lorentz, G. G., *On a problem of additive number theory*. Proc. Amer. Math. Soc.\n  (1954), 838-841.\n","FormalConjectures.ErdosProblems.«321»":"# Erdős Problem 321\n\n*Reference:* [erdosproblems.com/321](https://www.erdosproblems.com/321)\n","FormalConjectures.ErdosProblems.«322»":"# Erdős Problem 322\n\n*References:*\n- [erdosproblems.com/322](https://www.erdosproblems.com/322)\n- [Er36] Erdős, Paul, *On the Representation of an Integer as the Sum of k k-th Powers*. J. London\n  Math. Soc. (1936), 133-136.\n- [Er65b] Erdős, Paul, *Some recent advances and current problems in number theory*. Lectures on\n  Modern Mathematics, Vol. III (1965), 196-244.\n- [Gu04] Guy, Richard K., *Unsolved problems in number theory*. (2004), xviii+437.\n- [Ma36] Mahler, Kurt, *Note on Hypothesis K of Hardy and Littlewood*. J. London Math. Soc. (1936),\n  136-138.\n","FormalConjectures.ErdosProblems.«323»":"# Erdős Problem 323\n\n*Reference:* [erdosproblems.com/323](https://www.erdosproblems.com/323)\n","FormalConjectures.ErdosProblems.«324»":"# Erdős Problem 324\n\n*Reference:* [erdosproblems.com/324](https://www.erdosproblems.com/324)\n","FormalConjectures.ErdosProblems.«325»":"# Erdős Problem 325\n*Reference:* [erdosproblems.com/325](https://www.erdosproblems.com/325)\n","FormalConjectures.ErdosProblems.«326»":"# Erdős Problem 326\n\n*Reference:* [erdosproblems.com/326](https://www.erdosproblems.com/326)\n","FormalConjectures.ErdosProblems.«328»":"# Erdős Problem 328\n\n*References:*\n- [erdosproblems.com/328](https://www.erdosproblems.com/328)\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*.\n  Ann. Discrete Math. (1980), 89-115.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial\n  number theory*. Monographies de L'Enseignement Mathématique (1980).\n- [Er80e] Erdős, P., *Some applications of Ramsey's theorem to additive number theory*.\n  European J. Combin. (1980), 43-46.\n- [NeRo85] J. Nešetřil and V. Rödl, *Two proofs in combinatorial number theory*.\n  Proc. Amer. Math. Soc. (1985), 185-188.\n","FormalConjectures.ErdosProblems.«329»":"# Erdős Problem 329: Maximum Density of Sidon Sets\n\n*References:*\n- [erdosproblems.com/329](https://www.erdosproblems.com/329)\n- [AlMi25] B. Alexeev and D. G. Mixon, Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof. [arXiv:2510.19804](https://arxiv.org/abs/2510.19804) (2025).\n- [Ha47] Hall, Jr., Marshall, Cyclic projective planes. Duke Math. J. (1947), 1079--1090.\n","FormalConjectures.ErdosProblems.«32»":"# Erdős Problem 32\n\n*References:*\n* [erdosproblems.com/32](https://www.erdosproblems.com/32)\n* [Erd54] Erdős, Paul, Some results on additive number theory. Proc. Amer. Math. Soc. (1954),\n 847-853.\n* [Guy04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437\n* [Ru98c] Ruzsa, Imre Z., On the additive completion of primes. Acta Arith. (1998), 269-275.\n","FormalConjectures.ErdosProblems.«330»":"# Erdős Problem 330\n\n*Reference:* [erdosproblems.com/330](https://www.erdosproblems.com/330)\n","FormalConjectures.ErdosProblems.«331»":"# Erdős Problem 331\n\n*Reference:* [erdosproblems.com/331](https://www.erdosproblems.com/331)\n","FormalConjectures.ErdosProblems.«332»":"# Erdős Problem 332\n\n*Reference:* [erdosproblems.com/332](https://www.erdosproblems.com/332)\n","FormalConjectures.ErdosProblems.«333»":"# Erdős Problem 333\n\n*References:*\n- [erdosproblems.com/333](https://www.erdosproblems.com/333)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [ErNe77] Erdős, P. and Newman, D. J., *Bases for sets of integers*. J. Number Theory (1977),\n  420-425.\n","FormalConjectures.ErdosProblems.«337»":"# Erdős Problem 337\n\n*References:*\n- [erdosproblems.com/337](https://www.erdosproblems.com/337)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [ErGr80b] Erdős, P. and Graham, R. L., *On bases with an exact order*. Acta Arith. (1980),\n  201-207.\n- [RT85] Ruzsa, I. Z. and Turjányi, S., *A note on additive bases of integers*. Publ. Math.\n  Debrecen (1985), 101-104.\n- [Tu84] Turjányi, S., *A note on basis sequences*. Topics in classical number theory, Vol. I, II\n  (Budapest, 1981) (1984), 1571-1576.\n","FormalConjectures.ErdosProblems.«33»":"# Erdős Problem 33\n\n*Reference:* [erdosproblems.com/33](https://www.erdosproblems.com/33)\n","FormalConjectures.ErdosProblems.«340»":"# Erdős Problem 340\n\n*Reference:* [erdosproblems.com/340](https://www.erdosproblems.com/340)\n","FormalConjectures.ErdosProblems.«341»":"# Erdős Problem 341\n\n*References:*\n* [erdosproblems.com/341](https://www.erdosproblems.com/341)\n* [Ben Green's Open Problem 7](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.1)\n","FormalConjectures.ErdosProblems.«342»":"# Erdős Problem 342\n\n*References:*\n- [erdosproblems.com/342](https://www.erdosproblems.com/342)\n- [Ben Green's Open Problem 7](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.7)\n- [OEIS A002858](https://oeis.org/A002858)\n- [Gu04] Guy, Richard K., *Unsolved problems in number theory* (2004), xviii+437.\n","FormalConjectures.ErdosProblems.«346»":"# Erdős Problem 346\n\n*References:*\n - [erdosproblems.com/346](https://www.erdosproblems.com/346)\n - [Gr64d] Graham, R. L., A property of Fibonacci numbers. Fibonacci Quart. (1964), 1-10.\n - [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\n    theory. Monographies de L'Enseignement Mathematique (1980).\n -\n","FormalConjectures.ErdosProblems.«347»":"# Erdős Problem 347\n\n*Reference:* [erdosproblems.com/347](https://www.erdosproblems.com/347)\n","FormalConjectures.ErdosProblems.«348»":"# Erdős Problem 348\n\n*Reference:* [erdosproblems.com/348](https://www.erdosproblems.com/348)\n","FormalConjectures.ErdosProblems.«349»":"# Erdős Problem 349\n\n*Reference:* [erdosproblems.com/349](https://www.erdosproblems.com/349)\n","FormalConjectures.ErdosProblems.«34»":"# Erdős Problem 34\n\n*References:*\n- [erdosproblems.com/34](https://www.erdosproblems.com/34)\n- [Er77c] Erdős, Paul, *Problems and results on combinatorial number theory. III*. Number theory day\n  (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [He86] Hegyvári, N., *On consecutive sums in sequences*. Acta Math. Hungar. (1986),\n  193--200.\n- [Ko15] Konieczny, J., *On consecutive sums in permutations*. arXiv:1504.07156 (2015).\n","FormalConjectures.ErdosProblems.«350»":"# Erdős Problem 350\n\n*References:*\n- [erdosproblems.com/350](https://www.erdosproblems.com/350)\n- [BeEr74] Benkoski, S. J. and Erdős, P., On weird and pseudoperfect numbers. Math. Comp. (1974),\n  617-623.\n- [HSS77] Hanson, F. and Steele, J. M. and Stenger, F., Distinct sums over subsets. Proc. Amer.\n  Math. Soc. (1977), 179-180.\n","FormalConjectures.ErdosProblems.«351»":"# Erdős Problem 351\n\n*Reference:* [erdosproblems.com/351](https://www.erdosproblems.com/351)\n","FormalConjectures.ErdosProblems.«352»":"# Erdős Problem 352\n\n*Reference:* [erdosproblems.com/352](https://www.erdosproblems.com/352)\n","FormalConjectures.ErdosProblems.«353»":"# Erdős Problem 353\n\n*References:*\n- [erdosproblems.com/353](https://www.erdosproblems.com/353)\n- [Er83d] Erdős, Paul, *Some combinatorial, geometric and set theoretic problems in measure\n  theory*. Measure Theory, Oberwolfach 1983 (1984), 321-327.\n- [Ko23] Kovač, V., *Coloring and density theorems for configurations of a given volume*.\n  arXiv:2309.09973 (2023).\n- [KoPr24] Kovač, V. and B. Predojević, *Polygons of unit area with vertices in sets of infinite\n  planar measure*. arXiv:2412.11725 (2024).\n- [Ko25] J. Koizumi, *Isosceles trapezoids of unit area with vertices in sets of infinite planar\n  measure*. arXiv:2501.01914 (2025).\n","FormalConjectures.ErdosProblems.«354»":"# Erdős Problem 354\n*Reference:* [erdosproblems.com/354](https://www.erdosproblems.com/354)\n\n","FormalConjectures.ErdosProblems.«355»":"# Erdős Problem 355\n\n*References:*\n- [erdosproblems.com/355](https://www.erdosproblems.com/355)\n- [DoKo25] W. van Doorn and V. Kovač, Lacunary sequences whose reciprocal sums represent all\n  rationals in an interval. arXiv:2509.24971 (2025).\n","FormalConjectures.ErdosProblems.«357»":"# Erdős Problem 357\n\n*Reference:* [erdosproblems.com/357](https://www.erdosproblems.com/357)\n","FormalConjectures.ErdosProblems.«358»":"# Erdős Problem 358\n\n*References:*\n- [erdosproblems.com/358](https://www.erdosproblems.com/358)\n- [Ta26] T. Tao, [Erdős problem 358](https://terrytao.wordpress.com/wp-content/uploads/2026/02/erdos-358-2.pdf) (2026)\n","FormalConjectures.ErdosProblems.«359»":"# Erdős Problem 359\n\n*Reference:* [erdosproblems.com/359](https://www.erdosproblems.com/359)\n","FormalConjectures.ErdosProblems.«361»":"# Erdős Problem 361\n\n*Reference:* [erdosproblems.com/361](https://www.erdosproblems.com/361)\n","FormalConjectures.ErdosProblems.«363»":"# Erdős Problem 363\n\n*References:*\n- [erdosproblems.com/363](https://www.erdosproblems.com/363)\n- [BaBe07] Bauer, Mark and Bennett, Michael A., *On a question of Erd\\H{o}s\n  and Graham*. Enseign. Math. (2) (2007), 259--264.\n- [BeVL12] Bennett, Michael A. and Van Luijk, Ronald, *Squares from blocks of\n  consecutive integers: a problem of Erd\\H{o}s and Graham*. Indag. Math. (N.S.)\n  (2012), 123--127.\n- [Ul05] Ulas, Maciej, *On products of disjoint blocks of consecutive integers*.\n  Enseign. Math. (2) (2005), 331--334.\n","FormalConjectures.ErdosProblems.«364»":"# Erdős Problem 364\n\n*Reference:* [erdosproblems.com/364](https://www.erdosproblems.com/364)\n","FormalConjectures.ErdosProblems.«366»":"# Erdős Problem 366\n\n*Reference:* [erdosproblems.com/366](https://www.erdosproblems.com/366)\n","FormalConjectures.ErdosProblems.«367»":"# Erdős Problem 367\n\n*References:*\n- [erdosproblems.com/367](https://www.erdosproblems.com/367)\n- [ErGr80] P. Erdős and R. L. Graham, *Old and New Problems and Results in Combinatorial Number Theory*, L'Enseignement Mathématique (1980).\n","FormalConjectures.ErdosProblems.«369»":"# Erdős Problem 369\n\n*References:*\n- [erdosproblems.com/369](https://www.erdosproblems.com/369)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [EgSe76] Eggleton, R. B. and Selfridge, J. L., *Consecutive integers with no large prime factors*.\n  J. Austral. Math. Soc. Ser. A (1976), 1--11.\n- [BFMW20] Bober, J. W. and Fretwell, D. and Martin, G. and Wooley, T. D., *Smooth values of\n  polynomials*. J. Aust. Math. Soc. (2020), 245--261.\n- [BaWo98] Balog, Antal and Wooley, Trevor D., *On strings of consecutive integers with no large\n  prime factors*. J. Austral. Math. Soc. Ser. A (1998), 266-276.\n","FormalConjectures.ErdosProblems.«36»":"# Erdős Problem 36\n\n*References:*\n - [erdosproblems.com/36](https://www.erdosproblems.com/36)\n - [Wikipedial: Minimum overlap problem](https://en.wikipedia.org/wiki/Minimum_overlap_problem)\n","FormalConjectures.ErdosProblems.«370»":"# Erdős Problem 370\n\n*Reference:* [erdosproblems.com/370](https://www.erdosproblems.com/370)\n","FormalConjectures.ErdosProblems.«371»":"# Erdős Problem 371\n\n*Reference:* [erdosproblems.com/371](https://www.erdosproblems.com/371)\n","FormalConjectures.ErdosProblems.«372»":"# Erdős Problem 372\n\n*Reference:* [erdosproblems.com/372](https://www.erdosproblems.com/372)\n\nConjectured by Erdős and Pomerance. Proved by Balog, who showed the stronger quantitative result\nthat this holds for $\\gg \\sqrt{x}$ many $n\\leq x$, for all large $x$.\n","FormalConjectures.ErdosProblems.«373»":"# Erdős Problem 373\n\n*Reference:* [erdosproblems.com/373](https://www.erdosproblems.com/373)\n","FormalConjectures.ErdosProblems.«375»":"# Erdős Problem 375\n\n*References:*\n - [erdosproblems.com/375](https://www.erdosproblems.com/375)\n - [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\n    theory. Monographies de L'Enseignement Mathematique (1980).\n - [RST75] Ramachandra, K. and Shorey, T. N. and Tijdeman, R., On Grimm's problem relating to\n    factorisation of a block of consecutive integers. J. Reine Angew. Math. (1975), 109-124.\n -\n","FormalConjectures.ErdosProblems.«376»":"# Erdős Problem 376\n\n*Reference:* [erdosproblems.com/376](https://www.erdosproblems.com/376)\n","FormalConjectures.ErdosProblems.«377»":"# Erdős Problem 377\n\n*Reference:* [erdosproblems.com/377](https://www.erdosproblems.com/377)\n","FormalConjectures.ErdosProblems.«379»":"# Erdős Problem 379\n\n*Reference:* [erdosproblems.com/379](https://www.erdosproblems.com/379)\n","FormalConjectures.ErdosProblems.«383»":"# Erdős Problem 383\n\n*Reference:* [erdosproblems.com/383](https://www.erdosproblems.com/383)\n","FormalConjectures.ErdosProblems.«385»":"# Erdős Problem 385\n\n*Reference:* [erdosproblems.com/385](https://www.erdosproblems.com/385)\n","FormalConjectures.ErdosProblems.«386»":"# Erdős Problem 386\n*Reference:* [erdosproblems.com/386](https://www.erdosproblems.com/386)\n","FormalConjectures.ErdosProblems.«387»":"# Erdős Problem 387\n\n*References:*\n - [erdosproblems.com/387](https://www.erdosproblems.com/387)\n - [ErGr76b] Erdős, P. and Graham, R. L., *On the prime factors of\n   ${n \\choose k}$*. Fibonacci Quart. (1976), 348-352.\n - [Er78g] Erdős, Pál, *On prime factors of binomial coefficients. II*. Mat. Lapok\n   (1978/82), 307-316.\n - [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial\n   number theory*. Monographies de L'Enseignement Mathematique (1980).\n - [Sc58] Schinzel, A., *Sur un problème de P. Erdős*. Colloq. Math. (1958), 198-204.\n - [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n - [Fa66] Faulkner, M. \"On a theorem of Sylvester and Schur.\" Journal of the London Mathematical\n    Society 1.1 (1966): 107-110.\n - [BNPZ26] Bui, H., Naprienko, S., Pratt, K., and Zaharescu, A. Binomial coefficients with\n    divisors avoiding an interval. arXiv:2605.21221 (2026).\n","FormalConjectures.ErdosProblems.«389»":"# Erdős Problem 389\n\n*Reference:* [erdosproblems.com/389](https://www.erdosproblems.com/389)\n","FormalConjectures.ErdosProblems.«38»":"# Erdős Problem 38\n\n*Reference:*\n- [erdosproblems.com/38](https://www.erdosproblems.com/38)\n- [Er56](Erdős, P., Problems and results in additive number theory.\n  Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137.)\n","FormalConjectures.ErdosProblems.«390»":"# Erdős Problem 390\n\n*References:*\n - [erdosproblems.com/390](https://www.erdosproblems.com/390)\n - [EGS82] Erdős, P., R. K. Guy, and J. L. Selfridge. \"Another Property of 239 and some related\n    questions.\" Congr. Numer. 34 (1982): 243-257.\n -\n","FormalConjectures.ErdosProblems.«392»":"# Erdős Problem 392\n\n*Reference:* [erdosproblems.com/392](https://www.erdosproblems.com/392)\n","FormalConjectures.ErdosProblems.«394»":"# Erdős Problem 394\n\n*References:*\n- [erdosproblems.com/394](https://www.erdosproblems.com/394)\n- [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\n  theory. Monographies de L'Enseignement Mathematique (1980).\n- [ErHa78] Erdős, P. and Hall, R. R., On some unconventional problems on the divisors of integers.\n  J. Austral. Math. Soc. Ser. A (1978), 479--485.\n","FormalConjectures.ErdosProblems.«396»":"# Erdős Problem 396\n\n*Reference:* [erdosproblems.com/396](https://www.erdosproblems.com/396)\n","FormalConjectures.ErdosProblems.«397»":"# Erdős Problem 397\n\n*References:*\n- [erdosproblems.com/397](https://www.erdosproblems.com/397)\n- [MathOverflow] (https://mathoverflow.net/questions/138209/product-of-central-binomial-coefficients)\n","FormalConjectures.ErdosProblems.«398»":"# Erdős Problem 398\n\n*References:*\n - [erdosproblems.com/398](https://www.erdosproblems.com/398)\n - [Wikipedia: Brocard's problem](https://en.wikipedia.org/wiki/Brocard%27s_problem)\n","FormalConjectures.ErdosProblems.«399»":"# Erdős Problem 399\n\nIs it true that there are no solutions to $n! = x^k \\pm y^k$ with $x,y,n \\in \\mathbb{N}$,\nwith $xy > 1$ and $k > 2$?\n\n*References:*\n - [erdosproblems.com/399](https://www.erdosproblems.com/399)\n- [Br32] Breusch, Robert, Zur Verallgemeinerung des Bertrandschen Postulates, da\\ss zwischen $x$\n  und 2 $x$ stets Primzahlen liegen. Math. Z. (1932), 505--526.\n- [ErOb37] Erdős, P. and Obláth, R., \\\"Über diophantische Gleichungen der Form $n!=x^p+y^p$ und\n  $n!\\pmd m!=x^p$. Acta Litt. ac Sci. Reg. Univ. Hung. Fr.-Jos., Sect. Sci. Math. (1937), 241-255.\n- [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n- [PoSh73] Pollack, Richard M. and Shapiro, Harold N., The next to last case of a factorial\n  diophantine equation. Comm. Pure Appl. Math. (1973), 313-325.\n","FormalConjectures.ErdosProblems.«39»":"# Erdős Problem 39\n\n*Reference:* [erdosproblems.com/39](https://www.erdosproblems.com/39)\n","FormalConjectures.ErdosProblems.«3»":"# Erdős Problem 3\n\n*Reference:* [erdosproblems.com/3](https://www.erdosproblems.com/3)\n","FormalConjectures.ErdosProblems.«400»":"# Erdős Problem 400\n\n*Reference:* [erdosproblems.com/400](https://www.erdosproblems.com/400)\n","FormalConjectures.ErdosProblems.«401»":"# Erdős Problem 401\n\n*References:*\n- [erdosproblems.com/401](https://www.erdosproblems.com/401)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n","FormalConjectures.ErdosProblems.«402»":"# Erdős Problem 402\n\n*Reference:* [erdosproblems.com/402](https://www.erdosproblems.com/402)\n","FormalConjectures.ErdosProblems.«403»":"# Erdős Problem 403\n\n*References:*\n- [erdosproblems.com/403](https://www.erdosproblems.com/403)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial\n  number theory*. Monographies de L'Enseignement Mathématique (1980).\n- [Li76] Lin, S., *On two problems of Erdős concerning sums of distinct factorials*.\n  Bell Laboratories internal memorandum (1960).\n","FormalConjectures.ErdosProblems.«406»":"# Erdős Problem 406\n\n*Reference:* [erdosproblems.com/406](https://www.erdosproblems.com/406)\n","FormalConjectures.ErdosProblems.«409»":"# Erdős Problem 409\n\n*Reference:* [erdosproblems.com/409](https://www.erdosproblems.com/409)\n","FormalConjectures.ErdosProblems.«40»":"# Erdős Problem 40\n\n*Reference:* [erdosproblems.com/40](https://www.erdosproblems.com/40)\n","FormalConjectures.ErdosProblems.«410»":"# Erdős Problem 410\n\n*Reference:* [erdosproblems.com/410](https://www.erdosproblems.com/410)\n","FormalConjectures.ErdosProblems.«412»":"# Erdős Problem 412\n\n*Reference:* [erdosproblems.com/412](https://www.erdosproblems.com/412)\n\nReviewed by @b-mehta on 2025-05-27\n","FormalConjectures.ErdosProblems.«413»":"# Erdős Problem 413\n\n*References:*\n- [erdosproblems.com/413](https://www.erdosproblems.com/413)\n- [A5236](https://oeis.org/A5236)\n\nErdős called a natural number `n` a *barrier* for `ω`, the number of distinct prime divisors,\nif `m + ω(m) ≤ n` for all `m < n`. He believed there should be infinitely many such barriers, and\neven posed a relaxed variant asking whether there is some `ε > 0` for which infinitely many `n`\nsatisfy `m + ε · ω(m) ≤ n` for every `m < n`.\n","FormalConjectures.ErdosProblems.«414»":"# Erdős Problem 414\n\n*Reference:* [erdosproblems.com/414](https://www.erdosproblems.com/414)\n\n","FormalConjectures.ErdosProblems.«416»":"# Erdős Problem 416\n\n*Reference:* [erdosproblems.com/416](https://www.erdosproblems.com/416)\n","FormalConjectures.ErdosProblems.«417»":"# Erdős Problem 417\n\n*References:*\n- [erdosproblems.com/417](https://www.erdosproblems.com/417)\n- [Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number\n  theory. Number theory (Eger, 1996) (1998), 169-180.\n","FormalConjectures.ErdosProblems.«418»":"# Erdős Problem 418\n\n*References:*\n- [erdosproblems.com/418](https://www.erdosproblems.com/418)\n- [Wikipedia: Noncototient](https://en.wikipedia.org/wiki/Noncototient)\n- [BaLu05] Banks, William D. and Luca, Florian, Nonaliquots and {R}obbins numbers. Colloq. Math.\n  (2005), 27--32.\n- [BrSc95] Browkin, J. and Schinzel, A., On integers not of the form {$n-\\phi(n)$}. Colloq. Math.\n  (1995), 55-58.\n- [ChZh11] Chen, Yong-Gao and Zhao, Qing-Qing, Nonaliquot numbers. Publ. Math. Debrecen (2011),\n  439--442.\n- [Er73b] Erdős, P., \\\"Über die Zahlen der Form $\\sigma (n)-n$ und $n-\\phi(n)$. Elem. Math.\n  (1973), 83-86.\n- [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n- [PoPo16] Pollack, Paul and Pomerance, Carl, Some problems of Erdős on the sum-of-divisors\n  function. Trans. Amer. Math. Soc. Ser. B (2016), 1-26.\n","FormalConjectures.ErdosProblems.«419»":"# Erdős Problem 419\n\n*References:*\n- [erdosproblems.com/419](https://www.erdosproblems.com/419)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [EGIP96] Erdős, Paul and Graham, S. W. and Ivić, Aleksandar and Pomerance, Carl,\n  *On the number of divisors of $n!$*. (1996), 337--355.\n","FormalConjectures.ErdosProblems.«41»":"# Erdős Problem 41\n\n*Reference:* [erdosproblems.com/41](https://www.erdosproblems.com/41)\n","FormalConjectures.ErdosProblems.«421»":"# Erdős Problem 421\n\n*Reference:* [erdosproblems.com/421](https://www.erdosproblems.com/421)\n","FormalConjectures.ErdosProblems.«422»":"# Erdős Problem 422\n\n*Reference:* [erdosproblems.com/422](https://www.erdosproblems.com/422)\n","FormalConjectures.ErdosProblems.«423»":"# Erdős Problem 423\n\n*References:*\n- [erdosproblems.com/423](https://www.erdosproblems.com/423)\n- [Er77c] Erdős, P., *Problems and results on combinatorial number theory. III*,\n  Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), 1977, pp. 43–72.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial\n  number theory*, Monographies de L'Enseignement Mathématique (1980).\n- [Cu25] Cushman, A., *A Note on the Sum-Product Problem and the Convex Sumset Problem*.\n  arXiv:2512.13849 (2025).\n- [Ta26] Tang, Q., *The Hofstadter consecutive-sum sequence omits infinitely many positive\n  integers*. arXiv:2603.09939 (2026).\n- [Bolan] Bolan, M., *Hofstader–Ulam Sequence*,\n  https://github.com/mjtb49/HofstaderUlam/blob/main/HofstaderUlamSequence.pdf\n- [OEIS A005243](https://oeis.org/A005243)\n","FormalConjectures.ErdosProblems.«424»":"# Erdős Problem 424: Sequence generated by $a_i a_j - 1$\n\n*References:*\n - [erdosproblems.com/424](https://www.erdosproblems.com/424)\n - [A5244](https://oeis.org/A5244)\n - [Ben Green's Open Problem 63](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 63)\n","FormalConjectures.ErdosProblems.«426»":"# Erdős Problem 426\n\n*References:*\n- [erdosproblems.com/426](https://www.erdosproblems.com/426)\n- [Er76b] Erdős, P., *Problems and results in graph theory and combinatorial analysis*.\n  Proceedings of the Fifth British Combinatorial Conference (1976), 169-192.\n- [EnEr72] Entringer, R. C. and Erdős, Paul, *On the number of unique subgraphs of a graph*.\n  J. Combinatorial Theory Ser. B (1972), 112-115.\n- [HaSc73] Harary, Frank and Schwenk, Allen J., *On the number of unique subgraphs*.\n  J. Combinatorial Theory Ser. B (1973), 156-160.\n- [Br75] Brouwer, A. E., *Note: \"On the number of unique subgraphs of a graph\"\n  (J. Combinatorial Theory Ser. B 13 (1972), 112-115) by R. C. Entringer and P. Erdős*.\n  J. Combinatorial Theory Ser. B (1975), 184-185.\n- [BrCh24] Bradač, D. and Christoph, M., *Unique subgraphs are rare*. arXiv:2410.16233 (2024).\n","FormalConjectures.ErdosProblems.«427»":"# Erdős Problem 427\n\n*Reference:* [erdosproblems.com/427](https://www.erdosproblems.com/427)\n","FormalConjectures.ErdosProblems.«428»":"# Erdős Problem 428\n\n*Reference:* [erdosproblems.com/428](https://www.erdosproblems.com/428)\n","FormalConjectures.ErdosProblems.«429»":"# Erdős Problem 429\n\n*References:*\n- [erdosproblems.com/429](https://www.erdosproblems.com/429)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*.\n  Ann. Discrete Math. (1980), 89-115.\n- [We24] D. Weisenberg, *Sparse Admissible Sets and a Problem of Erdős and Graham*.\n  Integers (2024).\n","FormalConjectures.ErdosProblems.«42»":"# Erdős Problem 42: Maximal Sidon Sets and Disjoint Difference Sets\n\n*Reference:* [erdosproblems.com/42](https://www.erdosproblems.com/42)\n\nThis problem asks whether maximal Sidon sets can coexist with other Sidon sets that have\ndisjoint difference sets (apart from 0).\n","FormalConjectures.ErdosProblems.«431»":"# Erdős Problem 431\n\n*Reference:* [erdosproblems.com/431](https://www.erdosproblems.com/431)\n","FormalConjectures.ErdosProblems.«433»":"# Erdős Problem 433\n\n*References:*\n- [Erdős Problem 433](https://www.erdosproblems.com/433)\n- P. Erdős and R. L. Graham, *On a linear Diophantine problem of Frobenius*,\n  Acta Arithmetica 21 (1972), 399–408.\n- J. Dixmier, *Proof of a conjecture by Erdős and Graham on Frobenius' coin problem*,\n  Proceedings of the AMS 109 (1990), 567–577.\n","FormalConjectures.ErdosProblems.«434»":"# Erdős Problem 434\n\n*References:*\n- [erdosproblems.com/434](https://www.erdosproblems.com/434)\n- [Ki02] Kiss, G., On the extremal Frobenius problem in a new aspect. Ann. Univ. Sci.\n  Budapest. Eötvös Sect. Math. (2002), 139–142.\n","FormalConjectures.ErdosProblems.«435»":"# Erdős Problem 435\n\n*References:*\n- [erdosproblems.com/435](https://www.erdosproblems.com/435)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [HwSo24] W. Hwang and K. Song, *The Frobenius problem for Numerical Semigroups generated by\n  binomial coefficients*. arXiv:2412.17882 (2024).\n","FormalConjectures.ErdosProblems.«43»":"# Erdős Problem 43\n\n*Reference:* [erdosproblems.com/43](https://www.erdosproblems.com/43)\n","FormalConjectures.ErdosProblems.«442»":"# Erdős Problem 442\n\n*Reference:* [erdosproblems.com/442](https://www.erdosproblems.com/442)\n","FormalConjectures.ErdosProblems.«443»":"# Erdős Problem 443\n\n*References:*\n- [erdosproblems.com/443](https://www.erdosproblems.com/443)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [He25] Hegyvári, Norbert, *An elementary question of Erdős and Graham*.\n  arXiv:2503.24201 (2025).\n","FormalConjectures.ErdosProblems.«445»":"# Erdős Problem 445\n\n*References:*\n- [erdosproblems.com/445](https://www.erdosproblems.com/445)\n- [He00] Heath-Brown, D. R., Arithmetic applications of {K}loosterman sums. Nieuw Arch. Wiskd. (5)\n  (2000), 380--384.\n- [MathOverflow](https://mathoverflow.net/questions/69509/small-residue-classes-with-small-reciprocal)\n","FormalConjectures.ErdosProblems.«447»":"# Erdős Problem 447\n\n*References:*\n- [erdosproblems.com/447](https://www.erdosproblems.com/447)\n- [Er65b] Erdős, Paul, *Some recent advances and current problems in number theory*. Lectures on\n  Modern Mathematics, Vol. III (1965), 196-244.\n- [Kl71] Kleitman, Daniel, *Collections of subsets containing no two sets and their union*.\n  Proceedings of the LA Meeting AMS (1971), 153-155.\n","FormalConjectures.ErdosProblems.«448»":"# Erdős Problem 448\n\n*References:*\n- [erdosproblems.com/448](https://www.erdosproblems.com/448)\n- [OEIS A397433](https://oeis.org/A397433): the integer sequence $\\tau^+(n)$\n  (the constant $\\alpha$ in Ford's asymptotic is [OEIS A074738](https://oeis.org/A074738)).\n- [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.\n- [Er79e] Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82.\n- [HaTe88] Hall, Richard R. and Tenenbaum, Gérald, Divisors. (1988), xvi+167.\n- [ErTe81] Erdős, P., Tenenbaum, G., *Sur la structure de la suite des diviseurs d'un entier.*\n  Ann. Inst. Fourier (Grenoble) **31** (1981), 17–37.\n- [Fo08] Ford, Kevin, *The distribution of integers with a divisor in a given interval.*\n  Ann. of Math. (2) **168** (2008), 367–433.\n","FormalConjectures.ErdosProblems.«44»":"# Erdős Problem 44: Extending Sidon Sets\n\n*Reference:* [erdosproblems.com/44](https://www.erdosproblems.com/44)\n","FormalConjectures.ErdosProblems.«450»":"# Erdős Problem 450\n\n*Reference:* [erdosproblems.com/450](https://www.erdosproblems.com/450)\n","FormalConjectures.ErdosProblems.«452»":"# Erdős Problem 452\n\n*Reference:* [erdosproblems.com/452](https://www.erdosproblems.com/452)\n","FormalConjectures.ErdosProblems.«453»":"# Erdős Problem 453\n\n*References:*\n- [erdosproblems.com/453](https://www.erdosproblems.com/453)\n- [Er70b] Erdős, P., *Some applications of graph theory to number theory*. Proc. Second Chapel Hill\n  Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C.,\n  1970) (1970), 136-145.\n- [Er74b] Erdős, P., *Remarks on some problems in number theory*. Math. Balkanica (1974), 197-202.\n- [Er77c] Erdős, Paul, *Problems and results on combinatorial number theory. III*. Number theory day\n  (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*. Ann. Discrete Math.\n  (1980), 89-115.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [Gu04] Guy, Richard K., *Unsolved problems in number theory*. (2004), xviii+437.\n- [Po79] Pomerance, Carl, *The prime number graph*. Math. Comp. (1979), 399-408.\n","FormalConjectures.ErdosProblems.«454»":"# Erdős Problem 454\n\n*References:*\n - [erdosproblems.com/454](https://www.erdosproblems.com/454)\n - [Po79] Pomerance, Carl, The prime number graph. Math. Comp. (1979), 399-408.\n","FormalConjectures.ErdosProblems.«455»":"# Erdős Problem 455\n*References:*\n - [erdosproblems.com/455](https://www.erdosproblems.com/455)\n - [Ri76] Richter, Bernd, Über die Monotonie von Differenzenfolgen. Acta Arith. (1976), 225-227.\n","FormalConjectures.ErdosProblems.«456»":"# Erdős Problem 456\n\n*References:*\n- [erdosproblems.com/456](https://www.erdosproblems.com/456)\n- [Er79e] Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73--82.\n","FormalConjectures.ErdosProblems.«457»":"# Erdős Problem 457\n\n*Reference:* [erdosproblems.com/457](https://www.erdosproblems.com/457)\n","FormalConjectures.ErdosProblems.«458»":"# Erdős Problem 458\n*Reference:* [erdosproblems.com/458](https://www.erdosproblems.com/458)\n","FormalConjectures.ErdosProblems.«459»":"# Erdős Problem 459\n\n*References:*\n- [erdosproblems.com/459](https://www.erdosproblems.com/459)\n- [ErGr80] P. Erdős and R. L. Graham, *Old and new problems and results in combinatorial number\n  theory*, Monographies de L'Enseignement Mathématique 28 (1980), p.91.\n- [OEIS A289280](https://oeis.org/A289280)\n","FormalConjectures.ErdosProblems.«45»":"# Erdős Problem 45\n\n*References:*\n- [erdosproblems.com/45](https://www.erdosproblems.com/45)\n- [Er95] Erdős, Paul, *Some of my favourite problems in number theory, combinatorics, and geometry*.\n  Resenhas (1995), 165-186.\n- [Er96b] Erdős, Paul, *Some problems I presented or planned to present in my short talk*. Analytic\n  number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.\n- [Cr03] Croot, III, Ernest S., *On a coloring conjecture about unit fractions*. Ann. of Math. (2)\n  (2003), 545-556.\n- [Gu04] Guy, Richard K., *Unsolved problems in number theory*. (2004), xviii+437.\n","FormalConjectures.ErdosProblems.«462»":"# Erdős Problem 462\n\n*Reference:* [erdosproblems.com/462](https://www.erdosproblems.com/462)\n","FormalConjectures.ErdosProblems.«463»":"# Erdős Problem 463\n\n*Reference:* [erdosproblems.com/463](https://www.erdosproblems.com/463)\n","FormalConjectures.ErdosProblems.«464»":"# Erdős Problem 464\n\n*References:*\n- [erdosproblems.com/464](https://www.erdosproblems.com/464)\n- [Er75i] Erdős, P., *Répartition modulo $1$*. (1975), iv+258.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial\n  number theory*. Monographies de L'Enseignement Mathématique (1980).\n- [Er82e] Erdős, Paul, *Some of my favourite problems which recently have been solved*.\n  (1982), 59--79.\n- [AkMo04] Akhunzhanov, R. K. and Moshchevitin, N. G., *On the chromatic number of a distance\n  graph associated with a lacunary sequence*. Dokl. Akad. Nauk (2004), 295-296.\n- [Du06] Dubickas, Artūras, *On the fractional parts of lacunary sequences*. Math. Scand. (2006),\n  136-146.\n- [Ka01] Katznelson, Y., *Chromatic numbers of Cayley graphs on $\\mathbb{Z}$ and recurrence*.\n  Combinatorica (2001), 211-219.\n- [PeSc10] Peres, Yuval and Schlag, Wilhelm, *Two Erdős problems on lacunary sequences: chromatic\n  number and Diophantine approximation*. Bull. Lond. Math. Soc. (2010), 295-300.\n- [Po79b] Pollington, A. D., *On the density of sequence $\\{n_k\\xi\\}$*. Illinois J. Math. (1979),\n  511-515.\n- [dM80] de Mathan, B., *Numbers contravening a condition in density modulo $1$*. Acta Math.\n  Acad. Sci. Hungar. (1980), 237-241 (1981).\n","FormalConjectures.ErdosProblems.«469»":"# Erdős Problem 469\n\n*References:*\n- [erdosproblems.com/469](https://www.erdosproblems.com/469)\n- [Le25] Lewis, Z. J., *On the convergence of the reciprocal sum of primitive pseudoperfect\n  numbers*. Preprint (2025).\n","FormalConjectures.ErdosProblems.«46»":"# Erdős Problem 46\n\n*References:*\n- [erdosproblems.com/46](https://www.erdosproblems.com/46)\n- [Cr03] Croot, III, Ernest S., *On a coloring conjecture about unit fractions*. Ann. of Math. (2)\n  (2003), 545-556.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n","FormalConjectures.ErdosProblems.«470»":"# Erdős Problem 470\n\n*Reference:* [erdosproblems.com/470](https://www.erdosproblems.com/470)\n","FormalConjectures.ErdosProblems.«476»":"# Erdős Problem 476\n\n*References:*\n- [erdosproblems.com/476](https://www.erdosproblems.com/476)\n- [Er65b] Erdős, Paul, *Some recent advances and current problems in number theory*. Lectures on\n  Modern Mathematics, Vol. III (1965), 196-244.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [Gu04] Guy, Richard K., *Unsolved problems in number theory*. (2004), xviii+437.\n- [dSHa94] Dias da Silva, J. A. and Hamidoune, Y. O., *Cyclic spaces for Grassmann\n  derivatives and additive theory*. Bull. London Math. Soc. (1994), 140-146.\n","FormalConjectures.ErdosProblems.«477»":"# Erdős Problem 477\n\n*References:*\n- [erdosproblems.com/477](https://www.erdosproblems.com/477)\n- [Sek59](http://dml.cz/dmlcz/100376) Milan Sekanina, Замечания к фактoризации беcкoнечнoй цикличеcкoй группы, Czechoslovak Mathematical Journal, Vol. 9 (1959), No. 4, 485–495\n","FormalConjectures.ErdosProblems.«478»":"# Erdős Problem 478\n\n*References:*\n- [erdosproblems.com/478](https://www.erdosproblems.com/478)\n- [AnTa16] V. Andrejić and M. Tatarevic, *On distinct residues of factorials*. arXiv:1603.04086\n  (2016).\n- [GSSV24] Grebennikov, Alexandr and Sagdeev, Arsenii and Semchankau, Aliaksei and Vasilevskii,\n  Aliaksei, *On the sequence {$n! \\bmod p$}*. Rev. Mat. Iberoam. (2024), 637--648.\n- [Gu04] Guy, Richard K., *Unsolved problems in number theory*. (2004), xviii+437.\n- [KlMu17] Klurman, Oleksiy and Munsch, Marc, *Distribution of factorials modulo {$p$}*. J. Théor.\n  Nombres Bordeaux (2017), 169--177.\n- [RoSc60] Rokowska, B. and Schinzel, A., *Sur un problème de {M}. {E}rdős*. Elem. Math. (1960),\n  84--85.\n- [Tr13] T. Trudgian, *There are no socialist primes less than $10^9$*. arXiv:1310.6403 (2013).\n","FormalConjectures.ErdosProblems.«479»":"# Erdős Problem 479\n\n*Reference:* [erdosproblems.com/479](https://www.erdosproblems.com/479)\n","FormalConjectures.ErdosProblems.«47»":"# Erdős Problem 47\n\n*References:*\n- [erdosproblems.com/47](https://www.erdosproblems.com/47)\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*. Ann. Discrete Math.\n  (1980), 89-115.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [Er92c] Erdős, P., *Some of my forgotten problems in number theory*. Hardy-Ramanujan J. (1992),\n  34-50.\n- [Er95] Erdős, Paul, *Some of my favourite problems in number theory, combinatorics, and geometry*.\n  Resenhas (1995), 165-186.\n- [Er96b] Erdős, Paul, *Some problems I presented or planned to present in my short talk*. Analytic\n  number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.\n- [Er97c] Erdős, Paul, *Some of my favorite problems and results*. The mathematics of Paul Erdős, I\n  (1997), 47-67.\n- [Bl21] Bloom, T. F., *On a density conjecture about unit fractions*. arXiv:2112.03726\n  (2021).\n- [LiSa24] Liu, Y. and Sawhney, M., *On further questions regarding unit fractions*.\n  arXiv:2404.07113 (2024).\n","FormalConjectures.ErdosProblems.«480»":"# Erdős Problem 480\n\n*Reference:* [erdosproblems.com/480](https://www.erdosproblems.com/480)\n","FormalConjectures.ErdosProblems.«481»":"# Erdős Problem 481\n\n*References:*\n- [erdosproblems.com/481](https://www.erdosproblems.com/481)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980), p.96.\n- [Kl82] Klarner, David A., *A sufficient condition for certain semigroups to be free*.\n  J. Algebra (1982), 140-148.\n- [KoTa22] Kolpakov, Alexander and Talambutsa, Alexey, *On free semigroups of affine maps on the\n  real line*. Proc. Amer. Math. Soc. (2022), 2301-2307.\n","FormalConjectures.ErdosProblems.«484»":"# Erdős Problem 484\n\n*References:*\n- [erdosproblems.com/484](https://www.erdosproblems.com/484)\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*. Ann. Discrete Math.\n  (1980), 89-115.\n- [ESS89] Erdős, P., Sárközy, A., and Sós, V. T., *On a conjecture of Roth and some related\n  problems. I*. (1989), 47-59.\n","FormalConjectures.ErdosProblems.«486»":"# Erdős Problem 486: Logarithmic density for sets avoiding modular subsets\n\n*Reference:* [erdosproblems.com/486](https://www.erdosproblems.com/486)\n","FormalConjectures.ErdosProblems.«487»":"# Erdős Problem 487\n\n*References:*\n- [erdosproblems.com/487](https://www.erdosproblems.com/487)\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [Er65b] Erdős, Paul, *Some recent advances and current problems in number theory*. Lectures on\n  Modern Mathematics, Vol. III (1965), 196-244.\n- [DaEr36] Davenport, H. and Erdős, P., *On sequences of positive integers*. Acta Arithmetica\n  (1936), 147-151.\n- [Kl71] Kleitman, Daniel, *Collections of subsets containing no two sets and their union*.\n  Proceedings of the LA Meeting AMS (1971), 153-155.\n","FormalConjectures.ErdosProblems.«488»":"# Erdős Problem 488\n\n*Reference:* [erdosproblems.com/488](https://www.erdosproblems.com/488)\n","FormalConjectures.ErdosProblems.«489»":"# Erdős Problem 489\n\n*Reference:* [erdosproblems.com/489](https://www.erdosproblems.com/489)\n","FormalConjectures.ErdosProblems.«48»":"# Erdős Problem 48\n\n*Reference:* [erdosproblems.com/48](https://www.erdosproblems.com/48)\n","FormalConjectures.ErdosProblems.«493»":"# Erdős Problem 493\n\n*References:*\n- [erdosproblems.com/493](https://www.erdosproblems.com/493)\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.\n","FormalConjectures.ErdosProblems.«494»":"# Erdős Problem 494\n\n*References:*\n  - [erdosproblems.com/494](https://www.erdosproblems.com/494)\n  - [SeSt58] Selfridge, J. L. and Straus, E., On the determination of numbers by their sums\n      of a fixed order. Pacific Journal of Math. (1958), 847-856.\n  - [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n      221-254.\n  - [GFS62] Gordon, B. and Fraenkel, A. S. and Straus, E. G., On the determination of sets\n      by the sets of sums of a certain order. Pacific J. Math. (1962), 187--196.\n","FormalConjectures.ErdosProblems.«495»":"# Erdős Problem 495\n\n*Reference:* [erdosproblems.com/495](https://www.erdosproblems.com/495)\n","FormalConjectures.ErdosProblems.«497»":"# Erdős Problem 497\n\n*References:*\n- [erdosproblems.com/497](https://www.erdosproblems.com/497)\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [Kl69] Kleitman, Daniel, *On Dedekind's problem: The number of monotone Boolean functions*.\n  Proc. Amer. Math. Soc. (1969), 677-682.\n","FormalConjectures.ErdosProblems.«498»":"# Erdős Problem 498\n\n*References:*\n- [erdosproblems.com/498](https://www.erdosproblems.com/498)\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [Er45] Erdős, P., _On a lemma of Littlewood and Offord_. Bull. Amer. Math. Soc. (1945), 898-902.\n- [Kl65] Kleitman, Daniel J., _On a lemma of Littlewood and Offord on the distribution of certain\n  sums_. Math. Z. (1965), 251-259.\n- [Kl70] Kleitman, Daniel J., _On a lemma of Littlewood and Offord on the distributions of linear\n  combinations of vectors_. Advances in Math. (1970), 155-157.\n","FormalConjectures.ErdosProblems.«499»":"# Erdős Problem 499\n*Reference:* [erdosproblems.com/499](https://www.erdosproblems.com/499)\n","FormalConjectures.ErdosProblems.«4»":"# Erdős Problem 4\n\n*Reference:* [erdosproblems.com/4](https://www.erdosproblems.com/4)\n","FormalConjectures.ErdosProblems.«501»":"# Erdős Problem 501\n\n*References:*\n- [erdosproblems.com/501](https://www.erdosproblems.com/501)\n- [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6\n  (1961), 221-254.\n- [ErHa71] Erdős, Paul and Hajnal, András, Unsolved problems in set theory. Axiomatic Set\n  Theory, Proc. Sympos. Pure Math. XIII Part I (1971), 17-48.\n- [ErHa60] Erdős, Paul and Hajnal, András. On some combinatorial problems involving\n  complete graphs. Acta Math. Acad. Sci. Hungar. (1960), 395-424.\n- [Gl62] Gladysz, S. Some topological properties of independent sets. Colloq. Math. (1962).\n- [He72] Hechler, S. H. A dozen small uncountable cardinals. TOPO 72, Lecture Notes\n  in Math. (1972), 207-218.\n- [NPS87] Newelski, L., Pawlikowski, J., and Seredyński, F. Infinite independent sets in\n  the closed case. Acta Math. Acad. Sci. Hungar. (1987).\n","FormalConjectures.ErdosProblems.«502»":"# Erdős Problem 502\n\n*References:*\n- [erdosproblems.com/502](https://www.erdosproblems.com/502)\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [BBS83] Bannai, Eiichi and Bannai, Etsuko and Stanton, Dennis, *An upper bound for the\n  cardinality of an $s$-distance subset in real Euclidean space. II*. Combinatorica (1983),\n  147-152.\n- [PePo21] Petrov, Fedor and Pohoata, Cosmin, *A remark on sets with few distances in\n  $\\mathbb{R}^d$*. Proc. Amer. Math. Soc. (2021), 569-571.\n","FormalConjectures.ErdosProblems.«503»":"# Erdős Problem 503\n\n*Reference:* [erdosproblems.com/503](https://www.erdosproblems.com/503)\n","FormalConjectures.ErdosProblems.«505»":"# Erdős Problem 505\n\n*Reference:* [erdosproblems.com/505](https://www.erdosproblems.com/505)\n\n**Borsuk's conjecture** (1933): Is every bounded set of diameter 1 in $\\mathbb{R}^n$\nthe union of at most $n + 1$ sets of diameter strictly less than 1?\n\nErdős [Er44] suspected this is false for sufficiently large $n$. Confirmed\nby Kahn–Kalai [KK93], who disproved the conjecture for $n \\geq 2015$.\nThe current best is $n \\geq 64$ (Jenrich–Brouwer, 2014).\n\nThe conjecture is true for $n \\leq 3$ (Eggleston [Eg55] for $n = 3$).\n\n### References\n\n- [Bo33] Borsuk, K. (1933). *Drei Sätze über die n-dimensionale euklidische Sphäre*.\n  Fund. Math. 20, 177–190.\n- [Er44] Erdős, P. (1944). Remarks on a conjecture of Borsuk.\n- [Eg55] Eggleston, H. G. (1955). *Covering a three-dimensional set with sets of\n  smaller diameter*. J. London Math. Soc. 30, 11–24.\n- [KK93] Kahn, J., Kalai, G. (1993). *A counterexample to Borsuk's conjecture*.\n  Bull. Amer. Math. Soc. 29, 60–62.\n\n### AI disclosure\n\nLean 4 code in this file was drafted with assistance from Claude (Anthropic).\nThe mathematical content and references are the author's own work.\n","FormalConjectures.ErdosProblems.«506»":"# Erdős Problem 506\n\n*References:*\n- [erdosproblems.com/506](https://www.erdosproblems.com/506)\n- [El67] Elliott, P. D. T. A., *On the number of circles determined by $n$ points*, Acta Math.\n  Acad. Sci. Hungar. (1967), 181–188.\n- [BaBa94] Bálintová, A. and Bálint, V., *On the number of circles determined by $n$ points in the\n  Euclidean plane*, Acta Math. Hungar. (1994), 283–289.\n- [PuSm] Purdy and Smith. No reference found.\n","FormalConjectures.ErdosProblems.«507»":"# Erdős Problem 507\n\n*References:*\n- [erdosproblems.com/507](https://www.erdosproblems.com/507)\n- [CPZ23] Cohen, Alex, Cosmin Pohoata, and Dmitrii Zakharov. \"A new upper bound for the Heilbronn\n  triangle problem.\" arXiv preprint arXiv:2305.18253 (2023).\n- [CPZ24] Cohen, Alex, Cosmin Pohoata, and Dmitrii Zakharov. \"Lower bounds for incidences.\"\n  Inventiones mathematicae (2025): 1-74.\n- [KPS82] Komlós, János, János Pintz, and Endre Szemerédi. \"A lower bound for Heilbronn's problem.\"\n  Journal of the London Mathematical Society 2.1 (1982): 13-24.\n- [KPS81] Komlós, János, János Pintz, and Endre Szemerédi. \"On Heilbronn's triangle problem.\"\n  Journal of the London Mathematical Society 2.3 (1981): 385-396.\n","FormalConjectures.ErdosProblems.«508»":"# Erdős Problem 508\n\n*Reference:* [erdosproblems.com/508](https://www.erdosproblems.com/508)\n\nproven by considering the [Moser-Spindel graph]\nor the [Golomb graph]\n*At least 4 colors are required:* [Moser-Spindel graph](https://de.wikipedia.org/wiki/Moser-Spindel)\n*At least 4 colors are required:* [Golomb graph](https://en.wikipedia.org/wiki/Golomb_graph)\n*At least 5 colors are required:* [de Grey 2018](https://arxiv.org/abs/1804.02385)\n","FormalConjectures.ErdosProblems.«509»":"# Erdős Problem 509\n\n*Reference:* [erdosproblems.com/509](https://www.erdosproblems.com/509)\n","FormalConjectures.ErdosProblems.«50»":"# Erdős Problem 50\n\n*References:*\n* [erdosproblems.com/50](https://www.erdosproblems.com/50)\n* [Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry.\nResenhas (1995), 165-186.\n* [Sch38] Schoenberg, I. J. \"On asymptotic distributions of arithmetical functions.\"\nTransactions of the American Mathematical Society 39.2 (1936): 315-330.\n","FormalConjectures.ErdosProblems.«510»":"# Erdős Problem 510\n\n*References:*\n- [erdosproblems.com/510](https://www.erdosproblems.com/510)\n- [Ben Green's Open Problem 81](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.11)\n- [Ru04] Ruzsa, Imre Z., Negative values of cosine sums. Acta Arith. (2004), 179-186.\n- [Be25c] B. Bedert, Polynomial bounds for the Chowla Cosine Problem. arXiv:2509.05260 (2025).\n","FormalConjectures.ErdosProblems.«512»":"# Erdős Problem 512\n\n*References:*\n- [erdosproblems.com/512](https://www.erdosproblems.com/512)\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl.\n  (1961), 221-254.\n- [Ko81] Konyagin, S. V., *On the Littlewood problem*. Izv. Akad. Nauk SSSR Ser. Mat. (1981),\n  243-265, 463.\n- [MPS81] McGehee, O. Carruth and Pigno, Louis and Smith, Brent, *Hardy's inequality and the\n  $L^1$ norm of exponential sums*. Ann. of Math. (2) (1981), 613-618.\n","FormalConjectures.ErdosProblems.«513»":"# Erdős Problem 513\n\n*Reference:*\n - [erdosproblems.com/513](https://www.erdosproblems.com/513)\n - [ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math.\n  (1964), 143-186.\n","FormalConjectures.ErdosProblems.«516»":"# Erdős Problem 516\n*References:*\n - [erdosproblems.com/516](https://www.erdosproblems.com/516)\n - [Fu63] Fuchs, W. H. J., Proof of a conjecture of G. Pólya concerning gap series. Illinois J.\n    Math. (1963), 661--667.\n - [Ko65] Kövari, Thomas, A gap-theorem for entire functions of infinite order. Michigan Math. J.\n    (1965), 133--140.\n","FormalConjectures.ErdosProblems.«517»":"# Erdős Problem 517\n\n*References:*\n - [erdosproblems.com/517](https://www.erdosproblems.com/517)\n - [Bi28] Biernacki, Miécislas, Sur les équations algébriques contenant des paramétres arbitraires.\n    (1928), 145.\n","FormalConjectures.ErdosProblems.«519»":"# Erdős Problem 519\n\n*References:*\n- [erdosproblems.com/519](https://www.erdosproblems.com/519)\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [Er65b] Erdős, Paul, *Some recent advances and current problems in number theory*. Lectures on\n  Modern Mathematics, Vol. III (1965), 196-244.\n- [Bi94] Biró, A., *On a problem of Turán concerning sums of powers of complex numbers*. Acta Math.\n  Hungar. (1994), 209-216.\n- [Bi00] Biró, András, *An improved estimate in a power sum problem of Turán*. Indag. Math. (N.S.)\n  (2000), 343-358.\n- [Bi00b] Biró, A., *An upper estimate in Turán's pure power sum problem*. Indag. Math. (N.S.)\n  (2000), 499--508.\n- [At61b] Atkinson, F. V., *On sums of powers of complex numbers*. Acta Math. Acad.\n  Sci. Hungar. (1961), 185-188.\n","FormalConjectures.ErdosProblems.«51»":"# Erdős Problem 51\n\n*Reference:* [erdosproblems.com/51](https://www.erdosproblems.com/51)\n","FormalConjectures.ErdosProblems.«520»":"# Erdős Problem 520\n\n*Reference:* [erdosproblems.com/520](https://www.erdosproblems.com/520)\n","FormalConjectures.ErdosProblems.«521»":"# Erdős Problem 521\n\n*Reference:* [erdosproblems.com/521](https://www.erdosproblems.com/521)\n","FormalConjectures.ErdosProblems.«522»":"# Erdős Problem 522\n\n*Reference:* [erdosproblems.com/522](https://www.erdosproblems.com/522)\n","FormalConjectures.ErdosProblems.«52»":"# Erdős Problem 52\n\n*Reference:* [erdosproblems.com/52](https://www.erdosproblems.com/52)\n","FormalConjectures.ErdosProblems.«532»":"# Erdős Problem 532\n\n*References:*\n- [erdosproblems.com/532](https://www.erdosproblems.com/532)\n- [Er73] Erdős, P., *Problems and results on combinatorial number theory*. A survey of combinatorial theory (1973), 117-138.\n- [Er75b] Erdős, Paul, *Problems and results in combinatorial number theory*. Journées Arithmétiques de Bordeaux (1975), 295-310.\n- [Er77c] Erdős, Paul, *Problems and results on combinatorial number theory. III*. Number theory day (1977), 43-72.\n- [Hi74] Hindman, Neil, *Finite sums from sequences within cells of a partition of $\\mathbb{N}$*. J. Combinatorial Theory Ser. A (1974), 1-11.\n","FormalConjectures.ErdosProblems.«533»":"# Erdős Problem 533\n\n*References:*\n- [erdosproblems.com/533](https://www.erdosproblems.com/533)\n- [EHSSS94] P. Erdős, A. Hajnal, M. Simonovits, V. T. Sós and E. Szemerédi,\n  *Turán-Ramsey theorems and $K_p$-independence numbers*,\n  Combin. Probab. Comput. **3** (1994), 297–325.\n- [ErRo62] P. Erdős and C. A. Rogers, *The construction of certain graphs*,\n  Canad. J. Math. **14** (1962), 702–707.\n- [BaLe13] J. Balogh and J. Lenz, *On the Ramsey–Turán numbers of graphs and hypergraphs*,\n  Israel J. Math. **194** (2013), 45–68.\n- [LRSS21] H. Liu, C. Reiher, M. Sharifzadeh and K. Staden,\n  *Geometric constructions for Ramsey–Turán theory*,\n  [arXiv:2103.10423](https://arxiv.org/abs/2103.10423) (2021).\n","FormalConjectures.ErdosProblems.«535»":"# Erdős Problem 535\n\n*References:*\n- [erdosproblems.com/535](https://www.erdosproblems.com/535)\n- [Er64] P. Erdős, _On a problem in elementary number theory and a combinatorial problem_. Math.\n  Comp. (1964), 644–646.\n- [AbHa70] H. L. Abbott and D. Hanson, _An extremal problem in number theory_. Bull. London Math.\n  Soc. (1970), 324–326.\n- [Er73] P. Erdős, _Problems and results on combinatorial number theory_, in\n  *A Survey of Combinatorial Theory*, North-Holland, 1973.\n","FormalConjectures.ErdosProblems.«536»":"# Erdős Problem 536\n\n*Reference:* [erdosproblems.com/536](https://www.erdosproblems.com/536)\n","FormalConjectures.ErdosProblems.«537»":"# Erdős Problem 537\n\n*References:*\n- [erdosproblems.com/537](https://www.erdosproblems.com/537)\n- [Er73] Erdős, P., *Problems and results on combinatorial number theory*. A survey of combinatorial\n  theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n","FormalConjectures.ErdosProblems.«538»":"# Erdős Problem 538\n\n*Reference:* [erdosproblems.com/538](https://www.erdosproblems.com/538)\n","FormalConjectures.ErdosProblems.«539»":"# Erdős Problem 539\n\nIn this problem, a function $h : \\mathbb{N} \\to\\mathbb{N}$ is defined maximally by a specified\ncounting property.\n\nThe problem asks to estimate $h(n)$. This has been interpreted here as asking for $\\Theta(h(n))$.\nThe principal version includes `answer(sorry)` for an unknown function. On the other hand, the best\nknown upper bound is $n^{2/3}$ and the best known lower bound is $\\sqrt{n}$ so we\nalso provide these candidates as variants. Moreover, it suffices to show $O(h(n))$ and\n$O(\\sqrt{n})$ respectively for each, so further variants are provided for those.\n\nIn the source paper [Er73], Erdős also remarks that it should not be too difficult\nto determine $\\lim_{n\\to\\infty}\\log(h(n))/\\log(n)$. This does not appear on the website, and\nit is not clear whether this remains open, but we include it here either way.\n\n*References:*\n- [erdosproblems.com/539](https://www.erdosproblems.com/539)\n- [GR99] Granville, A., & Roesler, F. (1999). _The Set of Differences of a Given Set_. The American Mathematical Monthly, 106(4), 338–344.\n- [Er73] Erdős, P., _Problems and results on combinatorial number theory_. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n- [Sc+26] Schmitt, J., Gehrunger, T., Dekoninck, J., Bérczi, G., Kreitner, U., Price, L., & Holmes, D. (2026). _ProofCouncil: An LLM Agent for Solving Open Mathematical Problems_. [arXiv:2607.09474](https://arxiv.org/abs/2607.09474), Appendix A, Theorem A.1.\n","FormalConjectures.ErdosProblems.«540»":"# Erdős Problem 540\n\n*References:*\n- [erdosproblems.com/540](https://www.erdosproblems.com/540)\n- [Er65b] Erdős, Paul, *Some recent advances and current problems in number theory*. Lectures on\n  Modern Mathematics, Vol. III (1965), 196-244.\n- [Er73] Erdős, P., *Problems and results on combinatorial number theory*. A survey of combinatorial\n  theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [Ol68] Olson, John E., *An addition theorem modulo {$p$}*. J. Combinatorial Theory (1968), 45--52.\n- [Ba12] Balandraud, Éric, *An addition theorem and maximal zero-sum free sets in\n  $\\mathbb{Z}/p\\mathbb{Z}$*. Israel J. Math. (2012), 405-429.\n- [HaZe96] Hamidoune, Yahya Ould and Zémor, Gilles, *On zero-free subset sums*. Acta Arith. (1996),\n  143--152.\n- [Gu04] Guy, Richard K., *Unsolved problems in number theory*. (2004), xviii+437.\n- [ErHe64] Erdős, P. and Heilbronn, H., *On the addition of residue classes\n  mod $p$*. Acta Arith. (1964), 149--159.\n- [Sz70] Szemerédi, E., *On a conjecture of Erdős and Heilbronn*. Acta Arith.\n  (1970), 227-229.\n","FormalConjectures.ErdosProblems.«541»":"# Erdős Problem 541\n\n*References:*\n- [erdosproblems.com/541](https://www.erdosproblems.com/541)\n- [ErSz76] Erdős, E. and Szemerédi, E., On a problem of Graham. Publ. Math. Debrecen (1976),\n  123--127.\n- [GHW10] Gao, Weidong and Hamidoune, Yahya Ould and Wang, Guoqing, Distinct length modular zero-sum\n  subsequences: a proof of Graham's conjecture. J. Number Theory (2010), 1425--1431.\n","FormalConjectures.ErdosProblems.«544»":"# Erdős Problem 544\n\n*References:*\n- [erdosproblems.com/544](https://www.erdosproblems.com/544)\n","FormalConjectures.ErdosProblems.«545»":"# Erdős Problem 545\n\n*References:*\n- [erdosproblems.com/545](https://www.erdosproblems.com/545)\n- [ErGr75] Erdős, P. and Graham, R. L., On partition theorems for finite graphs.\n  Infinite and finite sets (1975), 515-527.\n- [Er84b] Erdős, P., On some problems in graph theory, combinatorial analysis and combinatorial\n  number theory. Graph theory and combinatorics (Cambridge, 1983) (1984), 1-17.\n","FormalConjectures.ErdosProblems.«546»":"# Erdős Problem 546\n\n*References:*\n- [erdosproblems.com/546](https://www.erdosproblems.com/546)\n- [Su11] Sudakov, B., A conjecture of Erdős on graph Ramsey numbers. Adv. Math. (2011), 3148-3155.\n- [AKS03] Alon, N., Krivelevich, M. and Sudakov, B., Turán numbers of bipartite graphs and Ramsey\n  graphs of bounded degree. Combin. Probab. Comput. (2003), 477-483.\n","FormalConjectures.ErdosProblems.«547»":"# Erdős Problem 547\n\n*References:*\n- [erdosproblems.com/547](https://www.erdosproblems.com/547)\n- [Bu74] Burr, S. A., Generalized Ramsey theory for graphs—a survey. Graphs and combinatorics\n  (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973) (1974), 52-75.\n- [Zh11] Zhao, Y., Proof of the $(n/2-n/2-n/2)$ conjecture for large $n$.\n  Electron. J. Combin. (2011), Paper 27, 61.\n","FormalConjectures.ErdosProblems.«548»":"# Erdős Problem 548\n\n*References:*\n- [erdosproblems.com/548](https://www.erdosproblems.com/548)\n- [BrDo96] Brandt, Stephan and Dobson, Edward, *The Erdős-Sós conjecture for graphs of girth {$5$}*.\n  Discrete Math. (1996), 411-414.\n- [Er78] Erdős, Paul, *Problems and results in combinatorial analysis and combinatorial number\n  theory*. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and\n  Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40.\n- [ErGa59] Erdős, P. and Gallai, T., *On maximal paths and circuits of graphs*. Acta Math. Acad.\n  Sci. Hungar. (1959), 337-356 (unbound insert).\n- [SaWo97] Saclé, Jean-François and Woźniak, Mariusz, *The Erdős-Sós conjecture for graphs without\n  {$C_4$}*. J. Combin. Theory Ser. B (1997), 367-372.\n- [WLL00] Wang, Min and Li, Guo-jun and Liu, Ai-de, *A result of Erdős-Sós conjecture*. Ars Combin.\n  (2000), 123-127.\n- [YiLi04] Yin, Jian-hua and Li, Jiong-sheng, *The Erdős-Sós conjecture for graphs whose complements\n  contain no {$C_4$}*. Acta Math. Appl. Sin. Engl. Ser. (2004), 397-400.\n","FormalConjectures.ErdosProblems.«549»":"# Erdős Problem 549\n\n*References:*\n- [erdosproblems.com/549](https://www.erdosproblems.com/549)\n- [Bu74] Burr, S. A., Generalized Ramsey theory for graphs—a survey. Graphs and combinatorics\n  (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973) (1974), 52-75.\n- [NSZ16] S. Norin and Y. R. Sun and Y. Zhao, Asymptotics of Ramsey numbers of double stars.\n","FormalConjectures.ErdosProblems.«550»":"# Erdős Problem 550\n\n*References:*\n- [erdosproblems.com/550](https://www.erdosproblems.com/550)\n- [Ch77] Chvátal, V., Tree-complete graph Ramsey numbers. J. Graph Theory (1977), 93.\n","FormalConjectures.ErdosProblems.«551»":"# Erdős Problem 551\n\n*References:*\n- [erdosproblems.com/551](https://www.erdosproblems.com/551)\n- [BoEr73] Bondy, J. A. and Erdős, P., Ramsey numbers for cycles in graphs. J. Combin. Theory\n  Ser. B (1973), 46-54.\n- [Ni05] Nikiforov, V., The cycle-complete graph Ramsey numbers. Combin. Probab. Comput. (2005),\n  349-370.\n- [KLS21] Keevash, P., Long, E. and Skokan, J., Cycle-complete Ramsey numbers.\n  Int. Math. Res. Not. IMRN (2021), 277-302.\n","FormalConjectures.ErdosProblems.«552»":"# Erdős Problem 552\n\n*References:*\n- [erdosproblems.com/552](https://www.erdosproblems.com/552)\n- [BEFRS89] Burr, S. and Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Some\n  complete bipartite graph-tree Ramsey numbers. Graph theory in memory of G. A. Dirac (Sandbjerg,\n  1985) (1989), 79-89.\n","FormalConjectures.ErdosProblems.«562»":"# Erdős Problem 562\n\n*Reference:* [erdosproblems.com/562](https://www.erdosproblems.com/562)\n","FormalConjectures.ErdosProblems.«563»":"# Erdős Problem 563\n\n*References:*\n- [erdosproblems.com/563](https://www.erdosproblems.com/563)\n- [Er90b] Erdős, Paul, Problems and results on graphs and hypergraphs: similarities and differences.\n  Mathematics of Ramsey theory (1990), 12-28.\n","FormalConjectures.ErdosProblems.«564»":"# Erdős Problem 564\n\n*Reference:* [erdosproblems.com/564](https://www.erdosproblems.com/564)\n","FormalConjectures.ErdosProblems.«566»":"# Erdős Problem 566\n\n*References*:\n- [erdosproblems.com/566](https://www.erdosproblems.com/566)\n- [EFRS93] Erdős, Faudree, Rousseau and Schelp, _Ramsey size linear graphs_.\nCombin. Probab. Comput. (1993), 389-399.\n","FormalConjectures.ErdosProblems.«567»":"# Erdős Problem 567\n\nLet $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords\nto $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then\n$$ R(G,H) \\ll m? $$\n\nIn other words, is $G$ Ramsey size linear? A special case of Problem 566.\n\n*Reference:* [erdosproblems.com/567](https://www.erdosproblems.com/567)\n\n[EFRS93] Erdős, Faudree, Rousseau and Schelp, _Ramsey size linear graphs_.\nCombin. Probab. Comput. (1993), 389-399.\n","FormalConjectures.ErdosProblems.«568»":"# Erdős Problem 568\n\n*References:*\n- [erdosproblems.com/568](https://www.erdosproblems.com/568)\n- [EFRS93] Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear\n  graphs. Combin. Probab. Comput. (1993), 389-399.\n","FormalConjectures.ErdosProblems.«569»":"# Erdős Problem 569\n\n*References:*\n- [erdosproblems.com/569](https://www.erdosproblems.com/569)\n- [EFRS93] Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear\n  graphs. Combin. Probab. Comput. (1993), 389-399.\n","FormalConjectures.ErdosProblems.«56»":"# Erdős Problem 56\n\n*Reference:* [erdosproblems.com/56](https://www.erdosproblems.com/56)\n","FormalConjectures.ErdosProblems.«570»":"# Erdős Problem 570\n\n*References:*\n- [erdosproblems.com/570](https://www.erdosproblems.com/570)\n- [EFRS93] Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear\n  graphs. Combin. Probab. Comput. (1993), 389-399.\n- [GoKl94] Goddard, W. and Kleitman, D. J., An upper bound for the Ramsey numbers $r(K_3,G)$.\n  Discrete Math. (1994), 177-182.\n- [Si91] Sidorenko, A. F., An upper bound on the Ramsey number $r(K_3,G)$ depending only on\n  the size of the graph $G$. J. Graph Theory (1991), 15-17.\n- [Ja99] Jayawardene, C. J., Ramsey numbers related to small cycles. University of Memphis (1999).\n- [CFMPP26] Cambie, S., Freschi, A., Morawski, P., Petrova, K. and Pokrovskiy, A.,\n  Ramsey number of a cycle versus a graph of a given size. arXiv:2601.10238 (2026).\n","FormalConjectures.ErdosProblems.«571»":"# Erdős Problem 571\n\n*References:*\n- [erdosproblems.com/571](https://www.erdosproblems.com/571)\n- [BuCo18] Bukh, Boris and Conlon, David, *Rational exponents in extremal graph theory*. J. Eur.\n  Math. Soc. (JEMS) (2018), 1747-1757.\n- [CJL21] Conlon, David and Janzer, Oliver and Lee, Joonkyung, *More on the extremal number of\n  subdivisions*. Combinatorica (2021), 465-494.\n- [CoJa22] Conlon, David and Janzer, Oliver, *Rational exponents near two*. Adv. Comb. (2022), Paper\n  No. 9, 10.\n- [Er78] Erdős, Paul, *Problems and results in combinatorial analysis and combinatorial number\n  theory*. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and\n  Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40.\n- [JJM20] Jiang, Tao and Jiang, Zilin and Ma, Jie, *Negligible obstructions and Turán exponents*.\n  arXiv:2007.02975 (2020).\n- [JMY22] Jiang, Tao and Ma, Jie and Yepremyan, Liana, *On Turán exponents of bipartite graphs*.\n  Combin. Probab. Comput. (2022), 333-344.\n- [JiQi20] Jiang, Tao and Qiu, Yu, *Turán numbers of bipartite subdivisions*. SIAM J. Discrete Math.\n  (2020), 556-570.\n- [JiQi23] Jiang, Tao and Qiu, Yu, *Many Turán exponents via subdivisions*. Combin. Probab. Comput.\n  (2023), 134-150.\n- [KKL21] Kang, Dong Yeap and Kim, Jaehoon and Liu, Hong, *On the rational Turán exponents\n  conjecture*. J. Combin. Theory Ser. B (2021), 149-172.\n","FormalConjectures.ErdosProblems.«572»":"# Erdős Problem 572\n\n*References:*\n- [erdosproblems.com/572](https://www.erdosproblems.com/572)\n- [Er64c] Erdős, P., Extremal problems in graph theory. Theory of Graphs and its\n  Applications (1964), 29-36.\n- [BoSi74] Bondy, J. A. and Simonovits, M., Cycles of even length in graphs.\n  J. Combin. Theory Ser. B (1974), 97-105.\n- [LUW95] Lazebnik, F., Ustimenko, V. A. and Woldar, A. J., A new series of dense graphs\n  of high girth. Bull. Amer. Math. Soc. (N.S.) (1995), 73-79.\n","FormalConjectures.ErdosProblems.«579»":"# Erdős Problem 579\n\n*References:*\n- [erdosproblems.com/579](https://www.erdosproblems.com/579)\n- [EHSS83] P. Erdős, A. Hajnal, V. T. Sós and E. Szemerédi, *More results on Ramsey–Turán\n  type problems*, Combinatorica **3** (1983), 69–81.\n","FormalConjectures.ErdosProblems.«57»":"# Erdős Problem 57\n\n*References:*\n- [erdosproblems.com/57](https://www.erdosproblems.com/57)\n- [ErHa66] Erdős, P. and Hajnal, A., *On chromatic number of graphs and set-systems*.\n  Acta Math. Acad. Sci. Hungar. (1966), 61-99.\n- [LiMo20] Liu, Hong and Montgomery, Richard, *A solution to Erdős and Hajnal's odd cycle problem*.\n  arXiv:2010.15802 (2020).\n","FormalConjectures.ErdosProblems.«582»":"# Erdős Problem 582\n\n*References:*\n- [erdosproblems.com/582](https://www.erdosproblems.com/582)\n- [Er75b] Erdős, Paul, *Problems and results in combinatorial number theory*. Journées Arithmétiques\n  de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.\n- [Er69b] Erdős, P., *Problems and results in chromatic graph theory*. Proof Techniques in Graph\n  Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.\n- [Er75d] Erdős, Paul, *Problems and results on finite and infinite graphs*. Recent advances in\n  graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974) (1975), 183-192. (loose errata).\n- [FrRo86] Frankl, P. and Rödl, V., *Large triangle-free subgraphs in graphs without {$K_4$}*.\n  Graphs Combin. (1986), 135-144.\n- [Sp88] Spencer, Joel, *Three hundred million points suffice*. J. Combin. Theory Ser. A (1988),\n  210-217.\n- [Lu07] Lu, Linyuan, *Explicit construction of small Folkman graphs*. SIAM J. Discrete Math.\n  (2007), 1053-1060.\n- [DuRo08] Dudek, Andrzej and Rödl, Vojtěch, *On the Folkman number {$f(2,3,4)$}*. Experiment.\n  Math. (2008), 63-67.\n- [RaXu07] Radziszowski, Stanisław P. and Xu, Xiaodong, *On the most wanted Folkman graph*.\n  Geombinatorics (2007), 367-381.\n- [BiNe20] Bikov, Aleksandar and Nenov, Nedyalko, *On the independence number of\n  $(3,3)$-Ramsey graphs and the Folkman number $F_e(3,3;4)$*. Australas. J. Combin.\n  (2020), 35-50.\n- [ErHa67] Erdős, P. and Hajnal, A., *Research Problem 2.5*. J. Comb. Theory (1967).\n- [Fo70] Folkman, Jon, *Graphs with monochromatic complete subgraphs in every edge coloring*.\n  SIAM J. Appl. Math. (1970), 19-24.\n- [LRX14] Lange, Alexander R. and Radziszowski, Stanisław P. and Xu, Xiaodong, *Use of MAX-CUT\n  for Ramsey arrowing of triangles*. J. Combin. Math. Combin. Comput. (2014), 61-71.\n","FormalConjectures.ErdosProblems.«583»":"# Erdős Problem 583\n\n*References:*\n- [erdosproblems.com/583](https://www.erdosproblems.com/583)\n- [AnBa23] Anto, Nevil and Basavaraju, Manu, *Gallai's path decomposition for 2-degenerate graphs*.\n  Discrete Math. Theor. Comput. Sci. (2023), Paper No. 16, 11.\n- [BBB21] A. Blanché, M. Bonamy, and N. Bonichon, *Gallai's path decomposition in planar graphs*.\n  arXiv:2110.08870 (2021).\n- [BoPe19] Bonamy, Marthe and Perrett, Thomas J., *Gallai's path decomposition conjecture for graphs\n  of small maximum degree*. Discrete Math. (2019), 1293--1299.\n- [CFZ26] Chu, Yanan and Fan, Genghua and Zhou, Chuixiang, *Gallai's conjecture and the path number\n  of odd semi-cliques*. Discrete Math. (2026), Paper No. 114725, 6.\n- [Ch78] Chung, F. R. K., *On partitions of graphs into trees*. Discrete Math. (1978), 23-30.\n- [DeKo00] Dean, Nathaniel and Kouider, Mekkia, *Gallai's conjecture for disconnected graphs*.\n  Discrete Math. (2000), 43--54.\n- [Er71] Erdős, P., *Some unsolved problems in graph theory and combinatorial analysis*.\n  Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.\n- [Fa02] Fan, Genghua, *Subgraph coverings and edge switchings*. J. Combin. Theory Ser. B (2002),\n  54-83.\n- [Lo68] Lovász, L., *On covering of graphs*. Theory of Graphs (Proc. Colloq., Tihany, 1966) (1968),\n  231-236.\n- [Py96] Pyber, L., *Covering the edges of a connected graph by paths*. J. Combin. Theory Ser. B\n  (1996), 152-159.\n","FormalConjectures.ErdosProblems.«587»":"# Erdős Problem 587\n\n*Reference:* [erdosproblems.com/587](https://www.erdosproblems.com/587)\n","FormalConjectures.ErdosProblems.«58»":"# Erdős Problem 58\n\n*References:*\n- [erdosproblems.com/58](https://www.erdosproblems.com/58)\n- [Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.\n- [GaHuMa21] Gao, Jun and Huo, Qingyi and Ma, Jie, *A strengthening on odd cycles in graphs of given\n  chromatic number*. SIAM J. Discrete Math. (2021), 2317-2327.\n- [Gy92] Gyárfás, A., *Graphs with k odd cycle lengths*. Discrete Math. (1992), 41-48.\n","FormalConjectures.ErdosProblems.«590»":"# Erdős Problem 590\n\n*References:*\n - [erdosproblems.com/590](https://www.erdosproblems.com/590)\n - [Ch72] Chang, C. C., A partition theorem for the complete graph on {$\\omega\\sp{\\omega }$}. J. Combinatorial Theory Ser. A (1972), 396-452.\n - [Sp57] Specker, Ernst, Teilmengen von Mengen mit Relationen. Comment. Math. Helv. (1957), 302-314.\n - [La73] Larson, Jean A., A short proof of a partition theorem for the ordinal {$\\omega \\sp{\\omega }$}. Ann. Math. Logic (1973/74), 129-145.\n","FormalConjectures.ErdosProblems.«591»":"# Erdős Problem 591\n\n*References:*\n- [erdosproblems.com/591](https://www.erdosproblems.com/591)\n- [Sc10] Schipperus, Rene, Countable partition ordinals. Ann. Pure Appl. Logic (2010), 1195-1215.\n","FormalConjectures.ErdosProblems.«592»":"# Erdős Problem 592\n\n*Reference:* [erdosproblems.com/592](https://www.erdosproblems.com/592)\n","FormalConjectures.ErdosProblems.«593»":"# Erdős Problem 593\n\n*References:*\n- [erdosproblems.com/593](https://www.erdosproblems.com/593)\n- [EGH75] Erdős, Paul and Galvin, Fred and Hajnal, András, On set-systems having large\n  chromatic number and not containing prescribed subsystems.\n  Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th\n  birthday), Vol. I. Colloq. Math. Soc. János Bolyai 10, North-Holland (1975), 425–513.\n- [Er95d] Erdős, Paul, Some of my favourite problems in various branches of combinatorics.\n  Matematiche (Catania) 47 (1992), no. 2, 231–240 (1995).\n","FormalConjectures.ErdosProblems.«594»":"# Erdős Problem 594\n\n*References:*\n- [erdosproblems.com/594](https://www.erdosproblems.com/594)\n- [ErHa66] Erdős, P. and Hajnal, A., On chromatic number of graphs and set-systems.\n  Acta Math. Acad. Sci. Hungar. (1966), 61-99.\n- [Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in\n  Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968)\n  (1969), 27-35.\n- [EHS74] Erdős, P. and Hajnal, A. and Shelah, S., On some general properties of chromatic\n  numbers. Topics in topology (Proc. Colloq., Keszthely, 1972) (1974), 243-255.\n","FormalConjectures.ErdosProblems.«595»":"# Erdős Problem 595\n\n*References:*\n- [erdosproblems.com/595](https://www.erdosproblems.com/595)\n- [Er87] Erdős, Paul, Problems and results on set systems and hypergraphs. Extremal problems\n  for finite sets (Visegrád, 1991), Bolyai Soc. Math. Stud. (1994), 217-227.\n- [Fo70] Folkman, Jon, Graphs with monochromatic complete subgraphs in every edge coloring.\n  SIAM J. Appl. Math. (1970), 19:340-345.\n- [NeRo75] Nešetřil, Jaroslav and Rödl, Vojtěch, Type theory of partition problems of graphs.\n  Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974),\n  Academia, Prague (1975), 405-412.\n","FormalConjectures.ErdosProblems.«596»":"# Erdős Problem 596\n\n*References:*\n- [erdosproblems.com/596](https://www.erdosproblems.com/596)\n- [Er87] Erdős, *Some of my favourite problems in various branches of combinatorics*,\n  *Mat. Lapok* 1987.\n- [NeRo75] Nešetřil and Rödl, *The Ramsey property for graphs with forbidden complete\n  subgraphs*, *J. Combin. Theory* B **20** (1976), 243--249.\n","FormalConjectures.ErdosProblems.«598»":"# Erdős Problem 598\n\n*Reference:* [erdosproblems.com/598](https://www.erdosproblems.com/598)\n","FormalConjectures.ErdosProblems.«599»":"# Erdős Problem 599\n\n*References:*\n- [erdosproblems.com/599](https://www.erdosproblems.com/599)\n- [AhBe09] Aharoni, Ron and Berger, Eli, *Menger's theorem for infinite graphs*,\n  *Invent. Math.* **176** (2009), 1--62.\n","FormalConjectures.ErdosProblems.«5»":"# Erdős Problem 5\n\n*References:*\n- [erdosproblems.com/5](https://www.erdosproblems.com/5)\n- [BFM16] Banks, William D. and Freiberg, Tristan and Maynard, James, *On limit points of the\n  sequence of normalized prime gaps*. Proc. Lond. Math. Soc. (3) (2016), 515-539.\n- [Er55] Erdős, Paul, *Some remarks on number theory*. Riveon Lematematika (1955), 45-48.\n- [Er65b] Erdős, Paul, *Some recent advances and current problems in number theory*. Lectures on\n  Modern Mathematics, Vol. III (1965), 196-244.\n- [Er85c] Erdős, P., *On some of my problems in number theory I would most like to see solved*.\n  Number theory (Ootacamund, 1984) (1985), 74-84.\n- [Er97c] Erdős, Paul, *Some of my favorite problems and results*. The mathematics of Paul Erdős,\n  I (1997), 47-67.\n- [GPY09] Goldston, Daniel A. and Pintz, János and Yıldırım, Cem Y., *Primes in tuples. I*.\n  Ann. of Math. (2) (2009), 819-862.\n- [HiMa88] Hildebrand, Adolf and Maier, Helmut, *Gaps between prime numbers*. Proc. Amer. Math.\n  Soc. (1988), 1-9.\n- [Me20] Merikoski, Jori, *Limit points of normalized prime gaps*. J. Lond. Math. Soc. (2) (2020),\n  99-124.\n- [Pi16] Pintz, János, *Polignac numbers, conjectures of Erdős on gaps between primes, arithmetic\n  progressions in primes, and the bounded gap conjecture*. From arithmetic to zeta-functions\n  (2016), 367-384.\n- [Ri56] Ricci, Giovanni, *Recherches sur l'allure de la suite $\\{p_{n+1}-p_n/\\log p_n\\}$*.\n  Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 93-106.\n- [We31] Westzynthius, E., *Über die Verteilung der Zahlen, die zu den n ersten Primzahlen\n  teilerfremd sind*. Commentat. Phys. Math. (1931), 1-37.\n","FormalConjectures.ErdosProblems.«600»":"# Erdős Problem 600\n*Reference:*\n- [erdosproblems.com/600](https://www.erdosproblems.com/600)\n- [erdosproblems.com/80](https://www.erdosproblems.com/80)\n- [Er87] Erdős, P., _Some problems on finite and infinite graphs_. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228.\n- [RuSz78] Ruzsa, I. Z. and Szemerédi, E., _Triple systems with no six points carrying three triangles_. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II (1978), 939-945.\n","FormalConjectures.ErdosProblems.«602»":"# Erdős Problem 602\n\n*Reference:* [erdosproblems.com/602](https://www.erdosproblems.com/602)\n","FormalConjectures.ErdosProblems.«609»":"# Erdős Problem 609\n\n*References:*\n- [erdosproblems.com/609](https://www.erdosproblems.com/609)\n- [ErGr75] Erdős, P. and Graham, R. L., On partition theorems for finite graphs.\n  Colloq. Math. Soc. János Bolyai (1975).\n- [Ch97] Chung, F., Open problems of Paul Erdős in graph theory. J. Graph Theory (1997), 3-36.\n- [DaJo17] Day, A. N. and Johnson, J. R., Multicolour Ramsey numbers of odd cycles.\n  J. Combin. Theory Ser. B (2017), 56-63.\n- [GiHu24] Girão, A. and Hunter, Z., Monochromatic odd cycles in edge-coloured complete graphs.\n  arXiv:2412.07708 (2024).\n- [JaYi25] Janzer, O. and Yip, F., Short monochromatic odd cycles.\n  arXiv:2506.14910 (2025).\n","FormalConjectures.ErdosProblems.«60»":"# Erdős Problem 60\n\n*References:*\n- [erdosproblems.com/60](https://www.erdosproblems.com/60)\n- [HeMaYa21] He, J. and Ma, J. and Yang, T., *Some extremal results on 4-cycles*. Journal of\n  Combinatorial Theory B (2021).\n","FormalConjectures.ErdosProblems.«613»":"# Erdős Problem 613\n\n*Reference:* [erdosproblems.com/613](https://www.erdosproblems.com/613)\n","FormalConjectures.ErdosProblems.«615»":"# Erdős Problem 615\n\nA Ramsey–Turán problem of Erdős, Hajnal, Simonovits, Sós, and Szemerédi [EHSSS93]: does there\nexist a constant $c > 0$ such that every graph on $n$ vertices with at least $(1/8 - c)n^2$\nedges contains either a $K_4$ or an independent set on at least $n/\\log n$ vertices? In the\nnotation of Ramsey–Turán theory this asks whether\n$$\\mathrm{rt}(n; 4, n/\\log n) < (1/8 - c)n^2.$$\n\nThis was disproved by Fox, Loh, and Zhao [FLZ15], who showed that\n$\\mathrm{rt}(n; 4, ne^{-f(n)}) \\geq (1/8 - o(1))n^2$ whenever\n$f(n) = o(\\sqrt{\\log n/\\log\\log n})$. In the other direction Sudakov [Su03] had shown that\n$\\mathrm{rt}(n; 4, ne^{-f(n)}) = o(n^2)$ whenever $f(n)/\\sqrt{\\log n} \\to \\infty$.\n\n*References:*\n* [erdosproblems.com/615](https://www.erdosproblems.com/615)\n* [EHSSS93] Erdős, P., Hajnal, A., Simonovits, M., Sós, V. T., and Szemerédi, E.,\n  *Turán-Ramsey theorems and simple asymptotically extremal structures*. Combinatorica 13\n  (1993), 31--56.\n* [Su03] Sudakov, B., *A few remarks on Ramsey-Turán-type problems*. J. Combin. Theory Ser. B\n  88 (2003), 99--106.\n* [FLZ15] Fox, J., Loh, P.-S., and Zhao, Y., *The critical window for the classical\n  Ramsey-Turán problem*. Combinatorica 35 (2015), 435--476.\n","FormalConjectures.ErdosProblems.«617»":"# Erdős Problem 617\n\n*References:*\n- [erdosproblems.com/617](https://www.erdosproblems.com/617)\n- [ErGy99] Erdős, Paul and Gyárfás, András, Split and balanced colorings of complete graphs.\n  Discrete Math. (1999), 79-86.\n","FormalConjectures.ErdosProblems.«618»":"# Erdős Problem 618\n\n*References:*\n- [erdosproblems.com/618](https://www.erdosproblems.com/618)\n- [Er99] Erdős, Paul, *A selection of problems and results in combinatorics*. Combin. Probab.\n  Comput. (1999), 1-6.\n- [EGR98] Erdős, Paul and Gyárfás, András and Ruszinkó, Miklós, *How to decrease the diameter\n  of triangle-free graphs*. Combinatorica (1998), 493-501.\n","FormalConjectures.ErdosProblems.«619»":"# Erdős Problem 619\n\n*References:*\n- [erdosproblems.com/619](https://www.erdosproblems.com/619)\n- [EGR98] Erdős, Paul and Gyárfás, András and Ruszinkó, Miklós, *How to decrease the\n  diameter of triangle-free graphs*. Combinatorica **18** (1998), 493-501.\n- [Er99] Erdős, Paul, *A selection of problems and results in combinatorics*.\n  Combin. Probab. Comput. **8** (1999), 1-6.\n","FormalConjectures.ErdosProblems.«61»":"# Erdős Problem 61 -- Erdős–Hajnal Conjecture\n\n*Reference:* [erdosproblems.com/61](https://www.erdosproblems.com/61)\n","FormalConjectures.ErdosProblems.«621»":"# Erdős Problem 621\n\n*References:*\n- [erdosproblems.com/621](https://www.erdosproblems.com/621)\n- [EGT96] Erdős, Paul and Gallai, Tibor and Tuza, Zsolt, *Covering and independence in triangle\n  structures*. Discrete Math. (1996), 89-101.\n- [Er99] Erdős, Paul, *A selection of problems and results in combinatorics*.\n  Combin. Probab. Comput. (1999), 1-6.\n- [NoSu16] S. Norin and Y.-R. Sun, *Triangle-free independent sets vs. cuts*. arXiv:1602.04370\n  (2016).\n","FormalConjectures.ErdosProblems.«623»":"# Erdős Problem 623\n\n*Reference:* [erdosproblems.com/623](https://www.erdosproblems.com/623)\n","FormalConjectures.ErdosProblems.«624»":"# Erdős Problem 624\n\n*Reference:* [erdosproblems.com/624](https://www.erdosproblems.com/624)\n\n","FormalConjectures.ErdosProblems.«628»":"# Erdős Problem 628\n\n*References:*\n- [erdosproblems.com/628](https://www.erdosproblems.com/628)\n- [BKPS09] Balogh, József and Kostochka, Alexandr V. and Prince, Noah and Stiebitz, Michael,\n  *The Erdős-Lovász Tihany conjecture for quasi-line graphs*. Discrete Math. (2009), 3985-3991.\n- [BrJu69] Brown, W. G. and Jung, H. A., *On odd circuits in chromatic graphs*. Acta Math. Acad.\n  Sci. Hungar. (1969), 129-134.\n- [Er68b] Erdős, P., *Problem 2*. Theory of Graphs (1968), 361.\n- [So22] Song, Zi-Xia, *A survey on the Erdős-Lovász Tihany conjecture*. Adv. Math. (China)\n  (2022), 259--274.\n","FormalConjectures.ErdosProblems.«633»":"# Erdős Problem 633\n\n*Reference:*\n* [erdosproblems.com/633](https://www.erdosproblems.com/633)\n* [So09] Soifer, Alexander, How Does One Cut a Triangle? I\n* [So09c] Soifer, Alexander, Is there anything beyond the solution?\n","FormalConjectures.ErdosProblems.«639»":"# Erdős Problem 639\n\n*References:*\n- [erdosproblems.com/639](https://www.erdosproblems.com/639)\n- [Er97d] Erdős, Paul, *Some recent problems and results in graph theory*. Discrete Math.\n  (1997), 81-85.\n- [KeSu04] Keevash, Peter and Sudakov, Benny, *On the number of edges not covered by monochromatic\n  copies of a fixed graph*. J. Combin. Theory Ser. B (2004), 41-53.\n- [Py86] Pyber, L., *Clique covering of graphs*. Combinatorica (1986), 393-398.\n","FormalConjectures.ErdosProblems.«63»":"# Erdős Problem 63\n\n*References:*\n- [erdosproblems.com/63](https://www.erdosproblems.com/63)\n- [dBEr51] de Bruijn, N. G. and Erdős, P., *A colour problem for infinite graphs and a problem\n  in the theory of relations*. Indag. Math. (1951), 369--373.\n- [ErHa66] Erdős, P. and Hajnal, A., *On chromatic number of graphs and set-systems*.\n  Acta Math. Acad. Sci. Hungar. (1966), 61-99.\n- [LiMo20] Liu, Hong and Montgomery, Richard, *A solution to Erdős and Hajnal's odd cycle problem*.\n  arXiv:2010.15802 (2020).\n- [Re24] Reiher, C., *Graphs of large girth*. arXiv:2403.13571 (2024).\n","FormalConjectures.ErdosProblems.«645»":"# Erdős Problem 645\n\n*References:*\n- [erdosproblems.com/645](https://www.erdosproblems.com/645)\n- [BrLa99] Brown, Tom C. and Landman, Bruce M., Monochromatic arithmetic progressions with large\n  differences. Bull. Austral. Math. Soc. (1999), 21--35.\n","FormalConjectures.ErdosProblems.«646»":"# Erdős Problem 646\n\n*References:*\n- [erdosproblems.com/646](https://www.erdosproblems.com/646)\n- [ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number\n  theory*. Monographies de L'Enseignement Mathematique (1980).\n- [Er97e] Erdős, Paul, *Some of my favourite unsolved problems*. Math. Japon. (1997), 527-537.\n- [Be97] Berend, Daniel, *On the parity of exponents in the factorization of $n!$*.\n  J. Number Theory (1997), 13-19.\n","FormalConjectures.ErdosProblems.«647»":"# Erdős Problem 647\n\n*Reference:* [erdosproblems.com/647](https://www.erdosproblems.com/647)\n","FormalConjectures.ErdosProblems.«648»":"# Erdős Problem 648\n\n*References:*\n- [erdosproblems.com/648](https://www.erdosproblems.com/648)\n- [Er95c] Erdős, Paul, *Some problems in number theory*. Octogon Math. Mag. (1995), 3-5.\n- [Ca25b] S. Cambie, *On Erdős problem #648*. arXiv:2503.22691 (2025).\n","FormalConjectures.ErdosProblems.«649»":"# Erdős Problem 649\n\n*References:*\n- [erdosproblems.com/649](https://www.erdosproblems.com/649)\n- [Ma35] Mahler, Kurt, *Über den grössten Primteiler spezieller Polynome zweiten Grades*. Archiv\n  für math. og naturvid (1935).\n- [Ro64b] Rotkiewicz, André, *Sur les nombres naturels $n$ et $k$ tels que les nombres $n$ et $nk$\n  sont à la fois pseudopremiers*. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8)\n  (1964), 816-818.\n","FormalConjectures.ErdosProblems.«64»":"# Erdős Problem 64\n\n*Reference:* [erdosproblems.com/64](https://www.erdosproblems.com/64)\n","FormalConjectures.ErdosProblems.«650»":"# Erdős Problem 650\n\n*References:*\n- [erdosproblems.com/650](https://www.erdosproblems.com/650)\n- [Er78] Erdős, Paul, *Problems and results in combinatorial analysis and combinatorial number\n  theory*. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and\n  Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40.\n- [Er86c] Erdős, P., *Some problems on number theory*. (1986), 53-67.\n- [Er95c] Erdős, Paul, *Some problems in number theory*. Octogon Math. Mag. (1995), 3-5.\n- [ErSu59] Erdős, Pál and Surányi, János, *Bemerkungen zu einer Aufgabe eines mathematischen\n  Wettbewerbs*. Mat. Lapok (1959), 39-48.\n- [VLT26] W. Van Doorn, Y. Li, and Q. Tang, *Optimal bounds for an Erdős problem on matching\n  integers to distinct multiples*. arXiv:2603.28636 (2026).\n","FormalConjectures.ErdosProblems.«653»":"# Erdős Problem 653\n\n*Reference:* [erdosproblems.com/653](https://www.erdosproblems.com/653)\n","FormalConjectures.ErdosProblems.«655»":"# Erdős Problem 655\n\n*Reference:* [erdosproblems.com/655](https://www.erdosproblems.com/655)\n","FormalConjectures.ErdosProblems.«659»":"# Erdős Problem 659\n\n*References:*\n- [erdosproblems.com/659](https://www.erdosproblems.com/659)\n- [MoOs06] Moree, Pieter and Osburn, Robert, Two-dimensional lattices with few distances. Enseign. Math. (2) (2006), 361--380\n- [ErFi96] Erdős, Paul and Fishburn, Peter, Maximum planar sets that determine {$k$} distances. Discrete Math. (1996), 115--125.\n- [Gr26](https://arxiv.org/abs/2601.09102): Benjamin Grayzel, Solution to a Problem of Erdős Concerning Distances and Points\n","FormalConjectures.ErdosProblems.«65»":"# Erdős Problem 65\n\n*References:*\n- [erdosproblems.com/65](https://www.erdosproblems.com/65)\n- [GKS84] Gyárfás, A., Komlós, J. and Szemerédi, E., On the distribution of cycle lengths in graphs.\n  J. Graph Theory (1984), 441-462.\n- [LiMo20] Liu, H. and Montgomery, R., A solution to Erdős and Hajnal's odd cycle problem.\n  arXiv:2010.15802 (2020).\n","FormalConjectures.ErdosProblems.«660»":"# Erdős Problem 660\n\n*References:*\n- [erdosproblems.com/660](https://www.erdosproblems.com/660)\n- [Er97e] Erdős, Paul, *Some of my favorite problems and results*, The mathematics of Paul Erdős,\n  I (1997), 47–67.\n- [Al63] Altman, E., *On a problem of P. Erdős*, Amer. Math. Monthly (1963), 148–157.\n- [Er75f] Erdős, Paul, *On some problems of elementary and combinatorial geometry*, Ann. Mat. Pura\n  Appl. (4) (1975), 99–108.\n","FormalConjectures.ErdosProblems.«666»":"# Erdős Problem 666\n\n*References:*\n- [erdosproblems.com/666](https://www.erdosproblems.com/666)\n- [BDT93] Brouwer, A. E. and Dejter, I. J. and Thomassen, C., *Highly symmetric subgraphs of\n  hypercubes*. J. Algebraic Combin. (1993), 25-29.\n- [Ch92] Chung, Fan R. K., *Subgraphs of a hypercube containing no small even cycles*. J. Graph\n  Theory (1992), 273-286.\n- [Er91] Erdős, P., *Problems and results in combinatorial analysis and combinatorial number\n  theory*. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991),\n  397-406.\n","FormalConjectures.ErdosProblems.«66»":"# Erdős Problem 66\n\n*Reference:* [erdosproblems.com/66](https://www.erdosproblems.com/66)\n","FormalConjectures.ErdosProblems.«672»":"# Erdős Problem 672\n\n*Reference:* [erdosproblems.com/672](https://www.erdosproblems.com/672)\n","FormalConjectures.ErdosProblems.«674»":"# Erdős Problem 674\n\n*References:*\n- [erdosproblems.com/674](https://www.erdosproblems.com/674)\n- [Ko40] Ko, Chao, *Note on the Diophantine equation $x^xy^y=z^z$*. J. Chinese Math. Soc.\n  (1940), 31-39.\n","FormalConjectures.ErdosProblems.«677»":"# Erdős Problem 677\n*Reference:* [erdosproblems.com/677](https://www.erdosproblems.com/677)\n","FormalConjectures.ErdosProblems.«678»":"# Erdős Problem 678\n*References:*\n- [erdosproblems.com/678](https://www.erdosproblems.com/678)\n- [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138\n  (2024).\n- [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.\n- [Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka\n  (1992), 44-48.\n","FormalConjectures.ErdosProblems.«67»":"# Erdős Problem 67\n\n*References:*\n- [erdosproblems.com/67](https://www.erdosproblems.com/66)\n- [Ta16] Tao, Terence, The Erdős discrepancy problem. Discrete Anal. (2016), Paper No. 1, 29.\n","FormalConjectures.ErdosProblems.«680»":"# Erdős Problem 680\n\n*Reference:* [erdosproblems.com/680](https://www.erdosproblems.com/680)\n","FormalConjectures.ErdosProblems.«681»":"# Erdős Problem 681\n\n*Reference:* [erdosproblems.com/681](https://www.erdosproblems.com/681)\n","FormalConjectures.ErdosProblems.«683»":"# Erdős Problem 683\n\n*References:*\n- [erdosproblems.com/683](https://www.erdosproblems.com/683)\n- [Er34] Erdős, Paul, A Theorem of Sylvester and Schur. J. London Math. Soc. (1934), 282--288.\n- [Er55d] Erdős, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.\n- [Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.\n","FormalConjectures.ErdosProblems.«686»":"# Erdős Problem 686\n\n*References:*\n- [erdosproblems.com/686](https://www.erdosproblems.com/686)\n- [Er79d] Erdős, P., *Some unconventional problems in number theory*. Acta Math.\n  Acad. Sci. Hungar. (1979), 71-80.\n","FormalConjectures.ErdosProblems.«688»":"# Erdős Problem 688\n*Reference:*\n- [erdosproblems.com/688](https://www.erdosproblems.com/688)\n- [Er80] Erdős, Paul, _A survey of problems in combinatorial number theory_. Ann. Discrete Math. (1980), 89-115.\n","FormalConjectures.ErdosProblems.«689»":"# Erdős Problem 689\n*References:*\n* [erdosproblems.com/689](https://www.erdosproblems.com/689)\n* [Ben Green's Open Problem 45](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.45)\n","FormalConjectures.ErdosProblems.«68»":"# Erdős Problem 68\n\n*Reference:* [erdosproblems.com/68](https://www.erdosproblems.com/68)\n","FormalConjectures.ErdosProblems.«692»":"# Erdős Problem 692\n\n*References:*\n- [erdosproblems.com/692](https://www.erdosproblems.com/692)\n- [Er79e] Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82.\n- [Ob1] Erdős, P., *Oberwolfach Mathematical Problems, Volume 1*. Mathematisches\n  Forschungsinstitut Oberwolfach (Various).\n- [Fo08] Ford, Kevin, *The distribution of integers with a divisor in a given interval*.\n  Ann. of Math. (2) (2008), 367-433.\n- [Ca25] Cambie, S., *Resolution of Erdős' problems about unimodularity*.\n  arXiv:2501.10333 (2025).\n","FormalConjectures.ErdosProblems.«694»":"# Erdős Problem 694\n\n*Reference:* [erdosproblems.com/694](https://www.erdosproblems.com/694)\n","FormalConjectures.ErdosProblems.«695»":"# Erdős Problem 695\n*Reference:* [erdosproblems.com/695](https://www.erdosproblems.com/695)\n","FormalConjectures.ErdosProblems.«697»":"# Erdős Problem 697\n\n*Reference:*\n - [erdosproblems.com/697](https://www.erdosproblems.com/697)\n - [Ha92] Hall, R. R., On some conjectures of Erdős in Astérisque. I. J. Number Theory (1992),\n    313--319.\n","FormalConjectures.ErdosProblems.«698»":"# Erdős Problem 698\n\n*References:*\n- [erdosproblems.com/698](https://www.erdosproblems.com/698)\n- [ErSz78] Erdős, P. and Szekeres, G., *Some number theoretic problems on binomial\n  coefficients*. Austral. Math. Soc. Gaz. (1978), 97-99.\n- [Be11] Bergman, George M., *On common divisors of multinomial coefficients*. Bull. Aust.\n  Math. Soc. (2011), 138--157.\n","FormalConjectures.ErdosProblems.«699»":"# Erdős Problem 699\n\n*Reference:* [erdosproblems.com/699](https://www.erdosproblems.com/699)\n","FormalConjectures.ErdosProblems.«69»":"# Erdős Problem 69\n\n*Reference:* [erdosproblems.com/69](https://www.erdosproblems.com/69)\n","FormalConjectures.ErdosProblems.«6»":"# Erdős Problem 6\n\n*References:*\n- [erdosproblems.com/6](https://www.erdosproblems.com/6)\n- [BFT15] Banks, William D. and Freiberg, Tristan and Turnage-Butterbaugh, Caroline L., Consecutive primes in tuples. Acta Arith. (2015), 261-266.\n- [Ma15] Maynard, James, Small gaps between primes. Ann. of Math. (2) (2015), 383-413.\n","FormalConjectures.ErdosProblems.«700»":"# Erdős Problem 700\n\n*Reference:* [erdosproblems.com/700](https://www.erdosproblems.com/700)\n\nA problem of Erdős and Szekeres [ErSz78].\n\n*References:*\n * [ErSz78] Erdős, P. and Szekeres, G., _Some number theoretic problems on binomial coefficients_,\n   Austral. Math. Soc. Gaz. (1978), 97-99.\n * [OEIS A091963](https://oeis.org/A091963)\n * Guy, R. K., _Unsolved Problems in Number Theory_, B31, B33.\n","FormalConjectures.ErdosProblems.«701»":"# Erdős Problem 701\n\n*Reference:* [erdosproblems.com/701](https://www.erdosproblems.com/701)\n","FormalConjectures.ErdosProblems.«705»":"# Erdős Problem 705\n\n*References:*\n- [erdosproblems.com/705](https://www.erdosproblems.com/705)\n- [OD99] P. O'Donnell, High girth unit-distance graphs. PhD Dissertation, Rutgers University (1999).\n","FormalConjectures.ErdosProblems.«707»":"# Erdős Problem 707: Embedding Sidon Sets in Perfect Difference Sets\n\n*References:*\n- [erdosproblems.com/707](https://www.erdosproblems.com/707)\n- [arxiv/2510.19804](https://arxiv.org/abs/2510.19804) Boris Alexeev and Dustin G. Mixon, Forbidden\n  Sidon subsets of perfect difference sets, featuring a human-assisted proof (2025)\n- [Ha47] Marshall Hall, Jr., Cyclic projective planes, Duke Math. J. 14 (1947), 1079–1090.\n\nLet `A ⊆ ℕ` be a finite Sidon set. Is there some set `B` with `A ⊆ B` which is a perfect\ndifference set modulo `p^2 + p + 1` for some prime power `p`?\n\nThis problem is related to Erdős Problem 329 about the maximum density of Sidon sets.\nIf this conjecture is true, it would imply that the maximum density of Sidon sets is 1.\n","FormalConjectures.ErdosProblems.«70»":"# Erdős Problem 70\n\n*Reference:* [erdosproblems.com/70](https://www.erdosproblems.com/70)\n\nThe 3-uniform (triple) partition relation $\\mathfrak{c} \\to (\\beta, n)^3_2$\non the ordinal of the real numbers — the triple analogue of `OrdinalCardinalRamsey`\nused in Problems 590–592.\n","FormalConjectures.ErdosProblems.«713»":"# Erdős Problem 713\n\n*References:*\n- [erdosproblems.com/713](https://www.erdosproblems.com/713)\n- [Er67d] Erdős, P., *Some recent results on extremal problems in graph theory. {R}esults*. (1967),\n  117--123 (English); pp. 124--130 (French).\n- [ErSi70] Erdős, P. and Simonovits, M., *Some extremal problems in graph theory*. Combinatorial\n  theory and its applications, I-III (Proc. Colloq., Balatonfüred, 1969) (1970), 377-390.\n- [FrFu87] Frankl, P. and Füredi, Z., *Exact solution of some Turán-type problems*. J. Combin.\n  Theory Ser. A (1987), 226--262.\n- [FuGe21] Füredi, Zoltán and Gerbner, Dániel, *Hypergraphs without exponents*. J. Combin. Theory\n  Ser. A (2021), Paper No. 105517, 9.\n","FormalConjectures.ErdosProblems.«714»":"# Erdős Problem 714\n\n*References:*\n- [erdosproblems.com/714](https://www.erdosproblems.com/714)\n- [Br66] Brown, W. G., *On graphs that do not contain a Thomsen graph*. Canad. Math. Bull. (1966),\n  281-285.\n- [ERS66] Erdős, P. and Rényi, A. and Sós, V. T., *On a problem of graph theory*. Studia Sci. Math.\n  Hungar. (1966), 215--235.\n- [KST54] Kövari, T. and Sós, V. T. and Turán, P., *On a problem of K. Zarankiewicz*. Colloq. Math.\n  (1954), 50-57.\n","FormalConjectures.ErdosProblems.«71»":"# Erdős Problem 71\n\n*References:*\n- [erdosproblems.com/71](https://www.erdosproblems.com/71)\n- [Bo77] Bollobás, Béla, *Cycles modulo $k$*. Bull. London Math. Soc. (1977), 97-98.\n- [Er82e] Erdős, Paul, *Some of my favourite problems which recently have been solved*.\n  (1982), 59--79.\n- [Er95] Erdős, Paul, *Some of my favourite problems in number theory, combinatorics, and\n  geometry*. Resenhas (1995), 165-186.\n- [Er97b] Erdős, Paul, *Some old and new problems in various branches of combinatorics*.\n  Discrete Math. (1997), 227-231.\n","FormalConjectures.ErdosProblems.«723»":"# Erdős Problem 723: The prime power conjecture.\n\n*Reference:* [erdosproblems.com/723](https://www.erdosproblems.com/723)\n","FormalConjectures.ErdosProblems.«726»":"# Erdős Problem 726\n\n*References:*\n- [erdosproblems.com/726](https://www.erdosproblems.com/726)\n- [EGRS75] Erdős, P., and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of\n  $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n","FormalConjectures.ErdosProblems.«727»":"# Erdős Problem 727\n\n*Reference:* [erdosproblems.com/727](https://www.erdosproblems.com/727)\n","FormalConjectures.ErdosProblems.«728»":"# Erdős Problem 728\n\n*Reference:* [erdosproblems.com/728](https://www.erdosproblems.com/728)\n","FormalConjectures.ErdosProblems.«729»":"# Erdős Problem 729\n\n*References:*\n- [erdosproblems.com/729](https://www.erdosproblems.com/729)\n- [EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., *On the prime factors of $\\binom{2n}{n}$*. Math. Comp. (1975), 83-92.\n- [Er68c] Erdős, P., *Aufgabe 557*. Elemente Math. (1968), 111-113.\n","FormalConjectures.ErdosProblems.«730»":"# Erdős Problem 730\n\n*References:*\n  - [erdosproblems.com/730](https://www.erdosproblems.com/730)\n  - [A129515](https://oeis.org/A129515)\n","FormalConjectures.ErdosProblems.«73»":"# Erdős Problem 73\n\n*References:*\n- [erdosproblems.com/73](https://www.erdosproblems.com/73)\n- [Re99] Reed, B., Mangoes and Blueberries. Combinatorica (1999), 267-296.\n","FormalConjectures.ErdosProblems.«740»":"# Erdős Problem 740: Infinitary version of chromatic number and odd cycles\n\n*Reference:* [erdosproblems.com/740](https://erdosproblems.com/740)\n","FormalConjectures.ErdosProblems.«741»":"# Erdős Problem 741\n\n*References:*\n - [erdosproblems.com/741](https://www.erdosproblems.com/741)\n - [Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry.\n    Math. Pannon. (1994), 261-269.\n","FormalConjectures.ErdosProblems.«742»":"# Erdős Problem 742\n\n*References:*\n- [erdosproblems.com/742](https://www.erdosproblems.com/742)\n- [Pl75] Plesník, Ján, Critical graphs of given diameter. Acta Fac. Rerum Natur. Univ. Comenian.\n  Math. 30 (1975), 71-93.\n- [CaHa79] Caccetta, L. and Häggkvist, R., On diameter critical graphs.\n  Discrete Math. 28 (1979), 223-229.\n- [Fa87] Fan, Genghua, On diameter 2-critical graphs. Discrete Math. 67 (1987), 235-240.\n- [Fü92] Füredi, Zoltán, The maximum number of edges in a minimal graph of diameter 2.\n  J. Graph Theory 16 (1992), 81-98.\n","FormalConjectures.ErdosProblems.«749»":"# Erdős Problem 749\n\n*Reference:* [erdosproblems.com/749](https://www.erdosproblems.com/749)\n","FormalConjectures.ErdosProblems.«74»":"# Erdős Problem 74\n\n*Reference:* [erdosproblems.com/74](https://www.erdosproblems.com/74)\n","FormalConjectures.ErdosProblems.«750»":"# Erdős Problem 750\n\n*References:*\n- [erdosproblems.com/750](https://www.erdosproblems.com/750)\n- [EHS82] Erdős, P. and Hajnal, A. and Szemerédi, E., On almost bipartite large chromatic graphs.\n  Theory and practice of combinatorics (1982), 117-123.\n- [Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph\n  Theory (1969), 27-35.\n- [Er94b] Erdős, Paul, _Some problems in number theory, combinatorics and combinatorial geometry_.\n  Math. Pannon. (1994), 261-269.\n- [Er95d] Erdős, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd)\n  (N.S.) (1995), 61-65.\n- [ErHa67b] Erdős, P. and Hajnal, András, On chromatic graphs. Mat. Lapok (1967), 1--4.\n- [UlamErdos750](https://www.ulam.ai/research/erdos750.pdf)\n- [St85] Stiebitz, M., _Beiträge zur Theorie der färbungskritischen Graphen_. Habilitation,\n  TH Ilmenau (1985).\n- [SaSt89] Sachs, H. and Stiebitz, M., _On constructive methods in the theory of colour-critical\n  graphs_. Discrete Math. (1989), 287-296.\n- [MuSt19] Müller, T. and Stehlík, M., _Generalised Mycielski graphs and the Borsuk-Ulam theorem_.\n  Electron. J. Combin. (2019), P4.8.\n","FormalConjectures.ErdosProblems.«751»":"# Erdős Problem 751\n\n*References:*\n- [erdosproblems.com/751](https://www.erdosproblems.com/751)\n- [Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry.\n  Math. Pannon. (1994), 261--269.\n- [BoVi98] Bondy, J. A. and Vince, A., Cycles in a graph whose lengths differ by one or two.\n  J. Graph Theory (1998), 11--15.\n","FormalConjectures.ErdosProblems.«753»":"# Erdős Problem 753\n\n*References:*\n- [erdosproblems.com/753](https://www.erdosproblems.com/753)\n- [Al92] Alon, Noga, *Choice numbers of graphs: a probabilistic approach*. Combin. Probab.\n  Comput. (1992), 107-114.\n","FormalConjectures.ErdosProblems.«755»":"# Erdős Problem 755\n\n*References:*\n- [erdosproblems.com/755](https://www.erdosproblems.com/755)\n- [ErPu75] Erdős, Paul and Purdy, George, Some extremal problems in geometry. III.\n  (1975), 291--308.\n- [Er94b] Erdős, Paul, Some problems in number theory, combinatorics and\n  combinatorial geometry. Math. Pannon. (1994), 261--269.\n- [CDL25b] Clemen, Felix Christian, Dumitrescu, Adrian, and Liu, Dingyuan,\n  The number of regular simplices in higher dimensions. arXiv:2507.19841 (2025).\n","FormalConjectures.ErdosProblems.«756»":"# Erdős Problem 756\n\n*References:*\n- [erdosproblems.com/756](https://www.erdosproblems.com/756)\n- [Bh24] Bhowmick, K., *A problem of Erdős about rich distances*. arXiv:2407.01174 (2024).\n- [CDL25] Clemen, F., Dumitrescu, A. and Liu, D., *On multiplicities of interpoint distances*.\n  arXiv:2505.04283 (2025).\n- [Er97b] Erdős, Paul, *Some old and new problems in various branches of combinatorics*.\n  Discrete Math. (1997), 227-231.\n- [ErPa90] Erdős, P. and Pach, J., *Variations on the theme of repeated distances*.\n  Combinatorica (1990), 261-269.\n- [HoPa34] Hopf, H. and Pannwitz, E., *Aufgabe 167*. Jber. Deutsch. Math. Verein. (1934), 114.\n","FormalConjectures.ErdosProblems.«757»":"# Erdős Problem 757\n\n*References:*\n - [erdosproblems.com/757](https://www.erdosproblems.com/757)\n - [GyLe95] Gyárfás, András and Lehel, Jenő, Linear sets with five distinct differences among any\n    four elements. J. Combin. Theory Ser. B (1995), 108-118.\n","FormalConjectures.ErdosProblems.«75»":"# Erdős Problem 75\n\n*Reference:*\n* [erdosproblems.com/75] (https://www.erdosproblems.com/75)\n","FormalConjectures.ErdosProblems.«760»":"# Erdős Problem 760\n\n*References:*\n- [erdosproblems.com/760](https://www.erdosproblems.com/760)\n- [AKS97] Alon, Noga and Krivelevich, Michael and Sudakov, Benny, *Subgraphs with a large\n  cochromatic number*. J. Graph Theory (1997), 295-297.\n","FormalConjectures.ErdosProblems.«762»":"# Erdős Problem 762\n\n*References:*\n- [erdosproblems.com/762](https://www.erdosproblems.com/762)\n- [EGS90] Erdős, Paul and Gimbel, John and Straight, H. Joseph, *Chromatic number versus\n  cochromatic number in graphs with bounded clique number*. European J. Combin. (1990), 235-240.\n- [St24b] R. Steiner, *On the difference between the chromatic and cochromatic number*.\n  arXiv:2408.02400 (2024).\n","FormalConjectures.ErdosProblems.«769»":"# Erdős Problem 769\n\n*Reference:* [erdosproblems.com/769](https://www.erdosproblems.com/769)\n","FormalConjectures.ErdosProblems.«770»":"# Erdős Problem 770\n\n*References:*\n - [erdosproblems.com/770](https://www.erdosproblems.com/770)\n - [Er49d] Erdös, P. \"On the strong law of large numbers.\" Transactions of the American Mathematical\n    Society 67.1 (1949): 51-56.\n - [Ma66] Matsuyama, Noboru. \"On the strong law of large numbers.\" Tohoku Mathematical Journal,\n    Second Series 18.3 (1966): 259-269.\n","FormalConjectures.ErdosProblems.«773»":"# Erdős Problem 773\n\n*References:*\n- [erdosproblems.com/773](https://www.erdosproblems.com/773)\n- [AlEr85] Alon, Noga and Erdős, P., *An application of graph theory to additive number theory*.\n  European J. Combin. (1985), 201-203.\n- [Er80] Erdős, Paul, *A survey of problems in combinatorial number theory*. Ann. Discrete Math.\n  (1980), 89-115.\n- [LeTh95] Lefmann, Hanno and Thiele, Torsten, *Point sets with distinct distances*. Combinatorica\n  (1995), 379--408.\n","FormalConjectures.ErdosProblems.«774»":"# Erdős Problem 774\n\n*Reference:* [erdosproblems.com/774](https://www.erdosproblems.com/774)\n","FormalConjectures.ErdosProblems.«775»":"# Erdős Problem 775\n\n*References:*\n- [erdosproblems.com/775](https://www.erdosproblems.com/775)\n- [Ga25] Gao, J., *On cliques in hypergraphs*. arXiv:2510.14804 (2025).\n- [MoMo65] Moon, J. W. and Moser, L., *On cliques in graphs*. Israel J. Math. (1965), 23-28.\n- [Sp71] Spencer, J. H., *On cliques in graphs*. Israel J. Math. (1971), 419-421.\n","FormalConjectures.ErdosProblems.«779»":"# Erdős Problem 779\n\n*Reference:* [erdosproblems.com/779](https://www.erdosproblems.com/779)\n","FormalConjectures.ErdosProblems.«77»":"# Erdős Problem 77\n\n*References:*\n- [erdosproblems.com/77](https://www.erdosproblems.com/77)\n- [BBCGHMST24] Balister, P. and Bollobás, B. and Campos, M. and Griffiths, S. and Hurley, E. and\n  Morris, R. and Sahasrabudhe, J. and Tiba, M., Upper bounds for multicolour Ramsey numbers.\n  arXiv:2410.17197 (2024).\n- [CGMS23] Campos, Marcelo and Griffiths, Simon and Morris, Robert and Sahasrabudhe, Julian, An\n  exponential improvement for diagonal Ramsey. arXiv:2303.09521 (2023).\n- [Er88] Erdős, P, Problems and results in combinatorial analysis and graph theory. Discrete Math.\n  (1988), 81-92.\n- [Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones\n  Math. (1993), 333-350.\n- [GNNW24] Gupta, P. and Ndiaye, N. and Norin, S. and Wei, L., Optimizing the CGMS upper bound on\n  Ramsey numbers. arXiv:2407.19026 (2024).\n","FormalConjectures.ErdosProblems.«785»":"# Erdős Problem 785\n\n*References:*\n- [erdosproblems.com/785](https://www.erdosproblems.com/785)\n- [ChFa10] Fang, Jin-Hui and Chen, Yong-Gao, *On additive complements*. Proc. Amer. Math. Soc.\n  (2010), 1923-1927.\n- [ChFa11] Chen, Yong-Gao and Fang, Jin-Hui, *On additive complements. II*. Proc. Amer. Math. Soc.\n  (2011), 881-883.\n- [ChFa14] Fang, Jin-Hui and Chen, Yong-Gao, *On additive complements. III*. J. Number Theory\n  (2014), 83-91.\n- [ChFa15] Chen, Yong-Gao and Fang, Jin-Hui, *On a conjecture of Sárközy and Szemerédi*.\n  Acta Arith. (2015), 47-58.\n- [Da64] Danzer, L., *Über eine Frage von G. Hanani aus der additiven Zahlentheorie*.\n  J. Reine Angew. Math. (1964), 392-394.\n- [Er57] Erdős, Paul, *Some unsolved problems*. Michigan Math. J. (1957), 291-300.\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [Na59] Narkiewicz, Władysław, *Remarks on a conjecture of Hanani in additive number theory*.\n  Colloq. Math. (1959/60), 161-165.\n- [Ru17] Ruzsa, Imre Z., *Exact additive complements*. Q. J. Math. (2017), 227-235.\n- [SaSz94] Sárközy, A. and Szemerédi, E., *On a problem in additive number theory*.\n  Acta Math. Hungar. (1994), 237-245.\n","FormalConjectures.ErdosProblems.«786»":"# Erdős Problem 786\n\n*Reference:* [erdosproblems.com/786](https://www.erdosproblems.com/786)\n","FormalConjectures.ErdosProblems.«789»":"# Erdős Problem 789\n\nIn this problem, a function $h : \\mathbb{N} \\to\\mathbb{N}$ is defined maximally by\nsome counting property.\n\nThe problem asks to estimate $h(n)$. This has been interpreted here as asking for $\\Theta(h(n))$.\nThe principal version includes `answer(sorry)` for an unknown function. On the other hand, the best\nknown upper bound is $\\sqrt{n}$ and the best known lower bound is $(n\\log(n))^{1/3}$ so we\nalso provide these candidates as variants. Moreover, it suffices to show $O(h(n))$ and\n$O((n\\log(n))^{1/3})$ respectively for each, so further variants are provided for those.\n\n*References:*\n- [erdosproblems.com/789](https://www.erdosproblems.com/789)\n- [Str66] Straus, E. G., _On a problem in combinatorial number theory_. J. Math. Sci. (1966), 77--80.\n- [Er62c] Erdős, Pál, _Some remarks on number theory_. {III}. Mat. Lapok (1962), 28--38.\n- [Ch74b] Choi, S. L. G., _On an extremal problem in number theory_. J. Number Theory (1974), 105--111.\n","FormalConjectures.ErdosProblems.«794»":"# Erdős Problem 794\n\n*References:*\n- [erdosproblems.com/794](https://www.erdosproblems.com/794)\n- [Er69] Erdős, Paul, *Some applications of graph theory to number theory*. The Many Facets of\n  Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968) (1969), 77-82.\n- [FrFu84] Frankl, P. and Füredi, Z., *An exact result for $3$-graphs*. Discrete Math. (1984),\n  323-328.\n","FormalConjectures.ErdosProblems.«796»":"# Erdős Problem 796\n\n*Reference:* [erdosproblems.com/796](https://www.erdosproblems.com/796)\n","FormalConjectures.ErdosProblems.«798»":"# Erdős Problem 798\n\n*References:*\n- [erdosproblems.com/798](https://www.erdosproblems.com/798)\n- [Al91] Alon, N., *Economical coverings of sets of lattice points*. Geom. Funct. Anal. (1991),\n  224-230.\n","FormalConjectures.ErdosProblems.«79»":"# Erdős Problem 79\n\n*References:*\n- [erdosproblems.com/79](https://www.erdosproblems.com/79)\n- [EFRS93] Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs.\n  Combin. Probab. Comput. (1993), 389-399.\n- [Wi24] Wigderson, Y., Infinitely many minimally non-Ramsey size linear graphs.\n  arXiv:2409.05931 (2024).\n","FormalConjectures.ErdosProblems.«7»":"# Erdős Problem 7\n\n*Reference:* [erdosproblems.com/7](https://www.erdosproblems.com/7)\n","FormalConjectures.ErdosProblems.«800»":"# Erdős Problem 800\n\n*References:*\n- [erdosproblems.com/800](https://www.erdosproblems.com/800)\n- [Al94] Alon, N., Subdivided graphs have linear Ramsey numbers. J. Graph Theory (1994), 343-347.\n","FormalConjectures.ErdosProblems.«80»":"# Erdős Problem 80\n\n*References:*\n- [erdosproblems.com/80](https://www.erdosproblems.com/80)\n- [erdosproblems.com/600](https://www.erdosproblems.com/600), stated in\n  `FormalConjectures/ErdosProblems/600.lean`\n\n600 asks the same question from the other side. `Erdos600.eFunction n r` is the least edge\ncount forcing some edge into `r` triangles; `f c n` here is the largest book forced once the\nedge count is at least $cn^2$. So `r ≤ f c n` and `Erdos600.eFunction n r ≤ c * n^2` say the\nsame thing, and the two functions are inverse to each other in that sense. Both are built on\n`SimpleGraph.trianglesContaining`, which 600 introduced.\n","FormalConjectures.ErdosProblems.«812»":"# Erdős Problem 812\n\n*References:*\n- [erdosproblems.com/812](https://www.erdosproblems.com/812)\n- [BEFS89] Burr, S. A. and Erd\\H{o}s, P. and Faudree, R. J. and Schelp, R. H., On the difference\n  between consecutive {R}amsey numbers. Utilitas Math. (1989), 115--118.\n","FormalConjectures.ErdosProblems.«817»":"# Erdős Problem 817\n\n*Reference:* [erdosproblems.com/817](https://www.erdosproblems.com/817)\n","FormalConjectures.ErdosProblems.«818»":"# Erdős Problem 818\n\n*References:*\n- [erdosproblems.com/818](https://www.erdosproblems.com/818)\n- [So09d] Solymosi, József, *Bounding multiplicative energy by the sumset*. Adv. Math. (2009),\n  402-408.\n","FormalConjectures.ErdosProblems.«821»":"# Erdős Problem 821\n\n*References:*\n- [erdosproblems.com/821](https://www.erdosproblems.com/821)\n- [BaHa98] Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith.\n  (1998), 331--361.\n- [Er35b] Erdős, P., On the normal number of prime factors of $p-1$ and some related problems\n  concerning Euler's $\\varphi$-function. Quart. J. Math. (1935), 205-213.\n- [Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\n- [Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes\n  without large prime factors. arXiv:2211.09641 (2022).\n- [LuPo11] Luca, Florian and Pollack, Paul, An arithmetic function arising from {C}armichael's\n  conjecture. J. Théor. Nombres Bordeaux (2011), 697--714.\n","FormalConjectures.ErdosProblems.«822»":"# Erdős Problem 822\n\n*References:*\n- [erdosproblems.com/822](https://www.erdosproblems.com/822)\n- [GIL24] Gabdullin, Mikhail R. and Iudelevich, Vitalii V. and Luca,\n  Florian, Numbers of the form {$k+f(k)$}. J. Number Theory (2024), 58--85.\n","FormalConjectures.ErdosProblems.«825»":"# Erdős Problem 825\n\n*Reference:* [erdosproblems.com/825](https://www.erdosproblems.com/825)\n","FormalConjectures.ErdosProblems.«826»":"# Erdős Problem 826\n\n*Reference:* [erdosproblems.com/826](https://www.erdosproblems.com/826)\n","FormalConjectures.ErdosProblems.«828»":"# Erdős Problem 828\n\n*Reference:* [erdosproblems.com/828](https://www.erdosproblems.com/828)\n","FormalConjectures.ErdosProblems.«829»":"# Erdős Problem 829\n\n*References:*\n- [erdosproblems.com/829](https://www.erdosproblems.com/829)\n- [Er83] Erdős, P. and Dudley, U., _Some remarks and problems in number theory related to the\n  work of Euler_. Math. Mag. (1983), 292-298.\n","FormalConjectures.ErdosProblems.«82»":"# Erdős Problem 82\n\n*Reference:* [erdosproblems.com/82](https://www.erdosproblems.com/82)\n","FormalConjectures.ErdosProblems.«830»":"# Erdős Problem 830\n\n*Reference:* [erdosproblems.com/830](https://www.erdosproblems.com/830)\n","FormalConjectures.ErdosProblems.«835»":"# Erdős Problem 835\n\n*References:*\n - [erdosproblems.com/835](https://www.erdosproblems.com/835)\n - [MT25](https://github.com/QuanyuTang/erdos-problem-835/blob/main/On_Problem_835.pdf)\n","FormalConjectures.ErdosProblems.«839»":"# Erdős Problem 839\n\n*References:*\n- [erdosproblems.com/839](https://www.erdosproblems.com/839)\n- [Er78f] Erdős, P., *Problems in number theory and combinatorics*, Proc. Sixth Manitoba Conf. on\n  Numerical Math. (1978), 35-58.\n- [Er92c] Erdős, P., *Some of my favourite unsolved problems*, J. Combin. Theory Ser. A (1992).\n\nSee also [Erdős Problem 359](https://www.erdosproblems.com/359) and\n[Erdős Problem 867](https://www.erdosproblems.com/867).\n","FormalConjectures.ErdosProblems.«844»":"# Erdős Problem 844\n\n*References:*\n- [erdosproblems.com/844](https://www.erdosproblems.com/844)\n- [AMS25] Alexeev, B., Mixon, D. and Sawin, W., *The independence and clique cover numbers of the\n  squarefree graph*. arXiv:2507.01928 (2025).\n- [Ch74] Chvátal, V., *Intersecting families of edges in hypergraphs having the hereditary\n  property*. (1974), 61-66.\n","FormalConjectures.ErdosProblems.«845»":"# Erdős Problem 845\n\n*Reference:* [erdosproblems.com/845](https://www.erdosproblems.com/845)\n","FormalConjectures.ErdosProblems.«846»":"# Erdős Problem 846\n\n*Reference:* [erdosproblems.com/846](https://www.erdosproblems.com/846)\n","FormalConjectures.ErdosProblems.«847»":"# Erdős Problem 847\n\n*References:*\n- [erdosproblems.com/847](https://www.erdosproblems.com/847)\n- [RRS24] Reiher, Christian and R\\\"odl, Vojt\\v ech and Sales, Marcelo, Colouring versus density in integers and {H}ales-{J}ewett cubes. J. Lond. Math. Soc. (2) (2024)\n  [arXiv:2311.08556](https://arxiv.org/abs/2311.08556)\n","FormalConjectures.ErdosProblems.«848»":"# Erdős Problem 848\n\nIs the maximum size of a set $A \\subseteq \\{1, \\dots, N\\}$ such that $ab + 1$ is never\nsquarefree (for all $a, b \\in A$) achieved by taking those $n \\equiv 7 \\pmod{25}$?\n\n*References:*\n - [erdosproblems.com/848](https://www.erdosproblems.com/848)\n - [Er92b] Erdős, P. \"Some of my favourite problems in number theory, combinatorics,\n   and geometry.\" Resenhas do Instituto de Matemático e Estatística da Universidade\n   de São Paulo 2.2 (1995): 165-186.\n - [Sa25] Sawhney, M. \"Problem 848.\" (2025)\n   https://www.math.columbia.edu/~msawhney/Problem_848.pdf\n - Full formal proof of asymptotic result: https://github.com/The-Obstacle-Is-The-Way/erdos-banger\n","FormalConjectures.ErdosProblems.«849»":"# Erdős Problem 849\n\n*Reference:* [erdosproblems.com/849](https://www.erdosproblems.com/849)\n","FormalConjectures.ErdosProblems.«850»":"# Erdős Problem 850\n*Reference:* [erdosproblems.com/850](https://www.erdosproblems.com/850)\n","FormalConjectures.ErdosProblems.«851»":"# Erdős Problem 851\n\n*References:*\n- [erdosproblems.com/851](https://www.erdosproblems.com/851)\n- [Pr26] D. Price and GPT-5.2 Pro, [Erdős problem 851](https://www.overleaf.com/read/svgbjzpxxppv#4eea7e) (2026)\n","FormalConjectures.ErdosProblems.«853»":"# Erdős Problem 853\n\n*Reference:* [erdosproblems.com/853](https://www.erdosproblems.com/853)\n","FormalConjectures.ErdosProblems.«855»":"# Erdős Problem 855\n\n*Reference:* [erdosproblems.com/855](https://www.erdosproblems.com/855)\n\nThis is an \"eventually\" formulation of the Second Hardy–Littlewood conjecture.\n","FormalConjectures.ErdosProblems.«857»":"# Erdős Problem 857\n\n*Reference:* [erdosproblems.com/857](https://www.erdosproblems.com/857)\n\nFor fixed `n, k`, let `m(n, k)` be minimal such that every family of subsets of `[n]`\nof size at least `m(n, k)` contains a `k`-sunflower.\nThe problem asks to estimate `m(n, k)`, ideally asymptotically.\n","FormalConjectures.ErdosProblems.«859»":"# Erdős Problem 859\n\n*Reference:* [erdosproblems.com/859](https://www.erdosproblems.com/859)\n","FormalConjectures.ErdosProblems.«85»":"# Erdős Problem 85\n\n*Reference:* [erdosproblems.com/85](https://www.erdosproblems.com/85)\n","FormalConjectures.ErdosProblems.«862»":"# Erdős Problem 862\n\n*References:*\n- [erdosproblems.com/862](https://www.erdosproblems.com/862)\n- [Er92c] Erdős, P., *Some of my forgotten problems in number theory*. Hardy-Ramanujan J. (1992),\n  34-50.\n- [SaTh15] Saxton, David and Thomason, Andrew, *Hypergraph containers*. Invent. Math. (2015),\n  925-992.\n","FormalConjectures.ErdosProblems.«865»":"# Erdős Problem 865\n\n*References:*\n- [erdosproblems.com/865](https://www.erdosproblems.com/865)\n- [CES75] Choi, S. L. G. and Erdős, P. and Szemerédi, E., Some additive and multiplicative problems\n  in number theory. Acta Arith. (1975), 37--50.\n- [Ci26] R. Cipollini, [A sharp $5/8$ bound for an Erdős–Sós pairwise-sums problem](https://arxiv.org/html/2606.29361)\n  (2026).\n","FormalConjectures.ErdosProblems.«867»":"# Erdős Problem 867\n\n*References:*\n- [erdosproblems.com/867](https://www.erdosproblems.com/867)\n- [CoPh96] Coppersmith, Don and Phillips, Steven, *On a question of Erdős on subsequence sums*.\n  SIAM J. Discrete Math. (1996), 173-177.\n- [Fr93] Freud, R., *Adding numbers - on a problem of P. Erdős*. James Cook Mathematical\n  Notes (1993), 6199-6202.\n","FormalConjectures.ErdosProblems.«868»":"# Erdős Problem 868\n\n*References:*\n- [erdosproblems.com/868](https://www.erdosproblems.com/868)\n- [LaLa26] Larsen and Larsen, [Erdős problem 868](https://github.com/Larsen-Daniel/Erdos-868/blob/main/868.pdf) (2026)\n","FormalConjectures.ErdosProblems.«86»":"# Erdős Problem 86\n\n*References:*\n- [erdosproblems.com/86](https://www.erdosproblems.com/86)\n- [BHLL14] Balogh, József and Hu, Ping and Lidický, Bernard and Liu, Hong, *Upper bounds on the size\n  of 4- and 6-cycle-free subgraphs of the hypercube*. European J. Combin. (2014), 75-85.\n- [BHN95] Brass, Peter and Harborth, Heiko and Nienborg, Hauke, *On the maximum number of edges in a\n  {$C_4$}-free subgraph of {$Q_n$}*. J. Graph Theory (1995), 17--23.\n- [Ba12b] R. Baber, *Turán densities of hypercubes*. arXiv:1201.3587 (2012).\n- [Er91] Erdős, P., *Problems and results in combinatorial analysis and combinatorial number\n  theory*. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991),\n  397-406.\n","FormalConjectures.ErdosProblems.«871»":"# Erdős Problem 871\n\n*References:*\n- [erdosproblems.com/871](https://www.erdosproblems.com/871)\n- [ErNa88] Erdős, Paul and Nathanson, Melvyn B., *Partitions of bases into disjoint unions of bases*. J. Number Theory (1988), 1-9.\n- [ErNa89] Erdős, Paul and Nathanson, Melvyn B., *Additive bases with many representations*. Acta Arith. (1989), 399-406.\n","FormalConjectures.ErdosProblems.«872»":"# Erdős Problem 872\n\nThis file states Erdős Problem 872 for the primitive-set saturation game on $\\{2, \\dots, n\\}$.\nThe game value `L n` is defined by a finite minimax recursion: Prolonger moves first and maximizes\nthe final size of the claimed primitive set, while Shortener minimizes it.\n\nThe problem statement does not fix the turn order. This file fixes Prolonger to move first,\nfollowing the convention used in the forum discussion of the problem. The choice is not cosmetic:\ncomputational data suggests the Shortener-first value tracks $\\pi(n)$ while the Prolonger-first\nvalue grows linearly, and the questions below concern the Prolonger-first quantity.\n\n*References:*\n- [erdosproblems.com/872](https://www.erdosproblems.com/872)\n- [erdosproblems.com/forum/thread/872](https://www.erdosproblems.com/forum/thread/872)\n","FormalConjectures.ErdosProblems.«873»":"# Erdős Problem 873\n\n*Reference:* [erdosproblems.com/873](https://www.erdosproblems.com/873)\n","FormalConjectures.ErdosProblems.«87»":"# Erdős Problem 87\n\n*References:*\n- [erdosproblems.com/87](https://www.erdosproblems.com/87)\n- [Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry.\n  Resenhas (1995), 165-186.\n","FormalConjectures.ErdosProblems.«881»":"# Erdős Problem 881\n\n*Reference:* [erdosproblems.com/881](https://www.erdosproblems.com/881)\n","FormalConjectures.ErdosProblems.«883»":"# Erdős Problem 883\n\n*References:*\n- [erdosproblems.com/883](https://www.erdosproblems.com/883)\n- [ErSa97] Erdős, P. and Sárközy, G. N., On cycles in the coprime graph of integers.\n  Electron. J. Combin. (1997), Research Paper 8.\n- [Sa99] Sárközy, G. N., Complete tripartite subgraphs in the coprime graph of integers.\n  Discrete Math. (1999), 227-238.\n","FormalConjectures.ErdosProblems.«884»":"# Erdős Problem 884\n\n*References:*\n- [erdosproblems.com/884](https://www.erdosproblems.com/884)\n- [Tao25](https://terrytao.wordpress.com/wp-content/uploads/2025/09/erdos-884.pdf)\n- [Larsen](https://github.com/Larsen-Daniel/Erdos-884/blob/main/884.pdf)\n","FormalConjectures.ErdosProblems.«885»":"# Erdős Problem 885\n\n*References:*\n- [erdosproblems.com/885](https://www.erdosproblems.com/885)\n- [ErRo97] Erdős, P. and Rosenfeld, M., The factor-difference set of integers. (1997)\n- [Ji99] Jiménez-Urroz, J., A note on a conjecture of Erdős and {R}osenfeld. (1999)\n- [Br19] Bremner, A., On a problem of Erdős related to common factor differences. (2019)\n","FormalConjectures.ErdosProblems.«886»":"# Erdős Problem 886\n\n*References:*\n- [erdosproblems.com/886](https://www.erdosproblems.com/886)\n- [ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith.\n  (1997), 353--359.\n","FormalConjectures.ErdosProblems.«887»":"# Erdős Problem 887\n\n*References:*\n* [erdosproblems.com/887](https://www.erdosproblems.com/887)\n* [ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.\n","FormalConjectures.ErdosProblems.«888»":"# Erdős Problem 888\n\n*References:*\n- [erdosproblems.com/888](https://www.erdosproblems.com/888)\n- [Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number\n  theory. Number theory (Eger, 1996) (1998), 169-180.\n","FormalConjectures.ErdosProblems.«889»":"# Erdős Problem 889\n\n*Reference:* [erdosproblems.com/889](https://www.erdosproblems.com/889)\n","FormalConjectures.ErdosProblems.«890»":"# Erdős Problem 890\n\n*Reference:*\n- [erdosproblems.com/890](https://www.erdosproblems.com/890)\n- [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive\n  integers. Illinois J. Math. (1967), 428--430.\n","FormalConjectures.ErdosProblems.«891»":"# Erdős Problem 891\n\n*References:*\n- [erdosproblems.com/891](https://www.erdosproblems.com/891)\n- [Po18] Pólya, Georg, Zur arithmetischen {U}ntersuchung der {P}olynome. Math. Z. (1918), 143--148.\n- [Wikipedia] https://en.wikipedia.org/wiki/Dickson%27s_conjecture\n","FormalConjectures.ErdosProblems.«893»":"# Erdős Problem 893\n\n*References:*\n- [erdosproblems.com/893](https://www.erdosproblems.com/893)\n- [KoLu25] V. Kovač and F. Luca, On the number of divisors of Mersenne numbers. arXiv:2506.04883 (2025).\n","FormalConjectures.ErdosProblems.«897»":"# Erdős Problem 897\n\n*References:*\n- [erdosproblems.com/897](https://www.erdosproblems.com/897)\n- [Ar25] Archivara Math Research Agent, [An Additive Counterexample: Erdős Problem 897](https://archivara.org/paper/df04f023-6ef0-4c52-bd12-18cdaa8f0741) (2025)\n- [ArWu25] Aristotle, operated mostly by L. Wu, [Lean formalisation of Erdős problem 897](https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos897.lean) (2025)\n- [Wi70] E. Wirsing, A characterization of $\\log n$ as an additive arithmetic function.\n  Symposia Math. (1970), 45-57.\n- [Wi81] E. Wirsing, Additive and completely additive functions with restricted growth.\n  Recent progress in analytic number theory, Vol. 2 (Durham, 1979), 231--280 (1981).\n","FormalConjectures.ErdosProblems.«898»":"# Erdős Problem 898\n\n*References:*\n- [erdosproblems.com/898](https://www.erdosproblems.com/898)\n- [Er82e] Erdős, Paul, *Some of my favourite problems which recently have been solved*.\n  (1982), 59--79.\n- [Wikipedia](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Mordell_inequality)\n","FormalConjectures.ErdosProblems.«899»":"# Erdős Problem 899\n\n*Reference:* [erdosproblems.com/899](https://www.erdosproblems.com/899)\n","FormalConjectures.ErdosProblems.«89»":"# Erdős Problem 89\n\n*References:*\n- [erdosproblems.com/89](https://www.erdosproblems.com/89)\n- [Er46] Erdős, Paul. On sets of distances of $n$ points. Amer. Math. Monthly\n  53 (1946), 248--250.\n- [GuKa15] Guth, Larry and Katz, Nets Hawk. On the Erdős distinct distances\n  problem in the plane. Ann. of Math. (2) 181 (2015), 155--190.\n- [Mo52] Moser, Leo. On the different distances determined by $n$ points.\n  Amer. Math. Monthly 59 (1952), 85--91.\n\n### AI disclosure\n\nLean 4 code in this file was drafted with assistance from OpenAI Codex.\nThe mathematical content and references are the author's own work.\n","FormalConjectures.ErdosProblems.«904»":"# Erdős Problem 904\n\n*References:*\n- [erdosproblems.com/904](https://www.erdosproblems.com/904)\n- [BoEr75] Bollobás, B. and Erdős, P., *Unsolved Problems*. Proc. Fifth British Combinatorial Conf. (1975), 678-680.\n- [Er75] Erdős, P., *Some recent progress on extremal problems in graph theory*. Congr. Numer. (1975), 3-14.\n- [Ed78] Edwards, C. S., *Complete subgraphs with largest sum of vertex degrees*. (1978), 293-306.\n- [Fa92] Faudree, Ralph J., *Complete subgraphs with large degree sums*. J. Graph Theory (1992), 327-334.\n- [BoNi05] Bollobás, Béla and Nikiforov, Vladimir, *The sum of degrees in cliques*. Electron. J. Combin. (2005), Note 21, 10.\n","FormalConjectures.ErdosProblems.«905»":"# Erdős Problem 905\n\n*References:*\n- [erdosproblems.com/905](https://www.erdosproblems.com/905)\n- [KhNi79] Khadzhiivanov, N. G. and Nikiforov, S. V., *Solution of a problem of P. Erdős about the\n  maximum number of triangles with a common edge in a graph*. C. R. Acad. Bulgare Sci. (1979),\n  1315-1318.\n","FormalConjectures.ErdosProblems.«906»":"# Erdős Problem 906\n\n*Reference:* [erdosproblems.com/906](https://www.erdosproblems.com/906)\n","FormalConjectures.ErdosProblems.«907»":"# Erdős Problem 907\n\n*References:*\n- [erdosproblems.com/907](https://www.erdosproblems.com/907)\n- [dB51] de Bruijn, N. G., *Functions whose differences belong to a given class*. Nieuw Arch.\n  Wiskunde (2) (1951), 194-218.\n","FormalConjectures.ErdosProblems.«90»":"# Erdős Problem 90: The unit distance problem\n\n*Reference:* [erdosproblems.com/90](https://www.erdosproblems.com/90)\n\nThe conjecture asks whether every set of $n$ points in $\\mathbb{R}^2$ determines at most\n$n^{1 + O(1/\\log\\log n)}$ unit distances. It was **disproved** in May 2026: an internal model at\nOpenAI produced a construction beating the conjectured bound, with the proof digested and\nhuman-verified in two arXiv papers:\n\n* W. Sawin, [*An explicit lower bound for the unit distance problem*](https://arxiv.org/abs/2605.20579)\n  (2026), giving $u(n) \\ge n^{1.014114}/C$ for infinitely many $n$;\n* N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. Shankar, J. Tsimerman, V. Wang and\n  M. Matchett Wood, [*Remarks on the disproof of the unit distance conjecture*](https://arxiv.org/abs/2605.20695)\n  (2026), giving the qualitative form $u(n) \\ge n^{1+\\varepsilon}$ for some $\\varepsilon > 0$.\n\nThis file records the main statement (`erdos_90`), the two constructive disproof variants, the\nlogical implications between them, and the load-bearing reductions of Sawin's proof\n(`sawin_lattice_reduction` and `sawin_totally_real_tower`) as further benchmark challenges.\n","FormalConjectures.ErdosProblems.«912»":"# Erdős Problem 912\n\n*References:*\n - [erdosproblems.com/912](https://www.erdosproblems.com/912)\n - [Er82c] Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.\n","FormalConjectures.ErdosProblems.«913»":"# Erdős Problem 913\n\n*Reference:* [erdosproblems.com/913](https://www.erdosproblems.com/913)\n\nReviewed by @b-mehta on 2025-05-27\n","FormalConjectures.ErdosProblems.«914»":"# Erdős Problem 914\n\n*References:*\n- [erdosproblems.com/914](https://www.erdosproblems.com/914)\n- [CoHa63] Corrádi, K. and Hajnal, A., *On the maximal number of independent circuits in a graph*.\n  Acta Math. Acad. Sci. Hungar. (1963), 423-439.\n- [HaSz70] Hajnal, A. and Szemerédi, E., *Proof of a conjecture of P. Erdős*. (1970), 601-623.\n- [KiKo08] Kierstead, H. A. and Kostochka, A. V., *A short proof of the Hajnal-Szemerédi theorem on\n  equitable colouring*. Combin. Probab. Comput. (2008), 265-270.\n","FormalConjectures.ErdosProblems.«918»":"# Erdős Problem 918\n\n*References:*\n- [erdosproblems.com/918](https://www.erdosproblems.com/918)\n- [ErHa68b] Erdős, P. and Hajnal, A., On chromatic number of infinite graphs. (1968), 83--98.\n- [Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.\n","FormalConjectures.ErdosProblems.«91»":"# Erdős Problem 91\n\n*Reference:*\n- [Er87b] Erdős, P., Some combinatorial and metric problems in geometry.\n  Intuitive geometry (Siófok, 1985) (1987), 167-177.\n- [Ko24c] Z. Kovács, A note on Erdős's mysterious remark. arXiv:2412.05190 (2024).\n- [erdosproblems.com/91](https://www.erdosproblems.com/91)\n","FormalConjectures.ErdosProblems.«920»":"# Erdős Problem 920\n\n*References:*\n- [erdosproblems.com/166](https://www.erdosproblems.com/166)\n- [erdosproblems.com/920](https://www.erdosproblems.com/920)\n- [erdosproblems.com/986](https://www.erdosproblems.com/986)\n- [erdosproblems.com/1104](https://www.erdosproblems.com/1104)\n- [Br26] D. Brada\\v{c}, Nearly tight exponents for off-diagonal Ramsey numbers. arXiv:2605.28793\n  (2026).\n- [Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph\n  Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.\n- [GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's\n  theorem. J. Combinatorial Theory (1968), 125--175.\n- [MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).\n","FormalConjectures.ErdosProblems.«921»":"# Erdős Problem 921\n\n*References:*\n- [erdosproblems.com/921](https://www.erdosproblems.com/921)\n- [Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph\n  Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.\n- [Ga63] Gallai, T., Kritische Graphen. I. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1963), 165-192.\n- [KST84] Kierstead, H. A., Szemerédi, E. and Trotter, W. T., On coloring graphs with locally\n  small chromatic number. Combinatorica (1984), 183-185.\n","FormalConjectures.ErdosProblems.«923»":"# Erdős Problem 923\n\n*References:*\n- [erdosproblems.com/923](https://www.erdosproblems.com/923)\n- [Er69b] Erdős, P., *Problems and results in chromatic graph theory*. Proof Techniques in Graph Theory (1969), 27-35.\n- [Ro77] Rödl, V., *On the chromatic number of subgraphs of a given graph*. Proc. Amer. Math. Soc. (1977), 370-371.\n","FormalConjectures.ErdosProblems.«92»":"# Erdős Problem 92\n\nBoth questions here are disproved, by way of Erdős Problem 90. The source says so directly: this\nis a stronger form of the unit distance conjecture, so the disproof of that conjecture disproves\nthese too. See `FormalConjectures.ErdosProblems.«90»`.\n\n*Reference:* [erdosproblems.com/92](https://www.erdosproblems.com/92)\n","FormalConjectures.ErdosProblems.«930»":"# Erdős Problem 930\n\n*Reference:* [erdosproblems.com/930](https://www.erdosproblems.com/930)\n","FormalConjectures.ErdosProblems.«931»":"# Erdős Problem 931\n\n*Reference:* [erdosproblems.com/931](https://www.erdosproblems.com/931)\n","FormalConjectures.ErdosProblems.«932»":"# Erdős Problem 932\n\n*Reference:* [erdosproblems.com/932](https://www.erdosproblems.com/932)\n","FormalConjectures.ErdosProblems.«933»":"# Erdős Problem 933\n\n*References:*\n- [erdosproblems.com/933](https://www.erdosproblems.com/933)\n- [Er76d] Erdős, P, Problems and results on number theoretic properties of consecutive integers and\n  related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ.\n  Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n","FormalConjectures.ErdosProblems.«936»":"# Erdős Problem 936\n\n*Reference:* [erdosproblems.com/936](https://www.erdosproblems.com/936)\n","FormalConjectures.ErdosProblems.«937»":"# Erdős Problem 937\n\n*Reference:* [erdosproblems.com/937](https://www.erdosproblems.com/937)\n\n*References:*\n * [BBC24] Bajpai, P., Bennett, M. A. and Chan, T. H., _Arithmetic progressions in squarefull /\n   powerful numbers_, Int. J. Number Theory 20 (2024), 19-45.\n","FormalConjectures.ErdosProblems.«938»":"# Erdős Problem 938\n\n*Reference:* [erdosproblems.com/938](https://www.erdosproblems.com/938)\n","FormalConjectures.ErdosProblems.«939»":"# Erdős Problem 939\n\n*References:*\n- [erdosproblems.com/939](https://www.erdosproblems.com/939)\n- [Ni95] Nitaj, A., _On a conjecture of Erdős on 3-powerful numbers_. Bull. London Math. Soc.\n  (1995), 317-318.\n- [Co98] Cohn, J. H. E., _A conjecture of Erdős on 3-powerful numbers_. Math. Comp. (1998),\n  439-440.\n- [Wa24] Walsh, P., _A question of Erdős on 3-powerful numbers and an elliptic curve analogue\n  of the Ankeny-Artin-Chowla conjecture_. arXiv:2404.03970 (2024).\n- [LaPa67] Lander, L. J. and Parkin, T. R., _A counterexample to Euler's sum of powers\n  conjecture_. Math. Comp. (1967), 101-103.\n","FormalConjectures.ErdosProblems.«93»":"# Erdős Problem 93\n\n*References:*\n- [erdosproblems.com/93](https://www.erdosproblems.com/93)\n- [Er46b] Erdős, P., *On sets of distances of {$n$} points*. Amer. Math. Monthly (1946), 248--250.\n- [Er57] Erdős, Paul, *Some unsolved problems*. Michigan Math. J. (1957), 291-300.\n- [Er61] Erdős, Paul, *Some unsolved problems*. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961),\n  221-254.\n- [Er75f] Erdős, Paul, *On some problems of elementary and combinatorial geometry*. Ann. Mat. Pura\n  Appl. (4) (1975), 99-108.\n- [Er82e] Erdős, Paul, *Some of my favourite problems which recently have been solved*. (1982),\n  59--79.\n- [Er87b] Erdős, P., *Some combinatorial and metric problems in geometry*. Intuitive geometry\n  (Siófok, 1985) (1987), 167-177.\n- [Er90] Erdős, Paul, *Some of my favourite unsolved problems*. A tribute to Paul Erdős (1990),\n  467-478.\n- [Er92e] Erdős, Pál, *Some Unsolved problems in Geometry, Number Theory and Combinatorics*. Eureka\n  (1992), 44-48.\n- [Er95] Erdős, Paul, *Some of my favourite problems in number theory, combinatorics, and geometry*.\n  Resenhas (1995), 165-186.\n- [Er97e] Erdős, Paul, *Some of my favourite unsolved problems*. Math. Japon. (1997), 527-537.\n- [Er97f] Erdős, Paul, *Some unsolved problems*. Combinatorics, geometry and probability (Cambridge,\n  1993) (1997), 1-10.\n- [Al63] Altman, E., *On a problem of P. Erdős*. Amer. Math. Monthly (1963), 148-157.\n","FormalConjectures.ErdosProblems.«940»":"# Erdős Problem 940\n\n*References:*\n- [erdosproblems.com/940](https://www.erdosproblems.com/940)\n- [BaBr94] Baker, R. C. and Brüdern, J., _On sums of two squarefull numbers_. Math. Proc.\n  Cambridge Philos. Soc. (1994), 1-5.\n- [He88] Heath-Brown, D. R., _Ternary quadratic forms and sums of three square-full numbers_.\n  (1988), 137-163.\n","FormalConjectures.ErdosProblems.«942»":"# Erdős Problem 942\n\n*Reference:* [erdosproblems.com/942](https://www.erdosproblems.com/942)\n","FormalConjectures.ErdosProblems.«943»":"# Erdős Problem 943\n\n*Reference:* [erdosproblems.com/943](https://www.erdosproblems.com/943)\n","FormalConjectures.ErdosProblems.«944»":"# Erdős Problem 944\n\n*Reference:* [erdosproblems.com/944](https://www.erdosproblems.com/944)\n","FormalConjectures.ErdosProblems.«945»":"# Erdős Problem 945\n\n*References:*\n - [erdosproblems.com/945](https://www.erdosproblems.com/945)\n - [ErMi52] Erdős, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271.\n","FormalConjectures.ErdosProblems.«946»":"# Erdős Problem 946\n\n*References:*\n - [erdosproblems.com/946](https://www.erdosproblems.com/946)\n - [ErMi52] Erdős, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}.\n   Proc. London Math. Soc. (3) (1952), 257--271.\n - [Sp81] Spiro, C. A., The frequency with which an integral-valued, prime-independent,\n   multiplicative or additive function of n divides a polynomial function of n.\n - [He84] Heath-Brown, D. R., The divisor function at consecutive integers.\n   Mathematika 31 (1984), no. 2, 141--149.\n - [Hi85] Hildebrand, A., The divisor function at consecutive integers. Pacific J. Math.\n   (1987), 307--319\n - [EPS87] Erdős, P., Pomerance, C., and Sarkőzy, A., On locally repeated values of\n   arithmetic functions. III. Proc. Amer. Math. Soc. (1987), 1--7.\n","FormalConjectures.ErdosProblems.«949»":"# Erdős Problem 949\n\n*Reference:* [erdosproblems.com/949](https://www.erdosproblems.com/949)\n","FormalConjectures.ErdosProblems.«94»":"# Erdős Problem 94\n\n*References:*\n- [erdosproblems.com/94](https://www.erdosproblems.com/94)\n- [Er92e] Erdős, Pál, *Some Unsolved problems in Geometry, Number Theory and Combinatorics*. Eureka\n  (1992), 44-48.\n- [Er97c] Erdős, Paul, *Some of my favorite problems and results*. The mathematics of Paul Erdős, I\n  (1997), 47-67.\n- [LeTh95] Lefmann, Hanno and Thiele, Torsten, *Point sets with distinct distances*. Combinatorica\n  (1995), 379-408.\n","FormalConjectures.ErdosProblems.«950»":"# Erdős Problem 950\n\n*References:*\n- [erdosproblems.com/855](https://www.erdosproblems.com/855)\n- [erdosproblems.com/950](https://www.erdosproblems.com/950)\n- [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day\n  (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.\n- [mathoverflow/508491](https://mathoverflow.net/questions/508491)\n","FormalConjectures.ErdosProblems.«951»":"# Erdős Problem 951\n\n*References:*\n - [erdosproblems.com/951](https://www.erdosproblems.com/951)\n - [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\n    New York, 1976) (1977), 43-72.\n","FormalConjectures.ErdosProblems.«952»":"# Erdős Problem 952\n\n*References:*\n- [erdosproblems.com/952](https://www.erdosproblems.com/952)\n- [Wikipedia](https://wikipedia.org/wiki/Gaussian_moat)\n","FormalConjectures.ErdosProblems.«955»":"# Erdős Problem 955\n\n*References:*\n- [erdosproblems.com/955](https://www.erdosproblems.com/955)\n- [EGPS90] Erdős, P. and Granville, A. and Pomerance, C. and Spiro, C., On the normal behavior of\n  the iterates of some arithmetic functions. Analytic number theory (Allerton Park, IL, 1989)\n  (1990), 165-204.\n- [Er73b] Erdős, P., Über die Zahlen der Form $\\sigma(n) - n$ und $n - \\phi(n)$. Elem. Math.\n  (1973), 83--86.\n- [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n- [PPT18] Pollack, Paul and Pomerance, Carl and Thompson, Lola, Divisor-sum fibers. Mathematika\n  (2018), 330--342.\n- [Po14b] Pollack, Paul, Some arithmetic properties of the sum of proper divisors and the sum of\n  prime divisors. Illinois J. Math. (2014), 125--147.\n- [Tr15] Troupe, Lee, On the number of prime factors of values of the sum-of-proper-divisors\n  function. J. Number Theory (2015), 120--135.\n- [Tr20] Troupe, Lee, Divisor sums representable as the sum of two squares. Proc. Amer. Math. Soc.\n  (2020), 4189--4202.\n","FormalConjectures.ErdosProblems.«958»":"# Erdős Problem 958\n\n*References:*\n- [erdosproblems.com/958](https://www.erdosproblems.com/958)\n- [CDL25] Clemen, F., Dumitrescu, A. and Liu, D., *On multiplicities of interpoint distances*.\n  arXiv:2505.04283 (2025).\n","FormalConjectures.ErdosProblems.«959»":"# Erdős Problem 959\n\n*Reference:* [erdosproblems.com/959](https://www.erdosproblems.com/959)\n","FormalConjectures.ErdosProblems.«961»":"# Erdős Problem 961\n\n*References:*\n- [erdosproblems.com/961](https://www.erdosproblems.com/961)\n- [Ju74] Jutila, Matti, On numbers with a large prime factor. {II}. J. Indian Math. Soc. (N.S.) (1974), 125--130.\n- [RaSh73](https://eudml.org/doc/urn:eudml:doc:205214) Ramachandra, K. and Shorey, T. N., On gaps between numbers with a large prime factor. Acta Arith. (1973), 99--111.\n","FormalConjectures.ErdosProblems.«962»":"# Erdős Problem 962\n\n*References:*\n- [erdosproblems.com/962](https://www.erdosproblems.com/962)\n- [Er65] Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\n- [Er76e] Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.\n- [Tang](https://github.com/QuanyuTang/erdos-problem-962/blob/main/On_Erd%C5%91s_Problem_962.pdf)\n- [Tao](https://www.erdosproblems.com/forum/thread/962)\n","FormalConjectures.ErdosProblems.«965»":"# Erdős Problem 965\n\nFor every 2-coloring of ℝ, is there an uncountable set $A ⊆ ℝ$ such that\nall sums $a + b$ for $a, b ∈ A, a ≠ b$ have the same colour?\n\n*References:*\n- [erdosproblems.com/965](https://www.erdosproblems.com/965)\n- [Er75b] Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.\n- [HLS17] Hindman, Neil and Leader, Imre and Strauss, Dona, Pairwise sums in colourings of the reals. Abh. Math. Semin. Univ. Hambg. (2017), 275--287.\n- [Ko16] Komjáth, Péter, A certain 2-coloring of the reals. Real Anal. Exchange (2016), 227--231.\n- [SWCol] Sokoup Dániel and Weiss, William, Sums and Anti-Ramsey Colourings of ℝ. https://danieltsoukup.github.io/academic/finset_colouring.pdf\n","FormalConjectures.ErdosProblems.«966»":"# Erdős Problem 966\n\n*References:*\n- [erdosproblems.com/966](https://www.erdosproblems.com/966)\n- [Er75b] Erdős, Paul, *Problems and results in combinatorial number theory*. Journées\n  Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.\n","FormalConjectures.ErdosProblems.«967»":"# Erdős Problem 967\n\n*References:*\n- [erdosproblems.com/967](https://www.erdosproblems.com/967)\n- [ErIn64] Erdős, P. and Ingham, A. E., *Arithmetical Tauberian theorems*. Acta Arith. (1964),\n  341-356.\n- [Yi25] Yip, F., *On a problem of Erdős and Ingham*. arXiv:2512.16528 (2025).\n","FormalConjectures.ErdosProblems.«968»":"# Erdős Problem 968\n\nLet `uₙ = pₙ / n`, where `pₙ` is the `n`th prime. Does the set of `n` such that `uₙ < uₙ₊₁`\nhave positive lower density?\n\nErdős and Prachar also proved that `∑_{pₙ < x} |uₙ₊₁ - uₙ| ≍ (log x)^2`, and that the set of `n`\nsuch that `uₙ > uₙ₊₁` has positive lower density. Erdős also asked whether there are infinitely many\nincreasing triples `uₙ < uₙ₊₁ < uₙ₊₂` or decreasing triples `uₙ > uₙ₊₁ > uₙ₊₂`.\n\n*Reference:* [erdosproblems.com/968](https://www.erdosproblems.com/968)\n\n[ErPr61] Erdős, P. and Prachar, K., _Sätze und Probleme über pₖ/k_. Abh. Math. Sem. Univ. Hamburg\n(1961/62), 251–256.\n","FormalConjectures.ErdosProblems.«96»":"# Erdős Problem 96\n\n*Reference:* [erdosproblems.com/96](https://www.erdosproblems.com/96)\n","FormalConjectures.ErdosProblems.«970»":"# Erdős Problem 970\n\n*References:*\n- [erdosproblems.com/970](https://www.erdosproblems.com/970)\n- [FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence,\n  *Long gaps between primes*. J. Amer. Math. Soc. (2018), 65-105.\n- [Iw78] Iwaniec, Henryk, *On the problem of {J}acobsthal*. Demonstratio Math. (1978), 225--231.\n","FormalConjectures.ErdosProblems.«971»":"# Erdős Problem 971\n\n*Reference:* [erdosproblems.com/971](https://www.erdosproblems.com/971)\n","FormalConjectures.ErdosProblems.«972»":"# Erdős Problem 972\n\n*Reference:* [erdosproblems.com/972](https://www.erdosproblems.com/972)\n","FormalConjectures.ErdosProblems.«973»":"# Erdős Problem 973\n\n*References:*\n- [erdosproblems.com/973](https://www.erdosproblems.com/973)\n- [Er92f] Erdős, L., On some problems of {P}. Turán concerning power sums of\n  complex numbers. Acta Math. Hungar. (1992), 11--24.\n- [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n- [Tu84b] Turán, Paul, On a new method of analysis and its applications. (1984), xvi+584.\n","FormalConjectures.ErdosProblems.«974»":"# Erdős Problem 974\n\n*References:*\n- [erdosproblems.com/974](https://www.erdosproblems.com/974)\n- [Ti66] Tijdeman, R., *On a conjecture of Turán and Erdős*. Indag. Math. (1966), 374-383.\n","FormalConjectures.ErdosProblems.«975»":"# Erdős Problem 975\n\n*References:*\n - [erdosproblems.com/975](https://www.erdosproblems.com/975)\n - [Va39] van der Corput, J. G., Une in\\'egalit\\'e{} relative au nombre des diviseurs. Nederl. Akad. Wetensch., Proc. (1939), 547--553.\n - [Er52b] Erd\\\"os, P., On the sum {$\\sum^x_{k=1} d(f(k))$}. J. London Math. Soc. (1952), 7--15.\n - [Ho63] Hooley, Christopher, On the number of divisors of a quadratic polynomial. Acta Math. (1963), 97--114.\n - [Mc95] McKee, James, On the average number of divisors of quadratic polynomials. Math. Proc. Cambridge Philos. Soc. (1995), 389--392.\n - [Mc97] McKee, James, A note on the number of divisors of quadratic polynomials. (1997), 275--281.\n - [Mc99] McKee, James, The average number of divisors of an irreducible quadratic polynomial. Math. Proc. Cambridge Philos. Soc. (1999), 17--22.\n - [T] T. Tao, Erdos' divisor bound, https://terrytao.wordpress.com/2011/07/23/erdos-divisor-bound/\n","FormalConjectures.ErdosProblems.«978»":"# Erdős Problem 978\n\n*Reference:*\n - [erdosproblems.com/978](https://www.erdosproblems.com/978)\n - [Ho67] Hooley, C., On the power free values of polynomials. Mathematika (1967), 21--26.\n - [Br11] Browning, T. D., Power-free values of polynomials. Arch. Math. (Basel) (2011), 139--150.\n - [Er53] Erdős, P., Arithmetical properties of polynomials. J. London Math. Soc. (1953), 416--425.\n","FormalConjectures.ErdosProblems.«979»":"# Erdős Problem 979\n\n*Reference:* [erdosproblems.com/979](https://www.erdosproblems.com/979)\n","FormalConjectures.ErdosProblems.«97»":"# Erdős Problem 97\n\n*Reference:* [erdosproblems.com/97](https://www.erdosproblems.com/97)\n","FormalConjectures.ErdosProblems.«982»":"# Erdős Problem 982\n\n*Reference:* [erdosproblems.com/982](https://www.erdosproblems.com/982)\n","FormalConjectures.ErdosProblems.«985»":"# Erdős Problem 985\n\n*Reference:* [erdosproblems.com/985](https://www.erdosproblems.com/985)\n","FormalConjectures.ErdosProblems.«986»":"# Erdős Problem 986\n\n*References:*\n- [erdosproblems.com/986](https://www.erdosproblems.com/986)\n- [ChGr98] Chung, F. and Graham, R., *Erdős on Graphs: His Legacy of Unsolved Problems*.\n  A K Peters, Ltd. (1998).\n- [Br26] Bradač, D., Off-diagonal Ramsey numbers. arXiv:2605.28793 (2026).\n- [Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.\n- [MaVe23] Mattheus, S. and Verstraëte, J., The asymptotics of $r(4,t)$. Ann. of Math. (2024),\n  941-965.\n","FormalConjectures.ErdosProblems.«987»":"# Erdős Problem 987\n\n*References:*\n- [erdosproblems.com/987](https://www.erdosproblems.com/987)\n- [APSSV26b] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant,\n  [Short proofs in combinatorics, probability, and number theory II](https://arxiv.org/abs/2604.06609).\n  arXiv:2604.06609 (2026).\n- [Cl67] Clunie, J., On a problem of Erdős. J. London Math. Soc. (1967), 133--136.\n- [Er64b] Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964),\n  52-65.\n- [Er65b] Erdős, P., Some remarks on number theory. Israel J. Math. (the actual reference\n  cited by Clunie 1967 as [2]; the erdosproblems.com bibliography points to a different\n  Erdős 1965 paper, \"Some recent advances and current problems in number theory\" (Lectures\n  on Modern Mathematics III, 1965, 196-244), which does not appear to contain the\n  exponential-sum log-bound proof).\n- [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n- [Li69] Lindström, B., An inequality for $B_2$-sequences. J. Combinatorial Theory (1969), 211-212.\n","FormalConjectures.ErdosProblems.«98»":"# Erdős Problem 98\n\n*References:*\n- [[Er75f](https://mathscinet.ams.org/mathscinet/relay-station?mr=411984)]\n  Erdős, Paul, On some problems of elementary and combinatorial geometry.\n  Ann. Mat. Pura Appl. (4) (1975), 99-108.\n- [[Er83c](https://mathscinet.ams.org/mathscinet/relay-station?mr=706025)]\n  Erdős, Paul, Combinatorial problems in geometry.\n  Math. Chronicle (1983), 35-54.\n- [[Er87b](https://mathscinet.ams.org/mathscinet/relay-station?mr=910710)]\n  Erdős, P., Some combinatorial and metric problems in geometry.\n  Intuitive geometry (Siófok, 1985) (1987), 167-177.\n- [[Er90](https://mathscinet.ams.org/mathscinet/relay-station?mr=1117038)]\n  Erdős, Paul, Some of my favourite unsolved problems.\n  A tribute to Paul Erdős (1990), 467-478.\n- [[Er92b](https://mathscinet.ams.org/mathscinet/relay-station?mr=1275857)]\n  Erdős, Paul, Some of my favourite problems in various branches of combinatorics.\n  Matematiche (Catania) (1992), 231-240.\n- [[EFPR93](https://mathscinet.ams.org/mathscinet/relay-station?mr=1210096)]\n  Erdős, Paul and Füredi, Zoltán and Pach, János and Ruzsa, Imre Z.,\n  The grid revisited. Discrete Math. (1993), 189-196.\n- [[Er94b](https://mathscinet.ams.org/mathscinet/relay-station?mr=1304854)]\n  Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry.\n  Math. Pannon. (1994), 261-269.\n- [[Er97e](https://mathscinet.ams.org/mathscinet/relay-station?mr=1487304)]\n  Erdős, Paul, Some of my favourite unsolved problems.\n  Math. Japon. (1997), 527-537.\n- [erdosproblems.com/98](https://www.erdosproblems.com/98)\n","FormalConjectures.ErdosProblems.«990»":"# Erdős Problem 990\n\n*References:*\n- [erdosproblems.com/990](https://www.erdosproblems.com/990)\n- [APSSV26b] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant, *Short proofs in\n  combinatorics, probability, and number theory II*. arXiv:2604.06609 (2026).\n- [ErTu50] Erdős, P. and Turán, P., *On the distribution of roots of polynomials*. Ann. of Math. (2)\n  (1950), 105-119.\n- [Ha72b] Hayman, W. K., *Angular value distribution of power series with gaps*. Proc. London Math.\n  Soc. (3) (1972), 590-624.\n","FormalConjectures.ErdosProblems.«996»":"# Erdős Problem 996\n\n*Reference:*\n - [erdosproblems.com/996](https://www.erdosproblems.com/996)\n - [Er49d] Erdös, P. \"On the strong law of large numbers.\" Transactions of the American Mathematical\n    Society 67.1 (1949): 51-56.\n - [Ma66] Matsuyama, Noboru. \"On the strong law of large numbers.\" Tohoku Mathematical Journal,\n    Second Series 18.3 (1966): 259-269.\n","FormalConjectures.ErdosProblems.«997»":"# Erdős Problem 997\n\n*References:*\n- [erdosproblems.com/997](https://www.erdosproblems.com/997)\n- [APSSV26] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant,\n  [Short proofs in combinatorics and number theory](https://arxiv.org/abs/2603.29961).\n  arXiv:2603.29961 (2026).\n- [CLLW24] J. Champagne, T. Le, Y.-R. Liu, and T. D. Wooley, Well-distribution modulo one and the\n  primes. arXiv:2406.19491 (2024).\n- [Er64b] Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964),\n  52-65.\n- [Er85e] Erdős, P., Some problems and results in number theory. Number theory and combinatorics.\n  Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.\n- [Hl55] Hlawka, Edmund, Zur formalen {T}heorie der {G}leichverteilung in kompakten {G}ruppen. Rend.\n  Circ. Mat. Palermo (2) (1955), 33--47.\n- [Mo26] P. Monticone, [Lean formalisation of Erdős problem 997](https://live.lean-lang.org/#project=mathlib-v4.28.0&url=https://gist.githubusercontent.com/pitmonticone/016f2ed66b4cd1c4c4b9998095170e60/raw/b7dfc05c525ae385b5835f89f1ada721443e4305/Erdos997.lean) (2026)\n","FormalConjectures.ErdosProblems.«99»":"# Erdős Problem 99\n\n*References:*\n* [erdosproblems.com/99](https://www.erdosproblems.com/99)\n* [BeFo99] Bezdek, Andr\\'{a}s and Fodor, Ferenc, Minimal diameter of certain sets in the plane. J. Combin. Theory Ser. A (1999), 105-111.\n* [Er94b] Erd\\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\n","FormalConjectures.ErdosProblems.«9»":"# Erdős Problem 9\n\n*Reference:* [erdosproblems.com/9](https://www.erdosproblems.com/9)\n","FormalConjectures.GreensOpenProblems.«12»":"# Green's Open Problem 12\n\nReferences:\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.12)\n","FormalConjectures.GreensOpenProblems.«14»":"# Ben Green's Open Problem 14\n\n*References:*\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.14)\n- [AKS14] Ahmed, Tanbir, Oliver Kullmann, and Hunter Snevily. \"On the van der Waerden numbers\n  w (2; 3, t).\" Discrete Applied Mathematics 174 (2014): 27-51.\n- [KeMe23] Kelley, Zander, and Raghu Meka. \"Strong bounds for 3-progressions.\" 2023 IEEE 64th\n  Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2023.\n- [Hu22] Hunter, Zach. \"Improved lower bounds for van der Waerden numbers.\" Combinatorica 42.\n  Suppl 2 (2022): 1231-1252.\n- [Gr21] Green, Ben. \"New lower bounds for van der Waerden numbers.\" Forum of Mathematics,\n  Pi. Vol. 10. Cambridge University Press, 2022.\n- [Sc20] Schoen, Tomasz. \"A subexponential upper bound for van der Waerden numbers W (3, k).\"\n  arXiv preprint arXiv:2006.02877 (2020).\n- [BLR08] Brown, Tom, Bruce M. Landman, and Aaron Robertson. \"Bounds on some van der Waerden\n  numbers.\" Journal of Combinatorial Theory, Series A 115.7 (2008): 1304-1309.\n- [LiSh10] Li, Yusheng, and Jinlong Shu. \"A lower bound for off-diagonal van der Waerden numbers.\"\n  Advances in Applied Mathematics 44.3 (2010): 243-247.\n","FormalConjectures.GreensOpenProblems.«15»":"# Green's Open Problem 15\n\nReferences:\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.15)\n- [BJP14] T. Brown, V. Jungić and A. Poelstra, \"On double 3-term arithmetic progressions\",\n  Integers 14 (2014), Paper No. A43.\n- [CCS14] J. Cassaigne, J. D. Currie, L. Schaeffer and J. Shallit, \"Avoidance of additive cubes and\n  related results\", Adv. in Appl. Math. 56 (2014), 25–66.\n","FormalConjectures.GreensOpenProblems.«16»":"# Ben Green's Open Problem 16\n\n*References:*\n* [Ben Green's Open Problem 16](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.16)\n* [Ruzsa](I. Z. Ruzsa, Solving a linear equation in a set of integers. I. Acta Arith. 65 (1993), no. 3, 259–282.)\n* [Schoen and Sisask](T. Schoen and O. Sisask, Roth’s theorem for four variables and additive structures in sums of sparse sets Forum of Mathematics, Sigma (2016), Vol. 4, e5, 28 pages.)\n* [Yufei Zhao](Via Personal Communication with Ben Green)\n","FormalConjectures.GreensOpenProblems.«18»":"# Ben Green's Open Problem 18\n\n*Reference:*\n- [Gr26] [Ben Green's Open Problem 18](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.18)\n- [Au16] Austin, Tim. \"Ajtai–Szemerédi theorems over quasirandom groups.\" Recent trends in\n  combinatorics. Cham: Springer International Publishing, 2016. 453-484.\n- [So13] Solymosi, Jozsef. \"Roth-type theorems in finite groups.\" European Journal of Combinatorics\n  34.8 (2013): 1454-1458.\n- [Go01] Gowers, William T. \"A new proof of Szemerédi's theorem.\" Geometric & Functional Analysis\n  GAFA 11.3 (2001): 465-588.\n","FormalConjectures.GreensOpenProblems.«19»":"# Ben Green's Open Problem 19\n\n*References:*\n- [Gr26] [Ben Green's Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.19)\n- [FSS20] Fox, Jacob, et al. \"Triforce and corners.\" Mathematical Proceedings of the Cambridge\n  Philosophical Society. Vol. 169. No. 1. Cambridge University Press, 2020.\n- [Ma21] Mandache, Matei. \"A variant of the Corners theorem.\" Mathematical Proceedings of the\n  Cambridge Philosophical Society. Vol. 171. No. 3. Cambridge University Press, 2021.\n- [Ch11] Chu, Qing. \"Multiple recurrence for two commuting transformations.\" Ergodic Theory and\n  Dynamical Systems 31.3 (2011): 771-792.\n","FormalConjectures.GreensOpenProblems.«1»":"# Ben Green's Open Problem 1\n\n*Reference:* [Ben Green's Open Problem 1](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.1 Problem 1)\n","FormalConjectures.GreensOpenProblems.«21»":"# Ben Green's Open Problem 21\n\n*References:*\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.21)\n- [Ra33] Rado, Richard, *Studien zur Kombinatorik*. Math. Zeit. 36 (1933), 242-280.\n- [FoKl06] Fox, Jacob and Kleitman, Daniel, *On Rado's boundedness conjecture*. J. Combin. Theory\n  Ser. A 113 (2006), no. 1, 84-100.\n- [ElJo23] Ellis, David and Johnson, Robert (editors), *A collection of open problems in\n  celebration of Imre Leader's 60th birthday*. arXiv preprint arXiv:2310.18163 (2023).\n","FormalConjectures.GreensOpenProblems.«22»":"# Green's Open Problem 22\n\n*References:*\n- [Gr26] [Ben Green's Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.22)\n- [Mo17] Moreira, Joel. \"Monochromatic sums and products in N.\" Annals of Mathematics 185.3 (2017):\n  1069-1090.\n- [GrSa25] Green, Ben, and Mehtaab Sawhney. \"Bounds for monochromatic solutions to\n  $\\{x+ y, xy\\} $.\" arXiv preprint arXiv:2511.09365 (2025).\n- [Ri25] Richter, Florian K. \"Sums and products in sets of positive density.\" arXiv preprint\n  arXiv:2507.00515 (2025).\n- [BoSa24] Bowen, Matt, and Marcin Sabok. \"Monochromatic products and sums in the rationals.\" Forum\n  of Mathematics, Pi. Vol. 12. Cambridge University Press, 2024.\n- [Bo25] Bowen, Matt. \"Monochromatic products and sums in 2-colorings of N.\" Advances in Mathematics\n  462 (2025): 110095.\n- [Al23] Alweiss, Ryan. \"Monochromatic Sums and Products over $\\mathbb {Q} $.\" arXiv preprint\n  arXiv:2307.08901 (2023).\n","FormalConjectures.GreensOpenProblems.«23»":"# Green's Open Problem 23\n\nReferences:\n- [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.23)\n- [FrKlMo25] Frantzikinakis, N., O. Klurman, and J. Moreira. \"Partition regularity of Pythagorean pairs.\" Forum of Mathematics, Pi 13. Cambridge University Press (2025).\n","FormalConjectures.GreensOpenProblems.«24»":"# Green's Open Problem 24\n\nReferences:\n- [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.24)\n- [Aa19] Aaronson, James. \"Maximising the number of solutions to a linear equation in a set of integers.\"\n  Bulletin of the London Mathematical Society 51.4 (2019): 577-594.\n- [HaL28] Hardy, G. H., and J. E. Littlewood. \"Notes on the theory of series (VIII): an inequality.\"\n  Journal of the London Mathematical Society 1.2 (1928): 105-110.\n","FormalConjectures.GreensOpenProblems.«25»":"# Green's Open Problem 25\n\nReferences:\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.25)\n- [ESS89] Erdős, Pál, András Sárközy, and V. T. Sós. \"On a conjecture of Roth and some related\n  problems I.\" Irregularities of partitions. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989.\n  47-59.\n- [Ru04] Ruzsa, Imre Z. \"A problem on restricted sumsets.\" CONTEMPORARY MATHEMATICS 342 (2004):\n  245-248.\n","FormalConjectures.GreensOpenProblems.«26»":"# Green's Open Problem 26\n\nReferences:\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.26)\n- [JLP92] Jaeger, François, et al. \"Group connectivity of graphs—a nonhomogeneous analogue of\n  nowhere-zero flow properties.\" Journal of Combinatorial Theory, Series B 56.2 (1992): 165-182.\n- [ALM91] Alon, Noga, Nathan Linial, and Roy Meshulam. \"Additive bases of vector spaces over prime\n  fields.\" Journal of Combinatorial Theory, Series A 57.2 (1991): 203-210.\n- [Yu25] Yu, Yang. \"Note on the Additive Basis Conjecture.\" arXiv preprint arXiv:2510.01300 (2025).\n","FormalConjectures.GreensOpenProblems.«27»":"# Green's Open Problem 27\n\nReferences:\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.27)\n- [Be23] Bedert, Benjamin. \"On unique sums in Abelian groups.\" Combinatorica 44.2 (2024): 269-298.\n- [St76] Straus, E. G. \"Differences of residues (mod p).\" Journal of Number Theory 8.1 (1976): 40-42.\n","FormalConjectures.GreensOpenProblems.«28»":"# Green's Open Problem 28\n\nReferences:\n- [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.28)\n- [Mathoverflow/339137](https://mathoverflow.net/questions/339137/why-do-polynomials-with-coefficients-0-1-like-to-have-only-factors-with-0-1) asked by user [Sil](https://mathoverflow.net/users/136794/sil)\n- [MathStackexchange/3325163](https://math.stackexchange.com/questions/3325163/) asked by user [Emmanuel Amiot](https://math.stackexchange.com/users/403309/emmanuel-amiot)\n","FormalConjectures.GreensOpenProblems.«29»":"# Ben Green's Open Problem 29\n\n*References:*\n- [Ben Green's Open Problem 29](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.29)\n- [Gr12] Green, Ben. \"What is... an approximate group.\" Notices Amer. Math. Soc 59.5 (2012): 655-656.\n- [Br13] Breuillard, Emmanuel, Ben Green, and Terence Tao. \"Small doubling in groups.\"\n  Erdős Centennial. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. 129-151.\n- [Sa10] Sanders, Tom. \"On a nonabelian Balog–Szemerédi-type lemma.\" Journal of the Australian\n  Mathematical Society 89.1 (2010): 127-132.\n- [CrSi10] Croot, Ernie, and Olof Sisask. \"A probabilistic technique for finding almost-periods of\n  convolutions.\" Geometric and functional analysis 20.6 (2010): 1367-1396.\n","FormalConjectures.GreensOpenProblems.«2»":"# Ben Green's Open Problem 2\n\nReferences:\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.2)\n- [Er65] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII,\n  pages 181–189. Amer. Math. Soc., Providence, R.I., 1965.\n- [Sa21] Sanders, Tom. \"The Erdős–Moser Sum-free Set Problem.\" Canadian Journal of Mathematics 73.1\n  (2021): 63-107.\n- [Ru05] I. Z. Ruzsa, Sum-avoiding subsets. Ramanujan J., 9 (2005) (1-2):77–82.\n- [Ch71] S. L. G. Choi. On a combinatorial problem in number theory. Proc. London Math. Soc. (3),\n  23:629–642, 1971. doi:10.1112/plms/s3-23.4.629.\n- [BSS00] A. Baltz, T. Schoen, and A. Srivastav. Probabilistic construction of small strongly\n  sum-free sets via large Sidon sets. Colloq. Math., 86(2):171–176, 2000.\n  doi:10.4064/cm-86-2-171-176.\n","FormalConjectures.GreensOpenProblems.«31»":"# Ben Green's Open Problem 31\n\nWrite $F(N)$ for the largest Sidon subset of $[N]$.\nImprove, at least for infinitely many $N$, the bounds $N^{1/2} + O(1) \\le F(N) \\le N^{1/2} + N^{1/4} + O(1)$.\n\nNote: the upper bound was improved to $N^{1/2} + 0.98183 N^{1/4} + O(1)$ in [CHO25].\n\nRelated to Erdős Problem 30.\n\n*References:*\n- [Gr24] [Ben Green's Open Problem 31](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.7 Problem 31)\n- [Gr01] Green, Ben. \"The number of squares and $ B_h [g] $ sets.\"\n  Acta Arithmetica 100.4 (2001): 365-390.\n- [BFR23] Balogh, József, Zoltán Füredi, and Souktik Roy. \"An upper bound on the size of Sidon sets.\"\n  The American Mathematical Monthly 130.5 (2023): 437-445.\n- [CHO25] Carter, Daniel, Zach Hunter, and Kevin O’Bryant. \"On the diameter of finite Sidon sets.\"\n  Acta Mathematica Hungarica 175.1 (2025): 108-126.\n- [ET41] Erdos, Paul, and Pál Turán. \"On a problem of Sidon in additive number theory, and on some\n  related problems.\" J. London Math. Soc 16.4 (1941): 212-215.\n- [Li69] Lindström, Bernt. “A remark on B4-Sequences.” Journal of Combinatorial Theory,\n  Series A 7 (1969): 276-277.\n- [CLZ01] Cohen, G.D., Litsyn, S., & Zémor, G. (2001). Binary B2-Sequences : A New Upper Bound.\n  J. Comb. Theory A, 94, 152-155.\n","FormalConjectures.GreensOpenProblems.«32»":"# Green's Open Problem 32\n\n*Reference:*\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.32)\n- [Sh20] Shakan, George. \"A Large Gap in a Dilate of a Set.\" SIAM Journal on Discrete Mathematics\n  34.4 (2020): 2553-2555.\n","FormalConjectures.GreensOpenProblems.«33»":"# Ben Green's Open Problem 33\n\n*References:*\n- [Gr24] [Ben Green's Open Problem 33](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.33)\n- [CaHa20] Caprace, Pierre-Emmanuel, and Pierre de la Harpe. \"Groups with irreducibly unfaithful\n  subsets for unitary representations.\" Confluentes Mathematici 12.1 (2020): 31-68.\n- [CrLe07] Croot, Ernie, and Vsevolod F. Lev. \"Open problems in additive combinatorics.\"\n  Additive combinatorics 43.207-233 (2007): 1.\n","FormalConjectures.GreensOpenProblems.«35»":"# Ben Green's Open Problem 35\n\nEstimate the infimum of the $L^p$ norm of the self-convolution of a nonnegative integrable\nfunction supported on $[0,1]$ with total integral $1$.\n\nWe model a function `f : [0,1] → ℝ≥0` as a function `f : ℝ → ℝ` that is nonnegative, integrable,\nsupported on `[0,1]`, and has total integral `1`.\n\n*References:*\n- [Ben Green's Open Problem 35](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.35)\n- [Gr01](https://people.maths.ox.ac.uk/greenbj/papers/number-of-squares-and-Bh%5Bg%5D.pdf)\n  B. J. Green, *The number of squares and $B_h[g]$-sets*, Acta Arith. 100 (2001), no. 4, 365-390.\n- [CS17](https://arxiv.org/abs/1403.7988)\n  A. Cloninger and S. Steinerberger, *On suprema of autoconvolutions with an application to Sidon\n  sets*, Proc. Amer. Math. Soc. 145 (2017), no. 8, 3191-3200.\n- [MV10](https://arxiv.org/abs/0907.1379)\n  M. Matolcsi and C. Vinuesa, *Improved bounds on the supremum of autoconvolutions*,\n  J. Math. Anal. Appl. 372 (2010), 439-447.\n","FormalConjectures.GreensOpenProblems.«36»":"# Green's Open Problem 36\n\n*References:*\n* [Gr24] [Green's Open Problems #36](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.36)\n* [CKS05] Cohn, H., Kleinberg, R., Szegedy, B., and Umans, C. \"Group-theoretic Algorithms for\n  Matrix Multiplication\" (Problem 4.7)\n","FormalConjectures.GreensOpenProblems.«37»":"# Ben Green's Open Problem 37\n\nWhat is the smallest subset of `ℕ` containing, for each `d = 1, …, N`,\nan arithmetic progression of length `k` with common difference `d`?\n\n*References:*\n- [Ben Green's Open Problem 37](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.37)\n- [Green & Tao, *The primes contain arbitrarily long arithmetic progressions* (arXiv:math/0404188)](https://arxiv.org/abs/math/0404188)\n","FormalConjectures.GreensOpenProblems.«38»":"# Green's Open Problem 38\n\n*References:*\n- [100 open problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.38)\n- [La79] Lovász, László. \"On the Shannon capacity of a graph.\"\n  IEEE Transactions on Information theory 25.1 (1979): 1-7.\n- [Po20] Polak, Sven. \"New methods in coding theory: Error-correcting codes and the Shannon capacity.\"\n  arXiv preprint arXiv:2005.02945 (2020).\n","FormalConjectures.GreensOpenProblems.«39»":"# Green's Open Problem 39\n\n*References:*\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.39)\n- [BJR11] Bollobás, Béla, Svante Janson, and Oliver Riordan. \"On covering by translates of a set.\"\n  Random Structures & Algorithms 38.1‐2 (2011): 33-67.\n","FormalConjectures.GreensOpenProblems.«3»":"# Ben Green's Open Problem 3\n\n*Reference:* [Ben Green's Open Problem 3](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.3 Problem 3)\n","FormalConjectures.GreensOpenProblems.«40»":"# Ben Green's Open Problem 40\n\n*References:*\n- [Gr24] [Ben Green's Open Problem 40](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.40)\n- [Da90] Davydov, Alexander Abramovich. \"Construction of linear covering codes.\"\n  Problemy Peredachi Informatsii 26.4 (1990): 38-55.\n- [CHL97] Cohen, G., Honkala, I., Litsyn, S., & Lobstein, A. (1997). Covering codes (Vol. 54). Elsevier.\n- [St94] R. Struik, Covering codes, PhD Thesis, Eindhoven University of Technology, the Netherlands, 106 pp, 1994.\n\n","FormalConjectures.GreensOpenProblems.«41»":"# Ben Green's Open Problem 41\n\n*References*\n- [Gr24] [Ben Green's Open Problem 41](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.41)\n- [Ma15] Manners, Freddie. \"A solution to the pyjama problem.\" Inventiones mathematicae 202.1 (2015): 239-270.\n- [KrLe25] Kravitz, Noah, and James Leng. \"Quantitative pyjama.\" arXiv preprint arXiv:2510.17744 (2025).\n\n","FormalConjectures.GreensOpenProblems.«42»":"# Green's Open Problem 42\n\n*References:*\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.42)\n- [CoEl03] Cohn, Henry, and Noam Elkies. \"New upper bounds on sphere packings I.\"\n  Annals of Mathematics (2003): 689-714.\n- [Vi17] Viazovska, Maryna S. \"The sphere packing problem in dimension 8.\"\n  Annals of mathematics (2017): 991-1015.\n- [CKM17] Cohn, H., Kumar, A., Miller, S., Radchenko, D., & Viazovska, M. (2017).\n  The sphere packing problem in dimension 24. Annals of mathematics, 185(3), 1017-1033.\n- [Sa21] Sardari, Naser Talebizadeh. \"Higher Fourier interpolation on the plane.\"\n  arXiv preprint arXiv:2102.08753 (2021).\n\n","FormalConjectures.GreensOpenProblems.«44»":"# Green's Open Problem 44\n\n*References:*\n- [Gr24] [Ben Green's 100 Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.44)\n- [Er80] Erdős, Paul. \"A survey of problems in combinatorial number theory.\"\n  Annals of Discrete Mathematics 6 (1980): 89-115.\n","FormalConjectures.GreensOpenProblems.«45»":"# Ben Green's Open Problem 45\n\nCan we pick residue classes $a_p \\pmod{p}$, one for each prime $p \\leq N$,\nsuch that every integer $\\leq N$ lies in at least 10 of them?\n\n*References:*\n- [Ben Green's Open Problem 45](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.45)\n- [erdosproblems.com/689](https://www.erdosproblems.com/689)\n\nThis file points to the canonical formalization in `FormalConjectures.ErdosProblems.«689»`.\n","FormalConjectures.GreensOpenProblems.«46»":"# Ben Green's Open Problem 46\n\nWhat is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \\pmod{p}$, one for each prime $p \\leq x$?\n\n*References:*\n- [Gr24] [Ben Green's Open Problem 46](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.46)\n- [FGK18] Ford, K., Green, B., Konyagin, S., Maynard, J., & Tao, T. (2018). Long gaps between primes.\n  Journal of the American Mathematical Society, 31(1), 65-105.\n- [Iw78] Iwaniec, Henryk. \"On the problem of Jacobsthal.\" Demonstratio Mathematica 11.1 (1978): 225-232.\n","FormalConjectures.GreensOpenProblems.«47»":"# Green's Open Problem 47\n\n*References:*\n- [Gr24] [Ben Green's 100 Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.47)\n- [GH14] Green, Ben, and Adam J. Harper. \"Inverse questions for the large sieve.\"\n  Geometric and Functional Analysis 24.4 (2014): 1167-1203.\n- [HV09] Helfgott, Harald Andrés, and Akshay Venkatesh. \"How small must ill-distributed sets be.\"\n  Analytic number theory 2 (2009): 224-234.\n- [Wa12] Walsh, Miguel N. \"The inverse sieve problem in high dimensions.\" (2012): 2001-2022.\n- [Wa14] Walsh, Miguel N. \"The algebraicity of ill-distributed sets.\"\n  Geometric and Functional Analysis 24.3 (2014): 959-967.\n","FormalConjectures.GreensOpenProblems.«49»":"# Green's Open Problem 49\n\nAlso known as the *Marton's conjecture* or the *Polynomial Freiman-Ruzsa conjecture (PFR)*.\n\n*References:*\n- [Gr24] [Ben Green's 100 Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.49)\n- [Aa19] Aaronson, James. \"A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture.\" arXiv preprint arXiv:1902.00353 (2019).\n- [Fa00] I. Farah, Approximate homomorphisms. II. Group homomorphisms, Combinatorica 20 (2000), no. 1, 47–60.\n- [GGM25] W. T. Gowers, B. J. Green, F. Manners and T. C. Tao, On a conjecture of Marton, Ann. of Math. (2) 201 (2025), no. 2, 515–549.\n- [Gr05] B. J. Green, Finite field models in additive combinatorics, Surveys in combinatorics 2005, 1–27, London Math. Soc. Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, 2005.\n- [GrTa10] B. J. Green and T. C. Tao An equivalence between inverse sumset theorems and inverse conjectures for the U3 norm, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 1, 1–19.\n- [Lo12] S. Lovett, Equivalence of polynomial conjectures in additive combinatorics, Combinatorica 32 (2012), no. 5, 607–618.\n- [LoRe17] S. Lovett and O. Regev, A counterexample to a strong variant of the Polynomial Freiman Ruzsa conjecture in Euclidean space, Discrete Anal.(2017), Paper No. 8, 6 pp.\n- [Ma19] F. R. W. M. Manners, Formulations of the PFR conjecture over Z, Math. Proc. Cambridge Philos. Soc. 166 (2019), no. 2, 243–245.\n- [Sa12] T. Sanders, On the Bogolyubov-Ruzsa lemma, Anal. PDE 5 (2012), no. 3, 627–655.\n- [Ta08] T. C. Tao, A counterexample to a strong polynomial Freiman-Ruzsa conjecture, blog post November 2008, available at http://tinyurl.com/36j6hyxv.\n","FormalConjectures.GreensOpenProblems.«4»":"# Ben Green's Open Problem 4\n\n*Reference:* [Ben Green's Open Problem 4](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.4 Problem 4)\n","FormalConjectures.GreensOpenProblems.«50»":"# Ben Green's Open Problem 50\n\nSuppose that $A \\subset \\mathbb{F}_2^n$ is a set of density $\\alpha$. Does $10A$ contain a coset\nof some subspace of dimension at least $n - O(\\log(1/\\alpha))$?\n\nHere $kA$ denotes the $k$-fold iterated sumset, i.e., the set of all sums of $k$ elements from $A$\n(with repetition allowed). In `Mathlib`, this is denoted `k • A` using pointwise scalar\nmultiplication on sets.\n\n*Reference:* [Ben Green's Open Problem 50](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.6 Problem 50)\n","FormalConjectures.GreensOpenProblems.«51»":"# Green's Open Problem 51\n\n*References:*\n- [Gr24] [Green's Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.51)\n- [Gr13] B. J. Green, Restriction and Kakeya phenomena, notes from a 2003 course.\n  Available at http://people.maths.ox.ac.uk/greenbj/papers/rkp.pdf\n- [Sa11] Sanders, Tom. \"Green's sumset problem at density one half.\"\n  Acta Arithmetica 146.1 (2011): 91-101.\n- [Gr02] Green, Ben. \"Arithmetic progressions in sumsets.\"\n  Geometric & Functional Analysis GAFA 12.3 (2002): 584-597.\n- [Ruz91] Ruzsa, Imre Z. \"Arithmetic progressions in sumsets.\"\n  Acta Arithmetica 60.2 (1991): 191-202.\n","FormalConjectures.GreensOpenProblems.«52»":"# Green's Open Problem 52\n\n*Reference:* [Green's Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.52)\n\n","FormalConjectures.GreensOpenProblems.«53»":"# Green's Open Problem 53\n\n*References:*\n- [Gr24] [Green's Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.53)\n\n","FormalConjectures.GreensOpenProblems.«54»":"# Ben Green's Open Problem 54\n\n*References:*\n\n- [Ben Green's Open Problem 54](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.54)\n- Original formulation: M. Talagrand, *Are All Sets of Positive Measure Essentially Convex?*, in Operator Theory:\nAdvances and Applications, 77, 1995 Birkhäuser Verlag Basel/Switzerland.\n","FormalConjectures.GreensOpenProblems.«57»":"# Ben Green's Open Problem 57\n\n*Reference:* [Ben Green's Open Problem 57](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8)\n\nLet $G$ be a finite abelian group. Consider the space $\\Phi(G)$ of all functions on $G$ which\nare \"convex combinations\" (in the sense of complex coefficients $c_i$ with\n$\\sum |c_i| \\le 1$) of functions of the form\n$$\\phi(g) := \\mathbb{E}_{x_1 + x_2 + x_3 = g} f_1(x_2, x_3) f_2(x_1, x_3) f_3(x_1, x_2)$$\nwith $\\|f_i\\|_\\infty \\le 1$ (where $f_i : G \\times G \\to \\mathbb{C}$).\n\nLet $\\Phi'(G)$ be the space defined similarly, but with $f_3(x_1, x_2)$ required to be\na function of $x_1 + x_2$. Do $\\Phi(G)$ and $\\Phi'(G)$ coincide?\n\n**Note:** The \"convex combination\" here uses complex coefficients whose absolute values sum to\nat most 1 (cf. personal communication with B. Green, April 2026). Since the base sets are\nbalanced (closed under multiplication by unit complex numbers), this absolutely convex hull\nequals the real convex hull of the complex-valued base set.\n\n**Motivation:** $\\Phi(G)$ is a 'generalised convolution algebra' as considered by\nConlon–Fox–Zhao, whereas $\\Phi'(G)$ consists of Tao's $\\text{UAP}_2(G)$-functions.\n","FormalConjectures.GreensOpenProblems.«58»":"# Ben Green's Open Problem 58\n\n*Reference:* [Ben Green's Open Problem 58](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 58)\n","FormalConjectures.GreensOpenProblems.«5»":"# Ben Green's Open Problem 5\n\n*References:*\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.5)\n- [BaSo85] Babai L, Sós VT. Sidon sets in groups and induced subgraphs of Cayley graphs.\n  European Journal of Combinatorics. 1985 Jun 1;6(2):101-14.\n- [Ke97] Kedlaya, K. S., *Large product-free subsets of finite groups*, J. Combin. Theory\n  Ser. A 77 (1997), no. 2, 339–343.\n- [Ke09] Kedlaya, K. S., *Product-free subsets of groups, then and now*, Contemp. Math., 479,\n  American Mathematical Society, Providence, RI, 2009, 169–177.\n- [Go08] Gowers, W. T., *Quasirandom groups*, Combin. Probab. Comput. 17 (2008), no. 3,\n  363–387.\n","FormalConjectures.GreensOpenProblems.«60»":"# Ben Green's Open Problem 60\n\n*Reference:* [Ben Green's Open Problem 60](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 60)\n","FormalConjectures.GreensOpenProblems.«61»":"# Ben Green's Open Problem 61\n\n*Reference:* [Ben Green's Open Problem 61](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 61)\n\nThis problem was originally considered by Erdős and Newman.\n","FormalConjectures.GreensOpenProblems.«62»":"# Ben Green's Open Problem 62\n\nLet $p$ be a large prime, and let $A$ be the set of all primes less than $p$.\nIs every $x \\in \\{1, \\ldots, p-1\\}$ congruent to some product $a_1 a_2$ where $a_1, a_2 \\in A$?\n\nThis is a problem of Erdős, Odlyzko, and Sárközy [105] from 1987.\n\n*Reference:* [Ben Green's Open Problem 62](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.62)\n","FormalConjectures.GreensOpenProblems.«63»":"# Ben Green's Open Problem 63\n\nLet $A$ be the smallest set containing $2$ and $3$ and such that $a_1a_2 - 1 \\in A$\nif $a_1, a_2 \\in A$. Does $A$ have positive density?\n\n*References:*\n- [Ben Green's Open Problem 63](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8)\n- [erdosproblems.com/424](https://www.erdosproblems.com/424)\n\nThis file points to the canonical formalization in `FormalConjectures.ErdosProblems.«424»`.\n","FormalConjectures.GreensOpenProblems.«64»":"# Green's Open Problem 64\n\n*Reference:* [Ben Green's Open Problems](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.64)\n\nDo there exist infinitely many primes $p$ for which $p - 2$ has an odd number of prime factors,\ncounted with multiplicity?\n","FormalConjectures.GreensOpenProblems.«66»":"# Ben Green's Open Problem 66\n\n*Reference:* [Ben Green's Open Problem 66](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8)\n","FormalConjectures.GreensOpenProblems.«72»":"# Ben Green's Open Problem 72\n\nMore commonly known as the **no-three-in-line problem**.\n\nWhat is the largest subset of the grid $[N]^2$ with no three points in a line? In particular,\nfor $N$ sufficiently large, is it impossible to have a set of size $2N$ with this property?\n\nThe upper bound $2N$ is the easy half and is `allowedSetSize_le` below, by pigeonhole on the\ncolumns. The open content is whether $2N$ is attained. Green records that it is for $N$ up to\naround 50, that $(3/2 + o(1))N$ points are achievable for arbitrary $N$, and that his \"personal\nsuspicion is that this is optimal\". The Wikipedia reference points the same way: Guy and Kelly\nconjectured $c = \\sqrt[3]{2\\pi^2/3} \\approx 1.874$, and after an error in the heuristic was found\nGuy corrected it to $c = \\pi/\\sqrt3 \\approx 1.814$. Both are below $2$, so the expected answer to\nthe question above is yes.\n\n*References:*\n- [Ben Green's Open Problem 72](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.72)\n- [Wikipedia](https://en.wikipedia.org/wiki/No-three-in-line_problem)\n- [GK2025] Grebennikov, A. Kwan, M. No $(k + 1)$-in-line problem for large constant $k$.\n  https://arxiv.org/abs/2510.17743\n","FormalConjectures.GreensOpenProblems.«77»":"# Ben Green's Open Problem 77\n\n*Reference:*\n- [Ben Green's Open Problem 77](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.77)\n","FormalConjectures.GreensOpenProblems.«7»":"# Ben Green's Open Problem 7\n\nDoes Ulam's sequence have positive density?\nCan one explain the curious Fourier properties of Ulam's sequence?\n\n*References:*\n- [Ben Green's Open Problem 7](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.1)\n- [erdosproblems.com/342](https://www.erdosproblems.com/342)\n","FormalConjectures.GreensOpenProblems.«81»":"# Ben Green's Open Problem 81\n\nLet $A$ be a set of size $n$ integers. Is there some absolute constant $c > 0$ and $\\theta$\nsuch that $\\sum_{a \\in A} \\cos(a \\theta) \\leq - c \\sqrt{n}$?\n\n*References:*\n- [Ben Green's Open Problem 81](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.11)\n- [erdosproblems.com/510](https://www.erdosproblems.com/510)\n\nThis file points to the canonical formalization in `FormalConjectures.ErdosProblems.«510»`.\n","FormalConjectures.GreensOpenProblems.«82»":"# Ben Green's Open Problem 82\n\n*References:*\n- [Ben Green's Open Problem 82](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.82)\n- [An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials](https://arxiv.org/abs/2407.16075) (Bedert, 2025)\n- [Cosine polynomials with few zeros](https://arxiv.org/abs/2005.01695) (Juškevičius & Sahasrabudhe, 2020)\n","FormalConjectures.GreensOpenProblems.«85»":"# Green's Open Problem 85\n\n*Carbery’s rectangle problem*\n\nReferences:\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.85)\n- [CCW99] Carbery, Anthony, Michael Christ, and James Wright. \"Multidimensional van der Corput and sublevel set estimates.\" Journal of the American Mathematical Society 12.4 (1999): 981-1015 Section 6.\n- [Ke00] Keleti, Tamás. \"Density and covering properties of intervals of ℝn.\" Mathematika 47.1-2 (2000): 229-242.\n- [KKM02] Katz, Nets Hawk, Elliot Krop, and Mauro Maggioni. \"Remarks on the box problem.\" Mathematical Research Letters 9.4 (2002): 515-520.\n- [Mu02] Mubayi, Dhruv. \"Some exact results and new asymptotics for hypergraph Turán numbers.\" Combinatorics, Probability and Computing 11.3 (2002): 299-309 Conjecture 1.4.\n- [CPZ20] Conlon, David, Cosmin Pohoata, and Dmitriy Zakharov. \"Random multilinear maps and the Erd\\H {o} s box problem.\" arXiv preprint arXiv:2011.09024 (2020).\n","FormalConjectures.GreensOpenProblems.«94»":"# Ben Green's Open Problem 94\n\n*Reference:*\n- [Ben Green's Open Problem 94](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.94)\n- [erdosproblems.com/120](https://www.erdosproblems.com/120)\n","FormalConjectures.GreensOpenProblems.«9»":"# Green's Open Problem 9\n\nReferences:\n- [Gr24] [Green, Ben. \"100 open problems.\" (2024).](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.9)\n- [BlSi20] Bloom, Thomas F., and Olof Sisask. \"Breaking the logarithmic barrier in Roth's theorem on\n  arithmetic progressions.\" arXiv preprint arXiv:2007.03528 (2020).\n","FormalConjectures.HilbertProblems.«17»":"# Hilbert's 17th problem\n\nLet $f(x_1, \\dots, x_n)$ be a multivariable polynomial with real coefficients that takes only\nnonnegative values for all real inputs.\nHilbert's 17th problem asks whether there exist rational functions $g_1, \\dots, g_m$ such that\n$f = g_1^2 + g_2^2 + \\cdots + g_m^2$. Resolved affirmatively by Artin in 1927.\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Hilbert%27s_seventeenth_problem)\n- Motzkin, \"The arithmetic-geometric inequality\". In Shisha, Oved (ed.). Inequalities. Academic Press. pp. 205–224.\n","FormalConjectures.HilbertProblems.«5»":"# Hilbert's Fifth Problem and the Hilbert–Smith Conjecture\n\nThe **Hilbert–Smith conjecture** states that a locally compact topological group acting\ncontinuously and faithfully on a connected finite-dimensional topological manifold must be a\nLie group. It remains open in general; Pardon proved it for 3-manifolds in 2013.\nAn equivalent formulation: no p-adic integer group `ℤ_[p]` can act faithfully on any\nconnected finite-dimensional topological manifold.\n\n## Main statements\n\n- `hilbert_smith_conjecture`: the Hilbert–Smith conjecture.\n- `hilbert_smith_padic_formulation`: the equivalent formulation for `ℤ_[p]`.\n- `hilbert_smith_conjecture.variants.dimension_three`: Pardon's theorem for 3-manifolds.\n- `hilbert_smith_conjecture.variants.riemannian`: the case of isometric actions on Riemannian\n  manifolds.\n- `hilbert_fifth_problem`: Hilbert's fifth problem, solved by Gleason, Montgomery and Zippin.\n\n## Implementation notes\n\n`AdmitsLieGroupStructure G`, defined in\n`FormalConjecturesForMathlib.Geometry.Manifold.LieGroupPresentation`, says that `G` is\ncontinuously isomorphic to a finite-dimensional real-analytic Lie group. Lie groups are Hausdorff\nbut not assumed second countable: every discrete group is a `0`-dimensional Lie group\n(`admitsLieGroupStructure_of_discreteTopology`). This matters for the Hilbert–Smith conjecture,\nsince uncountable discrete groups act continuously and faithfully on connected manifolds, for\ninstance `ℝ` with the discrete topology acting on `ℝ` by translations.\n\nThe acting group is not assumed Hausdorff: a topological group acting continuously and\nfaithfully on a Hausdorff space is Hausdorff, because the closure of the identity acts trivially.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture)\n- [Tao's blog](https://terrytao.wordpress.com/2011/08/13/the-hilbert-smith-conjecture/)\n- [Pardon 2013] J. Pardon, *The Hilbert–Smith conjecture for three-manifolds*,\n  J. Amer. Math. Soc. 26 (2013), 879–899. https://doi.org/10.1090/S0894-0347-2013-00766-3,\n  [arXiv:1112.2324](https://arxiv.org/abs/1112.2324)\n- [Myers–Steenrod 1939] S. B. Myers, N. E. Steenrod, *The group of isometries of a Riemannian\n  manifold*, Ann. of Math. 40 (1939), 400–416. https://doi.org/10.2307/1968928\n- [van den Dries–Goldbring 2015] L. van den Dries, I. Goldbring, *Hilbert's 5th problem*,\n  Enseign. Math. 61 (2015), 3–43. https://doi.org/10.4171/LEM/61-1/2-2\n","FormalConjectures.Kourovka.«19_25»":"# Conjecture 19.25\n\nby B. Curtin, G. R. Pourgholi\n\n*Reference:* [The Kourovka Notebook](https://arxiv.org/abs/1401.0300v40)\n","FormalConjectures.Kourovka.«1_35c»":"# Conjecture 1.35(c)\n\nby A. I. Mal'cev and L. Fuchs\n\n*Reference:* [The Kourovka Notebook](https://arxiv.org/abs/1401.0300v46)\n","FormalConjectures.Kourovka.«1_40»":"# Conjecture 1.40\n\nby Sh. S. Kemkhadze\n\nIs a group a nilgroup if it is the product of two normal nilsubgroups?\n\nHere a nilgroup (Engel group) is a group in which every element is an Engel\nelement. This is the Engel-group analogue of Fitting's theorem, which\nguarantees that the product of two normal nilpotent subgroups is nilpotent.\n\n*Reference:* [The Kourovka Notebook](https://arxiv.org/abs/1401.0300v40)\n","FormalConjectures.Kourovka.«1_74»":"# Conjecture 1.74 (Tarski monster topologizability)\n\nby V. P. Platonov\n\nProblem 1.74 asks to describe all \"minimal topological groups\" in Platonov's\nsense: non-discrete Hausdorff topological groups all of whose proper closed\nsubgroups are discrete. A natural test case: does there exist a Tarski monster\ngroup admitting a non-discrete Hausdorff group topology? A Tarski monster\nwould be a minimal group in this sense, since all its proper subgroups are\nfinite (hence discrete in any Hausdorff group topology).\n\n*Reference:* [The Kourovka Notebook](https://arxiv.org/abs/1401.0300v40)\n","FormalConjectures.Kourovka.«20_76»":"# Conjecture 20.76\nby L. Pyber\n*Reference:* [The Kourovka Notebook](https://arxiv.org/abs/1401.0300v40)\n!","FormalConjectures.LittProblems.«1»":"# Lam--Litt conjecture\n\nA conjecture of Lam and Litt on algebraic solutions of algebraic ODEs.\n\nLet $g \\in \\mathbb{Q}(z, y_0, \\dots, y_{n-1})$ be a rational function in\n$n + 1$ variables. Let $f$ be a power series over $\\mathbb{Q}$ such that\n$f^{(n)}(z) = g(z, f(z), f'(z), \\dots, f^{(n-1)}(z))$.\nAlso, assume that $g(0, f(0), f'(0), \\dots, f^{(n-1)}(0))$ is defined.\nThen the following are equivalent:\n\n1) $f$ is algebraic over $\\mathbb{Q}[z]$.\n2) There exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\\mathbb{Z}[1/N]$.\n3) There exists an integer-valued function $\\omega$ on the set of primes with\n$\\lim_{p \\to \\infty} \\omega(p) / p = \\infty$ such that, for each prime $p$,\nthe rational numbers $a_0, a_1, \\dots, a_{\\omega(p)}$ are in $\\mathbb{Z}_{(p)}$.\n\nThe implication 1) => 2) is due to Eisenstein, and 2) => 3) is trivial.\n\n*References:*\n- [Litt's problem 1](https://www.problemsilike.com/1)\n- Yeuk Hay Joshua Lam, Daniel Litt, \"Algebraicity and integrality of solutions to differential equations\",\n  [arxiv/2501.13175](https://arxiv.org/abs/2501.13175)\n- Gotthold Eisenstein. \"Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen\",\n  Bericht der Königl. Preuss. Akademie der Wissenschaften zu Berlin, 1852\n\nTODO:\n- Lam-Litt conjecture implies Grothendieck p-curvature conjecture.\n- Examples in Remark 1.1.3 and 1.1.5 on the conditions of the conjecture.\n","FormalConjectures.Mathoverflow.«10799»":"# Optimal monotone families for the discrete isoperimetric inequality\n\n*References:*\n- [mathoverflow/10799](https://mathoverflow.net/questions/10799)\n  asked by user [*Gil Kalai*](https://mathoverflow.net/users/1532/gil-kalai)\n- [Optimal Monotone Families for the Discrete Isoperimetric Inequality](https://gilkalai.wordpress.com/ai/optimal-monotone-families-for-the-discrete-isoperimetric-inequality/)\n  by *Gil Kalai* (2026), a Polymath project with AI agents\n- [An Isoperimetric Inequality for the Hamming Cube and Integrality Gaps in Bounded-Degree\n  Graphs](https://arxiv.org/abs/math/0603218) by *Jeff Kahn* and *Gil Kalai* (2006)\n- [A Proof of the Kahn–Kalai Conjecture](https://arxiv.org/abs/2203.17207) by *Jinyoung Park*\n  and *Huy Tuan Pham* (2022)\n\n","FormalConjectures.Mathoverflow.«17560»":"# Mathoverflow 17560\n\n\n*Reference:* [mathoverflow/17560](https://mathoverflow.net/questions/17560)\nasked by user [Alon-Amit](https://mathoverflow.net/users/25/alon-amit)\n","FormalConjectures.Mathoverflow.«1973»":"# Mathoverflow 1973\n\nDoes the 6-sphere $S^6$ admit the structure of a complex manifold?\n\n*References:*\n- [mathoverflow/1973](https://mathoverflow.net/questions/1973/),\n  asked by user [*Fetchinson0234*](https://mathoverflow.net/users/41312/victor-ramos).\n- [Al26] L. Alpöge, [*A compact complex threefold fibred by tori over the projective line, and the six-sphere*](https://alpo.ge/s6.pdf) (2026),\n  originally [shared on X](https://x.com/__alpoge__/status/2091639597193368014).\n","FormalConjectures.Mathoverflow.«21003»":"# Mathoverflow 21003\n\nIs there any polynomial $f(x, y) \\in \\mathbb{Q}[x, y]$ such that\n$f : \\mathbb{Q} \\times \\mathbb{Q} \\rightarrow \\mathbb{Q}$ is a bijection?\n\n*Reference:* [mathoverflow/21003](https://mathoverflow.net/questions/21003)\nasked by user [*Z.H.*](https://mathoverflow.net/users/5098/z-h)\n","FormalConjectures.Mathoverflow.«235893»":"# Mathoverflow 235893\n\n*Reference:* [mathoverflow/235893](https://mathoverflow.net/questions/235893)\nasked by user [*Willie Wong*](https://mathoverflow.net/users/3948/willie-wong)\n","FormalConjectures.Mathoverflow.«31809»":"# Mathoverflow 31809\n\nSource:\n[Mathoverflow/31809](https://mathoverflow.net/questions/31809/pre-triangulated-category-that-isnt-triangulated)\n\n","FormalConjectures.Mathoverflow.«339137»":"# Mathoverflow 339137\n\nWhy do polynomials with coefficients 0,1\n like to have only factors with 0,1\n coefficients?\n\n*Reference:* [mathoverflow/339137](https://mathoverflow.net/questions/339137)\nasked by user [*Sil*](https://mathoverflow.net/users/136794/sil)\n","FormalConjectures.Mathoverflow.«34145»":"# Mathoverflow 34145\n\nCan the unit square be covered by $1/k$-by-$1/(k+1)$ rectangles (across $1 \\le k$ natural)?\n\nI am deliberately not requiring that the rotations can only be $0^\\circ, 90^\\circ, 180^\\circ, \\text{ or } 270^\\circ$.\n\nBecause of indexing, since `n : ℕ` starts at 0, we change the side lengths to $1 / (n + 1)$ and\n$1 / (n + 2)$, so that the first rectangle is $1/1$ by $1/2$, the second is $1/2$ by $1/3$, etc.\n\n*Reference:* [mathoverflow/34145](https://mathoverflow.net/q/34145)\nasked by user [*Kaveh*](https://mathoverflow.net/users/7507/kaveh)\n","FormalConjectures.Mathoverflow.«347178»":"# Mathoverflow 347178\n\n*Reference:* [mathoverflow/347178](https://mathoverflow.net/questions/347178)\nasked by user [*Biagio Ricceri*](https://mathoverflow.net/users/149235/biagio-ricceri)\n","FormalConjectures.Mathoverflow.«434111»":"# Are prime numbers among sums of prime numbers distributed as $\\frac n{2\\ln(n)}$?\n\n*Reference:*\n\n[mathoverflow.net/questions/434111](https://mathoverflow.net/questions/434111/are-prime-numbers-among-sums-of-prime-numbers-distributed-as-frac-n2-lnn)\n\n[Me18] Meštrović, R., *Curious Conjectures on the Distribution of Primes\nAmong the Sums of the First `2n` Primes*, [arXiv:1804.04198](https://arxiv.org/abs/1804.04198)\n(2018), Conjecture 3.3.\n","FormalConjectures.Mathoverflow.«486451»":"# Mathoverflow 486451\n\n*Reference:* [mathoverflow/486451](https://mathoverflow.net/questions/486451)\nasked by user [*Junyan Xu*](https://mathoverflow.net/users/3332/junyan-xu)\n","FormalConjectures.Mathoverflow.«507128»":"# Mathoverflow 507128\n\n*Reference:* [mathoverflow/507128](https://mathoverflow.net/questions/507128/embeddability-order-on-picard-groups)\nasked by user [*Junyan Xu*](https://mathoverflow.net/users/3332/junyan-xu)\n","FormalConjectures.Mathoverflow.«75792»":"# Mathoverflow 75792\n\nVarious questions about integer complexity, which is the minimum number of `1`s needed to express a natural number using addition, multiplication, and parentheses.\n\nLet `‖n‖` denote the integer complexity of `n > 0`.\n* It is known that `‖3^n‖ = 3n` for `n > 0`.\n* Is it true that `‖2^n‖ = 2n` for `n > 0`?\n* The corresponding conjecture for `5` is false, because\n  `5^6 = 15625 = 1 + 2^3 * 3^2 * (1 + 2^3 * 3^3)`!\n\nWe have chosen to formalise this using an inductive type.\n\n*References:*\n - [mathoverflow/75792](https://mathoverflow.net/a/75792) by user [Harry Altman](https://mathoverflow.net/users/5583)\n - http://arxiv.org/abs/1203.6462 by Jānis Iraids, Kaspars Balodis, Juris Čerņenoks, Mārtiņš Opmanis, Rihards Opmanis, Kārlis Podnieks\n - http://arxiv.org/abs/1207.4841 by Harry Altman, Joshua Zelinsky\n - https://oeis.org/A5245 : Mahler-Popken complexity.\n","FormalConjectures.Millenium.BSD":"# The Birch and Swinnerton-Dyer (BSD) Conjecture\n\n*References:*\n- [The Clay Institute](https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/),\n  official problem description by Andrew Wiles:\n  [PDF](https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf)\n- [BSD1965] B. J. Birch and H. P. F. Swinnerton-Dyer. \"Notes on elliptic curves. II.\"\n  Journal fur die reine und angewandte Mathematik 218 (1965), 79-108,\n  [doi](https://doi.org/10.1515/crll.1965.218.79)\n- [Tate1966] John Tate. \"On the conjectures of Birch and Swinnerton-Dyer and a geometric analog.\"\n  Seminaire Bourbaki, Vol. 9, Exp. No. 306 (1966), 415-440,\n  [numdam](https://www.numdam.org/item/SB_1964-1966__9__415_0/)\n- [Gross2011] Benedict H. Gross. \"Lectures on the conjecture of Birch and Swinnerton-Dyer.\"\n  Arithmetic of L-functions, IAS/Park City Math. Ser. 18, AMS (2011), 169-209,\n  [PDF](https://people.math.harvard.edu/~gross/preprints/lectures-pcmi.pdf)\n- [Ang2025] David Kurniadi Angdinata. \"L-functions of Dirichlet twists of elliptic curves:\n  computations and congruences.\" PhD thesis, University College London (2025),\n  [PDF](https://discovery.ucl.ac.uk/10223687/1/main-pages.pdf)\n- [Ada] Tom Adamczewski. \"Autoformalized conjectures\",\n  [Birch and Swinnerton-Dyer](https://tadamcz.com/autoformalization-results/#/p/wp-birch-and-swinnerton-dyer-conjecture)\n","FormalConjectures.Millenium.NavierStokes":"# Existence And Smoothness Of The Navier–Stokes Equation\n\nThis file formalizes the Clay Mathematics Institute millennium problem concerning\nthe existence and smoothness of solutions to the Navier-Stokes equations in three\nspatial dimensions. While the definitions are generalized to arbitrary dimension n,\nthe millennium problem specifically concerns the case n = 3.\n\n## References\n- [Wikipedia](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness)\n- [Clay Mathematics Institute](https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf)\n\n## Main Theorems (Clay Millennium Problem for n = 3)\n\nThe Clay Millennium Problem asks for a proof of one of the following four statements:\n\n- `navier_stokes_existence_and_smoothness_R3`: (A) Global existence on ℝ³\n- `navier_stokes_existence_and_smoothness_periodic`: (B) Global existence on ℝ³/ℤ³\n- `navier_stokes_breakdown_R3`: (C) Existence of breakdown scenario on ℝ³\n- `navier_stokes_breakdown_periodic`: (D) Existence of breakdown scenario on ℝ³/ℤ³\n\n## Variable conventions\n\nFefferman writes the velocity as $u(x,t)$, the initial velocity as $u^\\circ(x)$, the\npressure as $p(x,t)$, the force as $f(x,t)$, and the viscosity as $\\nu$. In Lean,\n`u₀ : ℝ^n → ℝ^n` denotes the initial velocity, while `v : ℝ^n → ℝ → ℝ^n`\ndenotes the solution velocity. The curried order `v x t`, `p x t`, and `f x t`\nkeeps the source convention that position comes before time.\n\nSince the Clay statement gives equation (1) on the closed time half-line $t \\ge 0$,\nthe time derivative is encoded with `derivWithin` relative to `Set.Ici 0`. The Clay\nPDF also includes errata; in particular, we include spatial 1-periodicity of the\npressure in the periodic case. The sign correction to the weak-solution identity in\nthe errata is not represented here, since this file formalizes the four prize\nalternatives rather than the later weak-solution discussion.\n","FormalConjectures.Millenium.Poincare":"# The Poincaré Conjecture\n\nReferences:\n- [Miln2022](https://www.claymath.org/wp-content/uploads/2022/06/poincare.pdf)\n- [Wang2017](https://annals.math.princeton.edu/2017/186-2/p03).\n- [mo296171](https://mathoverflow.net/questions/296171/unique-smooth-structure-on-3-manifolds)\n- [mathlib4](https://github.com/leanprover-community/mathlib4)\n\nThe formalisations in this file are based on the ones written by Junyan Xu in Mathlib4.\n","FormalConjectures.Millenium.PvsNP":"# Conjectures in Complexity Theory\n\nThis file contains formal statements of some of the main open conjectures\nin complexity theory, including\n\n- the P vs NP problem\n- the NP vs coNP problem\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/P_versus_NP_problem)\n- [The Clay Institute](https://www.claymath.org/millennium/p-vs-np/)\n","FormalConjectures.Millenium.RiemannHypothesis":"# Riemann Hypothesis and its generalizations\n\nThe Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function\n$\\zeta(s)$ have real part $\\frac{1}{2}$. The trivial zeros are the negative even integers\n$-2, -4, -6, \\ldots$. The hypothesis is one of the seven Millennium Prize Problems\nposed by the Clay Mathematics Institute.\n\nThe Generalized Riemann Hypothesis extends this to Dirichlet $L$-functions of primitive\nDirichlet characters.\n\nNote: the **Extended Riemann Hypothesis** (ERH) for Dedekind zeta functions is intentionally\n**not** stated here. Mathlib's `NumberField.dedekindZeta` is the naive Dirichlet series\n(`LSeries`), not a meromorphic continuation; outside the region of absolute convergence\n`tsum` returns junk `0`, producing spurious zeros that make the naive foramlisation of the\nconjecture provably false. The ERH should be added once Mathlib provides a meromorphic\ncontinuation of the Dedekind zeta function.\n\n*References:*\n- [The Clay Institute](https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf)\n- [Wikipedia: Riemann hypothesis](https://en.wikipedia.org/wiki/Riemann_hypothesis)\n- [Wikipedia: Generalized Riemann hypothesis](https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis)\n- [Wikipedia: Dedekind zeta function](https://en.wikipedia.org/wiki/Dedekind_zeta_function)\n- J. Neukirch, *Algebraic Number Theory*, Springer (Grundlehren 322), 1999, Chapter VII, §5.\n- D. A. Marcus, *Number Fields*, Springer (GTM 81), 1977, Chapter VII.\n","FormalConjectures.OEIS.«100434»":"# Expansion of g.f. $(1+x)(3+x)/(1+6x^2+x^4)$\n\nThis sequence is defined by the linear recurrence relation\n$a(n) = -6 a(n-2) - a(n-4)$ for $n \\ge 4$,\nwith initial values $a(0)=3$, $a(1)=4$, $a(2)=-17$, $a(3)=-24$.\n\n*References:*\n- [A100434](https://oeis.org/A100434)\n","FormalConjectures.OEIS.«100474»":"# Conjectures associated with A100474\n\n$a(1) = 1$; $a(n)$ is the smallest integer such that $a(n) + a(n-1)$ has the first $n$ distinct\nprime factors not used before in this construction.\n\n*References:*\n- [A100474](https://oeis.org/A100474)\n","FormalConjectures.OEIS.«100475»":"# Prime-th recurrence with reversal at each step\n\n$$a(n) = \\operatorname{reversal}(p_{a(n-1)})$$\nwith $a(0)=1$, where $p_k$ is the $k$-th prime number.\n\n*References:*\n- [A100475](https://oeis.org/A100475)\n","FormalConjectures.OEIS.«100478»":"# Pentanacci $\\pi$ sequence\n\nStart with $a(1)=a(2)=a(3)=a(4)=a(5)=1$ and\nfor $n>5$, $a(n) = \\pi(\\sum_{j=1}^5 a(n-j))$ where $\\pi = A000720$.\n\n*References:*\n- [A100478](https://oeis.org/A100478)\n","FormalConjectures.OEIS.«100800»":"# Conjectures associated with A100800\n\nLet $f(n) = n + \\text{sum of the digits of } n$. If $f(n)$ is multiple of $n$ then $a(n)= f(n)$\nelse $a(n) = f(f(f(n)))\\dots$ until one gets a multiple of $n$; $a(n) = 0$ if no such number\nexists.\n\n*References:*\n- [A100800](https://oeis.org/A100800)\n","FormalConjectures.OEIS.«101779»":"# Conjectures associated with A101779\n\n$a(n)$ is the least $k$ such that all of $k, 2k+1, 3k+2, ..., nk+n-1$ are primes,\nor $0$ if no such $k$ is found.\nIt is conjectured $k$ always exists.\n\n*References:*\n- [A101779](https://oeis.org/A101779)\n","FormalConjectures.OEIS.«102371»":"# Conjectures associated with A102371\n\nThe sequence $a(n)$ is defined by the recurrence relation $a(1)=1$,\nand for $n>1$, $a(n) = a(n-1) \\operatorname{XOR} (a(n-1) + n)$.\nThe conjecture asks if $a(n) = 2^n - 1 - \\operatorname{A105033}(n-1)$ for $n \\ge 1$.\n\n*References:*\n- [A102371](https://oeis.org/A102371)\n","FormalConjectures.OEIS.«102722»":"# Floor of sum of $\\{n/k\\}$\n\nGiven $n$, sum all division remainders $\\{n/k\\}$, with $k=1,\\dots,n$.\nThe value $a(n)$ is given by the floor of that sum. Note that $\\{x\\}:=x-[x]$.\nConjecture: a(n) ~ (1-EulerGamma)n.\n\n*References:*\n- [A102722](https://oeis.org/A102722)\n","FormalConjectures.OEIS.«102847»":"# $a(0) = 1$, $a(n) = a(n-1)a(n-1) + 2$\n\n*References:*\n- [A102847](https://oeis.org/A102847)\n","FormalConjectures.OEIS.«103151»":"# Number of decompositions of $2n+1$ into $2p+q$, where $p$ and $q$ are both odd primes\n\n*References:*\n- [A103151](https://oeis.org/A103151)\n","FormalConjectures.OEIS.«103311»":"# Fibonacci transform satisfying $|a(n)| = F(n+1)$\n\nThe sequence $a(n)$ satisfies the linear recurrence relation:\n$$a(n) = 3a(n-1) - 4a(n-2) + 2a(n-3) - a(n-4)$$\nwith initial terms $a(0)=0, a(1)=1, a(2)=1, a(3)=0$.\nThe sequence takes values in $\\mathbb{Z}$.\n\n*References:*\n- [A103311](https://oeis.org/A103311)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«103425»":"# $a(n) = 3a(n-1) + a(n-2) - 3a(n-3)$\n\n*References:*\n- [A103425](https://oeis.org/A103425)\n","FormalConjectures.OEIS.«103662»":"# Smallest power with base>1 and exponent $n$ without digit 0\n\nFor statistical reasons it is conjectured that the sequence is finite.\nAlso it is conjectured that $a(40)$ does not exist (i.e. the sequence is empty for $n=40$).\n\n*References:*\n- [A103662](https://oeis.org/A103662)\n","FormalConjectures.OEIS.«103885»":"# $a(n) = [x^{2n}] \\left(\\frac{1 + x}{1 - x}\\right)^n$\n\nThe sequence is given by the combinatorial identity:\n$a(n) = \\sum_{k = 0}^n \\binom{n}{k} \\binom{2n+k-1}{n-1}$\nwith $a(0) = 1$.\n\n*References:*\n- [A103885](https://oeis.org/A103885)\n","FormalConjectures.OEIS.«104320»":"# Number of zeros in ternary representation of $2^n$\n\n*References:*\n- [A104320](https://oeis.org/A104320)\n","FormalConjectures.OEIS.«105020»":"# Array read by upward antidiagonals\n\nArray read by upward antidiagonals: row $n$ ($n \\ge 0$) contains the numbers\n$m^2 - n^2$, $m \\ge n+1$.\n\n*References:*\n- [A105020](https://oeis.org/A105020)\n","FormalConjectures.OEIS.«105033»":"# Sloping binary numbers: read array of binary numbers (right-justified) along diagonals of slope $-1$\n\n*References:*\n- [A105033](https://oeis.org/A105033)\n","FormalConjectures.OEIS.«105210»":"# Conjectures associated with A105210\n\n$a(1) = 393$; for $n > 1$, $a(n) = a(n-1)$ + 1 + sum of distinct prime factors of $a(n-1)$\nthat are $< a(n-1)$.\n\n*References:*\n- [A105210](https://oeis.org/A105210)\n","FormalConjectures.OEIS.«105565»":"# Indicator sequence for 5 Fibonacci numbers with n digits\n\n$a(n) = 1$ if exactly 5 Fibonacci numbers exist with exactly $n$ digits, otherwise $0$.\nFor the partial sums $S(n) = \\sum_{k=1}^n a(k)$, it is conjectured that\n$\\beta-2 < S(n)-\\alpha n < \\beta-1$, where $\\alpha = \\log(10)/\\log(\\phi) - 4$\nand $\\beta = \\log(5)/(2\\log(\\phi)) - 1$.\n\n*References:*\n- [A105565](https://oeis.org/A105565)\n","FormalConjectures.OEIS.«105720»":"# Triangular matchstick numbers in the class of prime numbers\n\n$a(n) = \\sum_{k = n}^{2n} p_k$, where $p_k$ is the $k$-th prime.\n\n*References:*\n- [A105720](https://oeis.org/A105720)\n","FormalConjectures.OEIS.«105751»":"# Imaginary part of $\\prod_{k=0}^n (1 + k \\cdot i)$, $i = \\sqrt{-1}$\n\n*References:*\n- [A105751](https://oeis.org/A105751)\n","FormalConjectures.OEIS.«105801»":"# Fibonacci-Collatz sequence\n\nFibonacci-Collatz sequence: $a(1)=1, a(2)=2$; for $n > 2$, let $\\mathrm{fib} = a(n-1) + a(n-2)$;\nif $\\mathrm{fib}$ is odd then $a(n) = 3 \\cdot \\mathrm{fib} + 1$ else $a(n) = \\mathrm{fib}/2$.\n\n*References:*\n- [A105801](https://oeis.org/A105801)\n","FormalConjectures.OEIS.«107247»":"# Sum of squares of nonacci numbers\n\nSum of squares of nonacci numbers (Fibonacci 9-step numbers).\n\n*References:*\n- [A107247](https://oeis.org/A107247)\n","FormalConjectures.OEIS.«108081»":"# $a(n) = \\sum_{i=0}^n \\binom{2n-i}{n+i}$\n\nAlternatively the sequence `a` can be defined as\n$a(n) = \\sum_{k=0}^n \\binom{n+k-1}{k} F(n-k+1)$, where $F(m)$ is the $m$-th Fibonacci number.\nWe formalize a conjecture about the number of words of length $n$ in a set $X$ being related\nto this sequence.\n\n*References:*\n- [A108081](https://oeis.org/A108081)\n","FormalConjectures.OEIS.«108129»":"# Riesel Problem\n\nRiesel problem: let $k=2n-1$; then $a(n)$ is the smallest $m \\ge 1$ such that\n$k \\cdot 2^m-1$ is prime, or $-1$ if no such prime exists.\n\n*References:*\n- [A108129](https://oeis.org/A108129)\n","FormalConjectures.OEIS.«108211»":"# $a(n) = 16n^2 + 1$\n\n*References:*\n- [A108211](https://oeis.org/A108211)\n","FormalConjectures.OEIS.«108301»":"# Digital sum of the Fermat number $2^{2^n} + 1$\n\n`a n` is the digital sum of the Fermat number $2^{2^n} + 1$.\nThe conjecture asks if there are any prime numbers in this sequence beyond $n=11$.\n\n*References:*\n- [A108301](https://oeis.org/A108301)\n","FormalConjectures.OEIS.«108306»":"# A108306: Expansion of $(3x+1)/(1-3x-3x^2)$\n\nThis sequence satisfies the linear recurrence relation $a(0)=1$, $a(1)=6$,\nand $a(n) = 3a(n-1) + 3a(n-2)$ for $n \\ge 2$.\n\n*References:*\n- [A108306](https://oeis.org/A108306)\n","FormalConjectures.OEIS.«108569»":"# Numbers $n$ such that $\\phi(n) = \\phi(n + \\phi(n))$\n\n*References:*\n- [A108569](https://oeis.org/A108569)\n","FormalConjectures.OEIS.«108864»":"# Numbers $n$ such that the perfect deficiency of $n$ is $\\le 10$.\n\nThe perfect deficiency of $n$ (A109883) is the remainder after greedily subtracting\nfrom $n$ its divisors in increasing order, skipping any divisor larger than the\ncurrent remainder.\n\n*References:*\n- [A108864](https://oeis.org/A108864)\n- [A109883](https://oeis.org/A109883)\n","FormalConjectures.OEIS.«108866»":"# Numerator of $\\sum_{k=1}^n 2^k/k$.\n\nConjecture: for $n > 3$,\n$\\textrm{numerator}(-2/n + \\sum_{k=1}^{n} \\frac{2^k}{k}) == 0 (\\textrm{mod} n^2)$\nif and only if n is prime.\n\n*References:*\n- [A108866](https://oeis.org/A108866)\n","FormalConjectures.OEIS.«108»":"# Fractional parts of sums of reciprocals of Catalan numbers $C(n)$\n\nCatalan numbers $C(n) = \\frac{1}{n+1}\\binom{2n}{n}$.\n\nThe sum $\\sum_{i=j}^k \\frac{1}{a(i)}$ of reciprocals of Catalan numbers.\n\n*References:*\n- [A000108](https://oeis.org/A000108)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«109074»":"# Numerator of $\\binom{6n-2}{2n} / \\left(2 \\binom{4n-1}{2n}\\right)$\n\nConjecture: $\\binom{6n-2}{2n} / \\left(2 \\binom{4n-1}{2n}\\right) = A005156(n+1)/A005156(n)$\n\n*References:*\n- [A109074](https://oeis.org/A109074)\n","FormalConjectures.OEIS.«109227»":"# Conjectures associated with A109227\n\nBinary strings that have 1's where primes occur, 0's elsewhere and every term ends\nwith the $n$-th prime index.\n\nConjecture: $a(2)$ and $a(121)$ are primes. Are there any more?\n\n*References:*\n- [A109227](https://oeis.org/A109227)\n","FormalConjectures.OEIS.«109671»":"# Conjectures associated with A109671\n\n$a(1)=1$; thereafter, $a(2n)=a(n)$, $a(2n+1)$ is the smallest positive number\nsuch that $|a(2n+1)-a(2n-1)|=a(n)$.\nConjecture: Does the sequence contain every positive integer?\n\n*References:*\n- [A109671](https://oeis.org/A109671)\n","FormalConjectures.OEIS.«109845»":"# Conjectures associated with A109845\n\n$a(1) = 2$; $a(2n)$ = lcm of all previous terms + 1; $a(2n+1)$ = lcm of all previous terms - 1.\n\n*References:*\n- [A109845](https://oeis.org/A109845)\n","FormalConjectures.OEIS.«109905»":"# Conjectures associated with A109905\n\n$a(n)$ is the greatest prime of the form $k(n-k)+1$, where $k$ can take values from\n$1$ to $\\lfloor n/2 \\rfloor$. $a(n)=0$ if no such prime exists.\n\n*References:*\n- [A109905](https://oeis.org/A109905)\n","FormalConjectures.OEIS.«109908»":"# Conjectures associated with A109908\n\n$a(n)$ = greatest prime of the form $k(n-k)-1$, or $0$ if no such prime exists.\n\n*References:*\n- [A109908](https://oeis.org/A109908)\n","FormalConjectures.OEIS.«109909»":"# Conjectures associated with A109909\n\n$a(n)$ = number of primes of the form $k(n-k)-1$.\n\n*References:*\n- [A109909](https://oeis.org/A109909)\n","FormalConjectures.OEIS.«110475»":"# Number of symbols '*' and '^' to write the canonical prime factorization of n\n\nThe canonical prime factorization is $n = p_1^{e_1} p_2^{e_2} \\cdots p_k^{e_k}$.\nThe written form is $p_1^{\\wedge} e_1 * p_2^{\\wedge} e_2 * \\cdots * p_k^{\\wedge} e_k$,\nwhere the $\\wedge$ appears only if $e_i > 1$.\n$a(n) = (\\text{number of distinct prime factors}) - 1 +$\n$(\\text{number of distinct prime factors with exponent } > 1)$.\n\n*References:*\n- [A110475](https://oeis.org/A110475)\n","FormalConjectures.OEIS.«110566»":"# $a(n) = \\operatorname{lcm}\\{1,2,\\dots,n\\}/\\operatorname{denom}(H(n))$\n\n*References:*\n- [A110566](https://oeis.org/A110566)\n","FormalConjectures.OEIS.«110835»":"# Smallest $m > 0$ such that there are no primes between $nm$ and $n(m+1)$ inclusive.\n\nSierpinski's conjecture (1958) is precisely that a(n) >= n for all n.\n\n*References:*\n- [A110835](https://oeis.org/A110835)\n","FormalConjectures.OEIS.«110854»":"# Conjectures associated with A110854\n\n$a(n) = \\mathrm{prime}(2n+2) - \\mathrm{prime}(2n+1) - \\mathrm{prime}(2n) + \\mathrm{prime}(2n-1)$,\nwhere $\\mathrm{prime}(k)$ is the $k$-th prime number.\n\n*References:*\n- [A110854](https://oeis.org/A110854)\n","FormalConjectures.OEIS.«111114»":"# Integer part of $\\mathrm{prime}(n)/\\pi(n)$\n\nHere $\\mathrm{prime}(n)$ is the $n$-th prime number, and $\\pi(n)$ is the prime-counting function.\n\n*References:*\n- [A111114](https://oeis.org/A111114)\n","FormalConjectures.OEIS.«111291»":"# Number of refactorable numbers (A033950) $\\le 10^n$\n\nA number $k$ is refactorable if its number of divisors, $\\tau(k)$, divides $k$.\n\n*References:*\n- [A111291](https://oeis.org/A111291)\n","FormalConjectures.OEIS.«112521»":"# Sequence related to NOR bracketings\n\n$$a(n) = \\sum_{j=0}^{n-1} (-1)^j \\binom{2j}{j} \\binom{2n-j-2}{n-j-1}$$\n\n*References:*\n- [A112521](https://oeis.org/A112521)\n","FormalConjectures.OEIS.«112970»":"# A generalized Stern sequence\n\nA112970: A generalized Stern sequence, defined by the recurrence relations:\n$a(2n+1) = a(n)$ and $a(2n) = a(n) + a(n-2)$ with $a(0)=1$, $a(1)=1$ and $a(n)=0$ for $n \\le -1$.\n\n*References:*\n- [A112970](https://oeis.org/A112970)\n","FormalConjectures.OEIS.«113010»":"# Number of digits of n raised to the power of the sum of the digits of n\n\n*References:*\n- [A113010](https://oeis.org/A113010)\n","FormalConjectures.OEIS.«113019»":"# Number of digits of n raised to the power of the digital root of n\n\n*References:*\n- [A113019](https://oeis.org/A113019)\n","FormalConjectures.OEIS.«113213»":"# Smallest number $m$ such that $2^n - m$ and $2^n + m$ are primes\n\n\n*References:*\n- [A113213](https://oeis.org/A113213)\n","FormalConjectures.OEIS.«113250»":"# Expansion of g.f. $-(1 - 48x^2 - 256x^3) / ((1 - 4x)(1 + 4x)(1 + 4x + 16x^2))$\n\nCorresponds to m = 4 in a family of 4th-order linear recurrence sequences\n\nThis sequence is defined by the linear recurrence relation with signature $(-4, 0, 64, 256)$\nand initial values $a(0) = -1, a(1) = 4, a(2) = 32, a(3) = 64$.\nThe recurrence is $a(n) = -4 a(n-1) + 64 a(n-3) + 256 a(n-4)$.\n\n*References:*\n- [A113250](https://oeis.org/A113250)\n","FormalConjectures.OEIS.«113252»":"# Conjectures associated with A113252\n\nCorresponds to m = 6 in a family of 4th order linear recurrence sequences\n\nThis sequence is defined by the linear recurrence relation\na(n) = -4 a(n-1) + 144 a(n-3) + 1296 a(n-4) for n > 3.\nInitial values are a(0) = -1, a(1) = 4, a(2) = 92, a(3) = 784.\n\n*References:*\n- [A113252](https://oeis.org/A113252)\n","FormalConjectures.OEIS.«113254»":"# Square odd-indexed terms in the recurrence $a(n) = 8a(n-1) - 8a(n-2) + 8a(n-3) - a(n-4)$\n\nA113254: Corresponds to $m = 8$ in a family of 4th-order linear recurrence sequences.\n\nThe sequence $a(n)$ is defined by the initial conditions $a(0)=-1, a(1)=4, a(2)=176, a(3)=3136$,\nand the linear recurrence relation\n$a(n) = -4 * a (n-1) + 256 * a (n-3) + 4096 * a (n-4)$ for $n \\ge 4$.\n\n*References:*\n- [A113254](https://oeis.org/A113254)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«113255»":"# Conjectures associated with A113255\n\nCorresponds to m = 9 in a family of 4th-order linear recurrence sequences\n\nThis sequence is defined by the recurrence relation\n$a(n) = -4 a(n-1) + 324 a(n-3) + 6561 a(n-4)$ for n > 3.\nInitial values are $a(0) = -1, a(1) = 4, a(2) = 227, a(3) = 5329$.\n\n*References:*\n- [A113255](https://oeis.org/A113255)\n","FormalConjectures.OEIS.«113257»":"# Ascending descending base exponent transform of squares\n\na n is $\\sum_{i=1}^n (i^2)^((n-i+1)^2)$.\n\n*References:*\n- [A113257](https://oeis.org/A113257)\n","FormalConjectures.OEIS.«113258»":"# Ascending descending base exponent transform of factorials\n\n*References:*\n- [A113258](https://oeis.org/A113258)\n","FormalConjectures.OEIS.«113271»":"# Ascending descending base exponent transform of $2^n$\n\n*References:*\n- [A113271](https://oeis.org/A113271)\n","FormalConjectures.OEIS.«113609»":"# Number of prime powers $q<=n$ such that also $q+2$ is a prime power\n\n*References:*\n- [A113609](https://oeis.org/A113609)\n","FormalConjectures.OEIS.«114137»":"# Difference between first odd semiprime $> 2^n$ and $2^n$\n\n*References:*\n- [A114137](https://oeis.org/A114137)\n","FormalConjectures.OEIS.«114216»":"# Largest odd divisor of $a(n-1) + \\textrm{prime}(n)$\n\n$a(0)=0$; thereafter $a(n)$ = largest odd divisor of $a(n-1) + \\textrm{prime}(n)$.\n\n*References:*\n- [A114216](https://oeis.org/A114216)\n","FormalConjectures.OEIS.«114362»":"# Numerator of $\\zeta(4n)/\\zeta(2n)^2$ (with $a(0)=2$ instead of $-2$)\n\nThe ratio $\\zeta(4n)/\\zeta(2n)^2$ for $n \\ge 1$ is the rational number\n$$ Q_n = -2 \\frac{B_{4n}}{B_{2n}^2 \\binom{4n}{2n}} $$\nwhere $B_k$ is the $k$-th Bernoulli number. The sequence $a(n)$ is the numerator of $Q_n$,\nwith $a(0)$ defined as $2$.\n\n*References:*\n- [A114362](https://oeis.org/A114362)\n","FormalConjectures.OEIS.«1146»":"# $a(n) = 2^(2^n)$\n\n*References:*\n- [A001146](https://oeis.org/A001146)\n","FormalConjectures.OEIS.«114831»":"# Each term is previous term plus floor of harmonic mean of two previous terms.\n\n$a(1) = 1, a(2) = 2$ and\n$a(n) = a(n-1) + \\lfloor \\frac{2 a(n-1) a(n-2)}{a(n-1) + a(n-2)} \\rfloor$ for $n \\ge 3$.\n\n*References:*\n- [A114831](https://oeis.org/A114831)\n","FormalConjectures.OEIS.«115257»":"# Partial sums of $\\binom{2n}{n}^2$\n\n$$a(n) = \\sum_{k=0}^n \\binom{2k}{k}^2$$\n\n*References:*\n- [A115257](https://oeis.org/A115257)\n","FormalConjectures.OEIS.«115366»":"# $a(n)$ = the number of values of $k <= 10^n$ such that $\\sqrt{k(k+1)(k+2)(k+3)+1}$ is prime\n\nSince $\\sqrt{k(k+1)(k+2)(k+3)+1} = k^2 + 3k + 1$,\n$a(n) = \\#\\{k \\in \\mathbb{N} \\mid 1 \\le k \\le 10^n \\land (k^2 + 3k + 1) \\text{ is prime} \\}.$\n\n*References:*\n- [A115366](https://oeis.org/A115366)\n","FormalConjectures.OEIS.«11545»":"# $a(n)$ is the integer whose decimal digits are the first $n+1$ decimal digits of $\\pi$\n\n*References:*\n- [A011545](https://oeis.org/A011545)\n","FormalConjectures.OEIS.«1157»":"# Sum of squares of divisors of $n$\n\n*References:*\n- [A001157](https://oeis.org/A001157)\n","FormalConjectures.OEIS.«116150»":"# $a(n) = \\sum_{j=1}^{n} (3^j + (-2)^j)$\n\n*References:*\n- [A116150](https://oeis.org/A116150)\n","FormalConjectures.OEIS.«117027»":"# Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes\n\n$a(n) = p_{4n-3}p_{4n} - p_{4n-2}p_{4n-1}$ where $p_k$ is the k-th prime number (1-indexed).\n\n*References:*\n- [A117027](https://oeis.org/A117027)\n","FormalConjectures.OEIS.«117531»":"# Number of primes in $n$-th row of triangle $k^2 - k + p_n$\n\n$a(n)$ is the number of primes in the $n$-th row of the triangle $T(n, k) = k^2 - k + p_n$\nfor $1 \\le k \\le n$, where $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \\dots$).\n\n*References:*\n- [A117531](https://oeis.org/A117531)","FormalConjectures.OEIS.«117545»":"# Least $k$ such that cyclotomic polynomial $\\Phi_k(n)$ is prime\n\n$a(n) = \\min \\{k \\in \\mathbb{N} \\mid 0 < k \\wedge \\text{Prime}(|\\Phi_k(n)|) \\}$,\nwhere $\\Phi_k(n)$ is the $k$-th cyclotomic polynomial evaluated at $n$.\n\n*References:*\n- [A117545](https://oeis.org/A117545)","FormalConjectures.OEIS.«119563»":"# Sum of Fermat number and Mersenne number minus 1: $2^{2^n} + 2^n - 1$\n\nDefine $F(n) = 2^{2^n} + 1$ (the $n$-th Fermat number) and $M(n) = 2^n - 1$ (the $n$-th Mersenne\nnumber). Then $a(n) = F(n) + M(n) - 1 = 2^{2^n} + 2^n - 1$.\n\n*References:*\n- [A119563](https://oeis.org/A119563)","FormalConjectures.OEIS.«119591»":"# Least $k \\ge 1$ such that $2 \\cdot n^k - 1$ is prime\n\n$a(n) = \\min \\{k \\ge 1 \\mid \\text{Prime}(2 \\cdot n^k - 1)\\}$ for $n \\ge 2$.\n\n*References:*\n- [A119591](https://oeis.org/A119591)","FormalConjectures.OEIS.«120424»":"# Half-Fibonacci sequence\n\n$a(0) = 1, a(1) = 3$; for $n \\ge 2$, $a(n) = f(a(n-1)) + f(a(n-2))$ where $f(x) = x/2$ if $x$\nis even\nand $f(x) = x$ if $x$ is odd.\n\n*References:*\n- [A120424](https://oeis.org/A120424)","FormalConjectures.OEIS.«1223»":"# Prime gaps\n\nDifferences between consecutive primes: $a(n) = p_{n+1} - p_n$.\n\n*References:*\n- [A001223](https://oeis.org/A001223)\n","FormalConjectures.OEIS.«129365»":"# Ratio of product of GCDs to product of factorials of floor divisions\n\n$$a(n) = \\frac{\\prod_{j=1}^n \\prod_{k=1}^n \\gcd(j,k)}{\\prod_{k=1}^n (\\lfloor n/k \\rfloor!)^k}$$\n\n*References:*\n- [A129365](https://oeis.org/A129365)\n","FormalConjectures.OEIS.«130911»":"# Odious primes minus evil primes among first $n$ primes\n\n$a(n)$ is the number of primes with odd binary weight (odious primes) among the first $n$ primes\nminus the number with even binary weight (evil primes).\n\n*References:*\n- [A130911](https://oeis.org/A130911)","FormalConjectures.OEIS.«135508»":"# Recurrence involving LCM: $a(n) = x(n+1)/x(n) - 2$\n\n$a(n) = x(n+1)/x(n) - 2$ where $x(1)=1$ and $x(n) = 2 x(n-1) + \\operatorname{lcm}(x(n-1),n)$\nfor $n > 1$.\n\n*References:*\n- [A135508](https://oeis.org/A135508)","FormalConjectures.OEIS.«1359»":"# Lesser of twin primes\n\nPrimes $p$ such that $p+2$ is also prime.\n\n*References:*\n- [A001359](https://oeis.org/A001359)\n","FormalConjectures.OEIS.«141057»":"# Number of Abelian cubes of length $3n$ over an alphabet of size 3\n\nAn Abelian cube is a string of the form $x x' x''$ with $|x| = |x'| = |x''|$ and $x$ is a\npermutation of $x'$ and $x''$. The number of Abelian cubes of length $3n$ over an alphabet of\nsize 3 is given by\n$$a(n) = \\sum_{k=0}^n \\binom{n}{k}^3 \\sum_{j=0}^k \\binom{k}{j}^3.$$\n\n*References:*\n- [A141057](https://oeis.org/A141057)","FormalConjectures.OEIS.«145355»":"# Factorial distance to nearest square\n\nThe sequence is defined as\n$$a(n) =\n\\mathrm{round}\\left(\\frac{\\mathrm{round}(\\sqrt{n!})}{\\left|(\\mathrm{round}(\\sqrt{n!}))^2 -\nn!\\right|}\\right)$$\nfor $n \\ge 2$.\n\n*References:*\n- [A145355](https://oeis.org/A145355)\n","FormalConjectures.OEIS.«153330»":"# Collatz step differences\n\nDifferences in adjacent elements of the sequence quantifying the steps needed for $n$ to\nconverge to 1 in the Collatz Conjecture.\n$$a(n) = \\mathrm{A006577}(n+1) - \\mathrm{A006577}(n)$$\nfor $n > 0$.\n\n*References:*\n- [A153330](https://oeis.org/A153330)","FormalConjectures.OEIS.«157225»":"# Representations as $p + 2^x + 7 \\cdot 2^y$ with $p \\equiv 5 \\pmod 6$\n\nNumber of ways to write the $n$-th positive odd integer in the form $p + 2^x + 7 \\cdot 2^y$\nwith $p$ a prime congruent to $5 \\bmod 6$ and $x, y$ positive integers.\n$$a(n) = \\left|\\left\\{(p, x, y) : p + 2^x + 7 \\cdot 2^y = 2n - 1 \\text{ with } p \\text{ prime},\np \\equiv 5 \\pmod 6, x, y \\in \\mathbb{Z}^+\\right\\}\\right|.$$\n\n*References:*\n- [A157225](https://oeis.org/A157225)\n- Z.-W. Sun, \"Mixed sums of primes and other terms\", arXiv preprint\n  [arXiv:0901.3075](https://arxiv.org/abs/0901.3075) [math.NT], 2009.","FormalConjectures.OEIS.«157237»":"# Representations as $p + 2^x + 11 \\cdot 2^y$ with $p \\equiv 1 \\pmod 6$\n\nNumber of ways to write the $n$-th positive odd integer in the form $p + 2^x + 11 \\cdot 2^y$\nwith $p$ a prime congruent to $1 \\bmod 6$ and $x, y$ positive integers.\n$$a(n) = \\left|\\left\\{(p, x, y) : p + 2^x + 11 \\cdot 2^y = 2n - 1 \\text{ with } p \\text{ prime},\np \\equiv 1 \\pmod 6, x, y \\in \\mathbb{Z}^+\\right\\}\\right|.$$\n\n*References:*\n- [A157237](https://oeis.org/A157237)\n- Z.-W. Sun, \"Mixed sums of primes and other terms\", arXiv preprint\n  [arXiv:0901.3075](https://arxiv.org/abs/0901.3075) [math.NT], 2009.","FormalConjectures.OEIS.«159829»":"# Smallest $m$ such that $n^3 + m^3 + 1$ is prime\n\n$a(n)$ is the smallest natural number $m \\ge 1$ such that $n^3 + m^3 + 1$ is prime.\n\n*References:*\n- [A159829](https://oeis.org/A159829)","FormalConjectures.OEIS.«160324»":"# Number of ways to express $n$ as sum of square, pentagonal, and hexagonal numbers\n\n$$a(n) = |\\{(x, y, z) \\in \\mathbb{N}^3 : x^2 + p_5(y) + p_6(z) = n\\}|$$\n\n*References:*\n- [A160324](https://oeis.org/A160324)","FormalConjectures.OEIS.«166944»":"# Rowland-style prime-generating recurrence\n\nThe sequence is defined by $a(1) = 2$ and for $n \\ge 2$:\n$a(n) = a(n-1) + \\gcd(n, a(n-1))$ if $n$ is even, and\n$a(n) = a(n-1) + \\gcd(n-2, a(n-1))$ if $n$ is odd.\n\n*References:*\n- [A166944](https://oeis.org/A166944)\n- E. S. Rowland, \"A natural prime-generating recurrence\", *J. Integer Sequences* **11** (2008),\n  Article 08.2.8.\n- V. Shevelev, \"An infinite set of generators of primes based on the Rowland idea and\n  conjectures concerning twin primes\", arXiv preprint\n  [arXiv:0910.4676](https://arxiv.org/abs/0910.4676) [math.NT], 2009.","FormalConjectures.OEIS.«167604»":"# Chua's Euclidean prime sequence\n\nFor a product $n$ of the preceding terms, Chua's sequence chooses the least\nprime dividing $d + n / d$ for some divisor $d$ of $n$. The open question is\nwhether every prime occurs.\n\n*References:*\n- [OEIS A167604](https://oeis.org/A167604)\n- Andrew R. Booker,\n  [A variant of the Euclid--Mullin sequence containing every prime](https://arxiv.org/abs/1605.08929)\n","FormalConjectures.OEIS.«167918»":"# Smallest index $k > n$ such that $(p_k+p_{k+1})/(p_n+p_{n+1})$ is an integer $\\ge 2$\n\n*References:*\n- [A167918](https://oeis.org/A167918)","FormalConjectures.OEIS.«175386»":"# Denominators of $\\sum_{k=1}^n \\frac{1}{k 2^k}$\n\n$a(n)$ is the denominator of the sum\n$$\\sum_{i=1}^n \\frac{1}{i} \\binom{2n-i-1}{i-1}$$\n\n*References:*\n- [A175386](https://oeis.org/A175386)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«176477»":"# Recurrence with fourth powers of binomial coefficients\n\nThe sequence is defined by $a(1) = 2$, and for $n \\ge 2$,\n$$(2n+1)^3 a(n) = 32n^3 a(n-1) + (21n^3 + 22n^2 + 8n + 1) \\binom{2n-1}{n}^4.$$\n\n*References:*\n- [A176477](https://oeis.org/A176477)\n- Z.-W. Sun, \"Open Conjectures on Congruences\", arXiv preprint\n  [arXiv:0911.5665](https://arxiv.org/abs/0911.5665) [math.NT], 2009-2011.","FormalConjectures.OEIS.«17666»":"# Denominator of sum of reciprocals of divisors\n\nDenominator of sum of reciprocals of divisors of $n$:\n$$\\sum_{d \\mid n} \\frac{1}{d} = \\frac{\\sigma(n)}{n}$$\nin lowest terms.\n\n*References:*\n- [A017666](https://oeis.org/A017666)\n","FormalConjectures.OEIS.«179524»":"# Central binomial sum $a(n) = \\sum_{k=0}^n (-4)^k \\binom{n}{k}^2 \\binom{n-k}{k}^2$\n\nThe sequence is defined by\n$$a(n) = \\sum_{k=0}^n (-4)^k \\binom{n}{k}^2 \\binom{n-k}{k}^2.$$\n\n*References:*\n- [A179524](https://oeis.org/A179524)\n- Z.-W. Sun, \"Open Conjectures on Congruences\", arXiv preprint\n  [arXiv:0911.5665](https://arxiv.org/abs/0911.5665) [math.NT], 2009-2011.","FormalConjectures.OEIS.«179537»":"# Central binomial sum $a(n) = \\sum_{k=0}^n \\binom{n}{k}^2 \\binom{n-k}{k}^2 (-16)^k$\n\nThe sequence is defined by\n$$a(n) = \\sum_{k=0}^n \\binom{n}{k}^2 \\binom{n-k}{k}^2 (-16)^k.$$\n\n*References:*\n- [A179537](https://oeis.org/A179537)\n- Z.-W. Sun, \"Open Conjectures on Congruences\", arXiv preprint\n  [arXiv:0911.5665](https://arxiv.org/abs/0911.5665) [math.NT], 2009-2011.","FormalConjectures.OEIS.«180017»":"# Difference of digit sums in base 3 and base 2\n\nDifference of sums of digits of $n$ in ternary and in binary:\n$$a(n) = \\sum \\mathrm{digits}_3(n) - \\sum \\mathrm{digits}_2(n).$$\n\n*References:*\n- [A180017](https://oeis.org/A180017)","FormalConjectures.OEIS.«181546»":"# Sum of fourth powers of Fibonacci-like binomial coefficients\n\nThe sequence is defined by\n$$a(n) = \\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\binom{n-k}{k}^4.$$\n\n*References:*\n- [A181546](https://oeis.org/A181546)","FormalConjectures.OEIS.«1818»":"# Squares of double factorials\n\nSquares of double factorials: $a(n) = ((2n-1)!!)^2 = (1 \\cdot 3 \\cdot 5 \\cdots (2n-1))^2$.\n\n*References:*\n- [A001818](https://oeis.org/A001818)\n","FormalConjectures.OEIS.«182126»":"# Product of two consecutive primes modulo the next prime\n\nThe sequence is defined by\n$$a(n) = \\mathrm{prime}(n) \\cdot \\mathrm{prime}(n+1) \\bmod \\mathrm{prime}(n+2),$$\nwhere $\\mathrm{prime}(k)$ is the $k$-th prime number ($\\mathrm{prime}(1)=2$).\n\n*References:*\n- [A182126](https://oeis.org/A182126)","FormalConjectures.OEIS.«182510»":"# Recurrence with bitwise XOR\n\nThe sequence is defined by $a(0) = 0$, $a(1) = 1$, and for $n \\ge 0$,\n$$a(n+2) = (a(n+1) \\mathbin{\\mathrm{XOR}} (n+2)) - a(n),$$\nwhere $\\mathrm{XOR}$ is the bitwise exclusive-or operator on integers.\n\n*References:*\n- [A182510](https://oeis.org/A182510)","FormalConjectures.OEIS.«185150»":"# Number of odd primes between $n^2$ and $(n+1)^2$ with $(n/p) = 1$\n\n$a(n)$ is the number of odd primes $p$ between $n^2$ and $(n+1)^2$ such that the Legendre symbol\n$\\left(\\frac{n}{p}\\right) = 1$.\n\n*References:*\n- [A185150](https://oeis.org/A185150)\n- Z.-W. Sun, \"Conjectures involving primes and quadratic forms\", arXiv preprint\n  [arXiv:1211.1588](https://arxiv.org/abs/1211.1588) [math.NT], 2012.","FormalConjectures.OEIS.«185895»":"# Coefficients of $\\prod_{k>0} (1 - x^k/k!)$\n\nThe sequence $a(n)$ has exponential generating function\n$$E(x) = \\prod_{k=1}^\\infty \\left(1 - \\frac{x^k}{k!}\\right),$$\nso that $a(n) = n! [x^n] \\prod_{k=1}^n \\left(1 - \\frac{x^k}{k!}\\right)$.\n\n*References:*\n- [A185895](https://oeis.org/A185895)\n","FormalConjectures.OEIS.«194806»":"# Size of smallest subset of $\\{1, 2, \\dots, n\\}$ with distinct subset sums\n\nSize of the smallest subset $S$ of $T = \\{1,2,3,\\dots,n\\}$\nsuch that $S \\cdot S$ contains $T$,\nwhere $S \\cdot S$ is the set of all products of elements of $S$.\n\n*References:*\n- [A194806](https://oeis.org/A194806)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«211417»":"# Integrality and supercongruences of the factorial ratio $\\frac{(6n)! n!}{(3n)! (2n)!^2}$\n\nIntegral factorial ratio sequence:\n$$a(n) = \\frac{(30n)! n!}{(15n)! (10n)! (6n)!}$$\n\n*References:*\n- [A211417](https://oeis.org/A211417)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«22030»":"# A nonlinear recurrence sequence\n\nFor even $n$, $a(n+2)$ is the greatest integer such that $a(n+2)/a(n+1) < a(n+1)/a(n)$;\nfor odd $n$, the least integer such that $a(n+2)/a(n+1) > a(n+1)/a(n)$;\n$a(0) = 4, a(1) = 16$.\n\n*References:*\n- [A022030](https://oeis.org/A022030)\n","FormalConjectures.OEIS.«224515»":"# Existence of integers $k$ with $k^2 \\operatorname{XOR} (k+1)^2 = (2n+1)^2$\n\na: $a(n) = \\text{least } k \\text{ such that } \\sqrt{k^2 \\operatorname{XOR} (k+1)^2} = 2n+1$,\n$a(n) = -1 \\text{ if there is no such } k$.\nThis is equivalent to finding the smallest $k \\in \\mathbb{N}$\nsuch that $k^2 \\oplus (k+1)^2 = (2n+1)^2$.\nWe use the set infimum ($\\operatorname{sInf}$) to denote the least element\nof the set of natural numbers satisfying the condition.\nSince Mathlib's `sInf` on a subset of `ℕ` gives a result in `ℕ`, this definition\nis only completely faithful to the OEIS when the set is non-empty.\nThe OEIS definition implies that the set of k's is non-empty for all n.\n\n*References:*\n- [A224515](https://oeis.org/A224515)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«224»":"# Number of squares $\\bmod n$\n\nThe number of squares modulo $n$.\nThis is the cardinality of the set $\\{k^2 \\bmod n \\mid k \\in \\{0, 1, \\dots, n-1\\}\\}$.\n\n*References:*\n- [A000224](https://oeis.org/A000224)\n","FormalConjectures.OEIS.«227582»":"# Representation of sequence terms by harmonic numbers $\\lfloor \\frac{1}{2H(n) - H(n^2+n-1) - \\gamma} \\rfloor$\n\nA227582: Expansion of $(2+3x+2x^2+2x^3+3x^4+x^5-x^6)/(1-2x+x^2-x^5+2x^6-x^7)$.\n\nThe sequence satisfies the linear recurrence:\n$$a(n) = 2 a(n-1) - a(n-2) + a(n-5) - 2 a(n-6) + a(n-7)$$\nwith initial values $a(1) = 2, a(2) = 7, a(3) = 14, a(4) = 23, a(5) = 35, a(6) = 50, a(7) = 67$.\n\nThe sequence is 1-indexed in OEIS, so $a(n)$ is the $(n-1)$-th term of the 0-indexed solution.\n\n*References:*\n- [A227582](https://oeis.org/A227582)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«228143»":"# Eighth root of the generating function of an Apéry-like sequence\n\nThe auxiliary sequence used for the Hankel matrix, defined as\n$$\\sum_{k=0}^n \\binom{n}{k}^2 \\binom{n+k}{k}^2$$\n\nDeterminant of the $(n+1) \\times (n+1)$ Hankel-type matrix with\n$(i,j)$-entry equal to A005259$(i+j)$ for all $i,j = 0,\\dots,n$.\nThe entry function A005259 is taken to be $\\sum_{k=0}^n \\binom{n}{k}^2 \\binom{n+k}{k}^2$.\n\n*References:*\n- [A228143](https://oeis.org/A228143)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n- [A005259](https://oeis.org/A005259)\n","FormalConjectures.OEIS.«228828»":"# Numbers $n$ such that $n^2 + \\pi(n)$ is prime\n\n*References:*\n- [A228828](https://oeis.org/A228828)\n","FormalConjectures.OEIS.«231201»":"# Sum of two numbers with prime conditions\n\nNumber of ways to write $n = x+y$, for $x,y > 0$ such that $2^x + y$ is prime.\n\nZhi-Wei Sun has offered a \\$1000 prize for the first proof.\n\n*References:*\n- [A231201](https://oeis.org/A231201)\n- Zhi-Wei Sun, \"Table of n, a(n) for n = 1..10000\",\n  \"Write n = k + m with 2^k + m prime\", a message to Number Theory List, Nov. 16, 2013,\n  \"On a^n+ bn modulo m\", arXiv:1312.1166 [math.NT], 2013-2014,\n  \"Problems on combinatorial properties of primes\", arXiv:1402.6641 [math.NT], 2014-2015.\n","FormalConjectures.OEIS.«232174»":"# Representations with prime conditions\n\nAny integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and\n$x^2 + ny^2$ are prime.\n\nZhi-Wei Sun has offered a \\$200 prize for the first proof.\n\n*References:*\n- [A232174](https://oeis.org/A232174)\n- Z.-W. Sun, \"Conjectures on representations involving primes,\" in: M. Nathanson (ed.),\n  Combinatorial and Additive Number Theory II: CANT, Springer Proc. in Math. & Stat.,\n  Vol. 220, Springer, 2017, pp. 279-310. https://arxiv.org/abs/1211.1588\n- D.A. Cox, \"Primes of the Form x² + ny²,\" John Wiley & Sons, 1989.\n","FormalConjectures.OEIS.«2326»":"# Multiplicative order of 2 mod $2n+1$\n\nThe multiplicative order of 2 modulo $2n+1$.\nIn other words, the least $m > 0$ such that $2n+1$ divides $2^m - 1$.\n\n*References:*\n- [A002326](https://oeis.org/A002326)","FormalConjectures.OEIS.«237271»":"# Number of parts in the symmetric representation of $\\sigma(n)$\n\nNumber of parts in the symmetric representation of $\\sigma(n)$. $a(n)$ is $1$ plus the number of pairs $(d_k, d_{k+1})$ of consecutive divisors of $n$\nsuch that $d_{k+1}$ is odd and $d_{k+1} \\ge 2 d_k$.\n\nThe formula used is\n$1 + |\\{(d_k, d_{k+1}) \\in \\text{consecutive pairs of divisors of } n \\mid\nd_{k+1} \\text{ is odd and } d_{k+1} \\ge 2 d_k\\}|$,\nwhich is a known characterization of the sequence.\n\n*References:*\n- [A237271](https://oeis.org/A237271)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«239957»":"# Primitive roots of the form k² + 1\n\nEvery prime $p$ has a primitive root $0 < g < p$ of the form $k^2 + 1$, where $k$ is an integer.\n\nZhi-Wei Sun has offered a prize of RMB 2,000 for the first proof.\n\n*References:*\n- [A239957](https://oeis.org/A239957)\n- Z.-W. Sun, \"New observations on primitive roots modulo primes,\" arXiv:1405.0290 [math.NT], 2014.\n","FormalConjectures.OEIS.«2407»":"# Cuban Primes\n\nOEIS A002407 lists the primes that are differences of two consecutive positive cubes. The\nsequence is conjectured to be infinite.\n\n*References:*\n- [OEIS A002407](https://oeis.org/A002407)\n","FormalConjectures.OEIS.«2426»":"# Central trinomial coefficients\n\nCentral trinomial coefficients: largest coefficient of $(1 + x + x^2)^n$, which is the coefficient\nof $x^n$ in the expansion of $(1 + x + x^2)^n$.\n\n*References:*\n- [A002426](https://oeis.org/A002426)","FormalConjectures.OEIS.«243106»":"# Signed digit sums in bases $b \\ge 5$\n\nThe sequence is defined by\n$$a(n) = \\sum_{k=1}^n (-1)^{\\operatorname{isprime}(k)} 10^k$$\nwhere the sign is $-1$ if $k$ is prime, and $1$ if $k$ is not prime.\n\nGeneralizing the digit pattern observed in $a(n)$, it was conjectured on the OEIS entry that\nfor any base $b \\ge 5$ and any choice of signs $\\pm 1$, the absolute value of $\\sum_{k=1}^n \\pm b^k$\nonly contains digits in $\\{0, 1, b-2, b-1\\}$.\n\n*References:*\n- [A243106](https://oeis.org/A243106)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«24356»":"# Determinant of Hankel matrix of the first $2n-1$ prime numbers\n\nThe determinant of the $n \\times n$ Hankel matrix whose entries are the first $2n-1$ prime numbers.\nThe matrix $M$ has entries $M_{i, j} = p_{i+j}$ for $i, j \\in \\{0, \\dots, n-1\\}$,\nwhere $p_k = \\mathrm{Nat.nth\\;Nat.Prime} (k)$ is the $k$-th prime starting at $p_0=2$.\n$a(0)=1$ by convention.\n\n*References:*\n- [A024356](https://oeis.org/A024356)","FormalConjectures.OEIS.«2454»":"# Central factorial numbers: $((2n)!!)^2$\n\nCentral factorial numbers: $a(n) = 4^n (n!)^2 = ((2n)!!)^2$.\n\n*References:*\n- [A002454](https://oeis.org/A002454)","FormalConjectures.OEIS.«248802»":"# Smallest prime factors of $2^{2^n+2} + 3$\n\nSmallest prime factor of $2^{2^n+2} + 3$.\n\n*References:*\n- [A248802](https://oeis.org/A248802)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«256012»":"# Partitions of $n$ into distinct non-squarefree parts for $n > 23$\n\na: Number of partitions of $n$ into distinct parts that are not squarefree.\nThis is the number of finite subsets of positive integers $P$ such that\n$\\sum_{k \\in P} k = n$ and every element $k \\in P$ is not squarefree.\n\n*References:*\n- [A256012](https://oeis.org/A256012)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«258667»":"# Asymptotics of an inner sum formula for $a(n)$\n\nThe inner sum of the formula used in a:\n$$\\sum_{\\max(k-n+5, 0) \\le j \\le \\min(k,4)} \\binom{8-j}{j}\\binom{2n-k+j-10}{k-j}$$\n\na: A total of $n$ married couples, including a mathematician M and his wife, are to be seated\nat the $2n$ chairs around a circular table, with no man seated next to his wife. After the ladies\nare seated at every other chair, M is the first man allowed to choose one of the remaining chairs.\nThe sequence gives the number of ways of seating the other men, with no man seated next to his\nwife, if M chooses the chair that is 9 seats clockwise from his wife's chair.\n\n$$a(n) = \\begin{cases} 0 & \\text{if } n \\le 5 \\cr\n\\sum_{k=0}^{n-1}(-1)^k(n-k-1)! \\sum_{\\max(k-n+5, 0) \\le j \\le \\min(k,4)}\n\\binom{8-j}{j}\\binom{2n-k+j-10}{k-j} & \\text{if } n > 5 \\end{cases}$$\n\n*References:*\n- [A258667](https://oeis.org/A258667)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«260194»":"# A GCD-Driven Sequence with Universal Jumps\n\nOEIS A260194 begins with three ones and then adds the greatest common divisor of the current term\nand the term two places earlier. The open question asks whether every positive integer occurs as\nan adjacent difference.\n\n*References:*\n- [OEIS A260194](https://oeis.org/A260194)\n","FormalConjectures.OEIS.«267581»":"# Recurrence for Rule 167 cellular automaton sequence\n\nDecimal representation of the middle column of the \"Rule 167\" elementary cellular automaton\nstarting with a single ON (black) cell.\n\n*References:*\n- [A267581](https://oeis.org/A267581)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«271591»":"# Maximal run lengths in the second MSB of Tribonacci numbers $T(n)$\n\nThe Tribonacci numbers $T_n$ (A000073).\n$T_0=0, T_1=0, T_2=1$, and $T_n = T_{n-1} + T_{n-2} + T_{n-3}$ for $n \\ge 3$.\n\nSecond most significant bit of the tribonacci number A000073(n).\nThis is formalized by extracting the bit at position $\\lfloor \\log_2 T_n \\rfloor - 1$.\n\n*References:*\n- [A271591](https://oeis.org/A271591)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n- [A000073](https://oeis.org/A000073)\n","FormalConjectures.OEIS.«278070»":"# Congruences $a(n+k) \\equiv a(n) \\pmod k$ for hypergeometric values ${}_2F_1(n, -n; ; -1)$\n\na: $a(n) = \\text{hypergeometric}([n, -n], [], -1)$.\nThis is equivalent to the combinatorial sum:\n$$a(n) = \\sum_{k=0}^n \\binom{n}{k} \\binom{n+k-1}{k} k!$$\nThe expression uses $\\mathbb{N}$ arithmetic throughout, safely handling\nthe subtraction via `Nat.pred`.\n\n*References:*\n- [A278070](https://oeis.org/A278070)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«280831»":"# The 1680-Conjecture\n\nAny nonnegative integer can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative\nintegers such that $x^4 + 1680 y^3 z$ is a square.\n\nZhi-Wei Sun has offered a prize of 1,680 RMB for the first proof.\n\n*References:*\n- [A280831](https://oeis.org/A280831)\n- Z.-W. Sun, \"Refining Lagrange's four-square theorem,\" *J. Number Theory* **175** (2017), 167-190.\n- Z.-W. Sun, \"Refining Lagrange's four-square theorem,\" arXiv:1604.06723 [math.NT], 2016.\n","FormalConjectures.OEIS.«281976»":"# Sum of four squares with square conditions\n\nAny integer $n \\geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative\nintegers and $z \\leq w$, such that both $x$ and $x + 24y$ are squares.\n\nZhi-Wei Sun has offered a \\$2,400 prize for the first proof.\n\n*References:*\n- [A281976](https://oeis.org/A281976)\n- Z.-W. Sun, \"Refining Lagrange's four-square theorem,\" *J. Number Theory* **175** (2017), 167-190.\n  https://doi.org/10.1016/j.jnt.2016.11.008\n- Z.-W. Sun, \"Restricted sums of four squares,\" *arXiv:1701.05868* [math.NT], 2017.\n  https://arxiv.org/abs/1701.05868\n","FormalConjectures.OEIS.«282779»":"# Period of $p$-th powers modulo $n$\n\na: Period of cubes mod $n$.\nThe $n$-th term $a(n)$ is the smallest positive integer $T$ such that\n$\\forall k \\in \\mathbb{N}$, $(k+T)^3 \\equiv k^3 \\pmod n$.\n\nThe length of the minimal positive period of the sequence $k^p \\pmod n$.\n$a_p(n) = \\min \\{ T \\in \\mathbb{N}^+ \\mid \\forall k \\in \\mathbb{N}, (k+T)^p \\equiv k^p \\pmod n \\}$.\n\n*References:*\n- [A282779](https://oeis.org/A282779)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«287616»":"# Sum of a triangular number, a generalized pentagonal number, and a generalized heptagonal number\n\nAny nonnegative integer can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ with $x, y, z$\nnonnegative integers.\n\nZhi-Wei Sun has offered a USD 135 prize for the first proof of this conjecture.\n\n*References:*\n- [A287616](https://oeis.org/A287616)\n- Zhi-Wei Sun, \"Universal sums of three quadratic polynomials\", arXiv:1502.03056 [math.NT]\n","FormalConjectures.OEIS.«28859»":"# Combinatorial representation of the recurrence $a(n+2) = 2a(n+1) + 2a(n)$\n\nA028859 (OEIS): $a(n+2) = 2 \\cdot a(n+1) + 2 \\cdot a(n)$; $a(0) = 1$, $a(1) = 3$.\n\n*References:*\n- [A028859](https://oeis.org/A028859)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«289411»":"# Symmetry of digit sum differences $\\operatorname{sign}(S_5(k) - S_1(k))$\n\na: $\\mathrm{a}(n) = \\sum_{k=0}^n \\mathrm{sign}(\\mathrm{A007953}(5k) - \\mathrm{A007953}(k))$.\n$\\mathrm{A007953}(n)$ is the digital sum of $n$ in base 10.\nThe sequence is non-negative, so the sum over $\\mathbb{Z}$ is converted to $\\mathbb{N}$.\n\n*References:*\n- [A289411](https://oeis.org/A289411)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n- [A007953](https://oeis.org/A007953)\n","FormalConjectures.OEIS.«2897»":"# Coefficient of $(xyz)^n$ in $((x+y)(y+z)(z+x))^n$ equaling $\\binom{2n}{n}^3$\n\nThe sequence $a(n)$ is defined by $a(n) = \\binom{2n}{n}^3$.\n\n*References:*\n- [A002897](https://oeis.org/A002897)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«300997»":"# Step difference bound $a(n+1) - a(n) \\in \\{1, 2\\}$ in mass-redistribution cellular automata\n\n$a(n)$ is the number of steps needed to reach a stable configuration in the 1D cellular\nautomaton initialized with one cell with mass $n$ and based on the rule \"each cell gives half of\nits mass, rounded down, to its right neighbor\".\nThe stable configuration is $n$ cells with mass 1.\n\n*References:*\n- [A300997](https://oeis.org/A300997)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«303639»":"# Sum of two squares and two central-binomial-type terms\n\nA303639 counts the ways to write $n$ as $a^2 + b^2 + \\binom{2c+1}{c} + \\binom{2d+1}{d}$ with\n$a, b, c, d$ nonnegative integers, $a \\le b$ and $c \\le d$.\n\n*References:*\n- [A303639](https://oeis.org/A303639)\n- Z.-W. Sun, [\"Restricted sums of four squares\"](https://arxiv.org/abs/1701.05868),\n  arXiv:1701.05868 [math.NT], 2017-2018.\n- Z.-W. Sun, [\"Refining Lagrange's four-square theorem\"](https://doi.org/10.1016/j.jnt.2016.11.008),\n  *J. Number Theory* **175** (2017), 167-190.\n- S. Jeong, [A303639 Counterexample Project](https://github.com/DCLXAI/a303639-counterexample),\n  search, certificates and independent verifiers,\n  [DOI 10.5281/zenodo.21863025](https://doi.org/10.5281/zenodo.21863025). The counterexample is\n  recorded as an approved comment on the OEIS entry (Aug 10 2026).\n","FormalConjectures.OEIS.«303656»":"# Sum of two squares, a power of 3, and a power of 5\n\nAny integer $n > 1$ can be written as $a^2 + b^2 + 3^c + 5^d$ where $a, b, c, d$ are\nnonnegative integers.\n\nZhi-Wei Sun has offered a \\$3,500 prize for the first proof.\n\n*References:*\n- [A303656](https://oeis.org/A303656)\n- Z.-W. Sun, \"Restricted sums of four squares,\" arXiv preprint:\n  https://arxiv.org/abs/1701.05868v10\n- Z.-W. Sun, \"Refining Lagrange's four-square theorem,\" Journal of Number Theory:\n  http://maths.nju.edu.cn/~zwsun/RefineFourSquareTh.pdf\n- Z.-W. Sun, \"Restricted sums of three or four squares\":\n  http://maths.nju.edu.cn/~zwsun/Square-sum.pdf\n- Zhi-Wei Sun's 1-3-5 conjecture and variations:\n  https://www.aimspress.com/aimspress-data/era/2020/2/PDF/1935-9179_2020_2_589.pdf\n","FormalConjectures.OEIS.«306424»":"# Maximality of $k = 43$ with restricted digit counts in bases $3 \\le b < k$\n\nNumbers $k$ such that the base $b$ expansion of $k$ for each\n$b = 3..k-1$ never contains more than two distinct digits.\n\nThe sequence Numbers $k$ such that the base $b$ expansion of $k$ for each\n$b = 3..k-1$ never contains more than two distinct digits.\n\n*References:*\n- [A306424](https://oeis.org/A306424)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«306477»":"# The 2-4-6-8 Conjecture\n\nAny integer $n > 0$ can be written as $\\binom{w+2}{2} + \\binom{x+3}{4} + \\binom{y+5}{6} + \\binom{z+7}{8}$\nwith $w, x, y, z$ nonnegative integers.\n\nZhi-Wei Sun has offered a $2,468 prize for the first proof (or $2,468 RMB for a counterexample).\n\nThe conjecture has been verified for all $n$ up to $1.2 \\times 10^{12}$ by Yaakov Baruch (March 2019).\n\n*References:*\n- [A306477](https://oeis.org/A306477)\n- [mathoverflow/323541](https://mathoverflow.net/questions/323541): Z.-W. Sun, \"Positive integers written as C(w,2) + C(x,4) + C(y,6) + C(z,8) with w,x,y,z in {2,3,...},\", Feb. 19, 2019.\n","FormalConjectures.OEIS.«307865»":"# Vanishing of bases with $b^n \\equiv -1 \\pmod{2n+1}$ for absolute Euler pseudoprimes\n\n$a(n)$ is the number of natural bases $b < 2n+1$ such that $b^n \\equiv -1 \\pmod{2n+1}$.\n\nIf $2n+1$ is an absolute Euler pseudoprime, then $a(n) = 0$.\n\n*References:*\n- [A307865](https://oeis.org/A307865)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«308734»":"# Four-square conjecture with powers of 2, 3, and 5\n\nAny integer $n > 1$ can be written as $(2^a \\cdot 3^b)^2 + (2^c \\cdot 5^d)^2 + x^2 + y^2$\nwhere $a, b, c, d, x, y$ are nonnegative integers.\n\nZhi-Wei Sun has offered a \\$2,500 prize for the first proof.\n\n*References:*\n- [A308734](https://oeis.org/A308734)\n- Z.-W. Sun, \"Refining Lagrange's four-square theorem,\" *J. Number Theory* **175** (2017), 167-190.\n  https://doi.org/10.1016/j.jnt.2016.11.008\n- Z.-W. Sun, \"Restricted sums of four squares,\" *Int. J. Number Theory* **15** (2019), 1863-1893.\n- Z.-W. Sun, \"Various Refinements of Lagrange's Four-Square Theorem,\" Westlake Number Theory\n  Symposium, Nanjing University, China, 2020.\n- S. Banerjee, \"On a conjecture of Sun about sums of restricted squares,\" *J. Number Theory*\n  **256** (2024), 253-289.\n","FormalConjectures.OEIS.«309132»":"# Characterization of Carmichael numbers via squarefree denominators $a(n)$\n\n$a(n)$ is the denominator of $F(n) = \\operatorname{num}(B_{n-1})/n + \\operatorname{den}(B_{n-1})/n^2$.\n\nA composite number $n$ has squarefree $a(n)$ if and only if $n$ is a Carmichael number.\n\n*References:*\n- [A309132](https://oeis.org/A309132)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n- [A027641](https://oeis.org/A027641)\n- [A027642](https://oeis.org/A027642)\n","FormalConjectures.OEIS.«3161»":"# A binomial coefficient sum\n\nA binomial coefficient sum:\n$$a(n) = \\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\left( \\binom{n}{k} - \\binom{n}{k-1} \\right)^3$$\nwhere $\\binom{n}{-1} = 0$.\n\n*References:*\n- [A003161](https://oeis.org/A003161)","FormalConjectures.OEIS.«3162»":"# A binomial coefficient summation\n\nA binomial coefficient summation: $a(n) = S(3, n) / S(1, n)$, where for a positive integer $r$\nwe define\n$$S(r,n) = \\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\left( \\binom{n}{k} - \\binom{n}{k-1} \\right)^r$$\nwith $\\binom{n}{-1} = 0$.\n\n*References:*\n- [A003162](https://oeis.org/A003162)\n- H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171\n","FormalConjectures.OEIS.«317940»":"# Nonnegativity of the Dirichlet square root of A046644\n\nA317940 is the integer sequence whose value `a n` is the numerator of the\nrational sequence `f n`, where the Dirichlet convolution square of `f` is\nA046644. The auxiliary sequence A046644 is multiplicative and takes the value\n$2^{\\operatorname{A005187}(e)}$ on a prime power $p^e$.\n\nThe conjecture asks whether `f n` is nonnegative for every positive `n`.\nThe proof reduces the problem to prime powers and constructs positive rational\ncoefficients $c(e)$ satisfying\n\n$$\\sum_{i=0}^e c(i)c(e-i)=2^{\\operatorname{A005187}(e)}.$$\n\nThe coefficients are obtained from formal power series. A positive series\n$A$ is defined by the first-order differential equation\n$A' = \\frac12 D A$. A second explicitly defined series $B$ satisfies\n$B' = D B$ and has the same constant term as $A^2$; uniqueness of the\ncoefficient recurrence therefore gives $A^2=B$. After rescaling the\ncoefficients by $4^e$, this becomes the displayed prime-power convolution\nidentity.\n\nExtending $c(e)$ multiplicatively over prime factorizations gives a positive\narithmetic function `root` with `root * root = a046644`. Finally, the recursive\ndefinition of `f` is shown to be the unique Dirichlet square root with value\none at $1$, so `f = root` and every positive-index value of `f` is positive.\n\nThis route was found by first exploiting the multiplicativity recorded in the\nOEIS entry, reducing the recurrence to the exponent of a single prime, and\nthen recognizing the resulting coefficient identities as a formal-power-series\ndifferential equation.\n\n*References:*\n- [A317940](https://oeis.org/A317940)\n- [A046644](https://oeis.org/A046644)\n- [A005187](https://oeis.org/A005187)\n","FormalConjectures.OEIS.«323557»":"# Pronic indices for odd coefficients of $\\sum_{n \\ge 0} x^n \\frac{(1+x^n)^n}{(1+x^{n+1})^{n+1}}$\n\nCoefficients of G.f. $\\sum_{n\\ge 0} x^n \\cdot \\frac{(1 + x^n)^n}{(1 + x^{n+1})^{n+1}}$.\nThe $m$-th term $a(m)$ is the coefficient of $x^m$, which is explicitly given by the sum:\n$$ a(m) = \\sum_{n=0}^m \\sum_{k=0}^n \\binom{n}{k} (-1)^j \\binom{n+j}{j},$$\nwhere $j = \\frac{m - n(k+1)}{n+1}$, and the term is zero unless $j$ is a natural number.\n\n*References:*\n- [A323557](https://oeis.org/A323557)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«325046»":"# Pronic indices for odd coefficients of $\\sum_{n \\ge 0} x^n \\frac{(1+x^n)^n}{(1-x^{n+1})^{n+1}}$\n\nCoefficients of G.f. $\\sum_{n \\ge 0} x^n \\cdot \\frac{(1 + x^n)^n}{(1 - x^{n+1})^{n+1}}$.\n\nThe term $a(N)$ is the coefficient of $x^N$ in the generating function.\nExpanding the terms, we get a formula for $a(N)$:\n$$a(N) = \\sum_{n=0}^N \\sum_{k=0}^n \\mathbf{1}_{n + nk + (n+1)j = N} \\binom{n}{k} \\binom{n+j}{j}$$\nwhere $j = \\frac{N - n(k+1)}{n+1}$.\n\n*References:*\n- [A325046](https://oeis.org/A325046)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«340737»":"# Convergence of fraction numerators $a(n)/b(n)$ to $e$\n\na: Numerators of a sequence of fractions converging to $e$.\n$$a(1) = 3, a(2) = 5$$\nFor $n > 2$:\n$$a(n) = \\begin{cases} \\left(\\frac{n+2}{2}\\right) a(n-1) - a(n-2) -\n\\left(\\frac{n-2}{2}\\right) a(n-3)\n& \\text{if } n \\text{ is even} \\cr 2 a(n-1) + n a(n-2) & \\text{if } n \\text{ is odd} \\end{cases}$$\n\n*References:*\n- [A340737](https://oeis.org/A340737)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n- [b](https://oeis.org/b)\n","FormalConjectures.OEIS.«341254»":"# Bounds on fractional parts $n r^2 - a(n)$ for $r = (2+\\sqrt{5})/2$\n\n$a(n) = \\lfloor r \\cdot \\lfloor r \\cdot n \\rfloor \\rfloor$, where $r = (2 + \\sqrt{5})/2$.\n\n*References:*\n- [A341254](https://oeis.org/A341254)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«34693»":"# Smallest number $k$ such that $kn + 1$ is prime\n\n*References:*\n- [A34693](https://oeis.org/A34693)\n","FormalConjectures.OEIS.«34694»":"# Smallest prime $\\equiv 1 \\pmod n$\n\n$$a(n) = \\min \\{p \\in \\mathbb{P} \\mid p \\equiv 1 \\pmod n\\}$$\n\n*References:*\n- [A034694](https://oeis.org/A034694)","FormalConjectures.OEIS.«357513»":"# Numerator of a sum involving binomial coefficients\n\n$a(n)$ is the numerator of\n$\\sum_{k = 1}^n \\frac{1}{k^3} \\binom{n}{k}^2 \\binom{n+k}{k}^2$ for $n \\ge 1$\nwith $a(0) = 0$.\n\n*References:*\n- [A357513](https://oeis.org/A357513)\n","FormalConjectures.OEIS.«358684»":"# Factor bounds for Fermat numbers\n\n$a(n)$ is the minimum integer $k$ such that the smallest prime factor of the\n$n$-th Fermat number exceeds $2^{2^n - k}$.\n\n*References:*\n- [A358684](https://oeis.org/A358684)\n- [SA22](https://doi.org/10.26493/2590-9770.1473.ec5) Lorenzo Sauras-Altuzarra, *Some properties of the factors of Fermat numbers*, Art Discrete Appl. Math. (2022).\n","FormalConjectures.OEIS.«3625»":"# Primes congruent to $\\{3, 5, 6\\} \\pmod 7$\n\nPrimes congruent to $3, 5, \\text{ or } 6 \\pmod 7$.\n\n*References:*\n- [A003625](https://oeis.org/A003625)\n","FormalConjectures.OEIS.«363102»":"# Primality of continued fraction denominators $a(n)$ for $n \\ge 3$\n\nAuxiliary sequence A051403, defined as\n$$\\frac{(n+2) \\sum_{k=0}^n k!}{2}$$\n\nDenominator of the continued fraction $1/(2-3/(3-4/(4-5/(...(n-1)-n/(-2)))))$.\nThe sequence is defined by the formula:\n$$a(n) = \\frac{n^2 - 2}{\\gcd(n^2 - 2, 2 \\cdot A051403(n-3) + n \\cdot A051403(n-4))}$$\nThe formula is valid for $n \\ge 3$.\n\n*References:*\n- [A363102](https://oeis.org/A363102)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n- [A051403](https://oeis.org/A051403)\n","FormalConjectures.OEIS.«363347»":"# Realization of primes $p \\equiv \\pm 1 \\pmod{10}$ by continued fraction denominators\n\n$a(n)$ is the denominator of the finite continued fraction\n$$\\frac{1}{2 - \\frac{3}{3 - \\frac{4}{4 - \\frac{5}{\\dots - \\frac{n-1}{(n-1) - \\frac{n}{-4}}}}}}$$\n\n*References:*\n- [A363347](https://oeis.org/A363347)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«368692»":"# Integrality of the factorial ratio $\\frac{(12n+6)! (6n+9)!}{108 (4n+2)! (2n+3)! ((6n+5)!)^2}$\n\nThe factorial ratio\n$$a(n) = \\frac{(12n + 6)! \\cdot (6n + 9)!}{108 \\cdot (4n + 2)! \\cdot\n(2n + 3)! \\cdot ((6n + 5)!)^2}$$\nIt is conjectured that $a(n)$ are integers.\n\n*References:*\n- [A368692](https://oeis.org/A368692)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«37274»":"# Home primes (OEIS A037274)\n\nStarting from an integer $n\\geq 2$, list its prime factors in nondecreasing order with\nmultiplicity, concatenate their decimal representations, and repeat. The home-prime conjecture\nsays that this process always reaches a prime.\n\nFor example,\n\n$$25 \\longmapsto 55 \\longmapsto 511 \\longmapsto 773.$$\n\n*References:*\n* [OEIS A037274](https://oeis.org/A037274)\n* M. Herman and J. Schiffman, *Investigating home primes and their families*,\n  Mathematics Teacher 107 (2014), 606–614\n","FormalConjectures.OEIS.«372761»":"# Unique realization of odd primes $p \\notin \\{3, 5\\}$ by continued fraction values\n\n$a(n)$ is the denominator of the finite continued fraction\n$$\\frac{1}{2 - \\frac{3}{3 - \\frac{4}{4 - \\frac{5}{\\dots - \\frac{n-1}{(n-1) - \\frac{n}{n+4}}}}}}$$\n\n*References:*\n- [A372761](https://oeis.org/A372761)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«38098»":"# Number of primes $< n^3$\n\nNumber of primes strictly less than $n^3$.\n\n*References:*\n- [A038098](https://oeis.org/A038098)","FormalConjectures.OEIS.«38107»":"# Number of primes $< n^2$\n\nNumber of primes strictly less than $n^2$.\n\n*References:*\n- [A038107](https://oeis.org/A038107)","FormalConjectures.OEIS.«382590»":"# Periodicity of $k$-th prime factors in coupled nonlinear recurrence $a(n)$\n\nThe sequence $a(n)$ is defined by $a(n) = a(n-1)b(n-2) + a(n-2)b(n-1)$\nwhere $b(n) = a(n-1)b(n-2) - a(n-2)b(n-1)$, with $a(1)=1, a(2)=2, b(1)=1, b(2)=0$.\n\n*References:*\n- [A382590](https://oeis.org/A382590)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«38552»":"# Conjectures associated with A038552\n\nA038552 lists the largest squarefree number $k$ such that the imaginary quadratic field\n$\\mathbb{Q}(\\sqrt{-k})$ has class number $n$.\n\nThe conjectures state that:\n1. All terms are congruent to $19 \\pmod{24}$.\n2. This is also the largest absolute value of negative fundamental discriminant $d$ for\n   class number $n$.\n3. For even $n$, if $k$ is the largest odd number with $h(-k) = n$ and $k'$ is the largest\n   even number with $h(-k') = n$, then $k > k'$. The $n$-th term is the larger of $k$ and\n   $k'$, so this says that the $n$-th term is odd. Conjecture 1 implies it.\n\nThe squarefree condition in the definition is needed for the maximum to exist, since\n$\\mathbb{Q}(\\sqrt{-k}) = \\mathbb{Q}(\\sqrt{-4k})$.\n\nConjecture 2 is not a restatement of the definition. Both maxima range over the same imaginary\nquadratic fields, but they maximize different integers attached to those fields. A038552 uses\nthe squarefree radicand $k$, whereas the discriminant of $\\mathbb{Q}(\\sqrt{-k})$ is $-k$ for\n$k \\equiv 3 \\pmod 4$ and $-4k$ otherwise. The map $k \\mapsto |d|$ is not monotone: it sends $2$\nto $8$ and $3$ to $3$. So conjecture 2 says that the largest term $k$ satisfies\n$k \\equiv 3 \\pmod 4$, and that $4k' \\le k$ for every $k' \\equiv 1, 2 \\pmod 4$ with class\nnumber $n$.\n\n*References:*\n- [Sta67] Stark, Harold M. \"A complete determination of the complex quadratic fields of\n  class-number one.\" Michigan Mathematical Journal 14.1 (1967): 1-27.\n- [oeis.org/A038552](https://oeis.org/A038552)\n","FormalConjectures.OEIS.«38771»":"# Smallest composite $c$ such that $\\textrm{primorial}(n) + c$ is prime\n\n*References:*\n- [A038771](https://oeis.org/A038771)\n","FormalConjectures.OEIS.«40»":"# The prime numbers\n\nThe $n$-th prime number $p_n$, where $p_1 = 2$.\n\n*References:*\n- [A000040](https://oeis.org/A000040)\n","FormalConjectures.OEIS.«41»":"# No powers as partition numbers\n\nThere are no partition numbers $a(k)$ of the form $x^m$, with $x,m$ integers $>1$.\n\n*Reference:* [A41](https://oeis.org/A41)\n","FormalConjectures.OEIS.«4290»":"# Least positive multiple of $n$ in base 10 with digits 0 and 1\n\nLeast positive multiple of $n$ that when written in base 10 uses only 0's and 1's.\n\n*References:*\n- [A004290](https://oeis.org/A004290)\n","FormalConjectures.OEIS.«46969»":"# Denominators of coefficients in Stirling's expansion for $\\log(\\Gamma(z))$\n\nThe $n$-th term is the denominator of $\\frac{B_{2n}}{2n(2n-1)}$ where $B_{2n}$ is the $2n$-th\nBernoulli number.\n\n*References:*\n- [A046969](https://oeis.org/A046969)","FormalConjectures.OEIS.«48153»":"# $a(n) = \\sum_{k=1}^n (k^2 \\bmod n)$\n\n*References:*\n- [A048153](https://oeis.org/A048153)","FormalConjectures.OEIS.«49473»":"# Nearest integer to $n/\\sqrt{2}$\n\nNearest integer to $n/\\sqrt{2}$, defined by $\\lfloor n/\\sqrt{2} + 1/2 \\rfloor$.\n\n*References:*\n- [A049473](https://oeis.org/A049473)","FormalConjectures.OEIS.«51293»":"# Asymptotics of subsets of $\\{1, 2, \\dots, n\\}$ with integer average\n\nNumber of nonempty subsets of $\\{1, 2, 3, \\dots, n\\}$ whose elements have an integer average.\n\nBenoit Cloitre conjectured the asymptotic expansion:\n$$a(n) = \\frac{2^{n+1}}{n} \\left(1 + \\frac{1}{n} + \\frac{3}{n^2} + \\frac{13}{n^3} + \\frac{75}{n^4} + \\frac{541}{n^5} + o\\left(\\frac{1}{n^5}\\right)\\right)$$\nwhere the coefficients $1, 1, 3, 13, 75, 541, \\dots$ are the Fubini numbers (preferential arrangements, A000670).\n\n*References:*\n- [A051293](https://oeis.org/A051293)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n- [A000670](https://oeis.org/A000670)\n","FormalConjectures.OEIS.«5153»":"# Practical numbers\n\nA positive integer $n$ is called a *practical number* (or *panarithmic number*) if every positive\ninteger $m \\le n$ can be represented as a sum of distinct divisors of $n$.\n\n*References:*\n- [A005153](https://oeis.org/A005153)\n","FormalConjectures.OEIS.«51903»":"# Maximum exponent in the prime factorization of $n$\n\n*References:*\n- [A051903](https://oeis.org/A051903)","FormalConjectures.OEIS.«5258»":"# Apéry numbers\n\nApéry numbers:\n$$a(n) = \\sum_{k=0}^n \\binom{n}{k}^2 \\binom{n+k}{k}$$\n\n*References:*\n- [A005258](https://oeis.org/A005258)\n","FormalConjectures.OEIS.«52709»":"# Expansion of g.f. $(1-\\sqrt{1-4x-4x^2})/(2(1+x))$\n\nThe $n$-th term $a(n)$ is given by\n$$a(n) = \\sum_{k=0}^{n-1} \\frac{1}{k+1} \\binom{2k}{k} \\binom{k}{n-1-k}$$\n\n*References:*\n- [A052709](https://oeis.org/A052709)","FormalConjectures.OEIS.«53000»":"# $a(n) = (\\text{smallest prime} > n^2) - n^2$\n\nThe difference between the smallest prime strictly greater than $n^2$ and $n^2$.\n\n*References:*\n- [A053000](https://oeis.org/A053000)","FormalConjectures.OEIS.«53067»":"# Concatenation of the next $n$ numbers\n\n$a(n)$ is the concatenation of the next $n$ numbers: the integers from\n$\\frac{(n-1)n}{2} + 1$ up to $\\frac{n(n+1)}{2}$.\n\n*References:*\n- [A053067](https://oeis.org/A053067)","FormalConjectures.OEIS.«53175»":"# Catalan-Larcombe-French sequence\n\nThe Catalan-Larcombe-French sequence defined by $a(0)=1$, $a(1)=8$, and\n$$n^2 a(n) = 8(3n^2 - 3n + 1) a(n-1) - 128(n-1)^2 a(n-2)$$ for $n \\ge 2$.\n\n*References:*\n- [A053175](https://oeis.org/A053175)","FormalConjectures.OEIS.«55487»":"# Least $m$ such that $\\phi(m) = n!$\n\nThe smallest positive integer $m$ whose Euler totient equals $n!$.\n\n*References:*\n- [A055487](https://oeis.org/A055487)","FormalConjectures.OEIS.«56777»":"# Divisibility of $2^n + 1$ by $n$\n\nA56777 lists composite numbers $n$ satisfying both $\\varphi(n+12) = \\varphi(n) + 12$ and\n$\\sigma(n+12) = \\sigma(n) + 12$.\n\nThe conjectures state identities connecting A56777 and prime quadruples (A7530), as\nwell as congruences satisfied by the members of A56777.\n\n*References:*\n- [A56777](https://oeis.org/A56777)\n","FormalConjectures.OEIS.«60841»":"# Numerator of $1/\\det(M)$ for $M[i,j] = 1/\\operatorname{lcm}(i,j)$\n\nNumerator of $1/\\det(M)$ where $M$ is the $n \\times n$ matrix with\n$M[i,j] = 1/\\operatorname{lcm}(i,j)$.\n\n*References:*\n- [A060841](https://oeis.org/A060841)","FormalConjectures.OEIS.«60957»":"# Number of different products of subsets of $\\{1, 2, \\dots, n\\}$\n\nThe number of distinct products (including the empty product 1) of any subset\nof $\\{1, 2, \\dots, n\\}$.\n\n*References:*\n- [A060957](https://oeis.org/A060957)","FormalConjectures.OEIS.«62567»":"# Reversible multiples $a(3^n) = 10^{3^{n-2}} - 1$ for powers of $3$\n\nFirst multiple of $n$ whose reverse is also divisible by $n$,\nor 0 if no such multiple exists.\n\n*References:*\n- [A062567](https://oeis.org/A062567)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«63880»":"# Conjectures associated with A063880\n\nA063880 lists numbers $n$ such that $\\sigma(n) = 2 \\cdot \\text{usigma}(n)$, where $\\sigma(n)$ is the\nsum of all divisors and $\\text{usigma}(n)$ is the sum of unitary divisors.\n\nEquivalently, these are numbers whose unitary and non-unitary divisors have equal sum.\n\nThe conjectures state that all members satisfy $n \\equiv 108 \\pmod{216}$, and that all\nprimitive terms (those whose proper divisors aren't in the sequence) are powerful numbers,\nwith $108$ being the only primitive term.\n\n*References:*\n- [A063880](https://oeis.org/A063880)\n","FormalConjectures.OEIS.«64169»":"# Numerator - denominator in $n$-th harmonic number\n\n$a(n) = \\text{numerator}(H_n) - \\text{denominator}(H_n)$, where\n$H_n = 1 + 1/2 + \\dots + 1/n$.\n\n*References:*\n- [A064169](https://oeis.org/A064169)","FormalConjectures.OEIS.«64313»":"# Integer part of area of a regular polygon with $n$ sides each of length 1\n\nThe area of a regular $n$-gon with side length 1 is given by\n$\\frac{n}{4} \\cot(\\pi / n) = \\frac{n}{4 \\tan(\\pi / n)}$.\n\n*References:*\n- [A064313](https://oeis.org/A064313)\n","FormalConjectures.OEIS.«6697»":"# Subword complexity of the morphism a → aab, b → b\n\nLet $a_n$ be the number of distinct subwords (contiguous factors) of length $n$ in the infinite\nword generated by the morphism $\\sigma: a \\mapsto aab, b \\mapsto b$, starting from $a$.\nAs shown in the references, $a_n$ is given by the formula\n$$ a_n = \\sum_{i=0}^{n} \\min(2^i,n-i+1). $$\n\nThe conjectured generating function is:\n$$\\sum_{n \\geq 0} a_n x^n = \\frac{1}{1-x} + \\frac{x}{(1-x)^2}\\left(\\frac{1}{1-x} -\n  \\sum_{k \\geq 1} x^{2^k + k - 1}\\right).$$\n\n*References:*\n- [A6697](https://oeis.org/A6697)\n- J.-P. Allouche and J. Shallit, \"On the subword complexity of the fixed point of a → aab, b → b,\n  and generalizations,\" arXiv:1605.02361 [math.CO], 2016.\n- N. J. A. Sloane and Simon Plouffe, *The Encyclopedia of Integer Sequences*, Academic Press, 1995.\n","FormalConjectures.OEIS.«67599»":"# Decimal encoding of the prime factorization of $n$\n\nIf $n$ has prime factorization $p_1^{e_1} \\dots p_r^{e_r}$ with $p_1 < \\dots < p_r$,\nthen its decimal encoding is $p_1 e_1 \\dots p_r e_r$.\n\n*References:*\n- [A067599](https://oeis.org/A067599)\n","FormalConjectures.OEIS.«67720»":"# Conjectures associated with A067720\n\nA067720 lists numbers $k$ such that $\\varphi(k^2 + 1) = k \\cdot \\varphi(k + 1)$,\nwhere $\\varphi$ is Euler's totient function.\n\nThe sequence exhibits a strong connection to primes: for almost all terms $k$,\n$k + 1$ is prime. The conjecture states that $k = 8$ is the only exception.\n\n*References:*\n- [A067720](https://oeis.org/A067720)\n","FormalConjectures.OEIS.«67857»":"# Sum of $a(k)/k!$ over divisors equals harmonic number\n\nThe sequence $a(n)$ satisfies $\\sum_{k \\mid n} \\frac{a(k)}{k!} = \\sum_{j=1}^n \\frac{1}{j} = H_n$,\nwhere the sum on the left is over positive divisors $k$ of $n$. By Möbius inversion,\n$$a(n) = n! \\sum_{d \\mid n} \\mu(n/d) H_d$$\nwhere $H_d = \\sum_{j=1}^d \\frac{1}{j}$ is the $d$-th harmonic number.\n\n*References:*\n- [A067857](https://oeis.org/A067857)","FormalConjectures.OEIS.«69004»":"# Number of times $n^2 + s^2$ is prime for positive integers $s < n$\n\nThe sequence $a(n)$ counts the number of integers $s \\in \\{1, \\dots, n-1\\}$ such that\n$n^2 + s^2$ is prime:\n$$a(n) = \\sum_{s=1}^{n-1} [\\text{Prime}(n^2 + s^2)]$$\n\n*References:*\n- [A069004](https://oeis.org/A069004)","FormalConjectures.OEIS.«69922»":"# Number of primes $p$ such that $n^n \\le p \\le n^n + n^2$\n\nThe sequence $a(n)$ counts the number of prime numbers in the interval $[n^n, n^n + n^2]$:\n$$a(n) = |\\{p \\text{ prime} \\mid n^n \\le p \\le n^n + n^2\\}|$$\n\n*References:*\n- [A069922](https://oeis.org/A069922)","FormalConjectures.OEIS.«69923»":"# Number of primes $p$ such that $2^n \\le p \\le 2^n + \\mathrm{prime}(n)$\n\nThe sequence $a(n)$ counts the number of primes $p$ in the interval $[2^n, 2^n + p_n]$,\nwhere $p_n$ is the $n$-th prime ($p_1 = 2, p_2 = 3, \\dots$):\n$$a(n) = |\\{p \\text{ prime} \\mid 2^n \\le p \\le 2^n + p_n\\}|$$\nfor $n \\ge 1$, and $a(0) = 0$.\n\n*References:*\n- [A069923](https://oeis.org/A069923)","FormalConjectures.OEIS.«7013»":"# Catalan-Mersenne numbers\n\nCatalan-Mersenne numbers: $a(0) = 2$; for $n \\ge 0$, $a(n+1) = 2^{a(n)} - 1$.\n\n*References:*\n- [A007013](https://oeis.org/A007013)\n","FormalConjectures.OEIS.«70518»":"# Value of $n$-th cyclotomic polynomial at $n$\n\nThe sequence $a(n) = \\Phi_n(n)$ gives the value of the $n$-th cyclotomic polynomial evaluated\nat $n$:\n$$a(n) = |\\Phi_n(n)|$$\n\n*References:*\n- [A070518](https://oeis.org/A070518)","FormalConjectures.OEIS.«70823»":"# Recurrence $a(n+2) = |a(n+1)a(n) - a(n)a(n+1)|$ via concatenation\n\nThe sequence starts with $a(1) = 0, a(2) = 1$. For $n \\ge 1$,\n$$a(n+2) = |\\text{concat}(a(n+1), a(n)) - \\text{concat}(a(n), a(n+1))|$$\nwhere concatenation is in decimal representation.\n\n*References:*\n- [A070823](https://oeis.org/A070823)\n","FormalConjectures.OEIS.«71524»":"# Determinant of matrix with entries indicating primality of $i^2 + j^2$\n\nThe sequence $a(n)$ is the determinant of the $n \\times n$ matrix $M$ defined by\n$M(i,j) = 1$ if $i^2 + j^2$ is prime, and $0$ otherwise, where $1 \\le i, j \\le n$.\n\n*References:*\n- [A071524](https://oeis.org/A071524)","FormalConjectures.OEIS.«71532»":"# Alternating sum of signs of powers of $3/2$\n\nThe sequence $a(n) = -\\sum_{k=1}^n (-1)^{\\lfloor (3/2)^k \\rfloor}$.\n\n*References:*\n- [A071532](https://oeis.org/A071532)","FormalConjectures.OEIS.«72200»":"# Smallest factorial containing exactly $n$ 6's\n\nThe sequence $a(n)$ gives the smallest $k$ such that the decimal expansion of $k!$\ncontains exactly $n$ occurrences of the digit '6', or $0$ if no such $k$ exists.\n\n*References:*\n- [A072200](https://oeis.org/A072200)","FormalConjectures.OEIS.«7406»":"# Wolstenholme numbers\n\nWolstenholme numbers: numerator of $\\sum_{k=1}^n \\frac{1}{k^2}$.\n\n*References:*\n- [A007406](https://oeis.org/A007406)\n","FormalConjectures.OEIS.«7468»":"# Sum of next $n$ primes\n\nThe sum of the primes in the $n$-th row of the prime number triangle:\n$$a(n) = \\sum_{i = 1 + n(n-1)/2}^{n + n(n-1)/2} p_i$$\nwith $a(0) = 0$.\n\n*References:*\n- [A007468](https://oeis.org/A007468)\n","FormalConjectures.OEIS.«76141»":"# Number of times $n$ occurs as a binary sub-pattern of $n^2$\n\nThe sequence $a(n)$ is the number of times the binary expansion of $n$ appears as a contiguous\nsublist (infix) in the binary expansion of $n^2$.\n\n*References:*\n- [A076141](https://oeis.org/A076141)","FormalConjectures.OEIS.«76495»":"# Smallest $x$ such that $\\sigma(x) \\bmod x = n$\n\nThe sequence $a(n)$ is the smallest positive integer $x$ such that $\\sigma_1(x) \\bmod x = n$,\nor $0$ if no such $x$ exists.\n\n*References:*\n- [A076495](https://oeis.org/A076495)\n","FormalConjectures.OEIS.«77408»":"# Trajectory of 103 under the Reverse and Add! operation in base 3\n\nThe sequence $a(n)$ is the trajectory of $103$ under the Reverse and Add! operation carried out\nin base $3$, written in base $10$.\n$a(0) = 103$, and $a(n+1) = a(n) + \\text{rev}_3(a(n))$.\n\n*References:*\n- [A077408](https://oeis.org/A077408)","FormalConjectures.OEIS.«78590»":"# $a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)$\n\nThe sequence is defined by $a(1) = 1, a(2) = 1$, and for $n \\ge 3$,\n$$a(n) = \\frac{2^{a(n-1)} + 1}{a(n-2)}$$\n\n*References:*\n- [A078590](https://oeis.org/A078590)","FormalConjectures.OEIS.«78680»":"# Smallest $m > 0$ such that $n \\cdot 2^m + 1$ is prime\n\nThe sequence $a(n)$ is the smallest positive integer $m$ such that $n \\cdot 2^m + 1$ is prime,\nor $0$ if no such $m$ exists.\n\n*References:*\n- [A078680](https://oeis.org/A078680)","FormalConjectures.OEIS.«78729»":"# Least $k > 0$ such that $(k+1)(k+2)\\cdots(k+n) + 1$ is prime\n\nThe sequence $a(n)$ is the least positive integer $k$ such that $(k+1)(k+2)\\cdots(k+n) + 1$ is\nprime,\nif such $k$ exists; otherwise $a(n) = 0$.\n\n*References:*\n- [A078729](https://oeis.org/A078729)\n","FormalConjectures.OEIS.«7918»":"# Least prime $\\ge n$\n\nLeast prime $\\ge n$ (version 1 of the \"next prime\" function).\n\n*References:*\n- [A007918](https://oeis.org/A007918)\n","FormalConjectures.OEIS.«79727»":"# $a(n) = \\sum_{k=0}^n \\binom{2k}{k}^3$\n\nThe sequence $a(n) = 1 + \\binom{2}{1}^3 + \\binom{4}{2}^3 + \\cdots + \\binom{2n}{n}^3$:\n$$a(n) = \\sum_{k=0}^n \\binom{2k}{k}^3$$\n\n*References:*\n- [A079727](https://oeis.org/A079727)","FormalConjectures.OEIS.«80101»":"# Number of prime powers strictly between $n$-th prime and $(n+1)$-th prime\n\nThe sequence $a(n)$ is the number of prime powers $k$ strictly between the $n$-th prime $p_n$\nand the $(n+1)$-th prime $p_{n+1}$: $p_n < k < p_{n+1}$.\n\n*References:*\n- [A080101](https://oeis.org/A080101)","FormalConjectures.OEIS.«80170»":"# Conjecture relating two characterizations of a set of integers.\n\nInformal Statement:\nFor an integer $k \\ge 2$, the following are equivalent:\n\n1. The greatest common divisor of the binomial coefficients\n    $\\binom{2k}{k}, \\binom{3k}{k}, \\dots, \\binom{(k+1)k}{k} = 1$.\n\n2. Writing prime factorization of $k$ as\n    $k = \\prod p_i^{e_i}$, and let\n    $P = \\max_i p_i^{e_i}$,\n    one has $k / P > P$.\n\nThis conjecture asserts that the sequence defined by 1. is obtained by\ntaking 1 off each number in the sequence defined by 2.\n\n*References:*\n- [A80170](https://oeis.org/A80170)\n- [A51283](https://oeis.org/A51283)\n","FormalConjectures.OEIS.«80326»":"# Denominator of $\\sum_{k=1}^n k^{\\mu(k)}$\n\nThe sequence $a(n)$ is the denominator of $\\sum_{k=1}^n k^{\\mu(k)}$, where $\\mu$ is the\nMöbius function.\n\n*References:*\n- [A080326](https://oeis.org/A080326)","FormalConjectures.OEIS.«81091»":"# Primes of the form 2^n + 2^i + 1\n\nThere are infinite primes of the form $2^n + 2^i + 1$, with $0 < i < n$.\nSee Wagstaff (2001) where this conjecture is posed.\n\n*References:*\n- [A81091](https://oeis.org/A81091)\n- Samuel S. Wagstaff, Jr., [Prime Numbers with a fixed number of one bits or zero bits in their binary representation](http://projecteuclid.org/euclid.em/999188636), Exp. Math. vol. 10, issue 2 (2001) 267.\n","FormalConjectures.OEIS.«83753»":"# Smallest palindrome with exactly $n$ divisors\n\nThe sequence $a(n)$ is the smallest palindromic number with exactly $n$ divisors, or $0$\nif no such number exists.\n\n*References:*\n- [A083753](https://oeis.org/A083753)","FormalConjectures.OEIS.«84046»":"# Smallest prime $p$ such that $p + n$ is an $n$-th power\n\nSmallest prime $p$ such that $p + n$ is an $n$-th power, or $0$ if no such number exists.\nThat is, the smallest prime of the form $k^n - n$.\n\n*References:*\n- [A084046](https://oeis.org/A084046)","FormalConjectures.OEIS.«86766»":"# Smallest $r$ such that (concatenation of $n$, $r$ times) $\\cdot 10 + 1$ is prime\n\n$a(n)$ is the smallest $r$ where (concatenation of $n$, $r$ times with itself) $\\cdot 10 + 1$\nis a prime, or $0$ if no such number exists.\nThe number resulting from concatenating $n$, $r$ times, is\n$n \\cdot \\sum_{i=0}^{r-1} (10^d)^i$, where $d$ is the number of digits of $n$.\n\n*References:*\n- [A086766](https://oeis.org/A086766)","FormalConjectures.OEIS.«87207»":"# Binary representation of primes that divide a number, in decimal\n\nThe value $a(n)$ is given by\n$$ a(n) = \\sum_{p \\mid n, p \\text{ prime}} 2^{\\pi(p) - 1} $$\nwhere $\\pi(p) = \\mathrm{primeCounting}(p)$ gives the 1-based index of the prime $p$.\n\n*References:*\n- [A087207](https://oeis.org/A087207)","FormalConjectures.OEIS.«87455»":"# Expansion of $(1 - x)/(1 - 2 x + 3 x^2)$\n\nThis sequence is the expansion of $(1 - x)/(1 - 2 x + 3 x^2)$ in powers of $x$.\nIt satisfies the linear recurrence relation $a(n) = 2 a(n-1) - 3 a(n-2)$ for $n \\ge 2$,\nwith initial values $a(0)=1$ and $a(1)=1$.\n\n*References:*\n- [A087455](https://oeis.org/A087455)","FormalConjectures.OEIS.«87571»":"# Smallest prime formed by concatenation $n, n-1, \\dots, n-k$\n\n$a(n)$ is the smallest prime which has the form of the concatenation\n$n, n-1, n-2, \\dots, n-k$ for some $k < n$, or $0$ if no such prime exists.\n\n*References:*\n- [A087571](https://oeis.org/A087571)","FormalConjectures.OEIS.«87719»":"# Conjectures associated with A087719\n\nDefine $\\varsigma(n)$ the smallest prime factor of $n$ (`Nat.minFac`). Let $a_n$ be the least\nnumber such that the count of numbers $k \\le a_n$ with $k > \\varsigma(k)^n$ exceeds the count\nof numbers with $k \\le \\varsigma(k)^n$.\n\nThe conjecture states that $a_n = 3^n + 3 \\cdot 2^n + 6$ for $n \\ge 1$.\n\n*References:*\n- [A087719](https://oeis.org/A087719)\n","FormalConjectures.OEIS.«89026»":"# $a(n) = n$ if $n$ is prime, otherwise $a(n) = 1$\n\n*References:*\n- [A089026](https://oeis.org/A089026)\n","FormalConjectures.OEIS.«91591»":"# Number of pairs of twin primes between $n^2$ and $(n+1)^2$\n\n$a(n)$ is the number of pairs of twin primes between $n^2$ and $(n+1)^2$.\nThis counts the number of primes $p$ such that $p$ and $p+2$ are both prime,\nand the entire twin prime pair $(p, p+2)$ lies strictly between $n^2$ and $(n+1)^2$.\nThat is, $n^2 < p$ and $p + 2 < (n+1)^2$.\n\n*References:*\n- [A091591](https://oeis.org/A091591)","FormalConjectures.OEIS.«91669»":"# Primality and primitive root property from divisibility $n \\mid (a(n-1) + 2^{n-2})$\n\n$a(n) = \\frac{2^{n-1}}{n!} \\prod_{k=1}^{n-1} (2^k-1)$.\nThe sequence $a(n)$ is composed of natural numbers, thus we define it\nas a function $\\mathbb{N} \\to \\mathbb{N}$.\n\n*References:*\n- [A091669](https://oeis.org/A091669)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.OEIS.«92243»":"# Tug of war score between prime gap increases and decreases\n\nScore at stage $n$ in \"tug of war\" between prime gap increases vs. prime gap decreases:\nstart with score $= 0$ at $n = 1$ and at stage $k > 1$, increase (resp. decrease) the score by $1$\nif the $k$-th prime gap is greater (resp. less) than the previous prime gap.\n\n*References:*\n- [A092243](https://oeis.org/A092243)","FormalConjectures.OEIS.«93456»":"# Product of composite numbers in triangular intervals\n\nProduct of all composite numbers between $n(n-1)/2+1$ and $n(n+1)/2$ (including boundaries),\nwhere $n(n-1)/2 = \\binom{n}{2}$ and $n(n+1)/2 = \\binom{n+1}{2}$.\n\n*References:*\n- [A093456](https://oeis.org/A093456)","FormalConjectures.OEIS.«93818»":"# $\\gcd(\\mathrm{numerator}(H_n), n!)$\n\n$a(n) = \\gcd(\\mathrm{A001008}(n), n!)$, where $\\mathrm{A001008}(n)$ is the numerator of\nthe $n$-th harmonic number $H_n = \\sum_{i=1}^n \\frac{1}{i}$.\n\n*References:*\n- [A093818](https://oeis.org/A093818)","FormalConjectures.OEIS.«945»":"# Euclid-Mullin sequence\n\nThe Euclid-Mullin sequence starts with $a(1) = 2$. Each subsequent term is the smallest prime\nfactor of one plus the product of all preceding terms. We extend the sequence by $a(0) = 1$ and\nwrite $b(n)$ for the product of the first $n$ official terms.\n\n\n*References:*\n- [A000945](https://oeis.org/A000945)\n- [Mullin63] A. A. Mullin,\n  [\"Research Problem 8 (ii)\"](https://doi.org/10.1090/S0002-9904-1963-11017-4),\n  *Bull. Amer. Math. Soc.* **69** (1963), p. 737.\n- [Wagstaff93] S. S. Wagstaff, Jr.,\n  [\"Computing Euclid's primes\"](https://oeis.org/A000945/a000945_4.pdf),\n  *Bull. Institute Combin. Applications* **8** (1993), pp. 23-32.\n- [CrandallPomerance01] R. Crandall and C. Pomerance,\n  *Prime Numbers: A Computational Perspective*, Springer (2001), p. 6.\n- A. R. Booker, \"A variant of the Euclid-Mullin sequence containing every prime,\"\n  [arXiv:1605.08929](https://arxiv.org/abs/1605.08929), *Journal of Integer Sequences* **19**\n  (2016), Article 16.6.4.\n","FormalConjectures.OEIS.«96535»":"# Recurrence $a(n) = (a(n-1) + a(n-2)) \\pmod n$\n\n$a(0) = a(1) = 1$; $a(n) = (a(n-1) + a(n-2)) \\pmod n$.\n\n*References:*\n- [A096535](https://oeis.org/A096535)","FormalConjectures.OpenQuantumProblems.«13»":"# Open Quantum Problem 13: Mutually unbiased bases\n\n## Mathematical problem\n\nFor each integer $d \\ge 2$, determine the maximum number $k$ for which there exist\northonormal bases $\\mathcal{B}_1, \\dots, \\mathcal{B}_k$ of the complex Hilbert space\n$\\mathbb{C}^d$ such that any two distinct bases are mutually unbiased.\n\nConcretely, if\n$\\mathcal{B}_r = \\{ e_0^{(r)}, \\dots, e_{d-1}^{(r)} \\}$\nand\n$\\mathcal{B}_s = \\{ e_0^{(s)}, \\dots, e_{d-1}^{(s)} \\}$,\nthen $\\mathcal{B}_r$ and $\\mathcal{B}_s$ are mutually unbiased if for all $i, j$\nand all $r \\ne s$,\n$|\\langle e_i^{(r)}, e_j^{(s)} \\rangle| = d^{-1/2}$.\n\nThe problem is therefore to determine the maximal value\n$\\mu(d) := \\max \\{ k : \\text{there exist } k \\text{ pairwise mutually unbiased\northonormal bases in } \\mathbb{C}^d \\}$.\n\nIn this file, an orthonormal basis is represented by a unitary matrix whose columns are the\nbasis vectors. For two such bases `U` and `V`, the matrix `relativeUnitary U V`, which is\n$U^\\dagger V$, contains all cross-basis overlaps as its entries. Since Lean works more\nsmoothly with squared norms, we formalize mutual unbiasedness by requiring\n$\\| (relativeUnitary\\ U\\ V)_{ij} \\|^2 = 1 / d$\nfor all $i, j$, which is equivalent to\n$|\\langle e_i^{(r)}, e_j^{(s)} \\rangle| = d^{-1/2}$.\n\n## Background\n\nMutually unbiased bases are a basic structure in finite-dimensional quantum theory.\nThey arise in quantum state determination, quantum tomography, quantum cryptography,\nfinite geometry, and combinatorics.\n\nA general upper bound is $\\mu(d) \\le d + 1$.\nEquality is known when $d$ is a prime power, via constructions over finite fields.\nFor composite dimensions that are not prime powers, the exact value of $\\mu(d)$ is in\ngeneral open.\n\nThe smallest and most famous unresolved case is $d = 6$.\nThe IQOQI OQP page emphasizes this dimension in particular: although many equivalent\nreformulations are known, no construction yielding more than three mutually unbiased bases\nin dimension six is known.\n\n## What this file formalizes\n\nThis file is organized around the quantity `IsMaxMUBCount d k`, which expresses that\n$k$ is the maximum number of mutually unbiased orthonormal bases in dimension $d$.\n\n- the open theorem `mutuallyUnbiasedBases` expresses the full problem for all $d \\ge 2$;\n- the open theorem `mutuallyUnbiasedBases_dim6` expresses the especially important case\n  $d = 6$;\n- the solved theorem `mutuallyUnbiasedBases_dim2` proves the qubit case $\\mu(2) = 3$.\n\n## References\n\n*Primary source list entry:*\n- IQOQI Vienna Open Quantum Problems, problem 13:\n  https://oqp.iqoqi.oeaw.ac.at/mutually-unbiased-bases\n- Master list of open quantum problems:\n  https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems\n\n### Foundational papers\n- I. D. Ivanović,\n  *Geometrical description of quantal state determination*,\n  J. Phys. A 14, 3241-3245 (1981).\n- W. K. Wootters and B. D. Fields,\n  *Optimal state-determination by mutually unbiased measurements*,\n  Ann. Phys. 191, 363-381 (1989).\n\n### General constructions and surveys\n- A. Klappenecker and M. Rötteler,\n  *Constructions of mutually unbiased bases*,\n  in *Finite Fields and Applications*, LNCS 2948 (2004).\n\n### Dimension six and the maximal-number problem\n- M. Grassl,\n  *On SIC-POVMs and MUBs in Dimension 6*,\n  arXiv:quant-ph/0406175 (2004).\n- P. Butterley and W. Hall,\n  *Numerical evidence for the maximum number of mutually unbiased bases in dimension six*,\n  Phys. Lett. A 369, 5-8 (2007),\n  arXiv:quant-ph/0701122.\n- S. Brierley and S. Weigert,\n  *Maximal Sets of Mutually Unbiased Quantum States in Dimension Six*,\n  Phys. Rev. A 78, 042312 (2008),\n  arXiv:0808.1614.\n- P. Raynal, X. Lü, and B.-G. Englert,\n  *Mutually unbiased bases in dimension six: The four most distant bases*,\n  Phys. Rev. A 83, 062303 (2011),\n  arXiv:1103.1025.\n\n## Remark on the status of $d = 6$\n\nThe dimension-six case is not known to be solved. At present, the best-known general picture is:\n- $3 \\le \\mu(6) \\le 7$,\n- complete sets of $7$ MUBs are not known,\n- and several analytic and numerical works give strong evidence that one cannot go beyond $3$.\n\nThis is why the theorem `mutuallyUnbiasedBases_dim6` is marked as an open research statement.\n","FormalConjectures.OpenQuantumProblems.«23»":"# Open Quantum Problem 23: SIC-POVMs\n\n## Mathematical problem\nThe OQP page presents three increasingly strong formulations of this problem.\nIn this file we formalize the first one, closest to the physics terminology:\nexistence of a symmetric informationally complete POVM in every finite dimension.\n\nA SIC-POVM in dimension $d$ can be represented by a family of $d^2$ normalized\nvectors in $\\mathbb{C}^d$ whose pairwise squared overlaps are all equal to\n$(d + 1)^{-1}$. We encode such a family as a map `Fin (d ^ 2) → StateVector d`.\n\n## Background\nSIC-POVMs are a basic structure in finite-dimensional quantum information.\nThey are closely related to equiangular lines, tight frames, quantum state\nreconstruction, and finite-dimensional measurement theory.\nThe open problem asks whether such families exist in every dimension.\n\n## What this file formalizes\nThis file formalizes the existence problem for symmetric informationally complete\nPOVMs through the predicate `HasSICPOVM d`.\n\nMore precisely, it contains the following layers.\n\n### Core API\nThe main definitions formalized in this file are:\n- `StateVector d`: a state vector in `ℂ^d`;\n- `mkStateVector`: constructor from coordinates in the computational basis;\n- `IsNormalized ψ`: normalization predicate for a state vector;\n- `overlapSq φ ψ`: squared magnitude of the inner-product overlap;\n- `HasConstantOverlapSq c Φ`: constant pairwise squared-overlap condition;\n- `sicOverlapSq d`: the SIC overlap value `(d + 1)⁻¹`;\n- `IsSICFamily d Φ`: the predicate that a family of `d^2` vectors in `ℂ^d`\n  is a SIC family;\n- `HasSICPOVM d`: existence of a SIC family in dimension `d`.\n\nIn addition, the file includes explicit witness families and convenient\nconstructors used in the low-dimensional benchmark cases:\n- `vec2`, `vec3`;\n- `qubitSICFamily`;\n- `hesseFamily`;\n- `bb84Family`.\n\n### Complete open conjecture\nThe main open theorem is:\n- `sicPOVMs`, expressing the conjecture that for every `d ≥ 1`, there exists a\n  SIC-POVM in dimension `d`.\n\n### Special cases\nThe file also isolates several special cases:\n- solved low-dimensional benchmark cases:\n  `hasSICPOVM_zero`, `hasSICPOVM_one`, `hasSICPOVM_two`, `hasSICPOVM_three`;\n- a negative benchmark result:\n  `bb84Family_not_isSICFamily`, showing that the BB84 family in dimension `2`\n  does not form a SIC family;\n- selected open benchmark dimensions:\n  `hasSICPOVM_56`, `hasSICPOVM_58`, `hasSICPOVM_59`, `hasSICPOVM_60`,\n  `hasSICPOVM_64`, `hasSICPOVM_68`, `hasSICPOVM_69`, `hasSICPOVM_70`,\n  `hasSICPOVM_71`, `hasSICPOVM_72`, `hasSICPOVM_75`.\n\n### Test lemmas\nThe file includes the following test lemmas and benchmark-support statements:\n- `hasConstantOverlapSq_singleton`;\n- `sicOverlapSq_one`, `sicOverlapSq_two`, `sicOverlapSq_three`,\n  `sicOverlapSq_pos`;\n- `isSICFamily_singleton_iff`, `isSICFamily_one_of_normalized`;\n- `qubitSICFamily_normalized`, `qubitSICFamily_pairwise`;\n- `hesseFamily_normalized`, `hesseFamily_pairwise`;\n- `bb84Family_normalized`.\n\nAt present, these `@[category test, AMS 15 47 81]` results are included with\nplaceholder proofs `by sorry`; they are intended to be proved in the next PR.\n\n## References\n*Primary source list entry:*\n- IQOQI Vienna Open Quantum Problems, problem 23:\n  https://oqp.iqoqi.oeaw.ac.at/sic-povms-and-zauners-conjecture\n- Formal Conjectures issue #1823:\n  https://github.com/google-deepmind/formal-conjectures/issues/1823\n\n### Foundational references\n- J. M. Renes, R. Blume-Kohout, A. J. Scott, and M. C. Caves,\n  *Symmetric informationally complete quantum measurements*,\n  J. Math. Phys. 45, 2171-2180 (2004), arXiv:quant-ph/0310075.\n- G. Zauner,\n  *Quantum Designs: Foundations of a Noncommutative Design Theory*,\n  PhD thesis, University of Vienna (1999).\n","FormalConjectures.OpenQuantumProblems.«35»":"# Open Quantum Problem 35: existence of absolutely maximally entangled pure states\n\n**Problem:** For which numbers of parties $n$ and local dimensions $d$ does there\nexist a pure absolutely maximally entangled state $\\psi$?\n\nA pure state $\\psi$ on $n$ parties of local dimension $d$ is called\n**absolutely maximally entangled (AME)** if, for every subset of at most half\nof the parties, the corresponding reduced density matrix is maximally mixed.\n\n*References:*\n- Open Quantum Problems, Problem 35:\n  <https://oqp.iqoqi.oeaw.ac.at/existence-of-absolutely-maximally-entangled-pure-states>\n- Formal Conjectures issue #3452:\n  <https://github.com/google-deepmind/formal-conjectures/issues/3452>\n- W. Helwig, W. Cui, A. Riera, J. I. Latorre, and H.-K. Lo,\n  *Absolute Maximal Entanglement and Quantum Secret Sharing*,\n  Phys. Rev. A 86, 052335 (2012), arXiv:1204.2289.\n- D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. Życzkowski,\n  *Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices*,\n  Phys. Rev. A 92, 032316 (2015), arXiv:1506.08857.\n- A. Higuchi and A. Sudbery,\n  *How entangled can two couples get?*,\n  Phys. Lett. A 273, 213-217 (2000), arXiv:quant-ph/0005013.\n- A. J. Scott,\n  *Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum\n  evolutions*, Phys. Rev. A 69, 052330 (2004), arXiv:quant-ph/0310137.\n- F. Huber, O. Gühne, and J. Siewert,\n  *Absolutely maximally entangled states of seven qubits do not exist*,\n  Phys. Rev. Lett. 118, 200502 (2017), arXiv:1608.06228.\n- F. Huber and M. Grassl,\n  *Quantum Codes of Maximal Distance and Highly Entangled Subspaces*,\n  Quantum 4, 284 (2020), arXiv:1907.07733.\n- S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć,\n  A. Lakshminarayan, and K. Życzkowski,\n  *Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem*,\n  Phys. Rev. Lett. 128, 080507 (2022), arXiv:2104.05122.\n- G. Rajchel-Mieldzioć, R. Bistroń, A. Rico, A. Lakshminarayan,\n  and K. Życzkowski,\n  *Absolutely maximally entangled pure states of multipartite quantum systems*,\n  arXiv:2508.04777 (2025).\n\nThis file formalizes the problem of determining for which pairs $(n,d)$ there exists an\nabsolutely maximally entangled pure state $\\mathrm{AME}(n,d)$.\n\nWe represent an $n$-partite state of local dimension $d$ by the finite-dimensional Hilbert space\n`EuclideanSpace ℂ (Config n d)`, whose coordinates in the computational basis are amplitudes.\nThe helper `mkStateVector` turns an amplitude function into such a state, and normalization is\nimposed explicitly via `IsNormalized`, i.e. via the ambient $L^2$ norm.\n\nThe main reusable lemma is `reducedDensityFirst_of_completion`: if a state is a\nuniform superposition over the graph of an injective completion function\n`completion : Config m d → Config (n - m) d`,\nthen the reduced state on the first $m$ parties is maximally mixed.\n\nAs demonstration, we show that the Bell states with $n=2$ and GHZ states with $n=3$ are\nAME states, and the GHZ state with $n=4$ is not an AME state.\n","FormalConjectures.OptimizationConstants.«1a»":"# Tao's Optimization constant 1a / An autocorrelation constant related to Sidon sets\n\n*References:*\n- [Tao's optimization constant 1a](https://teorth.github.io/optimizationproblems/constants/1a.html)\n- [M2010] Matolcsi, Máté, and Carlos Vinuesa. \"Improved bounds on the supremum of autoconvolutions.\"\n  Journal of mathematical analysis and applications 372.2 (2010): 439-447. [arXiv:0907.1379](https://arxiv.org/abs/0907.1379)\n- [Y2026] Yuksekgonul, Mert et al., \"Learning to Discover at Test Time,\" 2026, [arXiv:2601.16175](https://arxiv.org/abs/2601.16175)\n","FormalConjectures.Other.BeaverMathOlympiad":"# Beaver Math Olympiad (BMO)\n\nThe Beaver Math Olympiad (BMO) is a set of mathematical reformulations of the halting/nonhalting\nproblem of specific Turing machines from all-0 tape. These problems came from studying small Busy\nBeaver values. Some problems are open and have a conjectured answer, some are open and don't have a\nconjectured answer, and, some are solved.\n\nAmong these problems is the Collatz-like *Antihydra* problem which is open and coming from a 6-state\nTuring machine, and a testament to the difficulty of knowing the sixth Busy Beaver value.\n\nFor some BMO problem, the equivalence between the mathematical formulation and the corresponding\nTuring machine non-termination has been formally proved in Rocq, we indicate it when done.\n\n*References:*\n\n- [bbchallenge.org](https://bbchallenge.org)\n- [Beaver Math Olympiad wiki page](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad)\n- [Antihydra web page](https://bbchallenge.org/antihydra)\n- [Antihydra wiki page](https://wiki.bbchallenge.org/wiki/Antihydra)\n","FormalConjectures.Other.EquationalTheories_677_255":"# Equational Theories\n\n*Reference:* [Equational Theories project site](https://teorth.github.io/equational_theories/implications/?677&finite)\n","FormalConjectures.Other.GerstenhaberThreeMatrices":"# The Gerstenhaber problem for three commuting matrices\n\nGerstenhaber proved in 1961 [Ger61] that the unital algebra generated by two commuting\n$n \\times n$ matrices over a field has dimension at most $n$. It is an open problem,\nknown as the **Gerstenhaber problem**, whether the same bound holds for three pairwise\ncommuting matrices. The bound fails for four or more pairwise commuting matrices.\n\n*References:*\n- [Ger61] M. Gerstenhaber, *On dominance and varieties of commuting matrices*.\n  Annals of Mathematics (2) 73 (1961), no. 2, 324–348.\n  https://doi.org/10.2307/1970336\n- [arXiv:2402.16334](https://arxiv.org/abs/2402.16334), M. Satriano and W. Zhang,\n  *On the algebra generated by three commuting matrices: combinatorial cases*.\n- [arXiv:2006.08588](https://arxiv.org/abs/2006.08588), J. Holbrook and M. Omladič,\n  *A computing strategy and programs to resolve the Gerstenhaber Problem for\n  commuting triples of matrices*.\n","FormalConjectures.Other.Rule30":"# The Rule 30 Prize Problems\n\n**Rule 30** is the elementary cellular automaton with local update $c' = l \\oplus (c \\lor r)$,\nthe Boolean rule numbered $30$ by Wolfram. Started from a single black cell on a bi-infinite row,\nits **center column** `t ↦ (state t) 0` looks random — it was long *Mathematica*'s default\npseudorandom generator — yet nothing about that randomness is proven. In 2019 Wolfram offered\nthe **Rule 30 Prizes** for three questions about it; we formalize the first two.\n\n* **Problem 1.** Is the center column non-periodic (never eventually periodic)?\n* **Problem 2.** Does each color occur on average equally often, i.e. does the running average\n  of the values converge to $1/2$?\n\nProblem 3 — whether computing the $n$-th cell requires at least $O(n)$ work — is omitted. It is\nmodel-relative and has no canonical model-independent phrasing. All three are open.\n\n*References:*\n- [Announcing the Rule 30 Prizes](https://writings.stephenwolfram.com/2019/10/announcing-the-rule-30-prizes/),\n  Stephen Wolfram, 2019.\n- [Rule 30 Prizes](https://rule30prize.org/).\n- [Wikipedia: Rule 30](https://en.wikipedia.org/wiki/Rule_30).\n","FormalConjectures.Other.SchurTruncatedExponential":"# Schur's theorem on Galois groups of truncated exponential polynomials\n\n*Reference:* (https://math.stackexchange.com/questions/2814220)\n\n*Reference* (https://mathoverflow.net/questions/477077)\n","FormalConjectures.Other.SuffixPrefixAvoidance":"# Suffix-prefix avoidance bound\n\nLet $A$ and $B$ be sets of words of length $n$ over an alphabet with $q$ letters. If no\n(nonempty) suffix of any word in $A$ coincides with a prefix of any word in $B$, then\n$$|A| \\cdot |B| \\leq \\frac{q^{2n}}{en}.$$\n\n*References:*\n- [X post by Dmitry Rybin](https://x.com/DmitryRybin1/status/2027278135847428577)\n- [Maximal sets of strings with no prefix-suffix overlap]\n  (https://mathoverflow.net/questions/508648/maximal-sets-of-strings-with-no-prefix-suffix-overlap)\n  by *Dmitry Rybin*, MathOverflow (2026)\n- [An isoperimetric inequality for word overlap](https://arxiv.org/abs/2602.20143)\n  by *Dmitrii Zakharov* (2026)\n","FormalConjectures.Other.VCDimConvex":"# VCₙ dimension of convex sets in ℝⁿ, ℝⁿ⁺¹, ℝⁿ⁺²\n\nIn the literature it is known that every convex set in ℝ² has VC dimension at most 3,\nand there exists a convex set in ℝ³ with infinite VC dimension (even more strongly,\nwhich shatters an infinite set).\n\nThis file states that every convex set in ℝⁿ has finite VCₙ dimension, constructs a convex set in\nℝⁿ⁺² with infinite VCₙ dimension (even more strongly, which n-shatters an infinite set),\nand conjectures that every convex set in ℝⁿ⁺¹ has finite VCₙ dimension.\n","FormalConjectures.Paper.BranchingVAS":"# Decidability of reachability for branching vector addition systems\n\nA *branching vector addition system* (BVAS) of dimension `d` is given by a finite\nlist of *axioms*, a finite list of *unary rules*, and a finite list of *binary\nrules*, each of which is a vector in `ℤ^d`. A *configuration* is a vector in `ℕ^d`,\nwhich here is represented as a vector `v ∈ ℤ^d` subject to `0 ≤ v`. The set of\n*reachable* configurations is defined inductively:\n\n* every axiom that is a valid configuration (i.e. lies in `ℕ^d`) is reachable;\n* if `v` is a reachable configuration, `r` is a unary rule and `v + r ∈ ℕ^d`, then\n  `v + r` is reachable;\n* if `v₁, v₂` are both reachable configurations, `r` is a binary rule and\n  `v₁ + v₂ + r ∈ ℕ^d`, then `v₁ + v₂ + r` is reachable.\n\nModelling axioms as vectors in `ℤ^d` and only counting the non-negative ones as\nreachable yields the same set of reachable configurations as the usual definition\nin which axioms are required to lie in `ℕ^d`.\n\nBranching vector addition systems are distinguished from ordinary vector addition systems (VAS) by allowing binary rules. VAS reachability is known to be decidable.\n\n*References:*\n- [The covering and boundedness problems for branching vector addition systems](https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2317)\n  (*Stéphane Demri, Marcin Jurdziński, Oded Lachish, Ranko Lazić*, FSTTCS 2009)\n  Gives a similar definition of BVAS, and compares it to related equivalent definitions.\n- [On the Reachability Problem for Two-Dimensional Branching VASS](https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.22)\n  by *Clotilde Bizière, Thibault Hilaire, Jérôme Leroux, Grégoire Sutre*, MFCS 2025, which\n  settles the two-dimensional case and states that \"the decidability status of the reachability\n  problem for BVASS remains open in higher dimensions\".\n- [The General Vector Addition System Reachability Problem by Presburger Inductive\n  Invariants](https://arxiv.org/abs/1009.1076) by *Jérôme Leroux* (2010), for the decidability\n  of reachability for ordinary VAS.\n- [Solving the Reachability Problem for Branching Vector Addition Systems via Semilinear\n  Inductive Invariants](https://arxiv.org/abs/2607.09558) by *Clotilde Bizière, Jérôme Leroux,\n  Grégoire Sutre* (2026), a recent preprint claiming a positive resolution to the conjecture:\n  reachability for branching vector addition systems is decidable.\n","FormalConjectures.Paper.CardinalityLindelof":"# Conjecture about cardinality of Lindelöf spaces\n\nThe conjecture asks for a Lindelöf space where all singletons are G_δ sets\nand which has cardinality > 𝔠.\n\nThis is Problem 1 in https://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf\n\n*Reference:*\n* [Selected Old Open Problems in General Topology](https://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf)\n  by A. V. Arhangel’skii\n\n","FormalConjectures.Paper.CasasAlvero":"# Casas-Alvero Conjecture\n\n*References:*\n* [The Casas-Alvero conjecture for infinitely many degrees](https://arxiv.org/pdf/math/0605090)\n* [MathOverflow](https://mathoverflow.net/questions/27851)\n\nThe Casas-Alvero conjecture states that if a univariate polynomial `P` of degree `d` over a field\nof characteristic zero shares a non-trivial factor with its Hasse derivatives up to order `d-1`,\nthen `P` must be of the form `(X - α)ᵈ` for some `α` in the field.\n\nThe conjecture has been proven for:\n* Degrees `d ≤ 8`\n* Degrees of the form `p^k` where `p` is prime\n* Degrees of the form `2p^k` where `p` is prime\n\nThe conjecture is false in positive characteristic `p` for polynomials of degree `p+1`.\n\nThe conjecture is now claimed to be proven in this paper:\n* [Proof of the Casas-Alvero conjecture: Soham Ghosh)](https://arxiv.org/pdf/2501.09272)\n\n","FormalConjectures.Paper.CatchUpConjecture":"# The Catch-Up game and conjecture\n\nThe game **Catch-Up** (Isaksen–Ismail–Brams–Nealen, 2015) is a two-player, perfect-information game\nplayed on a finite nonempty set `S` of positive integers. Each time a player removes a number from\n`S`, that number is added to the player’s score.\n\n**Rules.**\n* The scores start at `0`. Player `p1` starts by removing **exactly one** number from `S`.\n* After the first move, players alternate turns. On a turn, the current player removes **one or more**\n  numbers from `S`, one at a time, and must keep removing numbers until their score becomes\n  **at least** the opponent’s score; before the final pick they must remain **strictly behind**.\n* If the current player cannot catch up (in particular, even taking all remaining numbers would still\n  leave them behind), the game ends immediately: the current player receives all remaining numbers.\n\nWhen `S` is empty, the player with higher score wins; equal scores give a draw.\n\nIn this file we define:\n* `Player` and `Outcome`,\n* the recursive evaluator `value` (optimal play),\n* the conjecture `value_of_even_mul_succ_self_div_two`.\n\n## Example\nFor `S = {1,2,3,4}` one play is: `p1` takes `2`, `p2` takes `1` then `4`, and `p1` takes `3`,\nending with scores `(5,5)`.\n\n## References\nA. Isaksen, M. Ismail, S. J. Brams, A. Nealen,\n*Catch-Up: A Game in Which the Lead Alternates,* Game & Puzzle Design 1(2), 38–49 (2015).\n\n","FormalConjectures.Paper.Chvatal":"# Chvátal's Conjecture\n\n*References:*\n\n* [A Conjecture in Extremal Combinatorics](https://users.encs.concordia.ca/~chvatal/conjecture.html)\n* [Chvátal's Conjecture and Correlation Inequalities](https://arxiv.org/abs/1608.08954)\n","FormalConjectures.Paper.ClaudesCycles":"# Claude's Cycles\n\n*Reference:* [Claude's Cycles](https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cycles.pdf)\nby *Donald E. Knuth* (2026)\n\nFix `m ≥ 2`. Consider the directed graph with vertex set `(ZMod m)³`, where from each vertex\n`(i, j, k)` there are directed arcs to `(i+1, j, k)`, `(i, j+1, k)`, and `(i, j, k+1)`\n(arithmetic mod `m`). The goal is to partition all `3m³` directed arcs into three\nedge-disjoint directed Hamiltonian cycles (each of length `m³`).\n\nKnuth describes an explicit construction, found by Claude (Anthropic), that achieves this\ndecomposition for all odd `m ≥ 3`. The case `m = 2` is known to be impossible [Aub82].\nThe even case `m > 2` is also settled. Knuth's paper says so in its final section, added in the\n14 April 2026 revision: \"Breaking news: The problem for even values of m is no longer in doubt!\"\nHo Boon Suan's algorithm [Ho26] is proved correct for even `m ≥ 8` in [GPT26], and\nAquino-Michaels [AM26] gives a decomposition for the even case that is simpler. The statement\nbelow starts at `m = 4`, which is below the range [GPT26] covers, so it also rests on the\nexplicit solutions in [Kn26], whose header gives even `m ≥ 4`.\n\n## References\n\n- [Knu26] D. E. Knuth, \"Claude's Cycles\" (2026).\n- [Aub82] J. Aubert, B. Schneider, \"Graphes orientés indécomposables en circuits hamiltoniens\",\n  J. Combin. Theory Ser. B 32 (1982), 347–349.\n- [Ho26] Ho Boon Suan, closed-form construction for even `m`,\n  <https://cs.stanford.edu/~knuth/even_closed_form.c>\n- [GPT26] A proof that [Ho26] yields three `m³`-cycles for every even `m ≥ 8`, 14 pages,\n  <https://cs.stanford.edu/~knuth/even_closed_form_proof_final.pdf>\n- [AM26] K. Aquino-Michaels, \"Completing Claude's cycles: Multi-agent structured exploration on\n  an open combinatorial problem\", <https://github.com/no-way-labs/residue>\n- [Kn26] Explicit even solutions for even `m ≥ 4`, recorded with [Knu26],\n  <https://cs.stanford.edu/~knuth/even_solution.py>\n- [KM26] K. Morrison, a Lean formalisation of the odd case,\n  <https://github.com/kim-em/KnuthClaudeLean>\n","FormalConjectures.Paper.ConjugacyClassSizes":"# The $S_3$-conjecture (conjugacy classes of distinct sizes)\n\n*References:*\n* W. Zhou, I. Gorshkov, *On $\\{2,3,5\\}$-groups with conjugacy classes of distinct sizes*,\n  [arXiv:2606.22244](https://arxiv.org/abs/2606.22244) (2026).\n* F. M. Markel, *Groups with many conjugate elements*, J. Algebra **26** (1973), 69–74.\n  (Origin of the $S_3$-conjecture.)\n* R. Knörr, W. Lempken, B. Thielcke, *The $S_3$-conjecture for solvable groups*,\n  Israel J. Math. **91** (1995), 61–76.\n* J. Zhang, *Finite groups with many conjugate elements*, J. Algebra **170** (1994), 608–624.\n* Z. Arad, M. Muzychuk, A. Oliver, *On groups with conjugacy classes of distinct sizes*,\n  J. Algebra **280** (2004), 537–576.\n* [Conjugacy class](https://en.wikipedia.org/wiki/Conjugacy_class)\n\nA finite group in which distinct conjugacy classes have distinct cardinalities is called an\n*anti-homogeneous* group (or *ah-group*). The symmetric group $S_3$ is an ah-group: its three\nconjugacy classes have sizes $1$, $2$, and $3$. Markel's **$S_3$-conjecture** (1973) asserts that,\nup to isomorphism, $S_3$ is the only nontrivial finite ah-group. The conjecture has been proved\nfor all solvable groups (independently by Zhang and by Knörr–Lempken–Thielcke), but the general\nnon-solvable case remains open.\n","FormalConjectures.Paper.DeGiorgi":"# De Giorgi's conjecture\n\nThis file states a conjecture of De Giorgi about entire solutions to $Δ u + u - u^3 = 0$.\nThe conjecture is a rigidity theorem: in spatial dimension $n ≤ 8$, the level sets of bounded solutions\nwhich satisfy $∂₁u > 0$ everywhere are hyperplanes. It has been shown that the condition $n ≤ 8$ is sharp.\n\nThe main theorems are:\n- `DeGiorgi_le_eight`: the conjecture holds in dimension $n ≤ 8$.\n- `DeGiorgi_ge_nine`: the conclusion of the conjecture does not hold if $n ≥ 9$.\n\nThe cases $1 ≤ n ≤ 8$ are also listed individually to enable partial solutions.\nThe cases $1 ≤ n ≤ 3$ are solved, while $4 ≤ n ≤ 8$ remains open.\n\n## Existing results\n- The case $n = 1$ trivially holds ($u$ is injective since $∂_1 u > 0$).\n- The case $n = 2$ was proven by Ghoussoub and Gui.\n- The case $n = 3$ was proven by Ambrosio and Cabré.\n- The case $4 ≤ n ≤ 8$ was proven under an extra assumption by Savin.\n- The counterexample for $n ≥ 9$ was proven by Del Pino, Kowalczyk, and Wei.\n\n## References\n* [Ghoussoub, Gui](https://doi.org/10.1007/s002080050196),\n  Mathematische Annalen 311 (1998) proves the conjecture for $n = 2$.\n* [Ambrosio, Cabré](https://doi.org/10.1090/S0894-0347-00-00345-3),\n  Journal of the American Mathematical Society 13 (2000) proves the conjecture for $n = 3$.\n* [Savin](https://doi.org/10.4007/annals.2009.169.41),\n  Annals of Mathematics 169 (2009) proves the case $4 ≤ n ≤ 8$ under an additional assumption.\n* [Del Pino, Kowalczyk, Wei](http://dx.doi.org/10.4007/annals.2011.174.3.3),\n  Annals of Mathematics 174 (2011) shows that the condition $n ≤ 8$ is sharp.\n","FormalConjectures.Paper.DegreeSequencesTriangleFree":"Title: Degree sequences in triangle-free graphs\nAuthors: P. Erdős, S. Fajtlowicz and W. Staton,\nPublished in Discrete Mathematics 92 (1991) 85–88.\n","FormalConjectures.Paper.Dubner":"# Dubner's conjecture\n\n*Reference*: [Every even number greater than 4208 is the sum of two t-primes](https://scispace.com/pdf/twin-prime-conjectures-3icaxy6b0m.pdf)\nby *Harvey Dubner*.\n","FormalConjectures.Paper.FusibleNumber":"# Main conjecture on fusible numbers\n\n*References:*\n- [Fusible numbers and Peano Arithmetic](https://arxiv.org/abs/2003.14342),\n  by Jeff Erickson, Gabriel Nivasch, and Junyan Xu.\n- [Fusible numbers and Peano Arithmetic](https://doi.org/10.46298/lmcs-18%283%3A6%292022),\n  Logical Methods in Computer Science, Volume 18, Issue 3 (July 28, 2022).\n","FormalConjectures.Paper.Gourevitch":"# Gourevitch's series identity\n\n*References:*\n - [About a New Kind of Ramanujan-Type Series](https://doi.org/10.1080/10586458.2003.10504518) by *Jesús Guillera*\n - [G2003] Guillera, Jesús. \"About a new kind of Ramanujan-type series.\" Experimental Mathematics 12.4 (2003): 507-510.\n - [A2025] Au, Kam Cheong. \"Wilf-Zeilberger seeds and non-trivial hypergeometric identities.\" Journal of Symbolic Computation 130 (2025): 102421. [arXiv:2312.14051](https://arxiv.org/abs/2312.14051)\n","FormalConjectures.Paper.HartshorneConjecture":"# Hartshorne's conjecture on Vector Bundles\n\n*References:*\n* [Har1974] R. Hartshorne, [Varieties of small codimension in projective space](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-80/issue-6/Varieties-of-small-codimension-in-projective-space/bams/1183535999.full).\n* [MO2010] [Evidences on Hartshorne's conjecture? References?](https://mathoverflow.net/questions/13990/evidences-on-hartshornes-conjecture-references)\n","FormalConjectures.Paper.Homogenous":"# Conjectures around homogeneous topological spaces\n\nThis file formalizes the notion of a weakly first countable topological space and some conjectures\naround those.\n\n*References:*\n* [Ar2013] Arhangeliski, Alexandr. \"Selected old open problems in general topology.\"\n  Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica 73.2-3 (2013): 37-46.\n  https://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf\n","FormalConjectures.Paper.KotzigConjecture":"# Kotzig's Conjecture\n\n*Reference:* A. Kotzig, cited in A. Rosa, *On certain valuations of the vertices of a graph*,\nTheory of Graphs (Internat. Sympos., Rome, 1966), Gordon and Breach, 1967, pp. 349–355.\n\nKotzig conjectured that for every $n$, the complete graph $K_{2n+1}$ decomposes into copies of\nany $n$-edge tree via cyclic shifts of a single embedding. This is strictly stronger than\nRingel's conjecture; see `Paper/RingelConjecture.lean`. The large-$n$ case is proved by\nMontgomery–Pokrovskiy–Sudakov; see `Arxiv/2001.02665/RingelConjecture.lean`.\n","FormalConjectures.Paper.Kurepa":"# Kurepa's conjecture\n\n*Reference:* [On the left factorial function !N](https://oeis.org/A3422), by *Đuro Kurepa* Math. Balkanica 1, p. 147-153, 1971\n\n","FormalConjectures.Paper.LatinSquare":"# Conjectures about Latin Squares\n\nThis file formalizes some conjectures and theorems around latin squares.\n\n*References:*\n* [Wa2011] Wanless, Ian. \"Transversals in Latin Squares: A Survey.\"\n  Surveys in Combinatorics 2011, R. Chapman, Ed. Cambridge University Press, 2011, pp. 403–437.\n  https://users.monash.edu.au/~iwanless/papers/transurveyBCC.pdf\n* https://en.wikipedia.org/wiki/Problems_in_Latin_squares\n","FormalConjectures.Paper.LatinTableau":"# Latin Tableau Conjecture\n\nThe Latin Tableau Conjecture states that the graph associated\nto any (finite) Young diagram (i.e., whose vertices are the\ncells of the diagram, with edges between cells in the same row\nor column) is CDS-colorable, meaning that there exists a proper\ncoloring of the vertices of the graph such that for all k > 0, the\nnumber of vertices with color < k equals the maximum size of\nthe union of k independent sets of the graph.\n\n*References:*\n\n* [The Latin Tableau Conjecture](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v32i2p48)\n","FormalConjectures.Paper.MonochromaticQuantumGraph":"# Monochromatic quantum graphs (inherited vertex colorings)\n\nThis file studies the existence of *monochromatic quantum graphs*: edge-coloured, edge-weighted\ncomplete graphs whose perfect matchings induce vertex colourings, with the property that\n\n- every **non-monochromatic** inherited vertex colouring has total weight `0`, while\n- each of the `D` **monochromatic** colourings has total weight `1`.\n\nIn the quantum-optics motivation, such a construction corresponds to generating high-dimensional\nmultipartite GHZ-type states using probabilistic pair sources and linear optics (without additional\nresources), where interference patterns can be expressed as weighted sums over perfect matchings.\n\n## Main questions (informal)\n\n- For `N = 4` and `D ≥ 4`, does there exist such a graph/weighting?\n- For even `N ≥ 6` and `D ≥ 3`, does there exist such a graph/weighting?\n\n## Formalisation sketch\n\nA quantum graph with `N` vertices and `D` colours can be encoded by a weight function\n`W : EdgeN N D α → α` (for a coefficient domain `α`).\n\nFor each assignment of vertex indices `ι : V N → Fin D`, we define a perfect-matching sum\n`pmSumN N D W ι` (a sum over perfect matchings, where each matching contributes the product of the\ncorresponding edge weights determined by `ι`). The equation system `EqSystemN N D W` requires\n\n`pmSumN N D W ι = 1` iff `ι` is constant (all entries equal), and `0` otherwise.\n\nThe open conjectures in this file ask for non-existence/existence of such `W` over various\ncoefficient domains (e.g. `ℂ`, `ℝ`, `ℤ`, and restricted integer weights).\n\n## References\n\n* [Krenn2017] M. Krenn, X. Gu, A. Zeilinger,\n  \"Quantum Experiments and Graphs: Multiparty States as Coherent Superpositions of Perfect Matchings\",\n  *Physical Review Letters* 119(24), 240403 (2017).\n\n* [MO2018] [Vertex coloring inherited from perfect matchings (motivated by quantum physics)](https://mathoverflow.net/questions/311325),\n  MathOverflow question 311325.\n\n* [Gu2019] X. Gu, M. Erhard, A. Zeilinger, M. Krenn,\n  \"Quantum experiments and graphs II: Quantum interference, computation, and state generation\",\n  *PNAS* 116(10), 4147–4155 (2019).\n\n* [Krenn2019] [Questions on the Structure of Perfect Matchings inspired by Quantum Physics](https://arxiv.org/abs/1902.06023)\n  by *M. Krenn, X. Gu, U. Soltész*,\n  Proc. 2nd Croatian Combinatorial Days, 57–70 (2019).\n\n* [Chandran2022] [Edge-coloured graphs with only monochromatic perfect matchings and their connection to quantum physics](https://arxiv.org/abs/2202.05562)\n  by *N. Chandran, S. Gajjala* (2022).\n\n* [Chandran2024] [Krenn–Gu conjecture for sparse graphs](https://arxiv.org/abs/2407.00303)\n  by *N. Chandran, S. Gajjala, S. Illickan, M. Krenn*, MFCS 2024.\n","FormalConjectures.Paper.PrimeTuples":"# Prime Tuples Conjecture\n\n*Reference:* [FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a\ndivisor function series. Integers (2007).\n","FormalConjectures.Paper.ReedOmegaDeltaChi":"# Reed's omega, delta, and chi conjecture\n\n*References*:\n- [B. Reed,  ω Δ and χ, J. Graph Theory 27 (1998) 177-212.](https://onlinelibrary.wiley.com/doi/10.1002/(SICI)1097-0118(199804)27:4%3C177::AID-JGT1%3E3.0.CO;2-K)\n- [openproblemgarden](http://www.openproblemgarden.org/op/reeds_omega_delta_and_chi_conjecture)\n- [mathoverflow/37923](https://mathoverflow.net/questions/37923) asked by user [Andrew D. King](https://mathoverflow.net/users/4580/andrew-d-king)\n","FormalConjectures.Paper.RingelConjecture":"# Ringel's Conjecture\n\n*Reference:* G. Ringel, *Problem 25*, in *Theory of Graphs and its Applications*\n(Proc. Sympos. Smolenice, 1963), Academia, Prague, 1964.\n\nRingel's conjecture (1963): the complete graph $K_{2n+1}$ decomposes into copies of any tree\nwith $n$ edges. It remains open; the case of all sufficiently large $n$ is proved by\nMontgomery–Pokrovskiy–Sudakov, see `Arxiv/2001.02665/RingelConjecture.lean`.\n","FormalConjectures.Paper.Rupert":"# Is Every Convex Polyhedron Rupert?\n\nA polyhedron is Rupert if one can cut a hole in it and pass another\ncopy of the same polyhedron through that hole.\n\nMore formally: a convex body in ℝ³ is a compact, convex set with\nnonempty interior. A convex body X is said to be Rupert if there are\ntwo affine transforms T₁, T₂ ∈ SE(3) such that π(T₁(X)) ⊆\nint(π(T₂(X))), where π : ℝ³ → ℝ² is the evident projection, and int\ndenotes topological interior.\n\nNot all convex bodies are Rupert. For example,\n- the unit ball is not Rupert\n- the circular cylinder of unit diameter and height\n  closed on each end by disks is not Rupert\n\nHowever, many convex polyhedra are Rupert. All Platonic solids, and\nmost Archimedean and Catalan solids are known to be Rupert.\n\nQuestion: are all convex polyhedra with nonempty interior Rupert?\n\n*References:*\n\n* [Platonic Passages](https://www.researchgate.net/publication/314715434_Platonic_Passages),\n  R. P. Jerrard, J. E. Wetzel, and L. Yuan., Math. Mag., 90(2):87–98,\n  2017. conjectures (\"with a certain hesitancy\") that perhaps all\n  convex polyhedra are Rupert.\n\n* However, [An Algorithmic Approach to Rupert's Problem](https://arxiv.org/pdf/2112.13754#cite.JeWeYu17)\n  describes experimental evidence to suggest that three Archimedean\n  solids may not be Rupert.\n\n* [Optimizing for the Rupert property](https://arxiv.org/abs/2210.00601)\n  is the source of some of the Catalan solid results, and has more\n  results for Johnson polyhedra as well.\n\n* [This video by David Renshaw](https://www.youtube.com/watch?v=evKFok65t_E) visualizes\n  known results for Platonic, Archimedean, and Catalan solids.\n\n* This problem's name comes from the fact that it is a generalization\n  of [Prince Rupert's Cube](https://en.wikipedia.org/wiki/Prince_Rupert%27s_cube).\n\n* [A convex polyhedron without Rupert's property](https://arxiv.org/abs/2508.18475),\n  Jakob Steininger and Sergey Yurkevich, 2025. Constructs a convex polyhedron and\n  a proof that it is not Rupert, resolving the open question.\n\n","FormalConjectures.Paper.SerreUniformity":"# Serre's uniformity conjecture over the rationals\n\nIs there a bound $C$, independent of the non-CM elliptic curve $E/\\mathbb{Q}$, such that\nthe Galois action on $E[p]$ is surjective onto $\\mathrm{GL}_2(\\mathbb{F}_p)$ for every\nprime $p > C$? This is the $\\mathbb{Q}$ case of Serre's question [Ser72], recalled in\nthe introduction of [Lem17].\n\nWe express surjectivity without choosing a basis: every additive automorphism of $E[p]$\nmust be induced by an element of $G_{\\mathbb{Q}}$. For prime $p$, these automorphisms are\nexactly the $\\mathbb{F}_p$-linear automorphisms. The non-CM condition uses the classification\nof rational CM $j$-invariants.\n\n*References:*\n- [Ser72] J.-P. Serre, *Propriétés galoisiennes des points d'ordre fini des courbes\n  elliptiques*. Inventiones Mathematicae 15 (1972), 259–331.\n  https://doi.org/10.1007/BF01405086\n- [Lem17] P. Lemos, *Serre's uniformity conjecture for elliptic curves with rational\n  cyclic isogenies*, introduction. https://arxiv.org/abs/1702.01985\n","FormalConjectures.Paper.StrongSensitivityConjecture":"# Strong Sensitivity Conjecture (`bs(f) ≤ s(f)^2`)\n\nThis file formalizes the *strong* sensitivity conjecture, asserting:\n\nFor every Boolean function `f : {0,1}^n → {0,1}`,\n`bs(f) ≤ s(f)^2`,\nwhere bs(f) denotes block sensitivity and s(f) denotes sensitivity.\n\nHuang's theorem proves a *quartic* upper bound, `bs(f) ≤ s(f)^4`, thereby\nresolving the most widely known form of the sensitivity conjecture.\n\nWe now ask whether a stronger upper bound holds. Interestingly, the original\npaper of Nisan and Szegedy, where the sensitivity conjecture first appeared,\nalready speculated that a *quadratic* upper bound might be the correct\nrelation. On the lower bound side, Rubinstein\n(https://link.springer.com/article/10.1007/BF01200762) constructed Boolean functions\nexhibiting the first quadratic separation. The best currently\nknown gap, due to Ambainis and Sun (https://arxiv.org/abs/1108.3494), is\n`bs(f) ≥ (2/3)⋅s(f)^2`.\n\n*References:*\n* [Induced Subgraphs of Hypercubes and a Proof of the Sensitivity Conjecture](https://arxiv.org/abs/1907.00847)\n  by Hao Huang (see Section 3, Concluding Remarks)\n* [Variations on the Sensitivity Conjecture](https://arxiv.org/abs/1011.0354)\n  by Pooya Hatami, Raghav Kulkarni, and Denis Pankratov (see Question 3.1)\n* [On the Degree of Boolean Functions as Real Polynomials](https://link.springer.com/article/10.1007/BF01263419)\n  by Noam Nisan, and Mario Szegedy (see Section 4, Open Problems)\n","FormalConjectures.Paper.TuDengConjecture":"# Tu-Deng Conjecture\n\n*References:*\n* [Tu, Z., Deng, Y., *A conjecture about binary strings and its applications on constructing\n  Boolean functions with optimal algebraic immunity*, Des. Codes Cryptogr. **60** (2011),\n  1–14](https://doi.org/10.1007/s10623-010-9413-9)\n* [IACR ePrint 2009/272](https://eprint.iacr.org/2009/272)\n\nFor an integer $k \\ge 2$, identify the residues modulo $2^k - 1$ with the integers\n$0, 1, \\dots, 2^k - 2$, and let $w(a)$ denote the binary weight of $a$ (the number of ones in\nits binary expansion). The Tu-Deng conjecture states that for every residue $t \\ne 0$,\n\n$$\\#\\{(a, b) : a + b \\equiv t \\pmod{2^k - 1},\\ w(a) + w(b) \\le k - 1\\} \\le 2^{k-1}.$$\n\nTu and Deng showed that this combinatorial statement implies that their constructions of\nBoolean functions (a bent class and a balanced class) achieve optimal algebraic immunity,\nwhich is the motivation for the conjecture. They verified the conjecture numerically for\n$k \\le 29$ (Remark 3.1 of the ePrint version).\n","FormalConjectures.Paper.VoronovskajaTypeFormula":"# Voronovskaja-type Formula for the Bezier Variant of the Bernstein Operators\n\nThe Bézier-type Bernstein operators $B_{n,\\alpha}$ for $\\alpha > 0$ are defined for\n$f : [0,1] \\to \\mathbb{R}$ by\n$$\n(B_{n,\\alpha} f)(x)\n  = \\sum_{k=0}^n f\\!\\left(\\frac{k}{n}\\right)\n    \\left( J_{n,k}(x)^{\\alpha} - J_{n,k+1}(x)^{\\alpha} \\right),\n$$\nwhere\n$$\nJ_{n,k}(x) = \\sum_{j=k}^n p_{n,j}(x),\n\\qquad\np_{n,j}(x) = \\binom{n}{j} x^j(1-x)^{n-j},\n$$\nand $J_{n,n+1}(x) = 0$.\n\nIn the classical case $\\alpha = 1$, these operators reduce to the usual Bernstein operators.\nFor $f$ which are $C^2$ on $[0,1]$, one has the classical Voronovskaja\nasymptotic formula\n$$\n\\lim_{n \\to \\infty} n\\bigl( B_{n,1} f(x) - f(x) \\bigr)\n    = \\tfrac{1}{2} x(1-x) f''(x).\n$$\n\n## Known Results\n* For $\\alpha = 1$, the asymptotics are completely understood.\n* Numerical experiments indicate that for $\\alpha \\neq 1$ the quantity\n    $$\n        \\sqrt{n}\\,\\bigl( B_{n,\\alpha} f(x) - f(x) \\bigr)\n    $$\n    may converge to a non-zero limit.\n\n## The Problem\nDetermine the asymptotic behaviour of the Bézier-type Bernstein operators for $\\alpha > 0$,\n$\\alpha \\neq 1$:\n\\textbf{Existence of the limit:}\n    Prove (or disprove) the existence of the limit\n    $$\n        \\lim_{n \\to \\infty}\n        \\sqrt{n}\\,\\bigl( B_{n,\\alpha} f(x) - f(x) \\bigr),\n    $$\n    at least for sufficiently smooth functions $f$.\n    \\textbf{Explicit form of the limit:}\n    If the limit exists, determine an explicit expression for it in terms of $f$, $x$, and $\\alpha$.\n\n*References:*\n\n* [Voronovskaja-type Formula for the Bézier Variant of the Bernstein Operators](https://www.math.bas.bg/mathmod/Proceedings_CTF/CTF-2010/files_CTF-2010/Open_problems.pdf),\n  by *Ulrich Abel*, in *Constructive Theory of Functions, Sozopol 2010*.\n","FormalConjectures.Paper.WeakTiling":"# Weak tiling problems\n\nProblems 4.1, 4.2, and 4.3 from [arxiv/2506.23631](https://arxiv.org/abs/2506.23631).\n\n*Reference:*\n* [Geometric implications of weak tiling](https://arxiv.org/abs/2506.23631)\n\nSee also `FormalConjectures.Wikipedia.Fuglede` for Fuglede's spectral set conjecture, which\nmotivates the study of weak tilings.\n","FormalConjectures.Paper.WeaklyFirstCountable":"# Conjectures about Weakly First Countable spaces\n\nThis file formalizes the notion of a weakly first countable topological space and some conjectures\naround those.\n\n*References:*\n* [Ar2013] Arhangeliski, Alexandr. \"Selected old open problems in general topology.\"\n  Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica 73.2-3 (2013): 37-46.\n  https://www.math.md/files/basm/y2013-n2-3/y2013-n2-3-(pp37-46).pdf.pdf\n* [Ya1976] Yakovlev, N. N. \"On the theory of o-metrizable spaces.\"\n  Doklady Akademii Nauk. Vol. 229. No. 6. Russian Academy of Sciences, 1976.\n  https://www.mathnet.ru/links/016f74007f9f96fa3aadae05cbd98457/dan40570.pdf (in Russian)\n","FormalConjectures.Paper.ZagierMZV":"# Zagier's Conjecture on Multiple Zeta Values\n\n*References:*\n- [Za94] Zagier, Don. \"Values of zeta functions and their applications.\"\n  First European Congress of Mathematics Paris, July 6–10, 1992: Vol. II: Invited Lectures (Part 2). Basel: Birkhäuser Basel, 1994.\n- [Co18] Combariza, Germán AG. \"A few conjectures about the multiple zeta values.\"\n  ACM Communications in Computer Algebra 52.1 (2018): 11-20.\n- [Te02] T. Terasoma. Mixed Tate motives and multiple zeta values. Invent. Math., 149(2):339–369, 2002.\n- [DG05] P. Deligne and A. Goncharov. Groupes fondamentaux motiviques de Tate mixte. Ann. Sci.\n  Ecole Norm. Sup. (4), 38(1):1–56, 2005.\n- [OEIS A000931](https://oeis.org/A000931)\n","FormalConjectures.Subsets.FC100OpenSet1":"# FC100OpenSet1\n\nA random subset of 100 open research problems, drawn uniformly at random\nfrom all problems with the `category research open` tag.\n","FormalConjectures.Subsets.FC100SolvedSet1":"# FC100SolvedSet1\n\nA random subset of 100 non-open problems, drawn uniformly at random\nfrom all problems without the `category research open` tag\n(solved, test, API, etc.).\n","FormalConjectures.Wikipedia.ABC":"# *abc* conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Abc_conjecture)\n","FormalConjectures.Wikipedia.AgohGiuga":"# Agoh-Giuga conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Agoh-Giuga_conjecture)\n","FormalConjectures.Wikipedia.Agrawal":"# Agrawal's conjecture\n\nAgrawal's conjecture is a stronger version of the theorem that forms the basis\nof the AKS primality test. If true, it would significantly improve the\nefficiency of primality testing.\n\nThe conjecture states that for coprime $n$ and $r$, if the polynomial congruence\n$(X-1)^n \\equiv X^n-1 \\pmod{n, X^r-1}$ holds, then $n$ is either prime or $n^2 \\equiv 1 \\pmod{r}$.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Agrawal%27s_conjecture)\n- [AIM Math](https://aimath.org/WWN/primesinp/articles/html/50a/)\n- [Paper](https://eprint.iacr.org/2009/008.pdf)\n","FormalConjectures.Wikipedia.AlgebraicNormality":"# Normality of Irrational Algebraic Numbers\n\nIt is unknown whether every irrational algebraic real number is normal in any integer base.\nThe stronger conjecture that every irrational algebraic real number is absolutely normal is stated\nseparately: normality in one base and normality in every base are not equivalent definitions.\n\n*References:*\n- [Wikipedia: Normal number](https://en.wikipedia.org/wiki/Normal_number)\n- [BC01] Bailey, David H., and Richard E. Crandall. \"On the random character of fundamental constant\n  expansions.\" Experimental Mathematics 10.2 (2001): 175-190.\n  https://projecteuclid.org/journals/experimental-mathematics/volume-10/issue-2/On-the-random-character-of-fundamental-constant-expansions/em/999188630.full\n","FormalConjectures.Wikipedia.AlmostPerfectNumbers":"# Non-Power-of-2 Almost Perfect Numbers Conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Almost_perfect_number)\n- [mathworld](https://mathworld.wolfram.com/AlmostPerfectNumber.html)\n-\n","FormalConjectures.Wikipedia.AmicableNumbers":"# Amicable numbers\n\nTwo distinct positive integers form an amicable pair if each equals the sum of the\nproper divisors of the other. Equivalently, $(a, b)$ is an amicable pair if\n$\\sigma(a) = a + b$ and $\\sigma(b) = a + b$, where $\\sigma(n)$ denotes the sum of\nall positive divisors of $n$.\n\nSeveral open problems about amicable numbers are formalised here:\n\n* Do there exist relatively prime amicable numbers?\n* Are there infinitely many amicable pairs?\n* Do there exist amicable numbers with opposite parity (one even, one odd)?\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Amicable_numbers)\n- [MathWorld](https://mathworld.wolfram.com/AmicableNumbers.html)\n- [OEIS A063990](https://oeis.org/A063990)\n","FormalConjectures.Wikipedia.AndrewsCurtis":"# The Andrews-Curtis conjecture\n\nThe conjecture says that every normally generating `n`-tuple in the free group\non `n` generators is Andrews-Curtis equivalent to the standard free basis.\n\nReferences:\n\n* [Wikipedia, Andrews-Curtis conjecture](https://en.wikipedia.org/wiki/Andrews%E2%80%93Curtis_conjecture)\n* [A. D. Myasnikov, A. G. Myasnikov and V. Shpilrain,\n  *On the Andrews-Curtis equivalence*](https://arxiv.org/abs/math/0302080)\n","FormalConjectures.Wikipedia.Andrica":"# Andrica's conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Andrica%27s_conjecture)\n- [Luan Alberto Ferreira, *Real exponential sums over primes and prime gaps*](https://arxiv.org/abs/2307.08725)\n","FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture":"# Artin's conjecture on primitive roots\n\nArtin's conjecture predicts, given an integer $a$, densities of primes $p$ for which\n$a$ is a primitive root modulo $p$. Under certain conditions (when $a$ is not a\npower and its squarefree part is $1\\pmod{4}$) the density is given by Artin's constant\n$$\\prod_{p\\ \\text{prime}} \\left(1 - \\frac{1}{p(p - 1)}\\right).$$\nFor more general values of $a$, this constant must be corrected by certain factors.\n- When $a = b^m$, $m$ is a maximal odd power, the squarefree part of $b$ satisfies\n  $b_0 \\not\\equiv 1\\pmod{4}$. Then Artin's constant should be multiplied by\n  $$\\prod_{p \\mid m} \\frac{p(p - 2)}{p^2 - p - 1}.$$\n- When $a = b^m$, $m$ is a maximal power, the squarefree part of $b$ satisfies\n  $b_0\\equiv 1\\pmod{4}$. Then Artin's constant should be multiplied by the factor in\n  the above bullet, as well as an additional entanglement factor from the primes dividing\n  $\\gcd(b_0, m)$ and primes dividing $b_0$:\n  $$1 - \\prod_{p \\mid \\gcd(b_0, m)} \\frac{1}{2 - p}\n  \\prod_{p \\mid b_0, p\\nmid m} \\frac{1}{1 + p - p^2}.$$\n- When $a = -1$ or $a$ is a square, then the density is $0$.\n\nNote that Artin's conjecture has been proved subject to the Generalized Riemann Hypothesis\n[Ho67].\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots)\n- [A85397](https://oeis.org/A85397)\n- [LMS14](https://arxiv.org/pdf/1112.4816) Lenstra, H.W. et al. \"Character sums for primitive root densities\" _arXiv:1112.4816_ [math.NT] (2014).\n- [Ho67] Hooley, C. \"On Artin's conjecture.\" _Journal für die reine und angewandte Mathematik_ 225 (1967): 209-220.\n","FormalConjectures.Wikipedia.BalancedPrimes":"# Balanced prime conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Balanced_prime)\n- [OEIS A6562](https://oeis.org/A6562)\n","FormalConjectures.Wikipedia.BatemanHornConjecture":"# Bateman-Horn Conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture)\n","FormalConjectures.Wikipedia.BealConjecture":"# Beal conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Beal_conjecture)\n","FormalConjectures.Wikipedia.BeckFialaConjecture":"# Beck–Fiala theorem and conjecture\n\nDiscrepancy of bounded-degree set systems. Given sets $S_1, \\dots, S_m \\subseteq [n]$\nsuch that every element of $[n]$ belongs to at most $t$ of the sets (the system has\n*degree* at most $t$), one seeks a colouring $\\chi \\colon [n] \\to \\{-1, +1\\}$ making\nevery set as balanced as possible, i.e. minimizing the *discrepancy*\n$\\max_i \\left|\\sum_{j \\in S_i} \\chi(j)\\right|$.\n\nThe Beck–Fiala theorem (1981) states that every set system of degree at most $t \\ge 1$\nhas discrepancy at most $2t - 1$. The Beck–Fiala conjecture asserts that the truth is\nmuch stronger: the discrepancy of a degree-$t$ system is $O(\\sqrt{t})$, with a constant\nindependent of $n$, $m$ and $t$.\n\nDespite considerable attention the bound $2t - 1$ has been improved only slightly:\nBukh (2016) proved a bound of the form $2t - \\log^* t$ (where $\\log^*$ is the iterated\nlogarithm), and Banaszczyk's vector balancing theorem yields $O(\\sqrt{t \\log n})$.\nThe Komlós conjecture (see `KomlosConjecture.lean`) would imply the Beck–Fiala\nconjecture, since scaling the incidence vectors of a degree-$t$ system by $1/\\sqrt{t}$\nproduces vectors of Euclidean norm at most $1$.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Beck%E2%80%93Fiala_theorem)\n- [J. Beck and T. Fiala, *\"Integer-making\" theorems*,\n  Discrete Applied Mathematics **3** (1981), 1–8](https://doi.org/10.1016/0166-218X(81)90022-6)\n- [B. Bukh, *An improvement of the Beck–Fiala theorem*,\n  Combinatorics, Probability and Computing **25** (2016), 380–398](https://doi.org/10.1017/S0963548315000140)\n- [W. Banaszczyk, *Balancing vectors and Gaussian measures of n-dimensional convex bodies*,\n  Random Structures & Algorithms **12** (1998), 351–360](https://doi.org/10.1002/(SICI)1098-2418(199807)12:4%3C351::AID-RSA3%3E3.0.CO;2-S)\n","FormalConjectures.Wikipedia.BetrothedNumbers":"# Betrothed numbers\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Betrothed_numbers)\n- [OEIS A005276](https://oeis.org/A005276)\n","FormalConjectures.Wikipedia.BingBorsuk":"# The Bing-Borsuk Conjecture\n\nThe Bing-Borsuk conjecture states that every $n$-dimensional homogeneous absolute neighborhood\nretract is a topological $n$-manifold.\n\nThe conjecture has been verified in dimensions $1$ and $2$ but remains open in higher dimensions.\nA notable consequence is that if the $3$-dimensional case is true, it implies the Poincaré\nconjecture.\n\n*References:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Bing%E2%80%93Borsuk_conjecture)\n - [HR2008] Halverson, Denise M., and Dušan Repovš. \"The Bing-Borsuk and the Busemann\n   conjectures.\" Mathematical Communications 13.2 (2008): 163-184.\n   https://arxiv.org/abs/0811.0886\n","FormalConjectures.Wikipedia.Bloch":"# Bloch and Landau constants\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Bloch%27s_theorem_(complex_analysis))\n- [CP96] Chen, H., Gauthier, P. M. \"On Bloch's constant.\" Journal d'Analyse Mathématique 69 (1996),\n  275–291.\n- [AG37] Ahlfors, L. V., Grunsky, H. \"Über die Blochsche Konstante.\" Mathematische Zeitschrift 42\n  (1937), 671–673.\n- [Ya95] Yanagihara, H. \"On the locally univalent Bloch constant.\" Journal d'Analyse Mathématique\n  65 (1995), 1–17.\n- [Ra43] Rademacher, H. \"On the Bloch-Landau Constant.\"\" American Journal of Mathematics 65 (1943),\n  387–390.\n- [OptimizationConstants](https://teorth.github.io/optimizationproblems/constants/57c.html)\n- [Skin2009] Skinner, Brian. The univalent Bloch constant problem. Complex Variables and Elliptic\n  Equations 54 (2009), no. 10, 951–955.\n- [MathWorld](https://mathworld.wolfram.com/BlochConstant.html)\n- [Bhowmik–Sen](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/improved-bloch-and-landau-constants-for-meromorphic-functions/FD465D1F2CEF7E8C62AFF16C3E89B7B4)\n","FormalConjectures.Wikipedia.BoundedBurnsideProblem":"# Bounded Burnside problem\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Burnside_problem#Bounded_Burnside_problem)\n","FormalConjectures.Wikipedia.Brennanconjecture":"# Brennan's Conjecture\n\n*Reference:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Brennan_conjecture)\n- [arXiv:2409.15074](https://arxiv.org/abs/2409.15074)\n- [arXiv:2512.09330](https://arxiv.org/abs/2512.09330)\n","FormalConjectures.Wikipedia.BrocardConjecture":"# Brocard's Conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Brocard%27s_conjecture)\n- [Luan Alberto Ferreira, *Real exponential sums over primes and prime gaps*](https://arxiv.org/abs/2307.08725)\n","FormalConjectures.Wikipedia.BrocardProblem":"# Brocard's Problem\n\nBrocard's problem asks whether the only solutions to $n! + 1 = m^2$ are\n$n = 4, 5, 7$.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Brocard%27s_problem)\n\nThis file points to the canonical formalization in `FormalConjectures.ErdosProblems.«398»`.\n","FormalConjectures.Wikipedia.Buchi":"# Büchi's problem\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/B%C3%BCchi%27s_problem)\n","FormalConjectures.Wikipedia.Bunyakovsky":"# Bunyakovsky conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Bunyakovsky_conjecture)\n","FormalConjectures.Wikipedia.BusyBeaver":"# Busy Beaver\n\nThe Busy Beaver problem asks for the maximum number of steps that an n-state, 2-symbol Turing\nmachine can take before halting, when started on an empty tape.\n\n*References:*\n\n- [The Busy Beaver Challenge](https://wiki.bbchallenge.org/wiki/Main_Page)\n","FormalConjectures.Wikipedia.CarmichaelTotient":"# Carmichael's totient function conjecture\n\nFor every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that\n$φ(n) = φ(m)$ where $φ$ is the Euler totient function.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Carmichael%27s_totient_function_conjecture)\n- [F1998] Kevin Ford. The distribution of totients. https://arxiv.org/abs/1104.3264\n","FormalConjectures.Wikipedia.Catalan":"# Catalan's conjecture and related Diophantine equations\n\n*References:*\n- [Wikipedia - Catalan's conjecture](https://en.wikipedia.org/wiki/Catalan%27s_conjecture)\n- [arXiv:2507.12397](https://arxiv.org/abs/2507.12397) (Lebesgue-Nagell equation)\n","FormalConjectures.Wikipedia.CernyConjecture":"# Černý Conjecture\n\nA **synchronizing word** (also called a reset word) for a deterministic finite automaton (DFA)\n$M = (Q, \\Sigma, \\delta)$ is a word $w \\in \\Sigma^\\*$ such that reading $w$ from any state always\nleads to the same single state — formally, $\\exists p \\in Q, \\forall q \\in Q, \\delta^\\*(q, w) = p$.\n\nA DFA is called **synchronizing** if it admits at least one synchronizing word.\n\nThe **Černý conjecture** asserts that every synchronizing DFA with $n$ states has a\nsynchronizing word of length at most $(n - 1)^2$. This bound is sharp: the family of Černý\nautomata $C_n$ witnesses it, requiring exactly $(n - 1)^2$ steps.\n\n**Status:** Open. The best known upper bound is\n$\\left(\\frac{7}{48} + \\frac{2 \\cdot 15625}{1597536}\\right) n^3 + o(n^3) \\approx 0.1654\\,n^3$\n(Shitov, 2019). The bound $(n - 1)^2$ has been verified for small $n$ and for special classes of\nautomata (e.g., Eulerian, aperiodic, cyclic automata).\n\nWe use Mathlib's `DFA α σ` (from `Mathlib.Computability.DFA`), together with the auxiliary\n`DFA.IsSynchronizingWord` and `DFA.IsSynchronizing` predicates defined in\n`FormalConjecturesForMathlib.Computability.DFA`.\n\n*References:*\n- [Wikipedia: Synchronizing word](https://en.wikipedia.org/wiki/Synchronizing_word)\n- J. Černý, [*Poznámka k homogénnym experimentom s konečnými automatmi*](https://dml.cz/bitstream/handle/10338.dmlcz/126647/MathSlov_14-1964-3_2.pdf),\n  Matematicko-fyzikálny časopis, Vol. 14 (1964), No. 3, 208--216.\n- Y. Shitov, *An improvement to a recent upper bound for synchronizing words of finite automata*,\n  J. Autom. Lang. Comb. Vol. 24 (2019), 367--373.\n","FormalConjectures.Wikipedia.ClassNumberProblem":"# Class number problem for real quadratic fields\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Class_number_problem)\n","FormalConjectures.Wikipedia.CollatzConjecture":"# Collatz conjecture\n\n*References:*\n* [Wikipedia](https://en.wikipedia.org/wiki/Collatz_conjecture)\n* [erdosproblems.com/1135](https://www.erdosproblems.com/1135)\n* [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n* [La10] Lagarias, Jeffrey C., The {$3x+1$} problem: an overview. (2010), 3--29.\n* [La16] Lagarias, Jeffrey C., Erdős, Klarner, and the {$3x+1$} problem. Amer. Math. Monthly\n  (2016), 753--776.\n* [La85] Lagarias, Jeffrey C., The {$3x+1$} problem and its generalizations. Amer. Math. Monthly\n  (1985), 3--23.\n","FormalConjectures.Wikipedia.CongruentNumber":"# Congruent Number\n\nA natural number $n$ is called a congruent number if there exists a right triangle with rational\nsides $a$, $b$, and hypotenuse $c$ such that the area of the triangle is $\\frac{1}{2}ab = n$.\n\n*References:*\n- [Wikipedia (Congruent number)](https://en.wikipedia.org/wiki/Congruent_number)\n- [Wikipedia (Tunnell's theorem)](https://en.wikipedia.org/wiki/Tunnell%27s_theorem)\n- [Keith Conrad's note](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/congnumber.pdf)\n","FormalConjectures.Wikipedia.Conway99Graph":"# Conway's 99-graph problem\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Conway%27s_99-graph_problem)\n","FormalConjectures.Wikipedia.DedekindNumber":"# Dedekind Numbers\n\nA Dedekind number `M(n)` counts the number of monotone Boolean functions on `n` variables,\nor equivalently, the number of antichains (Sperner families) in the Boolean lattice `2^[n]`.\n\nFor example,\n$$M ( 0 ) = 2 , M ( 1 ) = 3 , M ( 2 ) = 6 , and M ( 3 ) = 20 .$$\nThe first few values grew slowly:\n$$M ( 4 ) = 168 , M ( 5 ) = 7581$$,\nbut then rapidly:\n$$M ( 6 ) = 7828354 , M ( 7 ) = 2414682040998 , M ( 8 ) = 56130437228687557907788$$, and\n$$M ( 9 ) = 286386577668298411128469151667598498812366$$\n(computed in 2023).\n\nWe formalize two definitions:\n- `M n`: the number of monotone Boolean functions `(Fin n → Bool) → Bool`\n- `M' n`: the number of antichains (Sperner families) of `Finset (Fin n)`\n\nWe prove their values for small `n` and show that the two definitions agree for all `n`.\n\nThe problem is to determine the exact values of $M(n)$ for $n ≥ 10$.\nIn particular, the value of $M(10)$ is currently unknown.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Dedekind_number)\n- [Oeis/A372](https://oeis.org/A000372)\n\n","FormalConjectures.Wikipedia.DeterminantalConjecture":"# Determinantal conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Determinantal_conjecture)\n","FormalConjectures.Wikipedia.DiameterSimpleFiniteGroups":"# Babai–Seress Conjectures on the Diameter of Finite Groups\n\n*References:*\n- [Wikipedia, *Diameter (group theory)*](https://en.wikipedia.org/wiki/Diameter_(group_theory))\n- [H. A. Helfgott and Á. Seress, *On the diameter of permutation groups*](https://arxiv.org/abs/1109.3550)\n- [L. Babai and Á. Seress, *On the diameter of permutation groups*,\n  European Journal of Combinatorics 13 (1992), 231–243](https://doi.org/10.1016/S0195-6698(05)80029-0)\n\nThis file contains two conjectures from the Babai–Seress paper:\n\n- **Conjecture 1.5**: $\\operatorname{diam}(A_n) < n^C$ for some absolute constant $C$,\n  where $A_n$ is the alternating group on $n$ elements.\n\n- **Conjecture 1.7**: $\\operatorname{diam}(G) < (\\log |G|)^C$ for some absolute constant $C$,\n  where $G$ ranges over all non-abelian finite simple groups.\n\nConjecture 1.7 generalises Conjecture 1.5, since for $G = A_n$ we have\n$\\log |A_n| \\approx n \\log n$, so a polylogarithmic bound in $|G|$ implies a polynomial\nbound in $n$.\n","FormalConjectures.Wikipedia.Dickson":"# Dickson's conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Dickson%27s_conjecture)\n- [PrimePages glossary](https://t5k.org/glossary/xpage/DicksonsConjecture.html)\n- [OEIS Wiki](https://oeis.org/wiki/Dickson%27s_conjecture)\n- [MathWorld](https://mathworld.wolfram.com/DicksonsConjecture.html)\n- [Leonard Eugene Dickson, *History of the Theory of Numbers, Vol. I: Divisibility and Primality*](https://archive.org/details/historyoftheoryo01dickuoft)\n- [Arxiv](https://arxiv.org/pdf/0906.3850)\n","FormalConjectures.Wikipedia.DiophantineTuple":"# Diophantine $m$-tuples\n\nA **Diophantine $m$-tuple** is a set of $m$ distinct positive integers\n$\\{a_1, \\dots, a_m\\}$ such that $a_i a_j + 1$ is a perfect square for every\n$i \\neq j$.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Diophantine_quintuple)\n- [Du16] Dujella, Andrej,\n  [*What is... a Diophantine m-tuple?*](https://www.ams.org/journals/notices/201607/rnoti-p772.pdf),\n  Notices Amer. Math. Soc. 63 (2016), 772--774.\n- [Du] Dujella, Andrej,\n  [*Diophantine quintuple conjecture*](https://web.math.pmf.unizg.hr/~duje/quint.html).\n- [HTZ19] He, Bo, Togbé, Alain and Ziegler, Volker,\n  [*There is no Diophantine quintuple*](https://arxiv.org/abs/1610.04020),\n  Trans. Amer. Math. Soc. 371 (2019), 6665--6709.\n- [BD69] Baker, Alan and Davenport, Harold, The equations $3x^2-2=y^2$ and $8x^2-7=z^2$.\n  Quart. J. Math. Oxford Ser. (2) 20 (1969), 129--137.\n- [Gi99] Gibbs, Philip,\n  [*Some rational Diophantine sextuples*](https://arxiv.org/abs/math/9902081), (1999).\n","FormalConjectures.Wikipedia.ElliottHalberstamConjecture":"# Elliott–Halberstam conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture)\n- [EH68] Elliott, Peter D. T. A. and Halberstam, Heini, *A conjecture in prime number\n  theory*, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), 59–72.\n- [FG89] Friedlander, John and Granville, Andrew, *Limitations to the equi-distribution of\n  primes I*, Ann. of Math. (2) 129 (1989), no. 2, 363–382.\n","FormalConjectures.Wikipedia.EllipticCurveRank":"# Some conjectures about ranks of elliptic curves over ℚ\n\n\n*References:*\n- [PPVW2016] Jennifer Park, Bjorn Poonen, John Voight, and Melanie Matchett Wood.\n    A heuristic for boundedness of ranks of elliptic curves,\n    https://ems.press/journals/jems/articles/16228\n- [BS2013] Manjul Bhargava and Arul Shankar. The average size of the 5-Selmer group of\n   elliptic curves is 6, and the average rank is less than 1, https://arxiv.org/pdf/1312.7859\n- [Goldfeld1979] Dorian Goldfeld. Conjectures on elliptic curves over quadratic fields,\n   Number Theory Carbondale 1979, Lecture Notes in Math. 751, 108-118,\n   https://doi.org/10.1007/BFb0062705\n- [Smith2025] Alexander Smith. The Birch and Swinnerton-Dyer conjecture implies Goldfeld's\n   conjecture, https://arxiv.org/abs/2503.17619\n- [Wikipedia](https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve)\n- [ICARM](https://elliptic-rank.icarm.cloud/curve/273)\n","FormalConjectures.Wikipedia.ErdosMoser":"# The Erdős–Moser equation\n\nFor positive integers $k$ and $m$, let\n\n$$S_k(m)=1^k+2^k+\\cdots+(m-1)^k.$$\n\nThe Erdős–Moser conjecture says that $S_k(m)=m^k$ has only the solution\n$(k,m)=(1,3)$.\n\n*References:*\n* [Wikipedia: Erdős–Moser equation](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Moser_equation)\n* B. C. Kellner,\n  [On stronger conjectures that imply the Erdős–Moser conjecture](https://arxiv.org/abs/1003.1646)\n","FormalConjectures.Wikipedia.ErdosRadoSunflowerConjecture":"# Erdős–Rado sunflower conjecture\n\nThis file is a Wikipedia-facing entry point for the formalization in\n`FormalConjectures.ErdosProblems.«20»`.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Sunflower_(mathematics))\n","FormalConjectures.Wikipedia.Euclid":"# Euclid Numbers conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Euclid_number)\n","FormalConjectures.Wikipedia.EulerBrick":"# Open questions regarding the existence of Euler bricks\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Euler_brick)\n- [stackexchange](https://math.stackexchange.com/questions/2264401/euler-bricks-and-the-4th-dimension)\n- [Sh12] Shapirov, Ruslan. Perfect cuboids and irreducible polynomials. https://arxiv.org/abs/1108.5348\n","FormalConjectures.Wikipedia.EulerSumOfPowers":"# Euler's sum of powers conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Euler's_sum_of_powers_conjecture)\n","FormalConjectures.Wikipedia.Exponentials":"# Exponentials conjectures and theorems\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Four_exponentials_conjecture)\n","FormalConjectures.Wikipedia.FactorialPrime":"# Factorial primes\n\nA factorial prime is a prime that is one more or one less than a factorial. It\nis conjectured that there are infinitely many factorial primes.\n\n*References:*\n- [Wikipedia, Factorial prime](https://en.wikipedia.org/wiki/Factorial_prime)\n- [OEIS A088054](https://oeis.org/A088054)\n","FormalConjectures.Wikipedia.Falconer":"# Falconer's distance set conjecture\n\nIf $E \\subseteq \\mathbb{R}^d$ is compact with $\\dim_H E > \\frac{d}{2}$, then the distance set\n$$\\{ |x - y| \\mid x, y \\in E \\}$$\nhas positive Lebesgue measure.\n\n## References\n\n* [K. Falconer, *On the Hausdorff dimensions of distance sets*](https://doi.org/10.1112/S0025579300010998)\n* [Wikipedia, *Falconer's conjecture*](https://en.wikipedia.org/wiki/Falconer%27s_conjecture)\n","FormalConjectures.Wikipedia.FeitThompsonPrimeConjecture":"# Feit-Thompson conjecture on primes\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Feit%E2%80%93Thompson_conjecture)\n","FormalConjectures.Wikipedia.Fermat":"# Open questions about Fermat numbers\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Fermat_number)\n","FormalConjectures.Wikipedia.FermatCatalanConjecture":"# Fermat-Catalan conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Fermat-Catalan_conjecture)\n","FormalConjectures.Wikipedia.FibonacciPrimes":"# Fibonacci Primes\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Fibonacci_prime)\n","FormalConjectures.Wikipedia.Firoozbakht":"# Firoozbakht's conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Firoozbakht%27s_conjecture)\n- [primepuzzles](https://www.primepuzzles.net/conjectures/conj_030.htm)\n","FormalConjectures.Wikipedia.FlintCooksonHills":"# Convergence of the Flint Hills and Cookson Hills series\n\n*References:*\n- [Wikipedia: Examples of numerical series](https://en.wikipedia.org/wiki/Series_(mathematics)#Examples_of_numerical_series)\n- [MathWorld: Flint Hills Series](https://mathworld.wolfram.com/FlintHillsSeries.html)\n- [Alekseyev, On the Flint Hills series](https://doi.org/10.48550/arXiv.1104.5100)\n- [MathWorld: Cookson Hills Series](https://mathworld.wolfram.com/CooksonHillsSeries.html)\n","FormalConjectures.Wikipedia.FortuneConjecture":"# Fortune's Conjecture\n\nA *Fortunate number* is the smallest integer $m > 1$ such that $p_n\\\\# + m$ is prime,\nwhere $p_n\\\\#$ denotes the primorial of the $n$-th prime — equivalently, the product\nof the first $n$ primes.\n\n**Fortune's Conjecture** asserts that every Fortunate number is prime — equivalently,\nthat no Fortunate number is composite.\n\nThe conjecture is named after the social anthropologist Reo Fortune, who proposed it.\nThe first few Fortunate numbers are $3, 5, 7, 13, 23, 17, 19, 23, 37, 61, \\ldots$\n(OEIS A005235); all known values are prime.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Fortunate_number)\n- [OEIS A005235](https://oeis.org/A005235)\n- [PrimePages glossary entry](https://primes.utm.edu/glossary/xpage/FortunateNumber.html)\n- [PlanetMath: Fortune's conjecture](https://planetmath.org/fortunesconjecture)\n","FormalConjectures.Wikipedia.Fuglede":"# Fuglede's conjecture in dimensions 1 and 2\n\n*References:*\n- [Fuglede's conjecture](https://en.wikipedia.org/wiki/Fuglede%27s_conjecture)\n","FormalConjectures.Wikipedia.GapConjecture":"# Gap conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Gromov%27s_theorem_on_groups_of_polynomial_growth#The_gap_conjecture)\n- [On the Gap Conjecture concerning group growth](https://arxiv.org/pdf/1202.6044) by\n  *Rostislav Grigorchuk*\n","FormalConjectures.Wikipedia.GaussCircleProblem":"# Gauss circle problem\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Gauss_circle_problem)\n","FormalConjectures.Wikipedia.Gilbreath":"# Gilbreath's conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Gilbreath%27s_conjecture)\n","FormalConjectures.Wikipedia.GoldbachConjecture":"# Goldbach's conjecture\n\n*References:*\n- [Landau Problems Wikipedia Page](https://en.wikipedia.org/wiki/Landau%27s_problems#Twin_prime_conjecture)\n- [Goldbach's Conjecture Wikipedia Page](https://en.wikipedia.org/wiki/Goldbach%27s_conjecture)\n","FormalConjectures.Wikipedia.Goormaghtigh":"# The Goormaghtigh conjecture\n\nA repunit is a number whose digits in some base are all $1$. Here a nontrivial representation has\nat least three digits. The Goormaghtigh conjecture says that $31$ and $8191$ are the only numbers\nhaving nontrivial repunit representations in two different bases.\n\n*References:*\n* [Wikipedia](https://en.wikipedia.org/wiki/Goormaghtigh_conjecture)\n* J. Grantham,\n  [No new Goormaghtigh primes up to $10^{700}$](https://math.colgate.edu/~integers/y98/y98.pdf)\n","FormalConjectures.Wikipedia.GracefulLabeling":"# Graceful Tree Conjecture (Ringel–Kotzig conjecture)\n\n*Reference:* [Wikipedia/Graceful_labeling](https://en.wikipedia.org/wiki/Graceful_labeling)\n\nConjectured by Ringel (1963) and Kotzig; formalized by Rosa (1967).\n","FormalConjectures.Wikipedia.Grimm":"# Grimm's conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Grimm%27s_conjecture)\n","FormalConjectures.Wikipedia.GromovPolynomialGrowth":"# Gromov's theorem on groups of polynomial growth\n\n*Reference:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Gromov%27s_theorem_on_groups_of_polynomial_growth)\n","FormalConjectures.Wikipedia.Hadamard":"# Hadamard's conjecture\n\n*References:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Hadamard_matrix#Hadamard_conjecture)\n - [Résolution d'une question relative aux déterminants](https://gallica.bnf.fr/ark:/12148/bpt6k486252g/f400.image.r) by *Jacques Hadamard*,  Bull. des sciences math., p.245, 1893\n","FormalConjectures.Wikipedia.HadwigerNelson":"# The Hadwiger-Nelson Problem\n\nThe Hadwiger-Nelson problem asks for the minimum number of colors required to\ncolor the plane such that no two points at unit distance from each other have\nthe same color.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem)\n\nThis file points to the canonical formalization in `FormalConjectures.ErdosProblems.«508»`.\n","FormalConjectures.Wikipedia.Hall":"# Hall's conjecture\n\nThere exists a positive number $C$ such that for any integer $x, y$ with $y^2 \\ne x^3$,\n$|y^2 - x^3| > C \\sqrt{|x|}$.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Hall%27s_conjecture)\n- L. Danilov, *The Diophantine equation $x^3 - y^2 = k$ and Hall's conjecture*, Mathematical notes of the Academy of Sciences of the USSR 32 (1982): 617-618\n","FormalConjectures.Wikipedia.HappyEndingProblem":"# Happy Ending Problem\n\nThe happy ending problem asks whether $f(n) = 2^{n-2} + 1$, where $f(n)$ is the\nsmallest number such that any $f(n)$ points in general position in the plane\ncontain $n$ that form a convex polygon.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Happy_ending_problem)\n\nThis file points to the canonical formalization in `FormalConjectures.ErdosProblems.«107»`.\n","FormalConjectures.Wikipedia.HardyLittlewood":"# First Hardy–Littlewood conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/First_Hardy%E2%80%93Littlewood_conjecture)\n","FormalConjectures.Wikipedia.HerzogSchonheimConjecture":"# Herzog–Schönheim conjecture\n\nThis file points to the canonical formalization in `FormalConjectures.ErdosProblems.«274»`.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Herzog%E2%80%93Sch%C3%B6nheim_conjecture)\n- [arXiv:1803.08301](https://arxiv.org/abs/1803.08301)\n- [arXiv:1803.03569](https://arxiv.org/abs/1803.03569)\n- [PMC7247885](https://pmc.ncbi.nlm.nih.gov/articles/PMC7247885/)\n- [arXiv:1804.11103](https://arxiv.org/abs/1804.11103)\n","FormalConjectures.Wikipedia.HilbertFifthProblem":"# Hilbert's Fifth Problem\n\nThe actual formalization is in `FormalConjectures.HilbertProblems.«5»`.\n\nHilbert's fifth problem asks whether every locally Euclidean topological group admits a Lie group\nstructure. This was resolved affirmatively by Gleason, Montgomery, and Zippin in 1952.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem)\n- [Gleason 1952](https://doi.org/10.2307/1969548)\n- [Montgomery–Zippin 1952](https://doi.org/10.2307/1969549)\n- [Tao's blog](https://terrytao.wordpress.com/2011/08/13/the-hilbert-smith-conjecture/)\n","FormalConjectures.Wikipedia.IdonealCompleteness":"# Idoneal numbers completeness conjecture\n\nAn integer $D>0$ is **idoneal** if every\ninteger that can be expressed in exactly one way (up to order and signs)\nas $x^2 + D y^2$ with gcd(x, Dy)=1 is a prime power or twice a prime power.\n\nThe Idoneal Numbers Completeness Conjecture asserts that the following list of\n65 numbers is complete:\n1,2,3,4,5,6,7,8,9,10,12,13,15,16,18,21,22,24,25,28,30,33,37,40,42,45,48,\n57,58,60,70,72,78,85,88,93,102,105,112,120,130,133,165,168,177,190,210,232,\n240,253,273,280,312,330,345,357,385,408,462,520,760,840,1320,1365,1848.\n*References:*\n- [Wikipedia: Idoneal number](https://en.wikipedia.org/wiki/Idoneal_number)\n- [OEIS A000926](https://oeis.org/A000926)\n","FormalConjectures.Wikipedia.InscribedSquare":"# Inscribed square problem\n\nThe *inscribed square problem* or *Toeplitz conjecture* asks whether every Jordan curve (i.e. simple\nclose curve in ℝ²) admits an inscribed square, i.e. a square whose vertices all lie on the curve.\nThere are several open and solved variants of this conjecture.\n\n*References:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Inscribed_square_problem)\n - [A Survey on the Square Peg Problem](https://www.researchgate.net/publication/274622766_A_Survey_on_the_Square_Peg_Problem)\n   by *Benjamin Matschke*\n - [arxiv/2005.09193](https://arxiv.org/abs/2005.09193)\n","FormalConjectures.Wikipedia.InvariantSubspaceProblem":"# Invariant Subspace Problem\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Invariant_subspace_problem),\n[Chalendar-Partington](https://arxiv.org/abs/2507.21834)\n","FormalConjectures.Wikipedia.InverseGalois":"# Inverse Galois problem\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Inverse_Galois_problem)\n","FormalConjectures.Wikipedia.Irrational":"# Open questions on irrationality of numbers\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Irrational_number#Open_questions)\n","FormalConjectures.Wikipedia.JacobianConjecture":"# Jacobian conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Jacobian_conjecture)\n","FormalConjectures.Wikipedia.Jacobson":"# Jacobson Conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Jacobson's_conjecture)\n- [He1965]: Herstein, I. N. (1965), \"A counterexample in Noetherian rings\",\n  Proceedings of the National Academy of Sciences of the United States of America, 54 (4): 1036–1037\n  https://pmc.ncbi.nlm.nih.gov/articles/PMC219788/\n- [Ja1956]: Jacobson, Nathan. Structure of rings. Vol. 37. American Mathematical Soc., 1956.\n- [Le1977]: Lenagan, T. H. (1977), \"Noetherian rings with Krull dimension one\",\n  J. London Math. Soc. Series 2, 15 (1): 41–47\n  https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-15.1.41\n","FormalConjectures.Wikipedia.JugglerConjecture":"# Juggler conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Juggler_sequence)\n","FormalConjectures.Wikipedia.Kakeya":"# Kakeya problem\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Kakeya_set)\n","FormalConjectures.Wikipedia.Kaplansky":"# Kaplansky's Conjectures\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Kaplansky%27s_conjectures)\n","FormalConjectures.Wikipedia.Koethe":"# Köthe conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/K%C3%B6the_conjecture)\n","FormalConjectures.Wikipedia.KomlosConjecture":"# Komlós conjecture\n\nThe Komlós conjecture in discrepancy theory: there is a universal constant $K$ such\nthat for all $n, m$ and all vectors $v\\_1, \\dots, v\\_n \\in \\mathbb{R}^m$ with\n$\\|v\\_i\\|\\_2 \\le 1$, there exist signs $\\varepsilon\\_i \\in \\{-1, +1\\}$ such that\n$$\\left\\|\\sum\\_{i=1}^n \\varepsilon\\_i v\\_i\\right\\|\\_\\infty \\le K.$$\n\nThe best known bound is due to Banaszczyk, who proved that one can always achieve\n$O(\\sqrt{\\log n})$. The Beck–Fiala theorem on the discrepancy of sparse set systems\nis a special case (up to scaling), and the conjecture would imply the Beck–Fiala\nconjecture that set systems of degree $t$ have discrepancy $O(\\sqrt{t})$.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Discrepancy_theory#Major_open_problems)\n- [W. Banaszczyk, *Balancing vectors and Gaussian measures of n-dimensional convex bodies*,\n  Random Structures & Algorithms **12** (1998), 351–360](https://doi.org/10.1002/(SICI)1098-2418(199807)12:4%3C351::AID-RSA3%3E3.0.CO;2-S)\n- [J. Spencer, *Six standard deviations suffice*,\n  Trans. Amer. Math. Soc. **289** (1985), 679–706](https://doi.org/10.1090/S0002-9947-1985-0784009-0)\n","FormalConjectures.Wikipedia.KummerVandiver":"# Kummer–Vandiver conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Kummer%E2%80%93Vandiver_conjecture)\n","FormalConjectures.Wikipedia.LanderParkinAndSelfridgeConjecture":"# Lander, Parkin, and Selfridge Conjecture\n\n**Reference:** https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture\n","FormalConjectures.Wikipedia.LegendreConjecture":"# Legendre's conjecture\n\n*References:*\n- [Landau Problems Wikipedia Page](https://en.wikipedia.org/wiki/Landau%27s_problems#Twin_prime_conjecture)\n- [Legendre Conjecture Wikipedia Page](https://en.wikipedia.org/wiki/Legendre%27s_conjecture)\n- [Luan Alberto Ferreira, *Real exponential sums over primes and prime gaps*](https://arxiv.org/abs/2307.08725)\n","FormalConjectures.Wikipedia.LehmerMahlerMeasureProblem":"# Lehmer's Mahler measure problem\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Lehmer%27s_conjecture)\n","FormalConjectures.Wikipedia.LehmerTotient":"# Lehmer's totient problem\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Lehmer%27s_totient_problem)\n","FormalConjectures.Wikipedia.LeinsterGroup":"# Leinster Groups\n\nA finite group is a Leinster group if the sum of the orders of all its normal subgroups\nequals twice the group's order.\n\n*References:*\n* [Wikipedia](https://en.wikipedia.org/wiki/Leinster_group)\n* Leinster, Tom (2001). \"Perfect numbers and groups\".\n  [arXiv:math/0104012](https://arxiv.org/abs/math/0104012)\n\nTODO: The following properties from the Wikipedia article can also be formalized:\n- There are no Leinster groups that are symmetric or alternating.\n- There is no Leinster group of order p²q² where p, q are primes.\n- No finite semi-simple group is Leinster.\n- No p-group can be a Leinster group.\n- All abelian Leinster groups are cyclic with order equal to a perfect number.\n","FormalConjectures.Wikipedia.Lemoine":"# Lemoine's conjectures\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/%C3%89mile_Lemoine#Lemoine's_conjecture_and_extensions)\n- [Ki85] Kiltinen, J. and Young P. (1985). Goldbach, Lemoine, and a Know/Don't Know Problem.\n","FormalConjectures.Wikipedia.LittlewoodConjecture":"# Littlewood conjectures\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Littlewood_conjecture)\n- [Bernard de Mathan and Olivier Teulié, *Problèmes diophantiens simultanés*](https://doi.org/10.1007/s00605-003-0199-y)\n","FormalConjectures.Wikipedia.LonelyRunnerConjecture":"# Lonely runner conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Lonely_runner_conjecture)\n","FormalConjectures.Wikipedia.LovaszPlummerConjecture":"# The Lovász–Plummer conjecture (proved 2011) and Sheehan's conjecture\n\n*References:*\n* [Wikipedia](https://en.wikipedia.org/wiki/Petersen%27s_theorem#Related_conjectures)\n* [LP86] Lovász, L. and Plummer, M. D. (1986). *Matching Theory.* North-Holland.\n* [EKKKN11] Esperet, L., Kardoš, F., King, A. D., Král', D. and Norine, S. (2011).\n  \"Exponentially many perfect matchings in cubic graphs.\" *Adv. Math.* 227, pp. 1646--1664.\n  [arXiv:1012.2878](https://arxiv.org/abs/1012.2878)\n* [Sh77] Sheehan, J. (1977). \"The multiplicity of Hamiltonian circuits in a graph.\" In *Recent\n  Advances in Graph Theory*, Academia, Prague, pp. 477--480.\n* [Th98] Thomassen, C. (1998). \"Independent dominating sets and a second Hamiltonian cycle in\n  regular graphs.\" *J. Combin. Theory Ser. B* 72, pp. 104--109.\n","FormalConjectures.Wikipedia.LychrelNumbers":"# Lychrel numbers in base 10\n\nA (base-10) *Lychrel number* is a positive integer which never becomes a palindrome under the\niteration\n\n$$a_{0} = n, \\qquad a_{k+1} = a_k + \\operatorname{rev}_{10}(a_k).$$\n\nOne commonly stated conjectural direction is that there are no Lychrel numbers in base 10.\nThe smallest widely studied open case is `196`.\n\n*References:*\n* [Wikipedia: Lychrel number](https://en.wikipedia.org/wiki/Lychrel_number)\n* [MathWorld: Lychrel Number](https://mathworld.wolfram.com/LychrelNumber.html)\n* [OEIS A023108](https://oeis.org/A023108)\n* [OEIS A023109](https://oeis.org/A023109)\n","FormalConjectures.Wikipedia.MagicSquares":"# Magic Squares\n\n*References:*\n\n* [Magic Square of Squares - Wikipedia](https://en.wikipedia.org/wiki/Magic_square_of_squares)\n* [multimagie.com](http://www.multimagie.com/English/SquaresOfSquaresSearch.htm)\n* [Semi-Magic Square of Cubes](https://unsolvedproblems.org/index_files/SquareofCubes.htm)\n* [Magic Square of Squares](https://static.nsta.org/pdfs/QuantumV6N3.pdf)\n","FormalConjectures.Wikipedia.Mahler32":"# Mahler's 3/2 Problem\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Mahler%27s_3/2_problem)\n","FormalConjectures.Wikipedia.Mandelbrot":"# Conjectures about the Mandelbrot and Multibrot sets\nThis file adds three conjectures about the Mandelbrot and Multibrot sets:\n- the *MLC conjecture*, stating that these sets are locally connected\n- the *density of hyperbolicity* conjecture, stating that parameters with attracting cycles are\n  dense in the Mandelbrot and Multibrot sets\n- the conjecture that the boundaries of these sets have zero area.\nThe first two conjectures are related in that the former implies the latter.\n\n*References:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Mandelbrot_set#Local_connectivity)\n - [arxiv/math/9902155](https://arxiv.org/abs/math/9902155)\n - [mathoverflow/37229](https://mathoverflow.net/questions/37229/)\n","FormalConjectures.Wikipedia.MeanValueProblem":"# Mean value problem\n\n*Reference:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Mean_value_problem)\n- [The fundamental theorem of algebra and complexity theory](https://www.ams.org/journals/bull/1981-04-01/S0273-0979-1981-14858-8/)\nby Steve Smale\n\nGiven a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$\nthere is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ K* |p'(z)|$ for $K=1$.\n\n\nThe conjecture has been proven for:\n* `K = 4`\n  [The fundamental theorem of algebra and complexity theory](https://www.ams.org/journals/bull/1981-04-01/S0273-0979-1981-14858-8/)\n  by *Steve Smale*\n* `K = (d-1)/d` if $p$ has real roots or all the roots of $p$ have the same norm.\n  [Critical points and values of complex polynomials](https://doi.org/10.1016/0885-064X(89)90019-8)\n  by *David Tischler*\n","FormalConjectures.Wikipedia.Mersenne":"# Conjectures about Mersenne primes\n\n*References:*\n- [Wikipedia: Mersenne conjectures](https://en.wikipedia.org/wiki/Mersenne_conjectures)\n- [Wikipedia: Catalan's Mersenne conjecture](https://en.wikipedia.org/wiki/Catalan%27s_Mersenne_conjecture)\n- [MathWorld: Catalan-Mersenne Number](https://mathworld.wolfram.com/Catalan-MersenneNumber.html)\n","FormalConjectures.Wikipedia.Mills":"# Mills' Theorem\n\nThere exists a real $A > 1$ such that\n$\\lfloor A^{3^n}\\rfloor$ is prime for every positive integer $n$, where $\\lfloor\\cdot\\rfloor$\ndenotes the floor function.\n\nThe least such $A$ is known as *Mills' constant*. It is irrational, and assuming the\nRiemann hypothesis it is approximately $1.3063778838\\ldots$.\n\n*References:*\n- [Wikipedia, Mills' constant](https://en.wikipedia.org/wiki/Mills%27_constant)\n- [A prime-representing function](https://www.ams.org/journals/bull/1947-53-06/S0002-9904-1947-08849-2/)\n  by *W. H. Mills*, Bull. Amer. Math. Soc. **53** (1947), 604.\n- [Mills' constant](https://mathworld.wolfram.com/MillsConstant.html) on Wolfram MathWorld.\n- [Mills' constant is irrational](https://doi.org/10.1112/mtk.70027) by *Kota Saito*,\n  Mathematika **71** (2025), no. 3, e70027, [arXiv:2404.19461](https://arxiv.org/abs/2404.19461).\n- [Determining Mills' Constant ..](https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.pdf)\n  by *Chris K. Caldwell and Yuanyou Cheng*, J. Integer Seq. **8** (2005), Article 05.4.1.\n- [OEIS A051021](https://oeis.org/A051021) (decimal expansion of Mills' constant)\n","FormalConjectures.Wikipedia.MinimalOverlapProblem":"# Minimum Overlap Problem\n\nThe minimum overlap problem asks for the limit of the minimum, over all\nsplittings of $\\{1, \\ldots, 2n\\}$ into two sets $A$ and $B$ of equal size, of\nthe maximum number of representations of any integer $k$ as $a - b$ with\n$a \\in A$, $b \\in B$, divided by $n$.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Minimum_overlap_problem)\n\nThis file points to the canonical formalization in `FormalConjectures.ErdosProblems.«36»`.\n","FormalConjectures.Wikipedia.ModularityConjecture":"# Modularity conjecture\n\nThe **Modularity conjecture** (also know as the Shimura-Taniyama-Weil conjecture) states that\nevery rational elliptic curve is modular, meaning that it can be\nassociated with a modular form. We state the `a_p` version of the conjecture, which relates the\ncoefficients of the modular form to the number of points on the elliptic curve over finite fields.\n\nSince we don't have the conductor of the elliptic curve, our definition of `a_p(E)` differs from\nthat in the literature at primes of bad reduction. For this reason, we state the conjecture with the\nassumption that `p ∤ N`, in order to give an equivalent statement.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Modularity_theorem)\n- [F. Diamond and J. Shurman, *A First Course in Modular Forms*](https://doi.org/10.1007/978-0-387-27226-9)\n\n","FormalConjectures.Wikipedia.MoserWorm":"# Moser's Worm\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Moser%27s_worm_problem)\n","FormalConjectures.Wikipedia.MovingSofa":"# Moving Sofa Problem\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Moving_sofa_problem)\n- [Ge92] Gerver, J. L., _On moving a sofa around a corner_. Geometriae Dedicata 42.3 (1992): 267-283.\n- [Ro18] Romik, D. _Differential equations and exact solutions in the moving sofa problem_. Experimental mathematics 27.3 (2018): 316-330.\n- [Ba24] Baek, J. _Optimality of Gerver's Sofa_. arXiv preprint arXiv:2411.19826 (2024).\n","FormalConjectures.Wikipedia.NoThreeInLineProblem":"# No-three-in-line problem\n\nThis file is a Wikipedia-facing entry point for the formalization in\n`FormalConjectures.GreensOpenProblems.«72»`.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/No-three-in-line_problem)\n","FormalConjectures.Wikipedia.NoetherProblem":"# Rational_variety\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Rational_variety)\n","FormalConjectures.Wikipedia.NormalityOfPi":"# Normality of mathematical constants\n\nDespite extensive empirical evidence—billions of digits have been computed for $\\pi$,\n$e$, and $\\sqrt{2}$, all showing near-uniform digit distribution—it is an open problem\nwhether any of the classical constants $\\pi$, $e$, $\\sqrt{2}$, $\\ln 2$, or $\\varphi$ is\nnormal in any base.\n\n*References:*\n* [Wikipedia (Normal number)](https://en.wikipedia.org/wiki/Normal_number)\n* [Wikipedia (Pi)](https://en.wikipedia.org/wiki/Pi)\n","FormalConjectures.Wikipedia.OddWeirdNumber":"# Existence of Odd Weird Numbers\n\n*References:*\n- [Wikipedia] (https://en.wikipedia.org/wiki/Weird_number)\n- [A006037] (https://oeis.org/A006037)\n","FormalConjectures.Wikipedia.Oppermann":"# Oppermann's Conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Oppermann%27s_conjecture)\n- [Luan Alberto Ferreira, *Real exponential sums over primes and prime gaps*](https://arxiv.org/abs/2307.08725)\n","FormalConjectures.Wikipedia.PebblingNumberConjecture":"# Pebbling number conjecture\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Graph_pebbling)\n- [Pebbling on Graph Products and Other Binary Graph Constructions](https://arxiv.org/abs/1801.07808)\n","FormalConjectures.Wikipedia.Pell":"# Infinitude of Pell number primes\n\n*References:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Pell_number#Primes_and_squares)\n - [A86383](https://oeis.org/A86383)\n\nThe Pell numbers $P_n$ are defined by $P_0 = 0$,\n$P_1 = 1$, $P_{n+2} = 2*P_{n+1} + P_n$. [OEIS A129](https://oeis.org/A129)\n\nThe conjecture says that there are infinitely many prime Pell numbers.\n","FormalConjectures.Wikipedia.PerfectNumbers":"# Perfect numbers\n\nA perfect number is a positive integer that equals the sum of its proper divisors\n(i.e., all its positive divisors excluding the number itself).\n\nFor example, 6 is perfect because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6.\nSimilarly, 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28.\n\nAll known perfect numbers are even. Several open problems about perfect numbers are\nformalised here:\n\n* Are there infinitely many perfect numbers?\n* Are there infinitely many even perfect numbers?\n* Do odd perfect numbers exist?\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Perfect_number)\n- [Wikipedia, Odd perfect numbers](https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers)\n","FormalConjectures.Wikipedia.PierceBirkhoff":"# Pierce–Birkhoff conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Pierce%E2%80%93Birkhoff_conjecture)\n\nThe Pierce-Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed\nas a maximum of finite minima of finite collections of polynomials. It was first stated in 1956\nby Garrett Birkhoff and Richard S. Pierce, though the modern rigorous formulation is due to\nMelvin Henriksen and John R. Isbell.\n\nThe conjecture has been proved for `n = 1` and `n = 2` by Louis Mahé.\n","FormalConjectures.Wikipedia.PierpontPrime":"# Pierpont primes\n\nA Pierpont prime is a prime of the form $2^a 3^b + 1$, where $a$ and $b$ are\nnonnegative integers. Marc Gleason conjectured that there are infinitely many.\n\n*References:*\n- [Wikipedia, Pierpont prime](https://en.wikipedia.org/wiki/Pierpont_prime)\n- [OEIS A005109](https://oeis.org/A005109)\n","FormalConjectures.Wikipedia.PollocksConjecture":"# Pollock's (tetrahedral numbers) conjecture\n\nEvery positive integer is the sum of at most 5 tetrahedral numbers.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Pollock%27s_conjectures)\n- [A797](https://oeis.org/A797)\n- L. E. Dickson, *History of the Theory of Numbers, Vol. II: Diophantine Analysis*, Dover (2005), pp. 22–23\n- Frederick Pollock, *On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant*, Abstracts of the Papers Communicated to the Royal Society of London **5** (1850), 922–924\n- H. E. Salzer and N. Levine, *Table of integers not exceeding 100000 that are not expressible as the sum of four tetrahedral numbers*, Math. Comp. **12** (1958), 141–144\n- [MathWorld: Pollock's Conjecture](https://mathworld.wolfram.com/PollocksConjecture.html)\n","FormalConjectures.Wikipedia.PolyTimeFunctions":"# Polynomial-time computability of factoring\n\nThis file formalizes the open problem of whether natural number / integer factorization can be computed in\npolynomial time on a deterministic classical Turing machine.\n\nMore precisely, it formalizes the following statement:\nCan the prime factorization of a natural number be computed in polynomial time\n(on a deterministic classical Turing machine)?\n\n*References:*\n- [Wikipedia: List of unsolved problems in computer science](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science)\n- [Wikipedia: Integer factorization](https://en.wikipedia.org/wiki/Integer_factorization)\n\n","FormalConjectures.Wikipedia.PompeiuProblem":"# The Pompeiu problem\n\nLet $\\Omega \\subset \\mathbb{R}^{N+1}$ be a bounded domain. A **rigid motion** is an isometry of\n$\\mathbb{R}^{N+1}$ (a composition of translations and rotations), modelled here by an\n`AffineIsometryEquiv` of Euclidean space with itself.\n\nThe domain $\\Omega$ has the **Pompeiu property** if the only continuous function\n$f : \\mathbb{R}^{N+1} \\to \\mathbb{R}$ with $\\int_{\\sigma(\\Omega)} f = 0$ for every rigid motion\n$\\sigma$ is $f \\equiv 0$.\n\nThe **Pompeiu problem** (Pompeiu's conjecture) asserted that, among bounded simply connected\ndomains with Lipschitz boundary, the Euclidean ball is the *only* domain that **fails** to have\nthe Pompeiu property. That a ball fails is classical: radial functions built from Bessel\nfunctions have vanishing integral over every congruent ball. The converse is false. Two\nindependent proofs of this negative conclusion are referenced below.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Pompeiu_problem)\n- [Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Pompeiu_problem)\n- [BST73] Brown, L., Schreiber, B. M. and Taylor, B. A., *Spectral synthesis and the Pompeiu\n  problem*, Ann. Inst. Fourier 23 (1973), 125–154. (Proves a ball fails the Pompeiu property;\n  for a ball of radius `R` the witness is `f(x) = sin(a x₁)` with `J_{n/2}(a R) = 0`.)\n- Zalcman, L., *A bibliographic survey of the Pompeiu problem*, in Approximation by\n  Solutions of Partial Differential Equations, NATO ASI Ser. 365 (1992), 185–194.\n- [CLD26] [Cao-Labora, G. and de Dios Pont, J., *Schiffer*](https://github.com/jaumededios/Schiffer),\n  a Lean 4 formalization of a non-disk planar counterexample.\n- [CS26] Colbrook, M. J. and Stepaniants, G., *A computer-assisted counterexample to the planar\n  Pompeiu and Schiffer conjectures*, [arXiv:2608.01579](https://arxiv.org/abs/2608.01579) (2026).\n","FormalConjectures.Wikipedia.PowerfulNumbersDensity":"# Asymptotic density of powerful numbers\n\nLet $Q(x)$ denote the number of powerful integers up to $x$. Erdős and Szekeres [ES35] proved\n$$Q(x) = \\frac{\\zeta(3/2)}{\\zeta(3)} x^{1/2} + O(x^{1/3}),$$\nand Bateman and Grosswald [BG58] sharpened this to\n$$Q(x) = \\frac{\\zeta(3/2)}{\\zeta(3)} x^{1/2} + \\frac{\\zeta(2/3)}{\\zeta(2)} x^{1/3} + O(x^{1/6}).$$\nImproving the exponent $1/6$ in the error term unconditionally remains open; conditional\nimprovements are known under the Riemann Hypothesis.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Powerful_number)\n- [ES35] Erdős, P. and Szekeres, G., *Über die Anzahl der Abelschen Gruppen gegebener Ordnung\n  und über ein verwandtes zahlentheoretisches Problem*, Acta Sci. Math. (Szeged) 7 (1935), 95–102.\n- [BG58] Bateman, P. T. and Grosswald, E., *On a theorem of Erdős and Szekeres*,\n  Illinois J. Math. 2 (1958), 88–98.\n","FormalConjectures.Wikipedia.PrimeTriplets":"# Prime Triplet Conjecture\n\n*Reference:* [Prime Triplet Wikipedia Page](https://en.wikipedia.org/wiki/Prime_triplet#Conjecture_on_prime_triplets)\n","FormalConjectures.Wikipedia.PrimesAndPerfectSquares":"# Primes and perfect squares\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Landau%27s_problems#Near-square_primes)\n","FormalConjectures.Wikipedia.QuasiperfectNumbers":"# Quasiperfect Numbers\n\n*Reference:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Quasiperfect_number)\n","FormalConjectures.Wikipedia.RamanujanTau":"# Ramanujan τ-function\n\nThere are two conjectures related to the Ramanujan τ-function:\n\n- Ramanujan-Petersson conjecture: For every prime `p`, the absolute value of the\n  Ramanujan τ-function at `p` is bounded by `2 * p^(11/2)`.\n- Lehmer's conjecture: The Ramanujan τ-function is never zero for any positive integer `n`.\n\n*References:*\n- [Ramanujan-Petersson conjecture](https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture)\n- [Lehmer's conjecture](https://en.wikipedia.org/wiki/Ramanujan_tau_function#Conjectures_on_the_tau_function)\n","FormalConjectures.Wikipedia.RamseyNumbers":"# Ramsey numbers\n\nThe (graph) Ramsey number $R(k,\\ell)$ is the least natural number $n$ such that every simple graph\non $n$ vertices contains either a clique of size $k$ or an independent set of size $\\ell$\n(equivalently, the complement graph contains a clique of size $\\ell$).\n\nWe formalize the classical open problem of determining $R(5,5)$, together with the currently best\nknown bounds $43 \\le R(5,5) \\le 46$.\n\nNote: the diagonal Ramsey number $R(n,n)$ can also be formulated in terms of 2-colorings of\n$2$-subsets, as `Combinatorics.hypergraphRamsey 2 n` (see `FormalConjecturesForMathlib/Combinatorics/Ramsey.lean`).\n\n*References:*\n- [Wikipedia: Ramsey number](https://en.wikipedia.org/wiki/Ramsey_number)\n- [Rad] S. P. Radziszowski, *Small Ramsey Numbers*, Electronic Journal of Combinatorics, Dynamic\n  Survey DS1. (Updated periodically.) https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1\n- [Exoo89] G. Exoo, *A lower bound for* $R(5,5)$, Journal of Graph Theory 13 (1989), 97–98.\n  DOI: 10.1002/jgt.3190130113\n- [AM24] V. Angeltveit and B. McKay, *$R(5,5) \\le 46$*, arXiv:2409.15709 (2024).\n- [OEIS A212954](https://oeis.org/A212954)\n- [MathWorld: Ramsey Number](https://mathworld.wolfram.com/RamseyNumber.html)\n","FormalConjectures.Wikipedia.RationalDistanceProblem":"# Rational distance problem\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Unit_square#Rational_distance_problem)\n- [mathoverflow/418260](https://mathoverflow.net/questions/418260/)\nasked by user [Yuan Yang](https://mathoverflow.net/users/177957/yuan-yang)\n- D19 in [Unsolved Problems in Number Theory](https://doi.org/10.1007/978-0-387-26677-0)\nby *Richard K. Guy*\n","FormalConjectures.Wikipedia.RegularPrimes":"# Infinite Regular Primes\n\nWe define the notion of regular primes, which are prime numbers that are coprime with the\ncardinality of the class group of the `p`-th cyclotomic field. We also state that there are\ninfinitely many regular primes.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Regular_prime)\n","FormalConjectures.Wikipedia.RiemannZetaValues":"# Particular values of the Riemann zeta function\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function)\n","FormalConjectures.Wikipedia.RudinsConjecture":"# Rudin's conjecture on squares in arithmetic progressions\n\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Rudin%27s_conjecture)\n- [Ru60] Rudin, W., *Trigonometric series with gaps*, J. Math. Mech. 9 (1960), 203–227.\n- González-Jiménez, E. and Xarles, X., *On a conjecture of Rudin on squares in arithmetic\n  progressions*, LMS J. Comput. Math. 17 (2014), 58–76.\n","FormalConjectures.Wikipedia.SatoTateConjecture":"# Sato–Tate conjecture\n\nThe **Sato–Tate conjecture** describes the distribution of the normalized Frobenius traces\n$a_p(E)/(2\\sqrt{p})$ of a non-CM elliptic curve $E$ over $\\mathbb{Q}$, as $p$ ranges over\nthe primes of good reduction: they equidistribute in $[-1,1]$ with respect to the\n**Sato–Tate measure**\n$$\n\\frac{2}{\\pi}\\sqrt{1-x^2}\\,dx.\n$$\n\nOriginally conjectured independently by Mikio Sato and John Tate around 1960, it is\nnow a theorem: the case of elliptic curves over totally real fields with nonintegral\n$j$-invariant was established through the work of Clozel, Harris, and Taylor [CHT08],\nTaylor [Tay08], and Harris, Shepherd-Barron, and Taylor [HST10]. The remaining non-CM\ncase was settled by Barnet-Lamb, Geraghty, Harris, and Taylor [BGHT11]. In particular,\nit holds unconditionally for every non-CM elliptic curve over $\\mathbb{Q}$.\n\nWe use `ModularityConjecture.WeierstrassCurve.ap`, which agrees with the Frobenius\ntrace at every prime where the supplied Weierstrass equation has integral\ncoefficients and nonsingular reduction. Only finitely many primes are exceptional.\nIncluding these primes does not change the limiting density, so the formal\nstatement averages over all primes below $N$.\n\nThe non-CM hypothesis is expressed using the classification of rational CM\n$j$-invariants. Complex multiplication here means complex multiplication over\n$\\overline{\\mathbb{Q}}$.\n\n## References\n\n- [CHT08] L. Clozel, M. Harris, R. Taylor, *Automorphy for some $l$-adic lifts of\n  automorphic mod $l$ Galois representations*, Publications Mathématiques de l'IHÉS\n  108 (2008), 1–181. https://doi.org/10.1007/s10240-008-0016-1\n- [Tay08] R. Taylor, *Automorphy for some $l$-adic lifts of automorphic mod $l$\n  Galois representations. II*, Publications Mathématiques de l'IHÉS\n  108 (2008), 183–239. https://doi.org/10.1007/s10240-008-0015-2\n- [HST10] M. Harris, N. Shepherd-Barron, R. Taylor, *A family of Calabi-Yau varieties\n  and potential automorphy*, Annals of Mathematics 171 (2010), no. 2, 779–813.\n  https://doi.org/10.4007/annals.2010.171.779\n- [BGHT11] T. Barnet-Lamb, D. Geraghty, M. Harris, R. Taylor, *A family of Calabi-Yau\n  varieties and potential automorphy II*, Publications of the Research Institute\n  for Mathematical Sciences 47 (2011), no. 1, 29–98.\n  https://doi.org/10.2977/PRIMS/31\n- [Wikipedia](https://en.wikipedia.org/wiki/Sato%E2%80%93Tate_conjecture)\n","FormalConjectures.Wikipedia.Schanuel":"# Schanuel's Conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Schanuel%27s_conjecture)\n","FormalConjectures.Wikipedia.Schinzel":"# Hypothesis H\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H)\n","FormalConjectures.Wikipedia.ScholzConjecture":"# Scholz conjecture on addition chains\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Scholz_conjecture)\n- [MathWorld](https://mathworld.wolfram.com/ScholzConjecture.html)\n- [Tall22](https://arxiv.org/abs/2210.13812) Amadou Tall. \"The Scholz conjecture on addition\n  chain is true for infinitely many integers with $\\ell(2n) = \\ell(n)$.\" _arXiv:2210.13812_ (2022).\n  Also available as [ePrint 2023/020](https://eprint.iacr.org/2023/020).\n- [OEIS A003313](https://oeis.org/A003313)\n","FormalConjectures.Wikipedia.Selfridge":"# Selfridge's conjectures\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/John_Selfridge#Selfridge's_conjecture_about_primality_testing)\n","FormalConjectures.Wikipedia.Sendov":"# Sendov's conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Sendov%27s_conjecture)\n\nTags: Sendov Conjecture, Ilieff's Conjecture.\n\n","FormalConjectures.Wikipedia.SidorenkoConjecture":"# Sidorenko's conjecture (1993)\n\n*References:*\n* [Wikipedia](https://en.wikipedia.org/wiki/Sidorenko%27s_conjecture)\n* [Si93] Sidorenko, A. (1993). \"A correlation inequality for bipartite graphs.\"\n  *Graphs Combin.* 9, pp. 201--204.\n* [CoFo10] Conlon, D. and Fox, J. (2010). \"Bounds for graph regularity and removal lemmas.\"\n  *Geom. Funct. Anal.* 22, pp. 1191--1256.\n* [KLL18] Kim, J.H., Lee, C., Lee, J. (2018). \"Two approaches to Sidorenko's conjecture.\"\n  *Trans. Amer. Math. Soc.* 370, pp. 8515--8552.\n* [ArXiv2605] [arXiv:2605.14138](https://arxiv.org/abs/2605.14138)\n* [BR65] Blakley, G. R. and Roy, P. (1965). \"A Hölder type inequality for symmetric matrices\n  with nonnegative entries.\" *Proc. Amer. Math. Soc.* 16, pp. 1244--1245.\n","FormalConjectures.Wikipedia.SierpinskiNumber":"# Sierpiński number\n\n*References:*\n- [Wikipedia, Sierpiński number](https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number)\n- [Si60] Sierpiński, W., Elementary Theory of Numbers. Państwowe Wydawnictwo Naukowe,\n  Warsaw (1960).\n\nA positive odd integer $k$ is a *Sierpiński number* if $k \\cdot 2^n + 1$ is composite for all\nnatural numbers $n$. In 1960, Sierpiński proved that there are infinitely many such numbers.\nJohn Selfridge proved in 1962 that 78557 is a Sierpiński number. It is conjectured to be the\nsmallest.\n\n## Sierpiński problem\n\nThe *Sierpiński problem* asks: is 78557 the smallest Sierpiński number?\n\n## Prime Sierpiński problem\n\nThe *prime Sierpiński problem* asks: is 271129 the smallest *prime* Sierpiński number?\n\n## Extended Sierpiński problem\n\nThe *extended Sierpiński problem* asks: is 271129 the second-smallest Sierpiński number?\n","FormalConjectures.Wikipedia.Singmaster":"# Singmaster's conjecture\n\nSingmaster's conjecture says that for any integer $t>1$, the number of solutions to the equation:\n\n$\\binom{n}{k} = t,\\quad 1 \\le k < n,$\n\nwith $\\binom{n}{k}$ being the numbers that appear in Pascal's triangle, is bounded by a global\nconstant $O(1)$.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Singmaster%27s_conjecture)\n","FormalConjectures.Wikipedia.SixStandardDeviations":"# Six standard deviations suffice (Spencer's theorem)\n\nSpencer's theorem in discrepancy theory: for every $n$ and every family of $n$ subsets\n$S_1, \\dots, S_n$ of $\\{1, \\dots, n\\}$, there is a colouring\n$\\chi : \\{1, \\dots, n\\} \\to \\{-1, +1\\}$ such that\n$$\\left|\\sum_{j \\in S_i} \\chi(j)\\right| \\le 6\\sqrt{n}$$\nfor every $i$.\n\nA uniformly random colouring only achieves the bound $O(\\sqrt{n \\log n})$, so Spencer's\ntheorem (\"six standard deviations suffice\") is a genuine improvement over the basic\nprobabilistic method; its proof uses a partial-colouring argument based on the entropy\nmethod, and the bound $\\Theta(\\sqrt{n})$ is tight up to the value of the constant.\nSpencer's original argument is non-constructive; polynomial-time algorithms achieving\n$O(\\sqrt{n})$ were later found by Bansal (2010) and Lovett–Meka (2012).\n\nThe Komlós conjecture (the subject of a companion file in this directory) would\nstrengthen this theorem: applied to the incidence vectors of the sets, scaled by\n$1/\\sqrt{n}$, it would give a discrepancy bound $O(\\sqrt{n})$ for any number of sets,\nnot just $n$ of them.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Discrepancy_of_hypergraphs#General_hypergraphs)\n- [J. Spencer, *Six standard deviations suffice*,\n  Trans. Amer. Math. Soc. **289** (1985), 679–706](https://doi.org/10.1090/S0002-9947-1985-0784009-0)\n- [N. Bansal, *Constructive algorithms for discrepancy minimization*,\n  FOCS 2010, 3–10](https://doi.org/10.1109/FOCS.2010.7)\n- [S. Lovett and R. Meka, *Constructive discrepancy minimization by walking on the edges*,\n  SIAM J. Comput. **44** (2015), 1573–1582](https://doi.org/10.1137/130929400)\n","FormalConjectures.Wikipedia.SnakeInTheBox":"# Snake in the box\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Snake-in-the-box)\n- [Hypercube](https://en.wikipedia.org/wiki/Hypercube_graph)\n- [xkcd](https://xkcd.com/3125/)\n","FormalConjectures.Wikipedia.SolitaryNumber":"# Solitary Numbers\n\n\n*References:*\n- [Solitary number (Wikipedia)](https://en.wikipedia.org/wiki/Solitary_number)\n- [Solitary number large clubs (Wikipedia)](https://en.wikipedia.org/wiki/Solitary_number#Large_clubs)\n","FormalConjectures.Wikipedia.SparseRuler":"# Sparse Ruler\n\nA sparse ruler of length $L$ is a sequence of marks $0 = a_1 < a_2 < \\dots < a_m = L$.\nA distance $k \\in \\mathbb{N}$ can be measured if there are $i, j \\in \\{1, \\dots, m\\}$, such that\n$k = a_j - a_i$.\n\nOne question concerns the structure of *optimal* rulers. Wichmann [Wi63] gave a\nparametric family of sparse rulers and conjectured that, beyond a small number of exceptions,\nevery optimal ruler is of his type. The known exceptions occur at lengths $1, 13, 17, 23, 58$;\nthe largest of these has $13$ segments, which motivates the bound below.\n\nThe asymptotic growth of the minimal number of marks of an optimal ruler of length $L$ — i.e.\nthe limit of $l(n)^2 / n$, conjectured to lie in $[2.434\\ldots, 3]$ — is the subject of\n`FormalConjectures.ErdosProblems.«170»` (there phrased via $F(N)/\\sqrt{N}$), and is not\nrestated here.\n\n*References:*\n- [Wi63] Wichmann, B. \"A note on restricted difference bases.\" Journal of the London\n  Mathematical Society 38 (1963): 465-466.\n- [Wikipedia](https://en.wikipedia.org/wiki/Sparse_ruler)\n","FormalConjectures.Wikipedia.SquarePacking":"# Packing\n\nThis file contains a number of open problems related to the minimal size of a square (or circle)\nthat can contain a given number of unit squares (or circles).\nIn each case, we provide a known upper bound, and ask for the least such size.\n\n*References:*\n- [Wikipedia on packing of squares](https://en.wikipedia.org/wiki/Square_packing)\n- [Wikipedia on packing of circles in a circle](https://en.wikipedia.org/wiki/Circle_packing_in_a_circle)\n- [Wikipedia on packing of circles in a square](https://en.wikipedia.org/wiki/Circle_packing_in_a_square)\n- Friedman, Erich (2009), \"Packing unit squares in squares: a survey and new results\",\n  Electronic Journal of Combinatorics, 1000, Dynamic Survey 7\n- Pirl, U. (1969),\n  [\"Der Mindestabstand von $n$ in der Einheitskreisscheibe gelegenen Punkten\"](https://doi.org/10.1002/mana.19690400110),\n  Mathematische Nachrichten, 40: 111–124\n- A website with visualizations of packings:\n  [link](https://erich-friedman.github.io/packing/)\n","FormalConjectures.Wikipedia.SteinerSystem":"# Steiner Systems\n\nA Steiner system $S(t, k, n)$ is a collection of $k$-element subsets (called blocks) of\nan $n$-element set such that every $t$-element subset is contained in exactly one block.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Steiner_system)\n- [Large Steiner Systems](https://epoch.ai/frontiermath/open-problems/large-steiner-systems)\n  by Kunal Marwaha\n","FormalConjectures.Wikipedia.SumOfThreeCubes":"# Sum of three cubes\n\nAn integer `n : ℤ` can be written as a sum of three cubes (of integers) if and only if\n`n` is not `4` or `5` mod `9`.\n\n*References:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Sums_of_three_cubes)\n - [mathoverflow/100324](https://mathoverflow.net/a/100324)\nasked by user [*David Feldman*](https://mathoverflow.net/users/10909/david-feldman)\n","FormalConjectures.Wikipedia.Superperfectnumbers":"# (m,k)-perfect numbers\n\nAn integer `n : ℤ` is `(m,k)-perfect` if `σᵐ(n) = kn` where `σᵐ` is the mᵗʰ iterate of the\nsum of divisors function.\n\n*References:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Superperfect_number#Generalizations)\n - [Wikipedia](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#General)\n","FormalConjectures.Wikipedia.SurjunctiveGroup":"# Gottschalk's surjunctivity conjecture\n\nA group $G$ is *surjunctive* if every injective, continuous, $G$-equivariant map\n$A^G \\to A^G$ (for any finite alphabet $A$) is surjective.\n\nHere equivariance is with respect to the left shift action of $G$ on $A^G$,\ndefined by $(g \\cdot x)(h) = x(g^{-1} h)$, and continuity is with respect to\nthe product topology on $A^G$ (where $A$ carries the discrete topology).\n\nGottschalk's conjecture (1973) states that every group is surjunctive.\n\n*References:*\n- [Wikipedia](https://en.wikipedia.org/wiki/Surjunctive_group)\n- Gottschalk, W. H. (1973), \"Some general dynamical notions\"\n","FormalConjectures.Wikipedia.Taxicab":"# Taxicab numbers\n\nA *taxicab number* for natural numbers $k, m, n$ is the\nsmallest number $x$ that can be expressed as a sum of $m$\npositive $k$-th powers in at least $n$ distinct ways. The\nmost famous taxicab number is\n$ 1729 = 1³ + 12³ = 9³ + 10³, $\nalso known as the Hardy–Ramanujan number.\n\nHowever, a taxicab number is not known for $k=5$, $m=2$, and any $n ≥ 2$:\nNo positive integer is known that can be written as the\nsum of two 5th powers in more than one way, and it is not\nknown whether such a number exists.\n\nIn particular, it is not known whether there exists a\ntaxicab number for $k=5$, $m=2$, and $n=2$.\n\n*References:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Taxicab_number)\n - [Generalized taxicab number](https://en.wikipedia.org/wiki/Generalized_taxicab_number)\n - [OEIS taxicab cubes](https://oeis.org/A001235)\n - [OEIS taxicab 4th powers](https://oeis.org/A018786)\n - [OEIS taxicab conjecture](https://oeis.org/A088703)\n","FormalConjectures.Wikipedia.Toronto":"# Toronto spaces\n\nA *Toronto space* is a topological space\nwhich is homeomorphic to all of its subspaces of same cardinality.\n\nIt is conjectured that every T2, Toronto space is discrete.\nW.R. Brian proved that this holds under GCH.\n\n*References:*\n - [Wikipedia](https://en.wikipedia.org/wiki/Toronto_space)\n - [The Toronto problem](https://wrbrian.wordpress.com/wp-content/uploads/2012/01/thetorontoproblem.pdf) by *W.R. Brian*\n","FormalConjectures.Wikipedia.Transcendental":"# Open questions on transcendence of numbers\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number)\n","FormalConjectures.Wikipedia.TwinPrimes":"# Twin prime conjecture\n\n*References:*\n- [Landau Problems Wikipedia Page](https://en.wikipedia.org/wiki/Landau%27s_problems#Twin_prime_conjecture)\n- [Twin Primes Conjecture Wikipedia Page](https://en.wikipedia.org/wiki/Twin_prime#Twin_prime_conjecture)\n","FormalConjectures.Wikipedia.UnionClosed":"# Union-closed sets conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Union-closed_sets_conjecture)\n\nIn this file, we:\n* state the conjecture\n* state three solved variants of the conjecture, without proof\n* prove two solved variants of the conjecture\n* prove the conjecture is sharp\n","FormalConjectures.Wikipedia.VaughtConjecture":"# Vaught conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Vaught_conjecture)\n","FormalConjectures.Wikipedia.VizingConjecture":"# Vizing's conjecture (1968)\n\n*References:*\n* [Wikipedia](https://en.wikipedia.org/wiki/Vizing%27s_conjecture)\n* [Vi68] Vizing, V. G. (1968). \"Some unsolved problems in graph theory.\"\n  *Uspekhi Mat. Nauk* 23, pp. 117--134.\n* [ClSu00] Clark, W. E. and Suen, S. (2000). \"An inequality related to Vizing's conjecture.\"\n  *Electron. J. Combin.* 7, N4.\n* [SuTa12] Suen, S. and Tarr, J. (2012). \"An improved inequality related to Vizing's conjecture.\"\n  *Electron. J. Combin.* 19, P8.\n* [BDGHHKR12] Brešar, B., Dorbec, P., Goddard, W., Hartnell, B. L., Henning, M. A., Klavžar, S.\n  and Rall, D. F. (2012). \"Vizing's conjecture: a survey and recent results.\"\n  *J. Graph Theory* 69, pp. 46--76.\n","FormalConjectures.Wikipedia.WallSunSun":"# Infinitude of Wall–Sun–Sun primes\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime)\n","FormalConjectures.Wikipedia.WieferichMirimanoffPrime":"# Can a prime $p$ satisfy $2^{p-1} \\equiv 1 \\pmod{p^2}$ and $3^{p-1} \\equiv 1 \\pmod{p^2}$?\n\nA prime $p$ with $2^{p-1} \\equiv 1 \\pmod{p^2}$ is a Wieferich prime (the only known examples\nare $1093$ and $3511$). A prime $p$ with $3^{p-1} \\equiv 1 \\pmod{p^2}$ is a Mirimanoff prime\n(the only known examples are $11$ and $1006003$). It is an open question whether a prime can\nsatisfy both congruences simultaneously. Lenstra gave a heuristic argument against this.\n\n*References:*\n* [Wikipedia, List of unsolved problems in mathematics](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics)\n* [Wikipedia, Wieferich prime](https://en.wikipedia.org/wiki/Wieferich_prime)\n* J. B. Dobson, [On Lerch's formula for the Fermat quotient](https://arxiv.org/abs/1103.3907v6)\n* F. G. Dorais and D. Klyve, [A Wieferich prime search up to $6.7 \\times 10^{15}$](https://cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.html),\n  J. Integer Seq. 14 (2011), Article 11.9.2.\n* [OEIS A001220](https://oeis.org/A001220) (Wieferich primes)\n* [OEIS A014127](https://oeis.org/A014127) (Mirimanoff primes)\n","FormalConjectures.Wikipedia.WieferichPrime":"# Wieferich primes\n\nA Wieferich prime is a prime $p$ with $2^{p-1} \\equiv 1 \\pmod{p^2}$. The only known examples are\n$1093$ and $3511$. It is conjectured that there are infinitely many Wieferich primes; a heuristic\nargument suggests that the number of Wieferich primes up to $x$ grows like $\\log \\log x$.\n\nMore generally, a prime $p$ is a Wieferich prime to base $a$ if $a^{p-1} \\equiv 1 \\pmod{p^2}$.\nWikipedia's list of unsolved problems also asks whether there are infinitely many Wieferich primes\nto every base $a > 0$, and whether there is any Wieferich prime to base $47$. It is also not known\nwhether there is any Wieferich prime besides $1093$ and $3511$.\n\n*References:*\n* [Wikipedia, List of unsolved problems in mathematics](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics)\n* [Wikipedia, Wieferich prime](https://en.wikipedia.org/wiki/Wieferich_prime)\n* [OEIS A001220](https://oeis.org/A001220)\n* P. Ribenboim, *Die Welt der Primzahlen*, 2nd ed., Springer (2006), pp. 242–243.\n* [PrimeGrid, Wieferich and Wall–Sun–Sun Prime Search](https://www.primegrid.com/stats_ww.php)\n","FormalConjectures.Wikipedia.WilsonPrime":"# Wilson primes\n\nA Wilson prime is a prime $p$ for which $p^2$ divides $(p-1)!+1$. The only known examples are\n$5$, $13$, and $563$. It is conjectured that infinitely many Wilson primes exist.\n\n*References:*\n* [Wikipedia, Wilson prime](https://en.wikipedia.org/wiki/Wilson_prime)\n* [OEIS A007540](https://oeis.org/A007540)\n* E. Costa, R. Gerbicz, and D. Harvey,\n  [A search for Wilson primes](https://arxiv.org/abs/1209.3436)\n","FormalConjectures.Wikipedia.WolstenholmePrime":"# Wolstenholme Prime\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Wolstenholme_prime)\n","FormalConjectures.Wikipedia.WoodalPrimes":"# Woodall Primes\n\nReferences:\n* [Wikipedia/Woodall Number](https://en.wikipedia.org/wiki/Woodall_number#Woodall_primes)\n* [A2234](https://oeis.org/A2234)\n\n","FormalConjectures.Wikipedia.conjecture_1_3_to_2_3":"# The $\\frac 1 3$–$\\frac 2 3$ conjecture\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/1/3%E2%80%932/3_conjecture)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture1":"# Written on the Wall II - Conjecture 1\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture100":"# Written on the Wall II - Conjecture 100\n\n**Verbatim statement (WOWII #100, status O):**\n> If G is a simple connected graph, then α(G) ≤ CEIL[(maximum of λ(v) + 0.5*length(Ḡ))/2]\n\n**Source:** http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj100\n\nThe WOWII HTML uses `length(Ḡ)` (the bar denotes graph complement); the\nextracted JSON in our private repo previously dropped the overline. The\nformal statement below uses the Euclidean norm of the degree sequence of `Gᶜ`.\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Definition of graph length\n\nThe WOWII definitions popup defines `length(H)` as the square root of the sum\nof the squares of the vertex degrees. This is `degreeL2Norm H` in Lean.\nCombined with the overline above, the inequality reads:\n  `α(G) ≤ ⌈(max_v l(v) + 0.5 · degreeL2Norm(Gᶜ)) / 2⌉`\nwhere `l(v) = indepNeighbors G v`.\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture101":"# Written on the Wall II - Conjecture 101\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Definitions\n\nThe **$\\alpha$-core** of a graph $G$, written `alphaCore G`, is the set of vertices\n$v$ such that removing $v$ strictly decreases the independence number:\n$$\\mathrm{alphaCore}(G) = \\{v \\mid \\alpha(G - v) < \\alpha(G)\\}$$\nwhere $G - v$ is the subgraph of $G$ induced on $V(G) \\setminus \\{v\\}$.\n\nThese vertices are also called \"critical vertices for independence.\"\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture103":"# Written on the Wall II - Conjecture 103\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture109":"# Written on the Wall II - Conjecture 109\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture13":"# Written on the Wall II - Conjecture 13\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture133":"# Written on the Wall II - Conjecture 133\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture141":"# Written on the Wall II - Conjecture 141\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture142":"# Written on the Wall II - Conjecture 142\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture143":"# Written on the Wall II - Conjecture 143\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture144":"# Written on the Wall II - Conjecture 144\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture145":"# Written on the Wall II - Conjecture 145\n\nThe WOWII HTML uses $\\lambda_{\\min}(\\overline{G})$ (the bar denotes graph complement).\nThe formal statement below uses the local-independence minimum of $G^c$.\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Definitions\n\nThe **local independence minimum** $\\mathrm{lMin}(G)$ is:\n$$\\mathrm{lMin}(G) = \\min_{v \\in V(G)} l(v)$$\nwhere $l(v) = \\mathrm{indepNeighborsCard}(G, v)$ is the independence number of the\nneighbourhood of $v$. This is the minimum over all vertices of the local\nindependence number.\n\nThe **boundary vertices** $B(G)$ of a connected graph are the vertices $v$ such\nthat the eccentricity of $v$ equals the diameter of $G$.\n\nThe **eccentricity of a set** $\\mathrm{ecc}(S) = \\max_{u \\notin S} \\min_{w \\in S}\n\\mathrm{dist}(u, w)$. In the conjecture below, $\\mathrm{ecc}(B)$ is the\neccentricity of the boundary set.\n\n**Conjecture 145:** $\\mathrm{tree}(G) \\ge 2 \\cdot \\mathrm{ecc}(B) /\n\\lambda_{\\min}(\\overline{G})$ where $\\mathrm{tree}(G)$ is `largestInducedTreeSize G`,\n$\\mathrm{ecc}(B)$ is the eccentricity of the boundary vertices, and\n$\\lambda_{\\min}(\\overline{G})$ is the local independence minimum of the complement\n$\\overline{G}$.\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture146":"# Written on the Wall II - Conjecture 146\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Definitions\n\nThe **square** of a graph $G$, denoted $G^2$, is the graph on the same vertex set\nwhere two distinct vertices are adjacent if and only if their distance in $G$ is at\nmost $2$.\n\nThe **radius of $G^2$** is the minimum eccentricity of any vertex in $G^2$, i.e.,\n$$\\mathrm{rad}(G^2) = \\min_{v \\in V} \\max_{u \\in V} \\mathrm{dist}_{G^2}(u, v).$$\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture16":"# Written on the Wall II - Conjecture 16\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture17":"# Written on the Wall II - Conjecture 17\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture18":"# Written on the Wall II - Conjecture 18\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture19":"# Written on the Wall II - Conjecture 19\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture194":"# Written on the Wall II - Conjecture 194\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Counterexample\n\nThe conjecture is false. Take a clique on vertices `0, ..., 10`, four additional vertices\n`11, ..., 14` adjacent to every clique vertex, and three leaves attached at `11`, `12`, and `14`.\nThis connected graph has independence number `4`, while the sum of the independence numbers of its\nvertex neighbourhoods is `54`, so their average is `3`. Thus it satisfies the conjecture's\nhypothesis with equality. However, its three leaves would all have to be endpoints of a Hamiltonian\npath, which is impossible.\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture198a":"# Written on the Wall II - Conjecture 198a\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture2":"# Written on the Wall II - Conjecture 2\n\n*References:*\n- [E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n- [arxiv/2605.22763](https://arxiv.org/abs/2605.22763) *Advancing Mathematics Research\n  with AI-Driven Formal Proof Search* by George Tsoukalas et al.\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture20":"# Written on the Wall II - Conjecture 20\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture200":"# Written on the Wall II - Conjecture 200\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Counterexample\n\nFor every integer $q \\geq 5$, let $C = \\{a,b\\} \\cup D$ have $q$ vertices and\ninduce $K_q$ minus the edge $ab$. Add two nonadjacent vertices $x,y$, each\ncomplete to $C$; a vertex $z$ adjacent exactly to $a,b$; and, for every\n$d \\in D$, a pendant vertex $p_d$ adjacent exactly to $d$.\n\nThe local-neighbourhood independence numbers are $2$ at $x,y,z$, $1$ at each\n$p_d$, and $3$ at every vertex of $C$. Their sum is $4q+4$, while the graph has\n$2q+1$ vertices, so\n$$\\left\\lceil 1 + l_{\\mathrm{avg}}(G)\\right\\rceil = 4.$$\nThe vertices $\\{a,x,y,z\\}$ induce a claw, and a case split on the number of\nvertices from $C$ shows that no induced tree has more than four vertices.\nThus $\\operatorname{tree}(G)=4$. Finally, the graph has $q-2 \\geq 3$ pendant\nvertices, but a Hamiltonian path has only two endpoints.\n\nThe smallest member of this family has 11 vertices and graph6 encoding\n`J??FFBRq}N_`.\nThis is a smallest counterexample overall: an exhaustive search over all\n11,989,760 connected graphs on 4 ≤ n ≤ 10 vertices (via nauty's `geng`;\ngraphs on at most 3 vertices are trivially traceable) found no graph\nsatisfying the premise without a Hamiltonian path.\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture217":"# Written on the Wall II - Conjecture 217\n\nPer the WOWII definitions popup linked from this conjecture:\n\n- $L(G)$ is the **maximum number of leaves of a spanning tree** of $G$\n  — i.e. `Ls G` in our invariant library.\n- $\\chi_{\\mathrm{residue}=2}(G)$ is the **characteristic function** for the\n  predicate $\\mathrm{residue}\\, G = 2$, i.e. $1$ when $\\mathrm{residue}\\, G = 2$\n  and $0$ otherwise. It is not a connected-component count of any 2-core.\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture23":"# Written on the Wall II - Conjecture 23\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture291":"# Written on the Wall II - Conjecture 291\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Definitions\n\nFor a vertex $v$ in a graph $G$, **$T(v)$** is the number of triangles (3-cliques)\nincident to $v$, i.e., the number of 3-element cliques in $G$ that contain $v$.\n\nThe **triangle-frequency of the minimum** is the number of vertices that achieve the\nminimum value of $T(v)$.\n\n**$k$** is the **first step in the Havel–Hakimi process at which a zero appears**.\nConcretely, starting from the descending degree sequence $s_0$ of $G$, we set\n$s_{i+1} = \\mathrm{havelHakimiStep}\\, s_i$ and let $k$ be the least $i \\ge 0$ such\nthat $s_i$ contains a zero entry (or, vacuously, has been emptied entirely). Since\neach step is sorted descending and never increases entries, this is equivalent\nto the last (smallest) entry of $s_i$ being $0$, or $s_i$ being $[]$.\n\nThis is **strictly weaker** than $n - \\mathrm{residue}(G)$: $n - \\mathrm{residue}(G)$\nis the *total* number of reduction steps until *every* entry is zero, whereas $k$\nonly requires that *some* entry has hit zero — and a $0$ typically appears well\nbefore the all-zero state is reached.\n\n**Conjecture 291:** For a simple connected graph $G$ with $n > 2$,\n$\\gamma_t(G) \\le k + \\mathrm{frequency}(t_{\\min}(v))$\nwhere $\\gamma_t(G)$ is the total domination number, $k$ is the Havel-Hakimi zero\nstep, and $\\mathrm{frequency}(t_{\\min}(v))$ is the number of vertices achieving\nthe minimum triangle count.\n\nThe conjecture is **false**: on July 23, 2026, Zyad Tamimi sent a 12-vertex\ncounterexample with $\\gamma_t(G) = 4$, $\\mathrm{frequency}(t_{\\min}(v)) = 1$ and $k = 2$,\nand the source now lists Conjecture 291 as refuted.\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture3":"# Written on the Wall II - Conjecture 3\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture31":"# Written on the Wall II - Conjecture 31\n\nThe WOWII page records this as **Chung's theorem**: proved in F. R. K. Chung,\n*The average distance and the independence number*, J. Graph Theory **12** (1988),\n229-235. We state it here as a theorem; the formal proof is left as `sorry`\npending a Lean port of Chung's argument.\n\nHere $\\mathrm{path}(G)$ is the floor of the average distance over ordered pairs\nof distinct vertices (definition `path` in `FormalConjecturesForMathlib`), and\n$\\mathrm{rad}(G)$ is the graph radius.\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture314":"# Written on the Wall II - Conjecture 314\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture315":"# Written on the Wall II - Conjecture 315\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Definitions\n\nThe notions `IsTotalDominatingSet`, `IsMinimalTotalDominatingSet`,\n`IsWellTotallyDominated`, and `pendantVertices` live in\n`FormalConjecturesForMathlib.Combinatorics.SimpleGraph.GraphConjectures.WellTotallyDominated`.\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture316":"# Written on the Wall II - Conjecture 316\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture32":"# Written on the Wall II - Conjecture 32\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture322":"# Written on the Wall II - Conjecture 322\n\n*Reference:*\n[E. DeLaViña, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.uhd.edu/faculty/delavinae/research/wowII/)\n\nThe source applies the local-independence hypothesis to the complement graph. Its conclusion\nalso follows from the characterization of minimal total dominating sets of complete multipartite\ngraphs in [M. Subramanian and A. Selvakumar, *Total Domination and Minimal Total Domination\nPolynomial of H-Join Graphs*](https://doi.org/10.2298/FIL2501267S).\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture327":"# Written on the Wall II - Conjecture 327\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture33":"# Written on the Wall II - Conjecture 33\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture34":"# Written on the Wall II - Conjecture 34\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture36":"# Written on the Wall II - Conjecture 36\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture4":"# Written on the Wall II - Conjecture 4\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture40":"# Written on the Wall II - Conjecture 40\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture5":"# Written on the Wall II - Conjecture 5\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture58":"# Written on the Wall II - Conjecture 58\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Counterexample\n\nThe conjecture is false. A counterexample is given by taking $K_{3,3}$\n(with bipartition $\\{0,1,2\\}$, $\\{3,4,5\\}$) and $K_{73}$ (on vertices $\\{6,\\ldots,78\\}$),\nthen adding edges between vertex $0$ and every vertex of $K_{73}$.\n\nThis graph $G$ has $n = 79$ vertices and satisfies:\n- $b(G) \\ge 7$: the set $\\{0,1,2,3,4,5,6\\}$ induces a bipartite subgraph\n- $f(G) \\le 6$: the largest induced forest has at most 6 vertices\n- $l_{\\mathrm{avg}}(G) = 92/79$\n- $\\lceil b(G) / l_{\\mathrm{avg}}(G) \\rceil \\ge 7 > 6 \\ge f(G)$\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture59":"# Written on the Wall II - Conjecture 59\n\n*References:*\n- [E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture6":"# Written on the Wall II - Conjecture 6\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture61":"# Written on the Wall II - Conjecture 61\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture63":"# Written on the Wall II - Conjecture 63\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Counterexample\n\nThe conjecture is false for the lexicographic product $C_5[K_4]$. This graph\nhas minimum even-distance count $9$, while its largest induced forest and\nlargest induced bipartite subgraph both have order $4$. The conjectured lower\nbound is therefore $\\lceil(9 + 4 + 1)/3\\rceil = 5 > 4$.\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture65":"# Written on the Wall II - Conjecture 65\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Counterexample\n\nThe conjecture is false. Start with the path $v_0-v_1-\\cdots-v_{12}$ and attach one\ntriangle at $v_1$ and another at $v_{11}$. The resulting graph has $17$ vertices.\nIts Graph_6 string is `PhCGGC@?G?_@?@O?G?G?G?@C`.\nIts only minimum-degree vertices are $v_0$ and $v_{12}$, at distance $12$, while its\nonly maximum-degree vertices are $v_1$ and $v_{11}$, at distance $10$. Thus the\nconjectured lower bound is $12 + \\lceil 10/3 \\rceil = 16$.\n\nEvery induced forest must omit at least one vertex from each of the two vertex-disjoint\ntriangles, so it has at most $15$ vertices. Conversely, deleting one non-path vertex\nfrom each triangle leaves a tree on $15$ vertices.\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture7":"# Written on the Wall II - Conjecture 7\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n","FormalConjectures.WrittenOnTheWallII.GraphConjecture85":"# Written on the Wall II - Conjecture 85\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Counterexample\n\nThe conjecture is false for the lexicographic product $C_5[K_4]$. This graph\nhas minimum even-distance count $9$ and largest induced-tree order $4$. The\nconjectured lower bound is therefore $\\lceil\\sqrt{1 + 2 \\cdot 9}\\rceil =\n\\lceil\\sqrt{19}\\rceil = 5 > 4$.\n","FormalConjectures.WrittenOnTheWallII.Test":"# Testing Graph Invariants\n\nThis file contains tests for graph invariants on 5 specific concrete graphs:\n1. `HouseGraph`: A graph on 5 vertices.\n2. `K4`: The complete graph on 4 vertices.\n3. `PetersenGraph`: The Petersen graph on 10 vertices.\n4. `C6`: The cycle graph on 6 vertices.\n5. `Star5`: The star graph with 5 leaves (6 vertices total).\n\nTests cover:\nindependence_number, dominationNumber, average_distance, diameter, radius,\ngirth, order, size, szeged_index, wiener_index, min_degree, max_degree,\naverage_degree, matching_number, residue, annihilation_number, cvetkovic.\n","FormalConjectures.WrittenOnTheWallII.«160»":"# Written on the Wall II - Conjecture 160\n\n*Reference:*\n[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\n## Definitions\n\nFor a vertex $v$ in $G$, **$T(v)$** is the number of triangles incident to $v$:\n$$T(v) = |\\{\\{u, w\\} \\subseteq N(v) \\mid u \\sim w\\}|$$\ni.e., the number of pairs of neighbors of $v$ that are themselves adjacent.\n\nThe invariant `maxTrianglesAtVertex G` is the maximum of $T(v)$ over all vertices.\n\nConjecture 160 uses both $\\max_v T(v)$ and $\\chi_{C_4}(G)$, the $C_4$-free\ncharacteristic function: it is `1` when $G$ contains no cycle of length four\nand `0` otherwise. The cycle need not be induced. These invariants lower-bound\nthe WOWII invariant $L_s(G)$, the maximum number of leaves over all spanning\ntrees of $G$ (exposed as `SimpleGraph.Ls G : ℝ`).\n\nThe earlier formalization used the number of induced four-cycles. The historical\nconjecture instead uses this binary $C_4$-free indicator.\n\n"},"problems":[{"answerKinds":[],"category":"research open","docstring":"Any T2, Toronto space is discrete. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Toronto","statement":"∀ (X : Type u_1) [inst : TopologicalSpace X] [T2Space X] [Toronto.TorontoSpace X], DiscreteTopology X","subjects":["54"],"theorem":"Toronto.DiscreteTopology.of_t2_of_torontoSpace"},{"answerKinds":[],"category":"research open","docstring":"Idoneal numbers completeness conjecture.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.IdonealCompleteness","statement":"True ↔ ∀ (n : ℕ), Idoneal.IsIdoneal n → n ∈ Idoneal.knownIdonealNumbers","subjects":["11"],"theorem":"Idoneal.idoneal_numbers_completeness"},{"answerKinds":[],"category":"test","docstring":"All 65 known idoneal numbers are indeed idoneal. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.IdonealCompleteness","statement":"∀ n ∈ Idoneal.knownIdonealNumbers, Idoneal.IsIdoneal n","subjects":["11"],"theorem":"Idoneal.knownIdonealNumbers_are_idoneal"},{"answerKinds":[],"category":"API","docstring":"**Homomorphism count of the star `K_{1,m}`.** The number of homomorphisms from the star\n`K_{1,m} = completeBipartiteGraph (Fin 1) (Fin m)` into `G` equals `∑_{c} (G.degree c)^m`.\n\n**Math.** A homomorphism `f : K_{1,m} →g G` is determined by the image `c := f` of the centre\n`Sum.inl 0` together with the images of the `m` leaves `Sum.inr j`, each of which must lie in\nthe neighbourhood `N(c)` (and there is no constraint between distinct leaves). So the data is a\nchoice of `c` and a function `Fin m → N(c)`, giving `∑_c |N(c)|^m = ∑_c (G.degree c)^m`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ (m : ℕ) [inst : DecidableRel (completeBipartiteGraph (Fin 1) (Fin m)).Adj] {W : Type u_3} [inst_1 : Fintype W]\n  [inst_2 : DecidableEq W] (G : SimpleGraph W) [inst_3 : DecidableRel G.Adj],\n  (completeBipartiteGraph (Fin 1) (Fin m)).homCount G = ∑ c, G.degree c ^ m","subjects":["5"],"theorem":"SidorenkoConjecture.homCount_star_eq_sum_degree_pow"},{"answerKinds":[],"category":"API","docstring":"**Homomorphism count of `K_2` into `G` equals `2 · #edgeFinset`.**\n\nA homomorphism `K_2 →g G` is the same data as an ordered pair `(f 0, f 1)` of distinct\nvertices with `G.Adj (f 0) (f 1)`. These are in bijection with the `Dart`s of `G`, and\n`#Darts(G) = 2 · #E(G)` by `dart_card_eq_twice_card_edges`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {W : Type u_3} [inst : Fintype W] [inst_1 : DecidableEq W] (G : SimpleGraph W) [inst_2 : DecidableRel G.Adj],\n  (SimpleGraph.completeGraph (Fin 2)).homCount G = 2 * G.edgeFinset.card","subjects":["5"],"theorem":"SidorenkoConjecture.homCount_completeGraph_fin_two_eq_two_mul_card_edgeFinset"},{"answerKinds":[],"category":"research solved","docstring":"**Case: `H = C_{2k}` is an even cycle (Sidorenko 1993, graphon version).**\n\nEvery even cycle $C_{2k}$ satisfies Sidorenko's inequality for all graphons on $[0, 1]$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ (k : ℕ),\n  1 ≤ k →\n    ∀ (W : LimitObjects.Graphon),\n      LimitObjects.graphonEdgeDensity W ^ (SimpleGraph.cycleGraph (2 * k)).edgeFinset.card ≤\n        LimitObjects.graphonHomDensity (SimpleGraph.cycleGraph (2 * k)) W","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_even_cycle_graphon"},{"answerKinds":[],"category":"API","docstring":"If the domain of `H` has at most one vertex, every function `V → W` is a homomorphism\n`H →g G`. Equivalently, `homCount H G = |W|^|V|`.\n\n**Proof.** The adjacency `H.Adj u v` is irreflexive; on a subsingleton it is uninhabited,\nso the homomorphism condition `H.Adj u v → G.Adj (f u) (f v)` is vacuous. Hence the\n`coe : (H →g G) → (V → W)` map is a bijection. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {V : Type u_3} {W : Type u_4} [inst : Fintype V] [inst_1 : Fintype W] [Subsingleton V] [inst_3 : DecidableEq V]\n  [inst_4 : DecidableEq W] (H : SimpleGraph V) (G : SimpleGraph W) [inst_5 : DecidableRel H.Adj]\n  [inst_6 : DecidableRel G.Adj], H.homCount G = Fintype.card W ^ Fintype.card V","subjects":["5"],"theorem":"SidorenkoConjecture.homCount_eq_of_subsingleton"},{"answerKinds":[],"category":"research solved","docstring":"**Case `H = K_{2,2}` (four-cycle, also called `C_4`): Sidorenko's conjecture holds, by\nCauchy–Schwarz.**\n\nThe textbook statement at `H = K_{2,2}` is\n`t(K_2, G)^{e(K_{2,2})} = t(K_2, G)^4 ≤ t(K_{2,2}, G)`.\n\n**Proof sketch.** Write `d(a) := G.degree a`. Then\n- `homCount(K_2, G) = ∑_a d(a) = 2·|E(G)|` (handshaking).\n- `homCount(K_{2,2}, G) = ∑_{b₀, b₁} |N(b₀) ∩ N(b₁)|²` (product structure of bipartite\n  homomorphism).\n- `∑_{b₀, b₁} |N(b₀) ∩ N(b₁)| = ∑_a d(a)²` (swap sums).\n- Cauchy–Schwarz #1: `(∑_{b₀, b₁} |N(b₀) ∩ N(b₁)|)² ≤ |W|² · ∑_{b₀, b₁} |N(b₀) ∩ N(b₁)|²`.\n- Cauchy–Schwarz #2: `(∑_a d(a))² ≤ |W| · ∑_a d(a)²`.\n- Chain: `(∑_a d(a))⁴ ≤ |W|⁴ · homCount(K_{2,2}, G)`.\n- Divide by `|W|^8` to get `t(K_2, G)^4 ≤ t(K_{2,2}, G)`.\n\nThe proof uses `Finset.sum_mul_sq_le_sq_mul_sq` (discrete Cauchy–Schwarz) from\n`Mathlib.Algebra.Order.BigOperators.Ring.Finset`.\n\n**Status (2026-04-22):** main theorem closed sorry-free. See [Si93].\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {W : Type} [inst : Fintype W] [inst_1 : DecidableEq W] (G : SimpleGraph W) [inst_2 : DecidableRel G.Adj],\n  (SimpleGraph.completeGraph (Fin 2)).homDensity G ^ (completeBipartiteGraph (Fin 2) (Fin 2)).edgeFinset.card ≤\n    (completeBipartiteGraph (Fin 2) (Fin 2)).homDensity G","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_K22"},{"answerKinds":[],"category":"research solved","docstring":"**Case: `H = K_{a,b}` is a complete bipartite graph (Sidorenko 1993, graphon version).**\n\nEvery complete bipartite graph $K_{a,b}$ satisfies Sidorenko's inequality for all graphons on $[0, 1]$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {A : Type u_3} {B : Type u_4} [inst : Fintype A] [inst_1 : Fintype B] [DecidableEq A] [DecidableEq B]\n  (W : LimitObjects.Graphon),\n  LimitObjects.graphonEdgeDensity W ^ (completeBipartiteGraph A B).edgeFinset.card ≤\n    LimitObjects.graphonHomDensity (completeBipartiteGraph A B) W","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_completeBipartiteGraph_graphon"},{"answerKinds":[],"category":"API","docstring":"Consequence of `homDensity_le_one`: both sides of the Sidorenko K_{2,2}\ninequality are bounded above by 1. This is a trivial consequence, recorded\nas a sanity check on the helper infrastructure in\n`FormalConjecturesForMathlib.Combinatorics.SimpleGraph.HomDensity`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {W : Type u_3} [inst : Fintype W] [Nonempty W] [inst_2 : DecidableEq W] (G : SimpleGraph W)\n  [inst_3 : DecidableRel G.Adj],\n  (SimpleGraph.completeGraph (Fin 2)).homDensity G ≤ 1 ∧ (completeBipartiteGraph (Fin 2) (Fin 2)).homDensity G ≤ 1","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_K22_both_sides_bounded"},{"answerKinds":[],"category":"research solved","docstring":"**Bipartiteness is necessary: the triangle is not a Sidorenko graph.**\n\nThe hypothesis `H.IsBipartite` in `sidorenko_conjecture` cannot be dropped. For the\ntriangle `H = K_3` (chromatic number `3`, hence not bipartite) and the single edge\n`G = K_2`, Sidorenko's inequality fails: the putative lower bound is\n`t(K_2, K_2)^{e(K_3)} = (1/2)^3 = 1/8`, while the actual density is `t(K_3, K_2) = 0`,\nbecause there is no graph homomorphism from `K_3` into the triangle-free graph `K_2`\n(such a homomorphism would be an injection `Fin 3 ↪ Fin 2`). So the lower bound `1/8`\nstrictly exceeds the true value `0`.\n\nMore generally, any `H` containing an odd cycle fails Sidorenko's inequality against a\nsuitable bipartite `G`, since then `t(H, G) = 0 < t(K_2, G)^{e(H)}`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"(SimpleGraph.completeGraph (Fin 3)).homDensity (SimpleGraph.completeGraph (Fin 2)) <\n  (SimpleGraph.completeGraph (Fin 2)).homDensity (SimpleGraph.completeGraph (Fin 2)) ^\n    (SimpleGraph.completeGraph (Fin 3)).edgeFinset.card","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_conjecture.variants.non_bipartite_necessary"},{"answerKinds":[],"category":"research solved","docstring":"**Case: `H` is a tree (Sidorenko 1993).**\n\nIf `H` is a finite tree then Sidorenko's inequality holds.\n\n**Proof idea (Sidorenko 1993):** by induction on the tree, applying Jensen / convexity. The\nnumber of homomorphisms from a tree `T` with `v` vertices and `v - 1` edges into `G` factors\nnicely in the degree sequence of `G`, and AM–GM / convexity gives the required lower bound.\n\n**Current status (2026-04-22, partial):**\n- The subsingleton base case (`|V(T)| ≤ 1`) is dispatched via `sidorenko_tree_subsingleton`.\n- The main inductive step (leaf extraction + AM–GM on the degree sequence) is deferred;\n  it requires a walk-parametrised homomorphism count for trees and a discrete Jensen\n  inequality. See [Si93].\n\nThe full proof is left as `sorry`; the subsingleton case is closed.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {V W : Type} [inst : Fintype V] [inst_1 : Fintype W] [inst_2 : DecidableEq V] [inst_3 : DecidableEq W] [Nonempty W]\n  (H : SimpleGraph V) (G : SimpleGraph W) [inst_5 : DecidableRel H.Adj] [inst_6 : DecidableRel G.Adj],\n  H.IsTree → (SimpleGraph.completeGraph (Fin 2)).homDensity G ^ H.edgeFinset.card ≤ H.homDensity G","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_tree"},{"answerKinds":[],"category":"research solved","docstring":"**Case: paths, the Blakley–Roy inequality (1965).**\n\nFor the path `P_n` on `n` vertices (`pathGraph n`), Sidorenko's inequality\n`t(K_2, G)^{e(P_n)} ≤ t(P_n, G)` is the Blakley–Roy inequality, established well before\nSidorenko's general conjecture. Paths are trees, so this is also a special case of\n`sidorenko_tree`.\n\n*Reference:* [BR65].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ (n : ℕ) [inst : DecidableRel (SimpleGraph.pathGraph n).Adj] {W : Type} [inst_1 : Fintype W] [inst_2 : DecidableEq W]\n  [Nonempty W] (G : SimpleGraph W) [inst_4 : DecidableRel G.Adj],\n  (SimpleGraph.completeGraph (Fin 2)).homDensity G ^ (SimpleGraph.pathGraph n).edgeFinset.card ≤\n    (SimpleGraph.pathGraph n).homDensity G","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_conjecture.variants.path_blakley_roy"},{"answerKinds":[],"category":"API","docstring":"`edgeCount` of `K_{2,2}` (complete bipartite graph on `Fin 2 + Fin 2`) is `4`.\n\nThe four edges are `{inl 0, inr 0}`, `{inl 0, inr 1}`, `{inl 1, inr 0}`, `{inl 1, inr 1}`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"(completeBipartiteGraph (Fin 2) (Fin 2)).edgeFinset.card = 4","subjects":["5"],"theorem":"SidorenkoConjecture.edgeCount_completeBipartiteGraph_fin_two"},{"answerKinds":[],"category":"research solved","docstring":"**Case: even cycles `C_{2k}` (Sidorenko 1993).**\n\nSidorenko's inequality holds for every even cycle `C_{2k}` with `k ≥ 2`\n(`cycleGraph (2 * k)`). Even cycles are bipartite; the `k = 2` case `C_4` coincides\nwith `K_{2,2}` (`sidorenko_K22`).\n\n*Reference:* [Si93].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ (k : ℕ),\n  2 ≤ k →\n    ∀ {W : Type} [inst : Fintype W] [inst_1 : DecidableEq W] [Nonempty W] (G : SimpleGraph W)\n      [inst_3 : DecidableRel G.Adj],\n      (SimpleGraph.completeGraph (Fin 2)).homDensity G ^ (SimpleGraph.cycleGraph (2 * k)).edgeFinset.card ≤\n        (SimpleGraph.cycleGraph (2 * k)).homDensity G","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_conjecture.variants.even_cycle"},{"answerKinds":[],"category":"research solved","docstring":"**Case `H = K_2` (single edge): Sidorenko's inequality holds trivially with equality.**\n\nWhen `H` is `K_2` (the single-edge graph on 2 vertices), `e(H) = 1`, so the RHS of Sidorenko's\ninequality is just `t(K_2, G)^1 = t(K_2, G) = t(H, G)`, which equals the LHS. Hence the\ninequality holds as equality.\n\nThe proof records that `(completeGraph (Fin 2)).edgeFinset.card = 1` and then reduces the claim\nto `t(K_2, G) ≤ t(K_2, G)`, which is `le_refl`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {W : Type} [inst : Fintype W] [inst_1 : DecidableEq W] (G : SimpleGraph W) [inst_2 : DecidableRel G.Adj],\n  (SimpleGraph.completeGraph (Fin 2)).homDensity G ^ (SimpleGraph.completeGraph (Fin 2)).edgeFinset.card ≤\n    (SimpleGraph.completeGraph (Fin 2)).homDensity G","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_K2"},{"answerKinds":[],"category":"research solved","docstring":"**TAS Trees Conjecture: The $(2,3,4)$-spider tree (Solved case).**\n\nThe $(2,3,4)$-spider tree satisfies the Tournament Anti-Sidorenko Trees Conjecture.\n\n*Reference:*\n* [ArXiv 2605.14138](https://arxiv.org/abs/2605.14138)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {V : Type u_3} [inst : Fintype V] [inst_1 : DecidableEq V] (T : SimpleGraph V) [inst_2 : DecidableRel T.Adj],\n  SidorenkoConjecture.IsSpider234 T →\n    ∃ D,\n      D.IsOrientation T ∧\n        ∀ {W : Type u_4} [inst_3 : Fintype W] [DecidableEq W] [Nonempty W] (G : Digraph W)\n          [inst_6 : DecidableRel G.Adj], G.IsTournament → D.homDensity G ≤ (1 / 2) ^ T.edgeFinset.card","subjects":["5"],"theorem":"SidorenkoConjecture.tournament_anti_sidorenko_spider234"},{"answerKinds":[],"category":"API","docstring":"**Edge count of the star `K_{1,m}`.** The star `completeBipartiteGraph (Fin 1) (Fin m)`\nhas exactly `m` edges (the centre `Sum.inl 0` joined to each of the `m` leaves). ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ (m : ℕ) [inst : DecidableRel (completeBipartiteGraph (Fin 1) (Fin m)).Adj],\n  (completeBipartiteGraph (Fin 1) (Fin m)).edgeFinset.card = m","subjects":["5"],"theorem":"SidorenkoConjecture.card_edgeFinset_star"},{"answerKinds":[],"category":"research solved","docstring":"**Case: `H` is a tree (Sidorenko 1993, graphon version).**\n\nIf `H` is a finite tree then Sidorenko's inequality holds for all graphons on $[0, 1]$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {V : Type u_3} [inst : Fintype V] [DecidableEq V] (H : SimpleGraph V) [inst_2 : DecidableRel H.Adj],\n  H.IsTree →\n    ∀ (W : LimitObjects.Graphon),\n      LimitObjects.graphonEdgeDensity W ^ H.edgeFinset.card ≤ LimitObjects.graphonHomDensity H W","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_tree_graphon"},{"answerKinds":[],"category":"API","docstring":"**`K_{2,2}` count via a re-indexed sum.** Swapping the order of summation, the\n`Hom(K_{2,2}, G)` count equals `∑_{a ∈ W} (G.degree a)²` summed over... wait, more\nprecisely: the sum `∑_{(b₀, b₁)} |N(b₀) ∩ N(b₁)|` (without the square) equals\n`∑_a (G.degree a)²`, by swapping `(∑_{b₀, b₁} ∑_a [a ~ b₀][a ~ b₁]) = ∑_a (∑_b [a ~ b])²`.\n\nThis version of the identity is what appears in the Cauchy-Schwarz step. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {W : Type u_3} [inst : Fintype W] [inst_1 : DecidableEq W] (G : SimpleGraph W) [inst_2 : DecidableRel G.Adj],\n  ∑ p, ↑(G.neighborFinset p.1 ∩ G.neighborFinset p.2).card = ∑ a, ↑(G.degree a) ^ 2","subjects":["5"],"theorem":"SidorenkoConjecture.sum_inter_card_eq_sum_degree_sq"},{"answerKinds":[],"category":"research open","docstring":"**Sidorenko's conjecture (1993).**\n\nFor every finite bipartite simple graph $H$ and every finite simple graph $G$:\n$t(H, G) \\ge t(K_2, G)^{e(H)}$, where $K_2$ denotes the single-edge graph on 2 vertices\n(i.e. `completeGraph (Fin 2)`).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"True ↔\n  ∀ {V W : Type} [inst : Fintype V] [inst_1 : Fintype W] [inst_2 : DecidableEq V] [inst_3 : DecidableEq W] [Nonempty W]\n    (H : SimpleGraph V) (G : SimpleGraph W) [inst_5 : DecidableRel H.Adj] [inst_6 : DecidableRel G.Adj],\n    H.IsBipartite → (SimpleGraph.completeGraph (Fin 2)).homDensity G ^ H.edgeFinset.card ≤ H.homDensity G","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**Tournament Anti-Sidorenko (TAS) Trees Conjecture (Tournamenton limit version).**\n\nFor every finite undirected tree $T$, there exists an orientation $\\vec{T}$ of its edges such that\nfor every tournamenton $W : [0, 1]^2 \\to [0, 1]$, the homomorphism density satisfies:\n$$ t_{\\vec{T}}(W) \\le 2^{-e(T)} $$\nwhere $e(T)$ is the total number of edges in $T$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"True ↔\n  ∀ {V : Type u_3} [inst : Fintype V] [DecidableEq V] (T : SimpleGraph V) [inst_2 : DecidableRel T.Adj],\n    T.IsTree →\n      ∃ D,\n        D.IsOrientation T ∧\n          ∀ (W : LimitObjects.Tournamenton), LimitObjects.tournamentonHomDensity D W ≤ (1 / 2) ^ T.edgeFinset.card","subjects":["5"],"theorem":"SidorenkoConjecture.tournament_anti_sidorenko_trees_conjecture_tournamenton"},{"answerKinds":[],"category":"research solved","docstring":"**Case: complete bipartite graphs `K_{s,t}` (Sidorenko 1993).**\n\nSidorenko's inequality holds when `H = K_{s,t}` is a complete bipartite graph. This is one\nof the earliest known cases; it follows from repeated applications of the Cauchy–Schwarz /\nHölder inequality. The `s = t = 2` instance is `sidorenko_K22`.\n\n*Reference:* [Si93].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ (s t : ℕ) [inst : DecidableRel (completeBipartiteGraph (Fin s) (Fin t)).Adj] {W : Type} [inst_1 : Fintype W]\n  [inst_2 : DecidableEq W] [Nonempty W] (G : SimpleGraph W) [inst_4 : DecidableRel G.Adj],\n  (SimpleGraph.completeGraph (Fin 2)).homDensity G ^ (completeBipartiteGraph (Fin s) (Fin t)).edgeFinset.card ≤\n    (completeBipartiteGraph (Fin s) (Fin t)).homDensity G","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_conjecture.variants.complete_bipartite"},{"answerKinds":[],"category":"research solved","docstring":"**TAS Trees Conjecture: Trees with a single even-degree vertex.**\n\nProven case of TAS Trees Conjecture: any tree containing exactly one vertex of even degree\npossesses an orientation satisfying the Tournament Anti-Sidorenko inequality $t_{\\vec{T}}(G) \\le 2^{-e(T)}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {V : Type u_3} [inst : Fintype V] [inst_1 : DecidableEq V] (T : SimpleGraph V) [inst_2 : DecidableRel T.Adj],\n  T.IsTree →\n    {v | Even (T.degree v)}.card = 1 →\n      ∃ D,\n        D.IsOrientation T ∧\n          ∀ {W : Type u_4} [inst_3 : Fintype W] [DecidableEq W] [Nonempty W] (G : Digraph W)\n            [inst_6 : DecidableRel G.Adj], G.IsTournament → D.homDensity G ≤ (1 / 2) ^ T.edgeFinset.card","subjects":["5"],"theorem":"SidorenkoConjecture.tournament_anti_sidorenko_single_even_degree_tree"},{"answerKinds":[],"category":"research solved","docstring":"**Base case of Sidorenko for trees.** When the tree `H` has at most one vertex,\nboth sides of Sidorenko's inequality evaluate to `1` (there are no edges, and every\nfunction is a homomorphism). Thus the inequality `1 ≤ 1` holds trivially.\n\nThis covers the `|V(T)| = 1` base case of the induction on tree size. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {V W : Type} [inst : Fintype V] [inst_1 : Fintype W] [Subsingleton V] [inst_3 : DecidableEq V]\n  [inst_4 : DecidableEq W] [Nonempty W] (H : SimpleGraph V) (G : SimpleGraph W) [inst_6 : DecidableRel H.Adj]\n  [inst_7 : DecidableRel G.Adj], (SimpleGraph.completeGraph (Fin 2)).homDensity G ^ H.edgeFinset.card ≤ H.homDensity G","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_tree_subsingleton"},{"answerKinds":[],"category":"research open","docstring":"**Tournament Anti-Sidorenko (TAS) Trees Conjecture.**\n\nFor every finite undirected tree $T$, there exists an orientation $\\vec{T}$ of its edges such that\nfor any finite tournament $G$, the homomorphism density satisfies:\n$$ t_{\\vec{T}}(G) \\le 2^{-e(T)} $$\nwhere $e(T)$ is the total number of edges in $T$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"True ↔\n  ∀ {V : Type u_3} [inst : Fintype V] [inst_1 : DecidableEq V] (T : SimpleGraph V) [inst_2 : DecidableRel T.Adj],\n    T.IsTree →\n      ∃ D,\n        D.IsOrientation T ∧\n          ∀ {W : Type u_4} [inst_3 : Fintype W] [DecidableEq W] [Nonempty W] (G : Digraph W)\n            [inst_6 : DecidableRel G.Adj], G.IsTournament → D.homDensity G ≤ (1 / 2) ^ T.edgeFinset.card","subjects":["5"],"theorem":"SidorenkoConjecture.tournament_anti_sidorenko_trees_conjecture"},{"answerKinds":[],"category":"API","docstring":"If the domain of `H` has at most one vertex, `H` has no edges.\n\n**Proof.** `SimpleGraph.Adj` is irreflexive, so any adjacency `H.Adj u v` forces\n`u ≠ v`; on a subsingleton that's impossible, so no edges exist. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {V : Type u_3} [inst : Fintype V] [Subsingleton V] (H : SimpleGraph V) [inst_2 : DecidableRel H.Adj],\n  H.edgeFinset.card = 0","subjects":["5"],"theorem":"SidorenkoConjecture.edgeCount_eq_zero_of_subsingleton"},{"answerKinds":[],"category":"API","docstring":"**The `Hom(K_{2,2}, G)` decomposition.** The number of homomorphisms from\n`K_{2,2}` to `G` equals `∑_{(a, b) ∈ W × W} |N(a) ∩ N(b)|^2`, where `N(v)` is the\nneighbourhood of `v` in `G`. Equivalently, summing over *ordered* pairs\n`(b₀, b₁) ∈ W × W` and counting common neighbours squared.\n\n**Math.** A homomorphism `K_{2,2} →g G` is an assignment `f : Fin 2 ⊕ Fin 2 → W`\nwith `G.Adj (f (inl i)) (f (inr j))` for all `i, j ∈ Fin 2`. Equivalently, choose\n`(a₀, a₁) := (f (inl 0), f (inl 1))` arbitrarily in `W × W` and require\n`(f (inr 0), f (inr 1))` to both lie in `N(a₀) ∩ N(a₁)`. The count is thus\n`∑_{(a₀, a₁)} |N(a₀) ∩ N(a₁)|²`.\n\n**Proof.** Construct an explicit bijection\n`(K_{2,2} →g G) ≃ Σ (p : W × W), (N(p.1) ∩ N(p.2)) × (N(p.1) ∩ N(p.2))`\nby sending a homomorphism `f` to `⟨(f (inl 0), f (inl 1)), ⟨f (inr 0), f (inr 1)⟩⟩`.\nThe total cardinality of the sigma-product is then\n`∑_p (Fintype.card (N(p.1) ∩ N(p.2)))² = ∑_p |N(p.1) ∩ N(p.2)|²`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ {W : Type u_3} [inst : Fintype W] [inst_1 : DecidableEq W] (G : SimpleGraph W) [inst_2 : DecidableRel G.Adj],\n  ↑((completeBipartiteGraph (Fin 2) (Fin 2)).homCount G) = ∑ p, ↑(G.neighborFinset p.1 ∩ G.neighborFinset p.2).card ^ 2","subjects":["5"],"theorem":"SidorenkoConjecture.homCount_completeBipartiteGraph_fin_two_eq_sum_inter_sq"},{"answerKinds":[],"category":"research open","docstring":"**Sidorenko's conjecture for graphons (1993).**\n\nFor every finite bipartite simple graph $H$ and every graphon $W$ on $[0, 1]$ with Lebesgue measure:\n$t(H, W) \\ge t(K_2, W)^{e(H)}$, where $t(K_2, W) = p(W)$ is the edge density of $W$,\nand $t(H, W)$ is the graphon homomorphism density of $H$ in $W$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"True ↔\n  ∀ {V : Type u_3} [inst : Fintype V] [DecidableEq V] (H : SimpleGraph V) [inst_2 : DecidableRel H.Adj],\n    H.IsBipartite →\n      ∀ (W : LimitObjects.Graphon),\n        LimitObjects.graphonEdgeDensity W ^ H.edgeFinset.card ≤ LimitObjects.graphonHomDensity H W","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_conjecture_graphon"},{"answerKinds":[],"category":"research solved","docstring":"**Case: stars `K_{1,m}` (Sidorenko).**\n\nSidorenko's inequality holds for every star `K_{1,m} = completeBipartiteGraph (Fin 1) (Fin m)`.\nThis is a fully proved instance of both `sidorenko_conjecture.variants.complete_bipartite` and\n`sidorenko_tree` (a star is a complete bipartite graph and a tree).\n\n**Proof.** With `d(c) := G.degree c` and `N := |W|`, the star has `m` edges, so the desired\ninequality is `(∑_c d(c) / N^2)^m ≤ (∑_c d(c)^m) / N^{m+1}` (using `t(K_2, G) = ∑_c d(c) / N^2`\nand `t(K_{1,m}, G) = ∑_c d(c)^m / N^{m+1}` via `homCount_star_eq_sum_degree_pow`). After clearing\ndenominators this is `(∑_c d(c))^m ≤ N^{m-1} · ∑_c d(c)^m`, the power-mean (Jensen) inequality\n`Finset.pow_sum_div_card_le_sum_pow`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SidorenkoConjecture","statement":"∀ (m : ℕ) [inst : DecidableRel (completeBipartiteGraph (Fin 1) (Fin m)).Adj] {W : Type} [inst_1 : Fintype W]\n  [inst_2 : DecidableEq W] [Nonempty W] (G : SimpleGraph W) [inst_4 : DecidableRel G.Adj],\n  (SimpleGraph.completeGraph (Fin 2)).homDensity G ^ (completeBipartiteGraph (Fin 1) (Fin m)).edgeFinset.card ≤\n    (completeBipartiteGraph (Fin 1) (Fin m)).homDensity G","subjects":["5"],"theorem":"SidorenkoConjecture.sidorenko_conjecture.variants.star"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The **Jacobian Conjecture**: any regular function\n(i.e. vector valued polynomial function from) `kⁿ → kᵐ`\nwhose Jacobian is a non-zero constant has an inverse that\nis given by a regular function, where `k` is a field of characteristic `0`.\n\nThis is false: `F` has Jacobian determinant `1` but identifies\ntwo distinct points, so it admits no inverse. This counterexample works in all characteristics. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.JacobianConjecture","statement":"∀ {k : Type} [inst : CommRing k] [Nontrivial k],\n  False ↔ ∀ {σ : Type} [inst_2 : Fintype σ] [inst_3 : DecidableEq σ], JacobianConjecture.JacobianConjectureProp k σ","subjects":["14"],"theorem":"JacobianConjecture.jacobian_conjecture"},{"answerKinds":[],"category":"API","docstring":"`G` identifies the two distinct points `(1, 0, 1)` and `(0, 3, -71)`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.JacobianConjecture","statement":"∀ (k : Type u_1) [inst : CommRing k],\n  (JacobianConjecture.G k).aeval ![1, 0, 1] = (JacobianConjecture.G k).aeval ![0, 3, -71]","subjects":["14"],"theorem":"JacobianConjecture.aeval_G_eq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.JacobianConjecture","statement":"∀ {k σ : Type} [inst : Fintype σ] [inst_1 : DecidableEq σ] [inst_2 : Field k],\n  JacobianConjecture.JacobianConjectureProp k σ →\n    ∃ G,\n      G.comp (MvPolynomial.RegularFunction.id k σ) = MvPolynomial.RegularFunction.id k σ ∧\n        (MvPolynomial.RegularFunction.id k σ).comp G = MvPolynomial.RegularFunction.id k σ","subjects":["14"],"theorem":"JacobianConjecture.jacobian_conjecture_identity"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.JacobianConjecture","statement":"∀ (k : Type u_1) [inst : CommRing k], (JacobianConjecture.G k).Jacobian.det = 1","subjects":["14"],"theorem":"JacobianConjecture.det_jacobian_G"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.JacobianConjecture","statement":"∀ (k : Type u_1) [inst : CommRing k], (JacobianConjecture.F k).Jacobian.det = -2","subjects":["14"],"theorem":"JacobianConjecture.det_jacobian_F"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.JacobianConjecture","statement":"∀ {k σ : Type} [inst : Fintype σ] [inst_1 : DecidableEq σ] [inst_2 : Field k] (F : MvPolynomial.RegularFunction k σ σ),\n  IsUnit F.Jacobian.det ↔ ∃ c, c ≠ 0 ∧ F.Jacobian.det = MvPolynomial.C c","subjects":["14"],"theorem":"JacobianConjecture.sanity_check_condition_1"},{"answerKinds":[],"category":"API","docstring":"`F` identifies the two distinct points `(0, 0, -1/4)` and `(1, -3/2, 13/2)`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.JacobianConjecture","statement":"∀ (k : Type u_1) [inst : Field k] [CharZero k],\n  (JacobianConjecture.F k).aeval ![0, 0, -1 / 4] = (JacobianConjecture.F k).aeval ![1, -3 / 2, 13 / 2]","subjects":["14"],"theorem":"JacobianConjecture.aeval_F_eq"},{"answerKinds":[],"category":"research open","docstring":"Does the Jacobian conjecture hold in the two variable case? ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.JacobianConjecture","statement":"True ↔ ∀ {k : Type} [inst : Field k] [CharZero k], JacobianConjecture.JacobianConjectureProp k (Fin 2)","subjects":["14"],"theorem":"JacobianConjecture.jacobian_conjecture_two_variables"},{"answerKinds":[],"category":"research open","docstring":"It is not known whether every Euclid number is a square-free number.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Euclid","statement":"True ↔ ∀ (n : ℕ), Squarefree (EuclidNumbers.Euclid n)","subjects":["11"],"theorem":"EuclidNumbers.euclid_numbers_are_square_free"},{"answerKinds":[],"category":"research open","docstring":"It is not known whether there is an infinite number of prime Euclid numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Euclid","statement":"True ↔ {n | Nat.Prime (EuclidNumbers.Euclid n)}.Infinite","subjects":["11"],"theorem":"EuclidNumbers.infinite_prime_euclid_numbers"},{"answerKinds":[],"category":"research open","docstring":"**Quasiperfect Numbers Conjecture.**\nDo quasiperfect numbers exist?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.QuasiperfectNumbers","statement":"True ↔ ∃ n, QuasiperfectNumbers.Quasiperfect n","subjects":["11"],"theorem":"QuasiperfectNumbers.exists_quasiperfect"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Pell","statement":"PellNumbers.pellNumber 1 = 1","subjects":["11"],"theorem":"PellNumbers.pellNumber_one"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many prime Pell numbers ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Pell","statement":"Infinite ↑{n | Prime (PellNumbers.pellNumber n)}","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"PellNumbers.infinite_pellNumber_primes"},{"answerKinds":[],"category":"textbook","docstring":"Similar to Fibonacci numbers, there exist numerous identities around Pell numbers, i.e.\nP_{2n+1} = P_n ^ 2 + P_{n+1} ^ 2 ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Pell","statement":"∀ (n : ℕ), PellNumbers.pellNumber (2 * n + 1) = PellNumbers.pellNumber n ^ 2 + PellNumbers.pellNumber (n + 1) ^ 2","subjects":["11"],"theorem":"PellNumbers.pellNumber_sq_add_pellNumber_succ_sq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Pell","statement":"PellNumbers.pellNumber 2 = 2","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"PellNumbers.pellNumber_two"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Pell","statement":"PellNumbers.pellNumber 5 = 29","subjects":["11"],"theorem":"PellNumbers.pellNumber_five"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Pell","statement":"PellNumbers.pellNumber 0 = 0","subjects":["11"],"theorem":"PellNumbers.pellNumber_zero"},{"answerKinds":[],"category":"textbook","docstring":"An explicit formula for Pell numbers, similar to Binet's formula ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Pell","statement":"∀ (n : ℕ), ↑(PellNumbers.pellNumber n) = ((1 + √2) ^ n - (1 - √2) ^ n) / (2 * √2)","subjects":["11"],"theorem":"PellNumbers.coe_pellNumber_eq"},{"answerKinds":[],"category":"research open","docstring":"`μ=M(X^10 + X^9 - X^7 - X^6 - X^5 - X^4 - X^3 + X + 1)` is the best value for `lehmer_mahler_measure_problem`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LehmerMahlerMeasureProblem","statement":"∀ (f : Polynomial ℤ),\n  LehmerMahlerMeasureProblem.mahlerMeasureZ f > 1 →\n    LehmerMahlerMeasureProblem.mahlerMeasureZ f ≥\n      LehmerMahlerMeasureProblem.mahlerMeasureZ LehmerMahlerMeasureProblem.lehmerPolynomial","subjects":["11"],"theorem":"LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem.variants.best"},{"answerKinds":[],"category":"research solved","docstring":"If $f$ is not reciprocal and $M(f) > 1$ then $M(f) \\ge M(X^3 - X - 1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LehmerMahlerMeasureProblem","statement":"∀ (f : Polynomial ℤ),\n  LehmerMahlerMeasureProblem.mahlerMeasureZ f > 1 →\n    f.reverse ≠ f →\n      LehmerMahlerMeasureProblem.mahlerMeasureZ f ≥\n        LehmerMahlerMeasureProblem.mahlerMeasureZ (Polynomial.X ^ 3 - Polynomial.X - 1)","subjects":["11"],"theorem":"LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem.variants.not_reciprocal"},{"answerKinds":[],"category":"research open","docstring":"Let `M(f)` denote the Mahler measure of `f`.\nThere exists a constant `μ>1` such that for any `f(x)∈ℤ[x], M(f)>1 → M(f)≥μ`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LehmerMahlerMeasureProblem","statement":"∃ μ,\n  ∀ (f : Polynomial ℤ),\n    μ > 1 ∧ (LehmerMahlerMeasureProblem.mahlerMeasureZ f > 1 → LehmerMahlerMeasureProblem.mahlerMeasureZ f ≥ μ)","subjects":["11"],"theorem":"LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem"},{"answerKinds":[],"category":"research solved","docstring":"If all the coefficients of $f$ are odd and $M(f) > 1$, then $M(f) \\ge M(X^2 - X - 1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LehmerMahlerMeasureProblem","statement":"∀ (f : Polynomial ℤ),\n  LehmerMahlerMeasureProblem.mahlerMeasureZ f > 1 →\n    LehmerMahlerMeasureProblem.Polynomial.HasOddCoeffs f →\n      LehmerMahlerMeasureProblem.mahlerMeasureZ f ≥\n        LehmerMahlerMeasureProblem.mahlerMeasureZ (Polynomial.X ^ 2 - Polynomial.X - 1)","subjects":["11"],"theorem":"LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem.variants.odd"},{"answerKinds":[],"category":"research open","docstring":"For real number $\\alpha$ and prime $p$,\n$$\n  \\liminf\\_{n \\to\\infty} n |n|\\_{p}\\||n\\alpha\\|| = 0\n$$\nwhere $\\||x\\|| := \\min(|x - \\lfloor x \\rfloor|, |x - \\lceil x \\rceil|)$ is the distance\nto the nearest integer, and $|x|\\_{p}$ is the $p$-adic norm.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LittlewoodConjecture","statement":"∀ (α : ℝ) (p : ℕ),\n  Nat.Prime p → Filter.liminf (fun n => ↑n * ↑(padicNorm p ↑n) * distToNearestInt (↑n * α)) Filter.atTop = 0","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"LittlewoodConjecture.padic_littlewood_conjecture"},{"answerKinds":[],"category":"research open","docstring":"For any two real numbers $\\alpha$ and $\\beta$,\n$$\n  \\liminf_{n\\to\\infty} n\\||n\\alpha\\||\\||n\\beta\\|| = 0\n$$\nwhere $\\||x\\|| := \\min(|x - \\lfloor x \\rfloor|, |x - \\lceil x \\rceil|)$ is the distance\nto the nearest integer.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LittlewoodConjecture","statement":"∀ (α β : ℝ), Filter.liminf (fun n => ↑n * distToNearestInt (↑n * α) * distToNearestInt (↑n * β)) Filter.atTop = 0","subjects":["11"],"theorem":"LittlewoodConjecture.littlewood_conjecture"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many Pierpont primes. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PierpontPrime","statement":"{p | PierpontPrime.IsPierpontPrime p}.Infinite","subjects":["11"],"theorem":"PierpontPrime.infinitely_many_pierpont_primes"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.PierpontPrime","statement":"PierpontPrime.IsPierpontPrime 2","subjects":["11"],"theorem":"PierpontPrime.two_isPierpontPrime"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.PierpontPrime","statement":"PierpontPrime.IsPierpontPrime 37","subjects":["11"],"theorem":"PierpontPrime.thirtySeven_isPierpontPrime"},{"answerKinds":[],"category":"research open","docstring":"Now form a sequence beginning with any positive integer, where each subsequent term is obtained\nby applying the operation defined above to the previous term.\nThe **Collatz conjecture** states that for any positive integer $n$, there exists a natural number\n$m$ such that the $m$-th term of the sequence is 1.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CollatzConjecture","statement":"∀ n > 0, ∃ m, CollatzConjecture.collatzStep^[m] n = 1","subjects":["11","37"],"theorem":"CollatzConjecture.collatz_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**Gilbreath's conjecture**\nGilbreath's conjecture states that every term in the sequence $d^k_0$ for $k > 0$ is equal to 1.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Gilbreath","statement":"∀ (k : ℕ+), Gilbreath.d (↑k) 0 = 1","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Gilbreath.gilbreath_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Consider $n$ runners on a circular track of unit length. At the initial time\n$t = 0$, all runners are at the same position and start to run; the runners'\nspeeds are constant, all distinct, and may be negative. A runner is said to be\nlonely at time $t$ if they are at a distance (measured along the circle) of at\nleast $\\frac 1 n$ from every other runner. The lonely runner conjecture states that each\nrunner is lonely at some time, no matter the choice of speeds.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LonelyRunnerConjecture","statement":"∀ (n : ℕ) (speed : Fin n ↪ ℝ) (lonely : Fin n → ℝ → Prop),\n  (∀ (r : Fin n) (t : ℝ), lonely r t ↔ ∀ (r2 : Fin n), r2 ≠ r → dist ↑(t * speed r) ↑(t * speed r2) ≥ 1 / ↑n) →\n    ∀ (r : Fin n), ∃ t ≥ 0, lonely r t","subjects":["11"],"theorem":"LonelyRunnerConjecture.lonely_runner_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"**Theorem 1.3 (Tao, 2017; arXiv:1701.02048).**\nThere exists an absolute constant $c > 0$ such that for all sufficiently large $n$,\nthe gap of loneliness satisfies\n$\\delta_n \\ge \\frac{1}{2n} + \\frac{c \\log n}{n^2 (\\log \\log n)^2}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LonelyRunnerConjecture","statement":"∃ c,\n  0 < c ∧\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      LonelyRunnerConjecture.deltaGap n ≥ 1 / (2 * ↑n) + c * Real.log ↑n / (↑n ^ 2 * Real.log (Real.log ↑n) ^ 2)","subjects":["11"],"theorem":"LonelyRunnerConjecture.lonely_runner_conjecture.variants.tao_2017"},{"answerKinds":[],"category":"research solved","docstring":"The Univalent Bloch constant is trivially bounded above by the Bloch radius of the identity\nfunction, which is $1$. This is the best upper bound we know according to [OptimizationConstants]. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Bloch","statement":"Bloch.univalentBlochConstant ≤ 1","subjects":["30"],"theorem":"Bloch.univalentBlochConstant_upper_bound"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Bloch","statement":"Bloch.blochRadius id = 1","subjects":["30"],"theorem":"Bloch.blochRadius_id_eq_one"},{"answerKinds":[],"category":"research solved","docstring":"It is proved in [Ya95] that the Landau constant is bounded below by $0.5 + 10 ^ {-335}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Bloch","statement":"0.5 + 10 ^ (-335) ≤ Bloch.landauConstant","subjects":["30"],"theorem":"Bloch.landauConstant_lower_bound"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Bloch","statement":"∀ (f : ℂ → ℂ), 0 ≤ Bloch.blochRadius f","subjects":["30"],"theorem":"Bloch.zero_le_blochRadius"},{"answerKinds":[],"category":"research solved","docstring":"It is proved in [AG37] that the Bloch constant is bounded above by\n$\\frac{1}{\\sqrt{1 + \\sqrt{3}}}\\frac{\\Gamma(1/3) \\Gamma(11/12)}{\\Gamma(1/4)}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Bloch","statement":"Bloch.blochConstant ≤ Real.Gamma (1 / 3) * Real.Gamma (11 / 12) / (Real.Gamma (1 / 4) * √(1 + √3))","subjects":["30"],"theorem":"Bloch.blochConstant_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"It is proved in [Ra43] that the Landau constant is bounded above by\n$\\frac{1}{\\sqrt{1 + \\sqrt{3}}}\\frac{\\Gamma(1/3) \\Gamma(5/6)}{\\Gamma(1/6)}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Bloch","statement":"Bloch.landauConstant ≤ Real.Gamma (1 / 3) * Real.Gamma (5 / 6) / Real.Gamma (1 / 6)","subjects":["30"],"theorem":"Bloch.landauConstant_upper_bound"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Bloch","statement":"∀ {X : Type u_1} {𝕜 : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup X] [NormedSpace 𝕜 X] [Nontrivial X]\n  {x y : X} {r d : ℝ}, 0 < r → Metric.ball x r ⊆ Metric.ball y d → dist x y + r ≤ d","subjects":["54"],"theorem":"Bloch.dis_add_radius_le_of_ball_subset_ball"},{"answerKinds":[],"category":"research open","docstring":"In [Ra43], Rademacher says that he strongly believed that this upper bound is the precise value\nof the Landau constant. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Bloch","statement":"Bloch.landauConstant = Real.Gamma (1 / 3) * Real.Gamma (5 / 6) / Real.Gamma (1 / 6)","subjects":["30"],"theorem":"Bloch.landauConstant_exact_value"},{"answerKinds":[],"category":"research solved","docstring":"It is proved in [Skin2009] that the Univalent Bloch constant is bounded below by $0.5708858$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Bloch","statement":"0.5708858 ≤ Bloch.univalentBlochConstant","subjects":["30"],"theorem":"Bloch.univalentBlochConstant_lower_bound"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Bloch","statement":"∀ {X : Type u_1} {𝕜 : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup X] [NormedSpace 𝕜 X] [Nontrivial X]\n  {x y : X} {r d : ℝ}, 0 < r → Metric.ball x r ⊆ Metric.ball y d → r ≤ d","subjects":["54"],"theorem":"Bloch.radius_le_of_ball_subset_ball"},{"answerKinds":[],"category":"research solved","docstring":"It is proved in [CP96] that the Bloch constant is bounded below by\n$\\sqrt{3}/4 + 2 \\times 10^{-4}$ ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Bloch","statement":"√3 / 4 + 2 * 10 ^ (-4) ≤ Bloch.blochConstant","subjects":["30"],"theorem":"Bloch.blochConstant_lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the\nBloch constant. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Bloch","statement":"Bloch.blochConstant = Real.Gamma (1 / 3) * Real.Gamma (11 / 12) / (Real.Gamma (1 / 4) * √(1 + √3))","subjects":["30"],"theorem":"Bloch.blochConstant_exact_value"},{"answerKinds":[],"category":"research open","docstring":"The New Mersenne Conjecture statement holds for odd primes. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mersenne","statement":"∀ (p : ℕ), Nat.Prime p → Odd p → Mersenne.NewMersenneConjectureStatement p","subjects":["11"],"theorem":"Mersenne.new_mersenne_conjecture.variants.prime"},{"answerKinds":[],"category":"textbook","docstring":"It suffices to check this conjecture for primes ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Mersenne","statement":"(∀ (p : ℕ), Nat.Prime p → Mersenne.NewMersenneConjectureStatement p) →\n  ∀ (p : ℕ), Odd p → Mersenne.NewMersenneConjectureStatement p","subjects":["11"],"theorem":"Mersenne.new_mersenne_conjecture_of_prime"},{"answerKinds":[],"category":"research open","docstring":"For any odd natural number `p` if two of the following conditions hold,\nthen all three must hold:\n1. $2^p-1$ is prime\n2. $(2^p+1)/3$ is prime\n3. Exists a number `k` such that $p = 2^k \\\\pm 1$ or $p = 4^k \\\\pm 3$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mersenne","statement":"∀ (p : ℕ), Odd p → Mersenne.NewMersenneConjectureStatement p","subjects":["11"],"theorem":"Mersenne.new_mersenne_conjecture"},{"answerKinds":[],"category":"research open","docstring":"The first five Catalan-Mersenne numbers $c_0, \\ldots, c_4$ are known to be prime.\nCatalan conjectured that they are prime \"up to a certain limit\".\nAre all Catalan-Mersenne numbers $c_n$ with $n \\geq 5$ prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mersenne","statement":"True ↔ ∀ n ≥ 5, Nat.Prime (Mersenne.catalanMersenne n)","subjects":["11"],"theorem":"Mersenne.catalans_mersenne_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many Mersenne primes?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mersenne","statement":"True ↔ {p | Nat.Prime (mersenne p)}.Infinite","subjects":["11"],"theorem":"Mersenne.infinitely_many_mersenne_primes"},{"answerKinds":[],"category":"research open","docstring":"**Schinzel conjecture (H hypothesis)**\nIf a finite set of polynomials $f_i$ satisfies both Schinzel and Bunyakovsky conditions,\nthere exist infinitely many natural numbers $n$ such that $f_i(n)$ are primes for all $i$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Schinzel","statement":"∀ (fs : Finset (Polynomial ℤ)),\n  (∀ f ∈ fs, BunyakovskyCondition f) →\n    SchinzelCondition fs → Infinite ↑{n | ∀ f ∈ fs, Nat.Prime (Polynomial.eval (↑n) f).natAbs}","subjects":["11"],"theorem":"Schinzel.schinzel_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**The Bateman-Horn Conjecture**\nGiven a finite collection of distinct irreducible polynomials non-constant $f_1, f_2, \\dots, f_k \\in \\mathbb{Z}[x]$\nwith positive leading coefficients that satisfy the Schinzel condition, the number\nof positive integers n ≤ x for which all polynomials $f_i$ are simultaneously prime is asymptotic to:\n$$C(f_1, f_2, \\dots, f_k) x / (log x)^k$$\nwhere $C$ is the Bateman-Horn constant given by the convergent infinite product:\n$$C = \\frac{1}{D}\\prod_{p\\in\\mathbb{P}} (1 - 1/p)^(-k) · (1 - \\omega_p/p)$$\nHere $\\omega_p/p$ is the number of residue classes modulo $p$ for which at least one polynomial vanishes.\n\nThe Schinzel condition ensures that for each prime $p$, there exists some integer $n$\nsuch that $p$ does not divide the product $f_(n) f_2(n) \\dotsb f_(n)$, which guarantees the\ninfinite product converges to a positive value.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BatemanHornConjecture","statement":"∀ (polys : Finset (Polynomial ℤ)),\n  polys.Nonempty →\n    (∀ f ∈ polys, BunyakovskyCondition f) →\n      SchinzelCondition polys →\n        Asymptotics.IsEquivalent Filter.atTop (fun x => ↑(BatemanHornConjecture.CountSimultaneousPrimes polys x))\n          fun x => BatemanHornConjecture.BatemanHornConstant polys * x / Real.log x ^ polys.card","subjects":["11","12"],"theorem":"BatemanHornConjecture.bateman_horn_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"The Pierce-Birkhoff conjecture holds for `n = 1`.\nThis was proved by Louis Mahé.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PierceBirkhoff","statement":"∀ (f : ℝ → ℝ),\n  PierceBirkhoff.IsPiecewisePolynomial f →\n    ∃ ι κ g, Finite ι ∧ Finite κ ∧ ∀ (x : ℝ), f x = ⨆ i, ⨅ j, Polynomial.eval x (g i j)","subjects":["13"],"theorem":"PierceBirkhoff.pierce_birkhoff_conjecture_dim_one"},{"answerKinds":[],"category":"research solved","docstring":"The Pierce-Birkhoff conjecture holds for `n = 2`.\nThis was proved by Louis Mahé.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PierceBirkhoff","statement":"∀ (f : (Fin 2 → ℝ) → ℝ),\n  PierceBirkhoff.IsPiecewiseMvPolynomial f →\n    ∃ ι κ g, Finite ι ∧ Finite κ ∧ ∀ (x : Fin 2 → ℝ), f x = ⨆ i, ⨅ j, (MvPolynomial.eval x) (g i j)","subjects":["13"],"theorem":"PierceBirkhoff.pierce_birkhoff_conjecture_dim_two"},{"answerKinds":[],"category":"research open","docstring":"The Pierce-Birkhoff conjecture states that for every real piecewise-polynomial function\n`f : ℝⁿ → ℝ`, there exists a finite set of polynomials `gᵢⱼ ∈ ℝ[x₁, ..., xₙ]` such that\n`f = supᵢ infⱼ(gᵢⱼ)`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PierceBirkhoff","statement":"∀ {n : ℕ} (f : (Fin n → ℝ) → ℝ),\n  PierceBirkhoff.IsPiecewiseMvPolynomial f →\n    ∃ ι κ g, Finite ι ∧ Finite κ ∧ ∀ (x : Fin n → ℝ), f x = ⨆ i, ⨅ j, (MvPolynomial.eval x) (g i j)","subjects":["13"],"theorem":"PierceBirkhoff.pierce_birkhoff_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Let $p_k$ be the $k$-th prime number.\nAre there infinitely many $n$ such that\n$p_n = \\dfrac{\\sum_{i = 1} ^ k p_{n - i} + p_{n + i}}{2*k}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BalancedPrimes","statement":"True ↔\n  ∀ k > 0,\n    {n |\n        k ≤ n ∧\n          2 * k * Nat.nth Prime n = ∑ i ∈ Finset.Ioc 0 k, (Nat.nth Prime (n - i) + Nat.nth Prime (n + i))}.Infinite","subjects":["11"],"theorem":"BalancedPrimes.balanced_primes_order"},{"answerKinds":[],"category":"research open","docstring":"Let $p_k$ be the $k$-th prime number.\nAre there infinitely many $n$ such that $(p_n + p_{n+2}) / 2$ is prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BalancedPrimes","statement":"True ↔ {n | Prime ((Nat.nth Prime n + Nat.nth Prime (n + 2)) / 2)}.Infinite","subjects":["11"],"theorem":"BalancedPrimes.balanced_primes"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Originally, on page 200 of [Ja1956], Jacobson asked if the Jacobson conjecture holds for all right\nNoetherian rings. However in [He1965] Herstein constructs a right Noetherian ring for which the\nJacobson conjecture does not hold. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Jacobson","statement":"False ↔ ∀ (R : Type) [inst : Ring R] [IsRightNoetherianRing R], Jacobson.JacobsonConjectureFor R","subjects":["16"],"theorem":"Jacobson.jacobson_conjecture_of_right_noetherian"},{"answerKinds":[],"category":"textbook","docstring":"For commutative rings this is the case as a consequence of Krull's intersection theorem. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Jacobson","statement":"∀ (R : Type u) [inst : CommRing R] [IsNoetherianRing R], Jacobson.JacobsonConjectureFor R","subjects":["13","16"],"theorem":"Jacobson.jacobson_conjecture_of_comm_ring"},{"answerKinds":[],"category":"research open","docstring":"The Jacobson conjecture (in its modern form):\nIn a (noncommutative) ring which is left and right Noetherian,\nthe intersection of the powers of the Jacobson ideal is trivial ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Jacobson","statement":"True ↔ ∀ (R : Type) [inst : Ring R] [IsNoetherianRing R] [IsRightNoetherianRing R], Jacobson.JacobsonConjectureFor R","subjects":["16"],"theorem":"Jacobson.jacobson_conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.GracefulLabeling","statement":"let T := ⊥;\nhave m := T.edgeFinset.card;\n∃ f,\n  Function.Injective f ∧\n    (∀ (v : Unit), f v ≤ m) ∧\n      Finset.image (fun e => Sym2.lift ⟨fun u v => (↑(f u) - ↑(f v)).natAbs, ⋯⟩ e) T.edgeFinset = Finset.Icc 1 m","subjects":["5"],"theorem":"GracefulLabeling.graceful_tree_one_vertex"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.GracefulLabeling","statement":"let T := ⊤;\nhave m := T.edgeFinset.card;\n∃ f,\n  Function.Injective f ∧\n    (∀ (v : Fin 2), f v ≤ m) ∧\n      Finset.image (fun e => Sym2.lift ⟨fun u v => (↑(f u) - ↑(f v)).natAbs, ⋯⟩ e) T.edgeFinset = Finset.Icc 1 m","subjects":["5"],"theorem":"GracefulLabeling.graceful_tree_two_vertex"},{"answerKinds":[],"category":"research open","docstring":"Every tree admits a graceful labeling.\n\nA graceful labeling of a tree $T$ with $m$ edges is an injective map $f : V \\to \\{0, \\dots, m\\}$\nsuch that the multiset of absolute differences $|f(u) - f(v)|$ over edges $\\{u,v\\}$ of $T$\nequals $\\{1, \\dots, m\\}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.GracefulLabeling","statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (T : SimpleGraph V) [inst_2 : DecidableRel T.Adj],\n  T.IsTree →\n    have m := T.edgeFinset.card;\n    ∃ f,\n      Function.Injective f ∧\n        (∀ (v : V), f v ≤ m) ∧\n          Finset.image (fun e => Sym2.lift ⟨fun u v => (↑(f u) - ↑(f v)).natAbs, ⋯⟩ e) T.edgeFinset = Finset.Icc 1 m","subjects":["5"],"theorem":"GracefulLabeling.graceful_tree_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**Roman B. Popovych Conjecture.**\nA stronger version of Agrawal's conjecture, which also considers the congruence\n$(X+2)^n \\equiv X^n + 2 \\pmod{n, X^r-1}$.\nIf both congruences hold, then $n$ is either prime or $n^2 \\equiv 1 \\pmod{r}$.\nThis variant was proposed by Roman B. Popovych in 2018.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Agrawal","statement":"∀ (n r : ℕ),\n  n > 1 →\n    r > 0 →\n      n.gcd r = 1 →\n        let R := Polynomial (ZMod n);\n        have X := Polynomial.X;\n        have I := Ideal.span {X ^ r - 1};\n        (Ideal.Quotient.mk I) ((X - 1) ^ n) = (Ideal.Quotient.mk I) (X ^ n - 1) →\n          (Ideal.Quotient.mk I) ((X + 2) ^ n) = (Ideal.Quotient.mk I) (X ^ n + 2) → Nat.Prime n ∨ ↑n ^ 2 = 1","subjects":["11"],"theorem":"AgrawalConjecture.agrawal_conjecture.variants.popovych"},{"answerKinds":[],"category":"research open","docstring":"**Agrawal's Primality Conjecture.**\n\nDoes the congruence $(X-1)^n \\equiv X^n - 1 \\pmod{n, X^r-1}$ imply\n$n$ is prime (with a specific exception for $n^2 \\equiv 1 \\pmod{r}$)?\n\nWhile the \"if\" direction is a known theorem, the \"only if\" direction\nremains a conjecture.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Agrawal","statement":"True ↔\n  ∀ (n r : ℕ),\n    n > 1 →\n      r > 0 →\n        n.gcd r = 1 →\n          let R := Polynomial (ZMod n);\n          have X := Polynomial.X;\n          have I := Ideal.span {X ^ r - 1};\n          (Ideal.Quotient.mk I) ((X - 1) ^ n) = (Ideal.Quotient.mk I) (X ^ n - 1) → Nat.Prime n ∨ ↑n ^ 2 = 1","subjects":["11"],"theorem":"AgrawalConjecture.agrawal_conjecture"},{"answerKinds":[],"category":"API","docstring":"Cyclic groups of perfect number order are Leinster groups.\n\nThis follows from the fact that for a cyclic group, all subgroups are normal and correspond\nto divisors of the group order, and a number is perfect if and only if the sum of its divisors\n(including itself) equals twice the number.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LeinsterGroup","statement":"∀ (G : Type u_1) [inst : Group G] [inst_1 : Fintype G] [IsCyclic G],\n  (Fintype.card G).Perfect → LeinsterGroup.IsLeinster G","subjects":["20"],"theorem":"LeinsterGroup.cyclic_of_perfect_is_leinster"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture:** Are there infinitely many Leinster groups?\n\nThis asks whether there exist infinitely many (non-isomorphic) finite groups that are\nLeinster groups.\n\nFormalized via the negation of \"Does there exist an n such that all Leinster groups have\norder less than n\".\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LeinsterGroup","statement":"True ↔ ¬∃ n, ∀ (G : Type) (x : Group G) (x_1 : Fintype G), LeinsterGroup.IsLeinster G → Fintype.card G < n","subjects":["20"],"theorem":"LeinsterGroup.infinitely_many_leinster_groups"},{"answerKinds":[],"category":"research solved","docstring":"Non-abelian Leinster groups exist.\n\nFor example, `S₃ × C₅` (order 30) and `A₅ × C₁₅₁₂₈` are Leinster groups.\n\nReference: Leinster, Tom (2001). \"Perfect numbers and groups\".\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LeinsterGroup","statement":"∃ G x x_1, LeinsterGroup.IsLeinster G ∧ ¬∀ (a b : G), a * b = b * a","subjects":["20"],"theorem":"LeinsterGroup.exists_nonabelian_leinster_group"},{"answerKinds":[],"category":"research solved","docstring":"An abelian group is a Leinster group if and only if it is cyclic with order equal\nto a perfect number.\n\nReference: Leinster, Tom (2001). \"Perfect numbers and groups\". Theorem 2.1.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LeinsterGroup","statement":"∀ (G : Type u_1) [inst : CommGroup G] [inst_1 : Fintype G],\n  LeinsterGroup.IsLeinster G ↔ IsCyclic G ∧ (Fintype.card G).Perfect","subjects":["20"],"theorem":"LeinsterGroup.abelian_is_leinster_iff_cyclic_perfect"},{"answerKinds":[],"category":"research solved","docstring":"The dihedral group `DihedralGroup n` (of order `2n`) is a Leinster group if and only if `n` is\nan odd perfect number. This gives a one-to-one correspondence between dihedral Leinster groups\nand odd perfect numbers.\n\nIn particular, the existence of odd perfect numbers is equivalent to the existence of\ndihedral Leinster groups.\n\nReference: Leinster, Tom (2001). \"Perfect numbers and groups\".\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LeinsterGroup","statement":"∀ (n : ℕ) [inst : NeZero n], LeinsterGroup.IsLeinster (DihedralGroup n) ↔ n.Perfect ∧ Odd n","subjects":["20"],"theorem":"LeinsterGroup.dihedral_is_leinster_iff_odd_perfect"},{"answerKinds":[],"category":"research solved","docstring":"Can every odd integer greater than 5 be written as the sum of three primes?\n(A prime may be used more than once.)\n\nNB. While Harald Helfgott's solution is not published in a peer-reviewed journal yet,\nhis results seem generally accepted.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.GoldbachConjecture","statement":"∀ (n : ℕ), 5 < n → Odd n → ∃ p q r, Nat.Prime p ∧ Nat.Prime q ∧ Nat.Prime r ∧ n = p + q + r","subjects":["11"],"theorem":"TernaryGoldbachConjecture.ternaryGoldbach"},{"answerKinds":[],"category":"research open","docstring":"Can every even integer greater than 2 be written as the sum of two primes?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.GoldbachConjecture","statement":"True ↔ ∀ (n : ℕ), 2 < n → Even n → ∃ p q, Prime p ∧ Prime q ∧ n = p + q","subjects":["11"],"theorem":"GoldbachConjecture.goldbach"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many primes p such that p + 2 is prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.TwinPrimes","statement":"True ↔ {p | Prime p ∧ Prime (p + 2)}.Infinite","subjects":["11"],"theorem":"TwinPrimes.twin_primes"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many prime numbers of the form `k * 2 ^ k - 1` for `k > 1`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WoodalPrimes","statement":"{k | 1 < k ∧ Nat.Prime (k * 2 ^ k - 1)}.Infinite","subjects":["11"],"theorem":"WoodallPrimes.infinitely_many_woodall_primes"},{"answerKinds":[],"category":"research open","docstring":"Salzer–Levine strengthening (as stated on Wikipedia/OEIS):\nthere are exactly $241$ integers that are not a sum of $4$ tetrahedral numbers, and the largest is $343867$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PollocksConjecture","statement":"IsGreatest PollocksConjecture.NotSumOfFourTetrahedral 343867","subjects":["11"],"theorem":"PollocksConjecture.pollock_tetrahedral.salzer_levine"},{"answerKinds":[],"category":"research open","docstring":"Pollock's (tetrahedral numbers) conjecture:\nevery integer is the sum of at most $5$ tetrahedral numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PollocksConjecture","statement":"∀ (N : ℕ), ∃ f, N = ∑ i, PollocksConjecture.tetrahedral (f i)","subjects":["11"],"theorem":"PollocksConjecture.pollock_tetrahedral"},{"answerKinds":[],"category":"textbook","docstring":"As stated on Wikipedia/OEIS (A797), the set of exceptions has cardinality $241$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PollocksConjecture","statement":"IsGreatest PollocksConjecture.NotSumOfFourTetrahedral 343867 ↔ PollocksConjecture.NotSumOfFourTetrahedral.ncard = 241","subjects":["11"],"theorem":"PollocksConjecture.pollock_tetrahedral.ncard_exceptions"},{"answerKinds":[],"category":"test","docstring":"`IsAmicable` is symmetric. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.AmicableNumbers","statement":"∀ {a b : ℕ}, IsAmicable a b → IsAmicable b a","subjects":["11"],"theorem":"AmicableNumbers.IsAmicable.symm"},{"answerKinds":[],"category":"research open","docstring":"**Relatively prime amicable numbers conjecture.**\nDo there exist amicable numbers $(a, b)$ with $\\gcd(a, b) = 1$?\n\nAll known amicable pairs share a common factor. It is an open question\nwhether a pair of relatively prime amicable numbers can exist.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Amicable_numbers)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AmicableNumbers","statement":"True ↔ ∃ a b, IsAmicable a b ∧ a ≠ b ∧ a.Coprime b","subjects":["11"],"theorem":"AmicableNumbers.relatively_prime_amicable"},{"answerKinds":["Prop"],"category":"research open","docstring":"**Infinitely many amicable numbers conjecture.**\n\nAre there infinitely many pairs of amicable numbers?\n\nWhile many amicable pairs are known, it remains open whether there are infinitely many.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Amicable_numbers),\n[erdosproblems.com/830](https://www.erdosproblems.com/830)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AmicableNumbers","statement":"sorry ↔ {(a, b) | IsAmicable a b}.Infinite","subjects":["11"],"theorem":"AmicableNumbers.infinitely_many_amicable"},{"answerKinds":[],"category":"research open","docstring":"**Amicable numbers with opposite parity conjecture.**\nDo there exist amicable numbers $(a, b)$ where one is even and the other is odd?\n\nAll known amicable pairs are either both even or both odd. It is widely believed\nthat mixed-parity amicable pairs do not exist, but this remains open.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Amicable_numbers)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AmicableNumbers","statement":"True ↔ ∃ a b, IsAmicable a b ∧ (Even a ↔ Odd b)","subjects":["11"],"theorem":"AmicableNumbers.opposite_parity_amicable"},{"answerKinds":[],"category":"test","docstring":"The classic amicable pair $(220, 284)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.AmicableNumbers","statement":"IsAmicable 220 284","subjects":["11"],"theorem":"AmicableNumbers.amicable_220_284"},{"answerKinds":[],"category":"research solved","docstring":"Upper bound $R(5,5) \\le 46$, i.e. every graph on $46$ vertices contains a $5$-clique or an\nindependent set of size $5$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RamseyNumbers","statement":"RamseyNumbers.IsGraphRamsey 46 5 5","subjects":["5"],"theorem":"RamseyNumbers.ramsey_number_five_five_upper_bound"},{"answerKinds":[],"category":"API","docstring":"Symmetry in the clique / independent set sizes. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.RamseyNumbers","statement":"∀ (n k l : ℕ), RamseyNumbers.IsGraphRamsey n k l ↔ RamseyNumbers.IsGraphRamsey n l k","subjects":["5"],"theorem":"RamseyNumbers.IsGraphRamsey.symm"},{"answerKinds":[],"category":"API","docstring":"The shared Ramsey number agrees with the clique-free formulation. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.RamseyNumbers","statement":"∀ (k l : ℕ), SimpleGraph.classicalRamsey k l = sInf {n | RamseyNumbers.IsGraphRamsey n k l}","subjects":["5"],"theorem":"RamseyNumbers.classicalRamsey_eq_sInf"},{"answerKinds":[],"category":"API","docstring":"Monotonicity in the number of vertices. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.RamseyNumbers","statement":"∀ (n k l : ℕ), RamseyNumbers.IsGraphRamsey n k l → RamseyNumbers.IsGraphRamsey (n + 1) k l","subjects":["5"],"theorem":"RamseyNumbers.IsGraphRamsey.succ"},{"answerKinds":[],"category":"research solved","docstring":"Lower bound $43 \\le R(5,5)$, equivalently: there exists a graph on $42$ vertices with no\n$5$-clique and no independent set of size $5$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RamseyNumbers","statement":"∃ G, G.CliqueFree 5 ∧ Gᶜ.CliqueFree 5","subjects":["5"],"theorem":"RamseyNumbers.ramsey_number_five_five_lower_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"The open problem: determine the Ramsey number $R(5,5)$.\n\nIt is known that $43 \\le R(5,5) \\le 46$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RamseyNumbers","statement":"SimpleGraph.classicalRamsey 5 5 = sorry","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"RamseyNumbers.ramsey_number_five_five"},{"answerKinds":[],"category":"research solved","docstring":"Euler's sum of powers conjecture is false for $k=4$ (counterexample exists). ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EulerSumOfPowers","statement":"¬∀ (n b : ℕ), 1 < n → ∀ (a : Fin n → ℕ), (∀ (i : Fin n), a i > 0) → ∑ i, a i ^ 4 = b ^ 4 → 4 ≤ n","subjects":["11"],"theorem":"EulerSumOfPowers.eulers_sum_of_powers_conjecture.false_for_k4"},{"answerKinds":[],"category":"research solved","docstring":"Euler's sum of powers conjecture is false for $k=5$ (counterexample exists). ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EulerSumOfPowers","statement":"¬∀ (n b : ℕ), 1 < n → ∀ (a : Fin n → ℕ), (∀ (i : Fin n), a i > 0) → ∑ i, a i ^ 5 = b ^ 5 → 5 ≤ n","subjects":["11"],"theorem":"EulerSumOfPowers.eulers_sum_of_powers_conjecture.false_for_k5"},{"answerKinds":[],"category":"research open","docstring":"Euler's sum of powers conjecture states that for integers $n > 1$ and $k > 1$,\nif the sum of $n$ positive integers each raised to the $k$-th power equals another integer\nraised to the $k$-th power, then $n ≥ k$.\n\nThe conjecture is known to be false for $k = 4$ and $k = 5$,\nbut remains open for $k ≥ 6$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EulerSumOfPowers","statement":"∀ (n k b : ℕ), 1 < n → 5 < k → ∀ (a : Fin n → ℕ), (∀ (i : Fin n), a i > 0) → ∑ i, a i ^ k = b ^ k → k ≤ n","subjects":["11"],"theorem":"EulerSumOfPowers.eulers_sum_of_powers_conjecture"},{"answerKinds":[],"category":"research open","docstring":"The Cookson Hills series summing $sec(n)^2 / n^3$ from $n=1$ to $\\infty$ converges.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.FlintCooksonHills","statement":"True ↔ Summable fun n => 1 / ((↑n + 1) ^ 3 * Real.cos (↑n + 1) ^ 2)","subjects":["40"],"theorem":"FlintCooksonHills.cookson_hills_series_converges"},{"answerKinds":[],"category":"research open","docstring":"The Flint Hills series summing $csc(n)^2 / n^3$ from $n=1$ to $\\infty$ converges.\n(Note that we 0-index the series below.)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.FlintCooksonHills","statement":"True ↔ Summable fun n => 1 / ((↑n + 1) ^ 3 * Real.sin (↑n + 1) ^ 2)","subjects":["40"],"theorem":"FlintCooksonHills.flint_hills_series_converges"},{"answerKinds":[],"category":"research open","docstring":"Is $\\ln(\\pi)$ irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Irrational","statement":"True ↔ Irrational (Real.log Real.pi)","subjects":["33"],"theorem":"Irrational.irrational_ln_pi"},{"answerKinds":[],"category":"research open","docstring":"Is the Catalan constant $$G = \\sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \\approx 0.91596$$ irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Irrational","statement":"True ↔ Irrational catalanConstant","subjects":["11","33"],"theorem":"Irrational.irrational_catalanConstant"},{"answerKinds":[],"category":"research open","docstring":"Is $\\pi ^ e$ irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Irrational","statement":"True ↔ Irrational (Real.pi ^ Real.exp 1)","subjects":["33"],"theorem":"Irrational.irrational_pi_to_e"},{"answerKinds":[],"category":"research open","docstring":"Is $\\pi ^ \\pi$ irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Irrational","statement":"True ↔ Irrational (Real.pi ^ Real.pi)","subjects":["33"],"theorem":"Irrational.irrational_pi_to_pi"},{"answerKinds":[],"category":"research open","docstring":"Is $e + \\pi$ irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Irrational","statement":"True ↔ Irrational (Real.exp 1 + Real.pi)","subjects":["33"],"theorem":"Irrational.irrational_e_plus_pi"},{"answerKinds":[],"category":"research open","docstring":"Is $e ^ e$ irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Irrational","statement":"True ↔ Irrational (Real.exp 1 ^ Real.exp 1)","subjects":["33"],"subsets":["FC100OpenSet1"],"theorem":"Irrational.irrational_e_to_e"},{"answerKinds":[],"category":"research open","docstring":"Are $e$ and $\\pi$ algebraically independent? ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Irrational","statement":"True ↔ AlgebraicIndependent ℚ ![Real.exp 1, Real.pi]","subjects":["33"],"theorem":"Irrational.algebraicIndependent_e_pi"},{"answerKinds":[],"category":"research open","docstring":"Is $e \\pi$ irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Irrational","statement":"True ↔ Irrational (Real.exp 1 * Real.pi)","subjects":["33"],"theorem":"Irrational.irrational_e_times_pi"},{"answerKinds":[],"category":"research open","docstring":"Is the Euler-Mascheroni constant $\\gamma$ irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Irrational","statement":"True ↔ Irrational Real.eulerMascheroniConstant","subjects":["33"],"theorem":"Irrational.irrational_eulerMascheroniConstant"},{"answerKinds":[],"category":"research open","docstring":"**Lebesgue-Nagell Equation Conjecture**\n\nFor any odd prime $p$, the only integer solutions $(x, y)$ to the equation $x^2 - 2 = y^p$\nare $(x, y) = (\\pm 1, -1)$.\n\n*Reference:* Ethan Katz and Kyle Pratt, \"On the Lebesgue-Nagell equation $x^2 - 2 = y^p$\",\n[arXiv:2507.12397](https://arxiv.org/abs/2507.12397)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Catalan","statement":"∀ (p : ℕ), Nat.Prime p → Odd p → ∀ (x y : ℤ), x ^ 2 - 2 = y ^ p ↔ (x = 1 ∨ x = -1) ∧ y = -1","subjects":["11"],"theorem":"LebesgueNagell.lebesgue_nagell"},{"answerKinds":[],"category":"test","docstring":"The pair $(1, -1)$ is a solution to $x^2 - 2 = y^p$ for any odd $p$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Catalan","statement":"∀ (p : ℕ), Odd p → 1 ^ 2 - 2 = (-1) ^ p","subjects":["11"],"theorem":"LebesgueNagell.lebesgue_nagell_solution_pos_one"},{"answerKinds":[],"category":"test","docstring":"The pair $(-1, -1)$ is a solution to $x^2 - 2 = y^p$ for any odd $p$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Catalan","statement":"∀ (p : ℕ), Odd p → (-1) ^ 2 - 2 = (-1) ^ p","subjects":["11"],"theorem":"LebesgueNagell.lebesgue_nagell_solution_neg_one"},{"answerKinds":[],"category":"research open","docstring":"For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the\nequation $ax^n - by^m = c$ where $(m, n) \\neq (2, 2)$ and $x, y > 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Catalan","statement":"∀ (a b c : ℕ),\n  0 < a →\n    0 < b → 0 < c → {(x, y, m, n) | 1 < x ∧ 1 < y ∧ 1 < m ∧ 1 < n ∧ (m, n) ≠ (2, 2) ∧ a * x ^ n - b * y ^ m = c}.Finite","subjects":["11"],"theorem":"Catalan.pillais_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"The only natural number solution to the equation $x^a - y^b = 1$ such that $a, b > 1$ and\n$x, y > 0$ is given by $a = 2$, $b = 3$, $x = 3$, and $y = 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Catalan","statement":"∀ (a b x y : ℕ), 1 < a → 1 < b → 0 < x → 0 < y → x ^ a - y ^ b = 1 → a = 2 ∧ b = 3 ∧ x = 3 ∧ y = 2","subjects":["11"],"theorem":"Catalan.catalans_conjecture"},{"answerKinds":[],"category":"test","docstring":"The prime $1006003$ is a Mirimanoff prime but not a Wieferich prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WieferichMirimanoffPrime","statement":"IsMirimanoffPrime 1006003 ∧ ¬IsWieferichPrime 1006003","subjects":["11"],"theorem":"WieferichMirimanoffPrime.isMirimanoffPrime_and_not_isWieferichPrime_1006003"},{"answerKinds":[],"category":"research open","docstring":"Are $11$ and $1006003$ the only Mirimanoff primes? They are the only known ones: Dorais and Klyve\nfound no other below $9.7 \\times 10^{14}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WieferichMirimanoffPrime","statement":"True ↔ ∀ (p : ℕ), IsMirimanoffPrime p ↔ p = 11 ∨ p = 1006003","subjects":["11"],"theorem":"WieferichMirimanoffPrime.isMirimanoffPrime_iff"},{"answerKinds":[],"category":"test","docstring":"Neither $2$ nor $3$ is a Mirimanoff prime, so no hypothesis excluding them is needed in the\nstatement of the problem. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WieferichMirimanoffPrime","statement":"¬IsMirimanoffPrime 2","subjects":["11"],"theorem":"WieferichMirimanoffPrime.not_isMirimanoffPrime_two"},{"answerKinds":[],"category":"test","docstring":"The prime $11$ is a Mirimanoff prime but not a Wieferich prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WieferichMirimanoffPrime","statement":"IsMirimanoffPrime 11 ∧ ¬IsWieferichPrime 11","subjects":["11"],"theorem":"WieferichMirimanoffPrime.isMirimanoffPrime_and_not_isWieferichPrime_11"},{"answerKinds":[],"category":"test","docstring":"The prime $3$ is not a Wieferich prime. For $2$, see `WieferichPrime.not_isWieferichPrime_two`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WieferichMirimanoffPrime","statement":"¬IsWieferichPrime 3","subjects":["11"],"theorem":"WieferichMirimanoffPrime.not_isWieferichPrime_three"},{"answerKinds":[],"category":"research open","docstring":"Can a prime $p$ satisfy $2^{p-1} \\equiv 1 \\pmod{p^2}$ and $3^{p-1} \\equiv 1 \\pmod{p^2}$\nsimultaneously? That is, does there exist a prime $p$ that is both a Wieferich prime and a\nMirimanoff prime? Wikipedia's list of unsolved problems poses this question, citing\nJ. B. Dobson, [On Lerch's formula for the Fermat quotient](https://arxiv.org/abs/1103.3907v6).\nLenstra gave a heuristic argument against the existence of such a prime (see Dobson, Section 9).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WieferichMirimanoffPrime","statement":"True ↔ ∃ p, IsWieferichPrime p ∧ IsMirimanoffPrime p","subjects":["11"],"theorem":"WieferichMirimanoffPrime.exists_isWieferichPrime_and_isMirimanoffPrime"},{"answerKinds":[],"category":"research open","docstring":"Does the determinant of the sum $A + B$ of two $n \\times n$ normal\ncomplex matrices $A$ and $B$ always lie in the convex hull\nof the $n!$ points $\\prod\\_i (\\lambda(A)\\_i + \\lambda(B)\\_{\\sigma(i)})$?\nHere the numbers $\\lambda(A)\\_i$ and $\\lambda(B)\\_i$ are\nthe eigenvalues of $A$ and $B$, and $\\sigma$ is an element of the symmetric\ngroup $S\\_n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DeterminantalConjecture","statement":"∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (d1 d2 : n → ℂ) (U1 U2 : ↥(unitary (Matrix n n ℂ))),\n  (↑U1 * Matrix.diagonal d1 * ↑(star U1) + ↑U2 * Matrix.diagonal d2 * ↑(star U2)).det ∈\n    (convexHull ℝ) {x | ∃ σ, ∏ i, (d1 i + d2 (σ i)) = x}","subjects":["15"],"theorem":"DeterminantalConjecture.determinantal_conjecture"},{"answerKinds":[],"category":"research open","docstring":"If a finitely generated group has superpolynomial growth, then with respect to any finite\ngenerating set its growth function is at least $e^{\\sqrt n}$ in Grigorchuk's preorder on\ngrowth functions, where the comparison is witnessed by linearly rescaling the radius.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.GapConjecture","statement":"∀ (G : Type) [inst : Group G] (S : Set G),\n  S.Finite →\n    Subgroup.closure S = ⊤ →\n      GromovPolynomialGrowth.HasSuperPolynomialGrowth G →\n        ∃ C, 0 < C ∧ ∀ᶠ (n : ℕ) in Filter.atTop, Real.exp √↑n ≤ ↑(GromovPolynomialGrowth.GrowthFunction S (C * n))","subjects":["20"],"theorem":"GapConjecture.gap_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**The extended Sierpiński problem.** Is 271129 the second-smallest Sierpiński number?\n\nEven if 78557 is confirmed as the smallest Sierpiński number, there could exist a composite\nSierpiński number $k$ with $78557 < k < 271129$. We formalize \"second-smallest\" as: the\nleast Sierpiński number $k$ such that there exists exactly one Sierpiński number below it.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SierpinskiNumber","statement":"True ↔ IsLeast {k | k.IsSierpinskiNumber ∧ ∃ k', k'.IsSierpinskiNumber ∧ k' < k} 271129","subjects":["11"],"theorem":"SierpinskiNumber.extended_sierpinski_problem"},{"answerKinds":[],"category":"research open","docstring":"**The Sierpiński problem (Selfridge's conjecture).** Is 78557 the smallest Sierpiński number?\n\nSelfridge conjectured that 78557 is the smallest Sierpiński number. He proved in 1962 that\n78557 is indeed a Sierpiński number by showing that all numbers of the form $78557 \\cdot 2^n + 1$\nhave a factor in the covering set $\\{3, 5, 7, 13, 19, 37, 73\\}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SierpinskiNumber","statement":"True ↔ IsLeast {k | k.IsSierpinskiNumber} 78557","subjects":["11"],"theorem":"SierpinskiNumber.selfridge_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"Selfridge proved in 1962 that 78557 is a Sierpiński number by showing that all numbers of the\nform $78557 \\cdot 2^n + 1$ have a factor in the covering set $\\{3, 5, 7, 13, 19, 37, 73\\}$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SierpinskiNumber","statement":"Nat.IsSierpinskiNumber 78557","subjects":["11"],"theorem":"SierpinskiNumber.selfridge_78557"},{"answerKinds":[],"category":"research open","docstring":"**The prime Sierpiński problem.** Is 271129 the smallest prime Sierpiński number?\n\nIn 1976, Nathan Mendelsohn determined that the second provable Sierpiński number is the prime\n$k = 271129$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SierpinskiNumber","statement":"True ↔ IsLeast {k | k.IsSierpinskiNumber ∧ Nat.Prime k} 271129","subjects":["11"],"theorem":"SierpinskiNumber.prime_sierpinski_problem"},{"answerKinds":[],"category":"research open","docstring":"For every positive real number `ε`, there exist only finitely many triples `(a, b, c)` of coprime positive integers, with `a + b = c`, such that `c > rad(abc)^(1+ε)`\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ABC","statement":"∀ (ε : ℝ),\n  0 < ε →\n    {(a, b, c) |\n        0 < a ∧\n          0 < b ∧ 0 < c ∧ {a, b, c}.Pairwise Nat.Coprime ∧ a + b = c ∧ ↑(ABC.radical (a * b * c)) ^ (1 + ε) < ↑c}.Finite","subjects":["11"],"theorem":"ABC.abc"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.ABC","statement":"ABC.radical 12 = 6","subjects":["11"],"theorem":"ABC.radical_12"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.ABC","statement":"ABC.radical 17 = 17","subjects":["11"],"theorem":"ABC.radical_17"},{"answerKinds":[],"category":"research open","docstring":"For every positive real number ε, there exists a constant `K_ε` such that for all triples (a, b, c) of coprime positive integers, with a + b = c we have `c < K_ε rad(abc)^(1+ε)`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ABC","statement":"∀ (ε : ℝ),\n  0 < ε →\n    ∃ K,\n      ∀ (a b c : ℕ),\n        0 < a →\n          0 < b → 0 < c → {a, b, c}.Pairwise Nat.Coprime → a + b = c → ↑c < K * ↑(ABC.radical (a * b * c)) ^ (1 + ε)","subjects":["11"],"theorem":"ABC.abc.variants.lt_constant_mul"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.ABC","statement":"ABC.radical 16 = 2","subjects":["11"],"theorem":"ABC.radical_16"},{"answerKinds":[],"category":"research open","docstring":"For every positive real number ε, there exist only finitely many triples `(a, b, c)` of coprime positive integers with `a + b = c` such that `q(a, b, c) > 1 + ε`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ABC","statement":"∀ (ε : ℝ),\n  0 < ε →\n    {(a, b, c) | 0 < a ∧ 0 < b ∧ 0 < c ∧ {a, b, c}.Pairwise Nat.Coprime ∧ a + b = c ∧ ABC.quality a b c > 1 + ε}.Finite","subjects":["11"],"theorem":"ABC.abc.variants.quality"},{"answerKinds":[],"category":"research open","docstring":"Now form a sequence beginning with any positive integer, where each subsequent term is obtained\nby applying the operation defined above to the previous term.\nThe **Juggler Conjecture** states that for any positive integer $n$, there exists a natural number\n$m$ such that the $m$-th term of the sequence is $1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.JugglerConjecture","statement":"∀ n > 0, ∃ m, JugglerConjecture.jugglerStep^[m] n = 1","subjects":["11","37"],"theorem":"JugglerConjecture.juggler_conjecture"},{"answerKinds":[],"category":"test","docstring":"Example: jugglerStep 36 = ⌊36^(1/2)⌋ = ⌊6⌋ = 6 (since 36 is even). ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.JugglerConjecture","statement":"JugglerConjecture.jugglerStep 36 = 6","subjects":["11"],"theorem":"JugglerConjecture.jugglerStep_36"},{"answerKinds":[],"category":"API","docstring":"The Wichmann ruler $W(r, s)$ has $4r + s + 2$ segments, hence $4r + s + 3$ marks [Wi63]. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SparseRuler","statement":"∀ (r s : ℕ), (SparseRuler.wichmannGaps r s).length = 4 * r + s + 2","subjects":["5"],"theorem":"SparseRuler.wichmannGaps_length"},{"answerKinds":[],"category":"API","docstring":"The Wichmann ruler $W(r, s)$ has length $4r(r + s + 2) + 3(s + 1)$ [Wi63]. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SparseRuler","statement":"∀ (r s : ℕ), (SparseRuler.wichmannGaps r s).sum = 4 * r * (r + s + 2) + 3 * (s + 1)","subjects":["5"],"theorem":"SparseRuler.wichmannGaps_sum"},{"answerKinds":[],"category":"research open","docstring":"**Wichmann's conjecture on optimal rulers.** Every optimal ruler with more than $13$\nsegments is a Wichmann ruler $W(r, s)$ (up to reflection, i.e. reversing the segment list).\nPosed by Wichmann [Wi63]; the finitely many known exceptions all have at most $13$ segments\n(lengths $1, 13, 17, 23, 58$), and no further exceptions are known up to length $213$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SparseRuler","statement":"∀ {g : List ℕ},\n  SparseRuler.IsOptimal g →\n    13 < g.length → ∃ r s, g = SparseRuler.wichmannGaps r s ∨ g = (SparseRuler.wichmannGaps r s).reverse","subjects":["5"],"theorem":"SparseRuler.wichmann_conjecture"},{"answerKinds":[],"category":"test","docstring":"The first few values of $\\ell(n)$. See [OEIS A003313](https://oeis.org/A003313). ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.ScholzConjecture","statement":"[additionChainLength 1, additionChainLength 2, additionChainLength 3, additionChainLength 4, additionChainLength 5,\n    additionChainLength 6, additionChainLength 7, additionChainLength 8, additionChainLength 9,\n    additionChainLength 10] =\n  [0, 1, 2, 2, 3, 3, 4, 3, 4, 4]","subjects":["11"],"theorem":"ScholzConjecture.additionChainLength_first_values"},{"answerKinds":[],"category":"research open","docstring":"The Scholz conjecture, also known as the Scholz-Brauer conjecture, asserts that\nfor every positive integer $n$, the addition-chain length of $2^n - 1$ is at most\n$n - 1 + \\ell(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ScholzConjecture","statement":"True ↔ ∀ (n : ℕ), 0 < n → additionChainLength (2 ^ n - 1) ≤ n - 1 + additionChainLength n","subjects":["11","68"],"theorem":"ScholzConjecture.scholz_conjecture"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many indices $i$, such that the $i$-th Fibonacci is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.FibonacciPrimes","statement":"{n | Nat.Prime (Nat.fib n)}.Infinite","subjects":["11"],"theorem":"FibonacciPrimes.fib_primes_infinite.variant"},{"answerKinds":[],"category":"test","docstring":"The two ways of phrasing the conjecture are equivalent.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FibonacciPrimes","statement":"{n | Nat.Prime (Nat.fib n)}.Infinite ↔ {n | (∃ m, Nat.fib m = n) ∧ Nat.Prime n}.Infinite","subjects":["11"],"theorem":"FibonacciPrimes.indices_infinite_iff_fib_primes_infinite"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many Fibonacci primes, i.e., Fibonacci numbers that are prime\nIt is also a barrier to defining a benchmark from this paper:\nhttps://arxiv.org/html/2505.13938v1 (see Figure 8).\n\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.FibonacciPrimes","statement":"{n | (∃ m, Nat.fib m = n) ∧ Nat.Prime n}.Infinite","subjects":["11"],"theorem":"FibonacciPrimes.fib_primes_infinite"},{"answerKinds":[],"category":"test","docstring":"Sanity check pinning `Q` to a concrete value: among the first `6` terms of `24 n + 1`, namely\n`1, 25, 49, 73, 97, 121`, exactly four are perfect squares (`1, 25, 49, 121`), so\n`Q 6 24 1 = 4`. This validates the definition of `Q` and matches the claim that `24 n + 1` is the\nextremal progression. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.RudinsConjecture","statement":"RudinsConjecture.Q 6 24 1 = 4","subjects":["11"],"theorem":"RudinsConjecture.Q_six_twentyfour_one"},{"answerKinds":[],"category":"research open","docstring":"The strongest form of Rudin's conjecture also asserts *uniqueness*: for $N \\ge 6$, any non-trivial\narithmetic progression attaining the maximum $Q(N)$ has common difference $24$. (Its initial term\nis then forced by $\\gcd(24, a) = 1$; the progression $24n + 1$ is the canonical representative.)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RudinsConjecture","statement":"∀ (N : ℕ),\n  6 ≤ N → ∀ (q a : ℕ), RudinsConjecture.IsNontrivial q a → RudinsConjecture.Q N q a = RudinsConjecture.Qmax N → q = 24","subjects":["11"],"theorem":"RudinsConjecture.rudins_conjecture_unique"},{"answerKinds":[],"category":"research open","docstring":"**Rudin's conjecture.** The maximal number of squares among the first $N$ terms of a non-trivial\narithmetic progression grows at most like $\\sqrt{N}$:\n$$Q(N) = O(\\sqrt{N}).$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RudinsConjecture","statement":"(fun N => ↑(RudinsConjecture.Qmax N)) =O[Filter.atTop] fun N => √↑N","subjects":["11"],"theorem":"RudinsConjecture.rudins_conjecture"},{"answerKinds":[],"category":"test","docstring":"Sanity check: the progression $24 n + 1$ is non-trivial. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.RudinsConjecture","statement":"RudinsConjecture.IsNontrivial 24 1","subjects":["11"],"theorem":"RudinsConjecture.isNontrivial_24_1"},{"answerKinds":[],"category":"research open","docstring":"A stronger form of Rudin's conjecture: for every $N \\ge 6$, the arithmetic progression\n$24 n + 1$ attains the maximum $Q(N)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RudinsConjecture","statement":"∀ (N : ℕ), 6 ≤ N → RudinsConjecture.Q N 24 1 = RudinsConjecture.Qmax N","subjects":["11"],"theorem":"RudinsConjecture.rudins_conjecture_strong"},{"answerKinds":[],"category":"test","docstring":"Sanity check: there are no squares among the first `0` terms of any progression. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.RudinsConjecture","statement":"∀ (q a : ℕ), RudinsConjecture.Q 0 q a = 0","subjects":["11"],"theorem":"RudinsConjecture.Q_zero"},{"answerKinds":[],"category":"textbook","docstring":"The Beal Conjecture implies Fermat's last theorem\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.BealConjecture","statement":"BealConjecture.bealConjecture → FermatLastTheorem","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"BealConjecture.flt_of_beal_conjecture"},{"answerKinds":[],"category":"research open","docstring":"The **Beal Conjecture**: if we are given positive integers $A, B, C, x, y, z$ such that\n$x, y, z > 2$ and $A^x + B^y = C^z$ then $A, B, C$ have a common divisor.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BealConjecture","statement":"BealConjecture.bealConjecture","subjects":["11"],"theorem":"BealConjecture.beal_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many tuples of three consecutive primes $(p, q, r)$ such that $r - p = 6$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PrimeTriplets","statement":"True ↔ {p | Prime p ∧ (Prime (p + 2) ∨ Prime (p + 4)) ∧ Prime (p + 6)}.Infinite","subjects":["11"],"theorem":"PrimeTriplets.prime_triplets"},{"answerKinds":[],"category":"test","docstring":"Sanity check: the zero function always integrates to `0` over every rigid-motion image, so the\nhypothesis in `HasPompeiuProperty` is satisfiable (the property is not vacuously false). ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.PompeiuProblem","statement":"∀ {N : ℕ} (Ω : Set (EuclideanSpace ℝ (Fin (N + 1)))) (σ : PompeiuProblem.RigidMotion N),\n  ∫ (_x : EuclideanSpace ℝ (Fin (N + 1))) in ⇑σ '' Ω, 0 = 0","subjects":["42"],"theorem":"PompeiuProblem.integral_zero_eq_zero"},{"answerKinds":[],"category":"research solved","docstring":"**Pompeiu's conjecture is false.** There is a bounded, simply connected Lipschitz domain in\n$\\mathbb{R}^2$ which fails to have the Pompeiu property but is not a Euclidean ball. This conclusion\nhas two independent proofs: the Lean-formalized construction of Cao-Labora and de Dios Pont\n[CLD26], and the computer-assisted construction of Colbrook and Stepaniants [CS26].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/jaumededios/Schiffer/blob/2938e277969c329caf154e48a3d8823f3635c7f1/Schiffer/FormalConjecturesSolution.lean#L446-L469"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PompeiuProblem","statement":"¬∀ (N : ℕ) (Ω : Set (EuclideanSpace ℝ (Fin (N + 1)))),\n    PompeiuProblem.IsAdmissibleDomain Ω → (¬PompeiuProblem.HasPompeiuProperty Ω ↔ PompeiuProblem.IsBall Ω)","subjects":["42"],"theorem":"PompeiuProblem.pompeiu_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"The hard direction of the Pompeiu problem is false: some admissible planar domain lacks the\nPompeiu property without being a ball. Two independent proofs are given in [CLD26] and [CS26].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/jaumededios/Schiffer/blob/2938e277969c329caf154e48a3d8823f3635c7f1/Schiffer/FormalConjecturesSolution.lean#L446-L469"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PompeiuProblem","statement":"¬∀ (N : ℕ) (Ω : Set (EuclideanSpace ℝ (Fin (N + 1)))),\n    PompeiuProblem.IsAdmissibleDomain Ω → ¬PompeiuProblem.HasPompeiuProperty Ω → PompeiuProblem.IsBall Ω","subjects":["42"],"theorem":"PompeiuProblem.not_hasPompeiuProperty_imp_ball"},{"answerKinds":[],"category":"research solved","docstring":"The classical (easy) direction, proved by Brown–Schreiber–Taylor [BST73]: a Euclidean ball fails\nto have the Pompeiu property. For a ball of radius `R`, an explicit witness is `f(x) = sin(a x₁)`\nwhere `a > 0` is chosen so that the Bessel function `J_{n/2}(a R) = 0`; its integral over every\ncongruent ball vanishes.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PompeiuProblem","statement":"∀ {N : ℕ} (Ω : Set (EuclideanSpace ℝ (Fin (N + 1)))), PompeiuProblem.IsBall Ω → ¬PompeiuProblem.HasPompeiuProperty Ω","subjects":["42"],"theorem":"PompeiuProblem.ball_not_hasPompeiuProperty"},{"answerKinds":[],"category":"research open","docstring":"**Bunyakovsky conjecture**\nIf a polynomial $f$ over integers satisfies both Schinzel and Bunyakovsky conditions,\nthere exist infinitely many natural numbers $m$ such that $f(m)$ is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Bunyakovsky","statement":"∀ (f : Polynomial ℤ),\n  BunyakovskyCondition f ∧ SchinzelCondition {f} → Infinite ↑{n | Nat.Prime (Polynomial.eval (↑n) f).natAbs}","subjects":["11"],"theorem":"Bunyakovsky.bunyakovsky_conjecture"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SurjunctiveGroup","statement":"∀ (G : Type u_1) [inst : Group G] {A : Type u_2} (g₁ g₂ : G) (x : G → A),\n  GottschalkSurjunctivity.shift G (g₁ * g₂) x =\n    GottschalkSurjunctivity.shift G g₁ (GottschalkSurjunctivity.shift G g₂ x)","subjects":["20","37"],"theorem":"GottschalkSurjunctivity.shift_mul"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SurjunctiveGroup","statement":"∀ (G : Type u_1) [inst : Group G] {A : Type u_2} (x : G → A), GottschalkSurjunctivity.shift G 1 x = x","subjects":["20","37"],"theorem":"GottschalkSurjunctivity.shift_one"},{"answerKinds":[],"category":"textbook","docstring":"Every finite group is surjunctive. This is a classical result: an injective\nendomorphism of a finite set is surjective. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SurjunctiveGroup","statement":"∀ (G : Type) [inst : Group G] [Finite G], GottschalkSurjunctivity.IsSurjunctive G","subjects":["20","37"],"theorem":"GottschalkSurjunctivity.isSurjunctive_of_finite"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SurjunctiveGroup","statement":"∀ (G : Type u_1) [inst : Group G] {A : Type u_2} (g : G) (x : G → A) (h : G),\n  GottschalkSurjunctivity.shift G g x h = x (g⁻¹ * h)","subjects":["20","37"],"theorem":"GottschalkSurjunctivity.shift_apply"},{"answerKinds":[],"category":"research open","docstring":"**Gottschalk's surjunctivity conjecture** (1973): every group is surjunctive.\nThat is, for every group `G` and every finite alphabet `A`, every injective cellular\nautomaton on `A^G` is surjective. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SurjunctiveGroup","statement":"∀ (G : Type) [inst : Group G], GottschalkSurjunctivity.IsSurjunctive G","subjects":["20","37"],"theorem":"GottschalkSurjunctivity.gottschalk_surjunctivity_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**Odd Perfect Number Conjecture.**\nThe Odd Perfect Number Conjecture states that all perfect numbers are even.\n\n*Reference:*\n[Wikipedia](https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PerfectNumbers","statement":"∀ (n : ℕ), n.Perfect → Even n","subjects":["11"],"theorem":"PerfectNumbers.odd_perfect_number_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"A known result: If an odd perfect number exists, it must be greater than $10^{1500}$\nand must have at least 101 prime factors (including multiplicities).\n\n*Reference:* Pascal Ochem, Michaël Rao (2012).\n\"Odd perfect numbers are greater than 10^1500\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PerfectNumbers","statement":"∀ (n : ℕ), Odd n → n.Perfect → n > 10 ^ 1500 ∧ n.primeFactorsList.length ≥ 101","subjects":["11"],"theorem":"PerfectNumbers.odd_perfect_number.lower_bound"},{"answerKinds":[],"category":"research open","docstring":"**Infinitely many perfect numbers conjecture.**\nAre there infinitely many perfect numbers?\n\n*Reference:*\n[Wikipedia](https://en.wikipedia.org/wiki/Perfect_number)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PerfectNumbers","statement":"True ↔ {n | n.Perfect}.Infinite","subjects":["11"],"theorem":"PerfectNumbers.infinitely_many_perfect"},{"answerKinds":[],"category":"research open","docstring":"**Infinitely many even perfect numbers conjecture.**\nAre there infinitely many even perfect numbers?\n\nThis is equivalent to asking whether there are infinitely many Mersenne primes,\nsince by the Euclid–Euler theorem an even number is perfect if and only if it\nhas the form $2^{p-1}(2^p - 1)$ where $2^p - 1$ is a Mersenne prime.\n\n*Reference:*\n[Wikipedia](https://en.wikipedia.org/wiki/Perfect_number)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PerfectNumbers","statement":"True ↔ {n | n.Perfect ∧ Even n}.Infinite","subjects":["11"],"theorem":"PerfectNumbers.infinitely_many_even_perfect"},{"answerKinds":[],"category":"research solved","docstring":"A known result: If an odd perfect number exists, it must be of the form\n$p^α * m^2$ where $p$ is prime, $p \\equiv 1 \\pmod{4}$, $\\alpha \\equiv 1 \\pmod{4}$,\nand $p \\nmid m$.\n\n*Reference:* Euler's theorem on odd perfect numbers.\n\nFormal proof linked here provided by AlphaProof.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mzhorvath1/formal-conjectures/blob/7deed78f7babe2ae9ea13969a8dfa26854982407/FormalConjectures/Wikipedia/PerfectNumbers.lean#L110"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PerfectNumbers","statement":"∀ (n : ℕ), Odd n → n.Perfect → ∃ p m α, Nat.Prime p ∧ ↑p ≡ 1 [ZMOD 4] ∧ ↑α ≡ 1 [ZMOD 4] ∧ ¬p ∣ m ∧ n = p ^ α * m ^ 2","subjects":["11"],"theorem":"PerfectNumbers.odd_perfect_number.euler_form"},{"answerKinds":[],"category":"test","docstring":"`IsBetrothed` is symmetric. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.BetrothedNumbers","statement":"∀ {m n : ℕ}, BetrothedNumbers.IsBetrothed m n → BetrothedNumbers.IsBetrothed n m","subjects":["11"],"theorem":"BetrothedNumbers.IsBetrothed.symm"},{"answerKinds":[],"category":"research open","docstring":"**Infinitude of betrothed numbers conjecture.**\nAre there infinitely many betrothed number pairs?\n\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BetrothedNumbers","statement":"True ↔ {p | p.1 < p.2 ∧ BetrothedNumbers.IsBetrothed p.1 p.2}.Infinite","subjects":["11"],"theorem":"BetrothedNumbers.infinitely_many_betrothed"},{"answerKinds":[],"category":"test","docstring":"The smallest known betrothed pair $(48, 75)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.BetrothedNumbers","statement":"BetrothedNumbers.IsBetrothed 48 75","subjects":["11"],"theorem":"BetrothedNumbers.betrothed_48_75"},{"answerKinds":[],"category":"research open","docstring":"**Same parity betrothed numbers conjecture.**\nDo there exist betrothed numbers $(m, n)$ where both have the same parity\n(both even or both odd)?\n\nAll known betrothed pairs consist of one even and one odd number.\n\nThe requirement $m \\neq n$ is part of the question: $\\mathrm{IsBetrothed}\\ n\\ n$ says\n$\\sigma(n) = 2n + 1$, i.e. that $n$ is quasiperfect, which is the separate open problem\n`QuasiperfectNumbers.exists_quasiperfect`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BetrothedNumbers","statement":"True ↔ ∃ m n, m ≠ n ∧ BetrothedNumbers.IsBetrothed m n ∧ (Even m ↔ Even n)","subjects":["11"],"theorem":"BetrothedNumbers.same_parity_betrothed"},{"answerKinds":[],"category":"research open","docstring":"Does every finite partially ordered set that is not totally ordered\ncontain two elements $x$ and $y$ such that the probability that\n$x$ appears before $y$ in a random linear extension is between $\\frac 1 3$ and $\\frac 2 3$?\n\nThe set of all total order extensions is represented as order preserving\nbijections $P$ of $1, ..., n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.conjecture_1_3_to_2_3","statement":"True ↔\n  ∀ (P : Type) [Finite P] [inst : PartialOrder P],\n    (¬Std.Total fun x1 x2 => x1 ≤ x2) →\n      ∀ (total_ext : Set (P →o ℕ)),\n        (∀ (σ : P →o ℕ), σ ∈ total_ext ↔ Set.range ⇑σ = Set.Icc 1 (Nat.card P)) →\n          ∃ x y, ↑{σ | σ ∈ total_ext ∧ σ x < σ y}.ncard / ↑total_ext.ncard ∈ Set.Icc (1 / 3) (2 / 3)","subjects":["6"],"theorem":"Conjecture_1_3_to_2_3.conjecture_1_3_to_2_3"},{"answerKinds":[],"category":"research open","docstring":"**The integer factorization problem**: Can the prime factorization of a positive integer\nbe computed in polynomial time?\n\nWe state the problem by asking if `Nat.primeFactorsList` is polynomial-time computable\n(assuming typical encodings of ℕ and List ℕ into bitstrings).\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Integer_factorization) ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PolyTimeFunctions","statement":"True ↔ ComplexityTheory.IsPolyTime Nat.primeFactorsList","subjects":["68"],"theorem":"PolyTime.isPolyTime_primeFactorsList"},{"answerKinds":[],"category":"textbook","docstring":"Oppermann's conjecture implies Brocard's conjecture. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Oppermann","statement":"∀ (n : ℕ),\n  1 ≤ n →\n    (∀ (x : ℕ),\n        2 ≤ x →\n          (∃ p ∈ Finset.Ioo (x * (x - 1)) (x ^ 2), Nat.Prime p) ∧ ∃ p ∈ Finset.Ioo (x ^ 2) (x * (x + 1)), Nat.Prime p) →\n      4 ≤ (Finset.filter Nat.Prime (Finset.Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2))).card","subjects":["11"],"theorem":"Oppermann.oppermann_implies_brocard"},{"answerKinds":[],"category":"research open","docstring":"For every integer $x \\ge 2$ there exists a prime between $x(x-1)$ and $x^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Oppermann","statement":"∀ (x : ℕ), 2 ≤ x → ∃ p ∈ Finset.Ioo (x * (x - 1)) (x ^ 2), Nat.Prime p","subjects":["11"],"theorem":"Oppermann.oppermann_conjecture.parts.i"},{"answerKinds":[],"category":"textbook","docstring":"Oppermann's conjecture implies Legendre's conjecture. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Oppermann","statement":"∀ (n : ℕ),\n  1 ≤ n →\n    (∀ (x : ℕ),\n        2 ≤ x →\n          (∃ p ∈ Finset.Ioo (x * (x - 1)) (x ^ 2), Nat.Prime p) ∧ ∃ p ∈ Finset.Ioo (x ^ 2) (x * (x + 1)), Nat.Prime p) →\n      ∃ p ∈ Finset.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p","subjects":["11"],"theorem":"Oppermann.oppermann_implies_legendre"},{"answerKinds":[],"category":"research open","docstring":"**Oppermann's Conjecture**:\nFor every integer $x \\ge 2$, the following hold:\n- There exists a prime between $x(x-1)$ and $x^2$.\n- There exists a prime between $x^2$ and $x(x+1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Oppermann","statement":"∀ (x : ℕ),\n  2 ≤ x → (∃ p ∈ Finset.Ioo (x * (x - 1)) (x ^ 2), Nat.Prime p) ∧ ∃ p ∈ Finset.Ioo (x ^ 2) (x * (x + 1)), Nat.Prime p","subjects":["11"],"theorem":"Oppermann.oppermann_conjecture"},{"answerKinds":[],"category":"research open","docstring":"For every integer $x \\ge 2$ there exists a prime between $x^2$ and $x(x+1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Oppermann","statement":"∀ (x : ℕ), 2 ≤ x → ∃ p ∈ Finset.Ioo (x ^ 2) (x * (x + 1)), Nat.Prime p","subjects":["11"],"theorem":"Oppermann.oppermann_conjecture.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Ferreira proved that Oppermann's conjecture is true for sufficiently large x.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Oppermann","statement":"∀ᶠ (x : ℕ) in Filter.atTop,\n  (∃ p ∈ Finset.Ioo (x * (x - 1)) (x ^ 2), Nat.Prime p) ∧ ∃ p ∈ Finset.Ioo (x ^ 2) (x * (x + 1)), Nat.Prime p","subjects":["11"],"theorem":"Oppermann.oppermann_conjecture.ferreira_large_x"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many primes $p$ such that $p - 1$ is a perfect square? In other words: Are there infinitely many primes of the form $n^2 + 1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PrimesAndPerfectSquares","statement":"True ↔ {n | Prime (n ^ 2 + 1)}.Infinite","subjects":["11"],"theorem":"PrimesAndPerfectSquares.infinite_prime_sq_add_one"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a point in the plane at rational distance from all four vertices of the unit square?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RationalDistanceProblem","statement":"True ↔ ∃ P, ∀ (i : Fin 4), ¬Irrational (dist P (RationalDistanceProblem.UnitSquareCorners i))","subjects":["11","51"],"theorem":"RationalDistanceProblem.rational_distance_problem"},{"answerKinds":[],"category":"research open","docstring":"There does not exist a $(2,5)$-perfect number ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Superperfectnumbers","statement":"¬∃ n, Superperfect.PerfectFor n 2 5","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Superperfect.twoFivePerfect"},{"answerKinds":[],"category":"research solved","docstring":"**Gromov's Polynomial Growth Theorem** : A finitely generated group has\npolynomial growth if and only if it is virtually nilpotent. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.GromovPolynomialGrowth","statement":"∀ (G : Type u_1) [inst : Group G] [Group.FG G],\n  GromovPolynomialGrowth.HasPolynomialGrowth G ↔ Group.IsVirtuallyNilpotent G","subjects":["20"],"theorem":"GromovPolynomialGrowth.GromovPolynomialGrowthTheorem"},{"answerKinds":[],"category":"test","docstring":"Infinite groups do not satisfy polynomial growth over `ℕ` for any degree `d` because when\n`d = 0` this reduces to the unbounded nature of `growthFunction` while `n = 0` works when `d ≠ 0`.\nThus a finitely-generated infinite nilpotent group would be a counter-example to\nGromov's theorem when quantifying over all of `ℕ`, and so `n = 0` should be excluded. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.GromovPolynomialGrowth","statement":"∀ {G : Type u_1} [inst : Group G] [Infinite G] {S : Set G},\n  S.Finite → Subgroup.closure S = ⊤ → ∀ {C : ℝ} (d : ℕ), ∃ n, C * ↑n ^ d < ↑(GromovPolynomialGrowth.GrowthFunction S n)","subjects":["20"],"theorem":"GromovPolynomialGrowth.growthFunction_not_polynomial_of_infinite"},{"answerKinds":[],"category":"test","docstring":"The prime $7$ is not a Wilson prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WilsonPrime","statement":"¬WilsonPrime.IsWilsonPrime 7","subjects":["11"],"theorem":"WilsonPrime.not_isWilsonPrime_seven"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many Wilson primes. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WilsonPrime","statement":"{p | WilsonPrime.IsWilsonPrime p}.Infinite","subjects":["11"],"theorem":"WilsonPrime.infinitely_many_wilson_primes"},{"answerKinds":[],"category":"test","docstring":"The prime $5$ is a Wilson prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WilsonPrime","statement":"WilsonPrime.IsWilsonPrime 5","subjects":["11"],"theorem":"WilsonPrime.isWilsonPrime_five"},{"answerKinds":[],"category":"test","docstring":"The primality condition excludes $1$, which satisfies the divisibility condition alone. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WilsonPrime","statement":"¬WilsonPrime.IsWilsonPrime 1","subjects":["11"],"theorem":"WilsonPrime.not_isWilsonPrime_one"},{"answerKinds":[],"category":"test","docstring":"The prime $13$ is a Wilson prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WilsonPrime","statement":"WilsonPrime.IsWilsonPrime 13","subjects":["11"],"theorem":"WilsonPrime.isWilsonPrime_thirteen"},{"answerKinds":[],"category":"research open","docstring":"The only Goormaghtigh numbers are $31$ and $8191$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Goormaghtigh","statement":"∀ (N : ℕ), Goormaghtigh.IsGoormaghtighNumber N → N = 31 ∨ N = 8191","subjects":["11"],"theorem":"Goormaghtigh.goormaghtigh_conjecture"},{"answerKinds":[],"category":"test","docstring":"Allowing two-digit repunits would make $13$ a representation in two distinct bases. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Goormaghtigh","statement":"Goormaghtigh.repunit 3 3 = 13 ∧ Goormaghtigh.repunit 12 2 = 13","subjects":["11"],"theorem":"Goormaghtigh.repunit_two_digits"},{"answerKinds":[],"category":"test","docstring":"The number $8191$ is a repunit in bases $2$ and $90$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Goormaghtigh","statement":"Goormaghtigh.IsGoormaghtighNumber 8191","subjects":["11"],"theorem":"Goormaghtigh.isGoormaghtighNumber_8191"},{"answerKinds":[],"category":"test","docstring":"The number $31$ is a repunit in bases $2$ and $5$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Goormaghtigh","statement":"Goormaghtigh.IsGoormaghtighNumber 31","subjects":["11"],"theorem":"Goormaghtigh.isGoormaghtighNumber_31"},{"answerKinds":[],"category":"research open","docstring":"The Vaught conjecture states that for a countable language L and a complete L-Theory T\nthe number of countable models of T (up to isomorphism) is finite, $\\aleph_0$ or $2^{\\aleph_0}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.VaughtConjecture","statement":"∀ {L : FirstOrder.Language},\n  Countable L.Symbols →\n    ∀ {T : L.Theory},\n      T.IsComplete →\n        VaughtConjecture.numberOfCountableModels T ≤ Cardinal.aleph0 ∨\n          VaughtConjecture.numberOfCountableModels T = Cardinal.continuum","subjects":["3"],"theorem":"VaughtConjecture.vaught_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"The Noether problem has a positive solution in the three indeterminate case.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.NoetherProblem","statement":"∀ {K L ι : Type} {G : Type} [inst : Field K] [inst_1 : Field L] [inst_2 : Fintype ι] [inst_3 : Algebra K L]\n  [inst_4 : NoetherProblem.IsRationalExtension K L ι], Fintype.card ι = 3 → NoetherProblem.HasNoetherProperty K L ι","subjects":["12","14"],"theorem":"NoetherProblem.noether_problem.variants.three"},{"answerKinds":[],"category":"research solved","docstring":"The Noether problem has a positive solution in the two indeterminate case.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.NoetherProblem","statement":"∀ {K L ι : Type} {G : Type} [inst : Field K] [inst_1 : Field L] [inst_2 : Fintype ι] [inst_3 : Algebra K L]\n  [inst_4 : NoetherProblem.IsRationalExtension K L ι], Fintype.card ι = 2 → NoetherProblem.HasNoetherProperty K L ι","subjects":["12","14"],"theorem":"NoetherProblem.noether_problem.variants.two"},{"answerKinds":[],"category":"research solved","docstring":"The Noether problem has a positive solution in the four indeterminate case.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.NoetherProblem","statement":"∀ {K L ι : Type} {G : Type} [inst : Field K] [inst_1 : Field L] [inst_2 : Fintype ι] [inst_3 : Algebra K L]\n  [inst_4 : NoetherProblem.IsRationalExtension K L ι], Fintype.card ι = 4 → NoetherProblem.HasNoetherProperty K L ι","subjects":["12","14"],"theorem":"NoetherProblem.noether_problem.variants.four"},{"answerKinds":[],"category":"test","docstring":"If the index set `ι` is empty, then `IsRationalExtension K L ι` means that\n`K, L` are isomorphic as `K` algebras. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.NoetherProblem","statement":"∀ (K : Type u_1) (L : Type u_2) (ι : Type u_3) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] [IsEmpty ι]\n  [NoetherProblem.IsRationalExtension K L ι], Nonempty (L ≃ₐ[K] K)","subjects":["12"],"theorem":"NoetherProblem.rationalExtension_empty_index"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The **Noether Problem**: let `L` be the field of rational functions in `n`\nindeterminates over `K`. Is it true that `L/K` has the Noether property?\n\nSolution: False.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.NoetherProblem","statement":"False ↔\n  ∀ (K L ι G : Type) [inst : Field K] [inst_1 : Field L] [inst_2 : Fintype ι] [inst_3 : Algebra K L]\n    [inst_4 : NoetherProblem.IsRationalExtension K L ι], NoetherProblem.HasNoetherProperty K L ι","subjects":["12","14"],"theorem":"NoetherProblem.noether_problem"},{"answerKinds":[],"category":"research solved","docstring":"One can find a counterexample to the Noether Problem's claim by considering a\nrational function field in 47 indeterminates.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.NoetherProblem","statement":"∃ K L ι G x x_1 x_2 x_3,\n  ∃ (x_4 : NoetherProblem.IsRationalExtension K L ι), Fintype.card ι = 47 ∧ ¬NoetherProblem.HasNoetherProperty K L ι","subjects":["12","14"],"theorem":"NoetherProblem.noether_problem.variants.forty_seven"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many Fermat primes?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Fermat","statement":"True ↔ Infinite ↑{n | Prime n.fermatNumber}","subjects":["11"],"theorem":"Fermat.infinite_fermat_primes"},{"answerKinds":[],"category":"research open","docstring":"Are Fermat numbers composite for all `n > 4`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Fermat","statement":"True ↔ ∀ n > 4, ¬Prime n.fermatNumber","subjects":["11"],"theorem":"Fermat.fermat_number_are_composite"},{"answerKinds":[],"category":"research open","docstring":"Are all Fermat numbers are square-free?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Fermat","statement":"True ↔ ∀ (n : ℕ), Squarefree n.fermatNumber","subjects":["11"],"theorem":"Fermat.all_fermat_squarefree"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many composite Fermat numbers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Fermat","statement":"True ↔ Infinite ↑{n | ¬Prime n.fermatNumber}","subjects":["11"],"theorem":"Fermat.infinite_fermat_composite"},{"answerKinds":[],"category":"research open","docstring":"**Babai–Seress Conjecture (Conjecture 1.5)**: There exists an absolute constant $C$ such\nthat the diameter of the alternating group $A_n$ satisfies\n$$\\operatorname{diam}(A_n) \\leq n^C.$$\n\n*Reference:* [L. Babai and Á. Seress, *On the diameter of permutation groups*,\nEuropean Journal of Combinatorics 13 (1992), Conjecture 1.5](https://doi.org/10.1016/S0195-6698(05)80029-0) ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DiameterSimpleFiniteGroups","statement":"∃ C, ∀ (n : ℕ), ↑(BabaiSeressConjectures.groupDiam ↥(alternatingGroup (Fin n))) ≤ ↑n ^ C","subjects":["5","20","68"],"subsets":["FC100OpenSet1"],"theorem":"BabaiSeressConjectures.babai_seress_conjecture_alternating"},{"answerKinds":[],"category":"test","docstring":"The symmetric group $S_2 \\cong \\mathbb{Z}/2\\mathbb{Z}$ has group diameter $1$: the unique\ngenerating set $\\{(01)\\}$ produces the complete graph $K_2$, since the single non-identity\nelement and its inverse (which are equal) cover the only other vertex. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DiameterSimpleFiniteGroups","statement":"BabaiSeressConjectures.groupDiam (Equiv.Perm (Fin 2)) = 1","subjects":["20"],"theorem":"BabaiSeressConjectures.groupDiam_perm_two"},{"answerKinds":[],"category":"test","docstring":"The alternating group $A_3 \\cong \\mathbb{Z}/3\\mathbb{Z}$ has group diameter $1$: every\nnon-trivial generating set produces a complete Cayley graph $K_3$, since any single non-identity\nelement and its inverse already reach the entire group. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DiameterSimpleFiniteGroups","statement":"BabaiSeressConjectures.groupDiam ↥(alternatingGroup (Fin 3)) = 1","subjects":["20"],"theorem":"BabaiSeressConjectures.groupDiam_alternating_three"},{"answerKinds":[],"category":"test","docstring":"For the trivial group (with one element), the group diameter is zero, since\nevery Cayley graph has only one vertex and hence diameter zero. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DiameterSimpleFiniteGroups","statement":"BabaiSeressConjectures.groupDiam ↥(alternatingGroup (Fin 0)) = 0","subjects":["20"],"theorem":"BabaiSeressConjectures.groupDiam_fin_one"},{"answerKinds":[],"category":"research open","docstring":"**Babai–Seress Conjecture (Conjecture 1.7)**: There exists an absolute constant $C$ such\nthat every finite simple non-abelian group $G$ satisfies\n$$\\operatorname{diam}(G) \\leq (\\log |G|)^C.$$\n\n*Reference:* [L. Babai and Á. Seress, *On the diameter of permutation groups*,\nEuropean Journal of Combinatorics 13 (1992), Conjecture 1.7](https://doi.org/10.1016/S0195-6698(05)80029-0) ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DiameterSimpleFiniteGroups","statement":"∃ C,\n  ∀ (G : Type) [inst : Group G] [inst_1 : Fintype G] [IsSimpleGroup G],\n    (∃ a b, a * b ≠ b * a) → ↑(BabaiSeressConjectures.groupDiam G) ≤ Real.log ↑(Fintype.card G) ^ C","subjects":["5","20","68"],"theorem":"BabaiSeressConjectures.babai_seress_conjecture"},{"answerKinds":[],"category":"research open","docstring":"The Lander–Parkin–Selfridge conjecture: if the sum of $n$ positive integer $k$-th powers\nequals the sum of $m$ positive integer $k$-th powers, with all values on the left distinct from\nall values on the right, then $n + m \\geq k$.\n\nFormally, for positive integers $k, n, m \\in \\mathbb{N}$ and sequences\n$x : \\{0, \\ldots, n-1\\} \\to \\mathbb{N}$ and $y : \\{0, \\ldots, m-1\\} \\to \\mathbb{N}$\nwith $x_i > 0$, $y_j > 0$, and $x_i \\neq y_j$ for all $i, j$, if\n$$\\sum_{i=0}^{n-1} x_i^k = \\sum_{j=0}^{m-1} y_j^k,$$\nthen $k \\leq n + m$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LanderParkinAndSelfridgeConjecture","statement":"∀ (k n m : ℕ) (x : Fin n → ℕ) (y : Fin m → ℕ),\n  0 < n →\n    0 < m →\n      (∀ (i : Fin n), 0 < x i) →\n        (∀ (j : Fin m), 0 < y j) → (∀ (i : Fin n) (j : Fin m), x i ≠ y j) → ∑ i, x i ^ k = ∑ j, y j ^ k → k ≤ n + m","subjects":["11"],"theorem":"LanderParkinSelfridge.lander_parkin_selfridge"},{"answerKinds":[],"category":"research open","docstring":"Special case of the Lander–Parkin–Selfridge conjecture: there is no solution in positive\nintegers to\n$$x_1^5 + x_2^5 + x_3^5 = y^5.$$\nThat is, for all $x_1, x_2, x_3, y \\in \\mathbb{N}$ with $x_1, x_2, x_3, y > 0$,\n$$x_1^5 + x_2^5 + x_3^5 \\neq y^5.$$\nThis corresponds to the case $k = 5$, $n = 3$, $m = 1$ of the general conjecture,\nwhere $n + m = 4 < 5 = k$ would be required to yield a counterexample. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LanderParkinAndSelfridgeConjecture","statement":"∀ (x₁ x₂ x₃ y : ℕ), 0 < x₁ → 0 < x₂ → 0 < x₃ → 0 < y → x₁ ^ 5 + x₂ ^ 5 + x₃ ^ 5 ≠ y ^ 5","subjects":["11"],"theorem":"LanderParkinSelfridge.lander_parkin_selfridge.variants.five_three"},{"answerKinds":[],"category":"research open","docstring":"**Four exponentials conjecture**\nLet $x_0, x_1$ and $y_0, y_1$ be $\\mathbb Q$-linearly independent pairs of complex numbers,\nthen some $e^{x_i y_j}$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Exponentials","statement":"∀ (x y : Fin 2 → ℂ), LinearIndependent ℚ x → LinearIndependent ℚ y → ∃ i j, Transcendental ℚ (Complex.exp (x i * y j))","subjects":["33"],"theorem":"Exponentials.four_exponentials_conjecture"},{"answerKinds":[],"category":"research open","docstring":"The four exponential conjecture would imply that for any irrational number $t$,\nat least one of the numbers $2^t$ and $3^t$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Exponentials","statement":"∀ (t : ℝ), Irrational t → Transcendental ℚ (2 ^ t) ∨ Transcendental ℚ (3 ^ t)","subjects":["11"],"theorem":"Exponentials.two_pow_three_pow_transcendental"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SumOfThreeCubes","statement":"SumOfThreeCubes.IsSumOfThreeCubes 42","subjects":["11"],"theorem":"SumOfThreeCubes.isSumOfThreeCubes_42"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SumOfThreeCubes","statement":"∀ (n : ℤ), SumOfThreeCubes.IsSumOfThreeCubes n → ¬(n ≡ 4 [ZMOD 9] ∨ n ≡ 5 [ZMOD 9])","subjects":["11"],"theorem":"SumOfThreeCubes.mod_9_of_isSumOfThreeCubes"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SumOfThreeCubes","statement":"SumOfThreeCubes.IsSumOfThreeCubes 2","subjects":["11"],"theorem":"SumOfThreeCubes.isSumOfThreeCubes_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SumOfThreeCubes","statement":"SumOfThreeCubes.IsSumOfThreeCubes 33","subjects":["11"],"theorem":"SumOfThreeCubes.isSumOfThreeCubes_33"},{"answerKinds":[],"category":"research open","docstring":"An integer `n : ℤ` can be written as a sum of three cubes (of integers) if and only if\n`n` is not `4` or `5` mod `9`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SumOfThreeCubes","statement":"True ↔ ∀ (n : ℤ), SumOfThreeCubes.IsSumOfThreeCubes n ↔ ¬(n ≡ 4 [ZMOD 9] ∨ n ≡ 5 [ZMOD 9])","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"SumOfThreeCubes.isSumOfThreeCubes_iff_mod_9"},{"answerKinds":[],"category":"research solved","docstring":"Any rational number is a sum of three rational cubes.\n\nFirst proved by Ryley in 1825, which can be found in [Ri1930].\nThe below parametrization is brought from the MSE answer [MSE].\n\n[Ri1930] Richmond, H. W. \"On Rational Solutions of $x^3 + y^3 + z^3 = R$.\" Proceedings of the Edinburgh Mathematical Society 2.2 (1930): 92-100.\n[MSE] Kieren MacMillan, Proving that any rational number can be represented as the sum of the cubes of three rational numbers, https://math.stackexchange.com/q/4480969\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SumOfThreeCubes","statement":"∀ (r : ℚ), SumOfThreeCubes.IsSumOfThreeCubes r","subjects":["11"],"theorem":"SumOfThreeCubes.isSumOfThreeCubesRat_any"},{"answerKinds":[],"category":"research solved","docstring":"Every finite abelian group is realizable.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InverseGalois","statement":"∀ {G : Type u_1} [Fintype G] [inst : CommGroup G], InverseGalois.IsRealizable ℚ G","subjects":["12"],"theorem":"InverseGalois.inverse_galois_problem.variants.abelian"},{"answerKinds":[],"category":"research solved","docstring":"Every finite group is realisable over the field of rational functions\nwith coefficients `K`, where `K` is any field of characteristic 0.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InverseGalois","statement":"∀ {G : Type u_1} {K : Type u_2} [inst : Field K] [CharZero K] [Fintype G] [inst_3 : Group G],\n  InverseGalois.IsRealizable (RatFunc K) G","subjects":["12"],"theorem":"InverseGalois.inverse_galois_problem.variants.complex_function_field"},{"answerKinds":[],"category":"research solved","docstring":"Every finite cyclic group is realizable.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InverseGalois","statement":"∀ {G : Type u_1} [Fintype G] [inst : Group G] [IsCyclic G], InverseGalois.IsRealizable ℚ G","subjects":["12"],"theorem":"InverseGalois.inverse_galois_problem.variants.cyclic"},{"answerKinds":[],"category":"research open","docstring":"The **Inverse Galois Problem**: every finite group is\nisomorphic to the Galois group of a Galois extension of the\nrationals.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InverseGalois","statement":"∀ {G : Type u_1} [Fintype G] [inst : Group G], InverseGalois.IsRealizable ℚ G","subjects":["12"],"theorem":"InverseGalois.inverse_galois_problem"},{"answerKinds":[],"category":"research solved","docstring":"Every finite symmetric group is realizable.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InverseGalois","statement":"∀ {S : Type u_1} [Fintype S], InverseGalois.IsRealizable ℚ (S ≃ S)","subjects":["12"],"subsets":["FC100SolvedSet1"],"theorem":"InverseGalois.inverse_galois_problem.variants.symmetric_group"},{"answerKinds":[],"category":"research solved","docstring":"Every finite group is realisable over the field of rational functions\nwith complex coefficients.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InverseGalois","statement":"∀ {G : Type u_1} [Fintype G] [inst : Group G], InverseGalois.IsRealizable (RatFunc ℂ) G","subjects":["12"],"theorem":"InverseGalois.inverse_galois_problem.variants.complex_rational_functions"},{"answerKinds":[],"category":"research solved","docstring":"Every normal linear operator `T : H → H` on a Hilbert space `H` of dimension at least 2 has a\nnon-trivial closed `T`-invariant subspace. If `T` is a multiple of the identity, one can tafrake any\nnon-trivial subspace . If not, one can take any nontrivial spectral subspace of `T`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InvariantSubspaceProblem","statement":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℂ H] [inst_2 : CompleteSpace H],\n  2 ≤ Module.rank ℂ H →\n    ∀ (T : H →L[ℂ] H) [IsStarNormal T], Nonempty (InvariantSubspaceProblem.ClosedInvariantSubspace T)","subjects":["47"],"theorem":"InvariantSubspaceProblem.Invariant_subspace_problem_normal_operator"},{"answerKinds":[],"category":"research solved","docstring":"There exists a bounded linear operator `T` on the l1 space `(lp (fun (_ : ℕ) => ℂ) 1))` without\nnon-trivial closed `T`-invariant subspace [Read 1985](https://doi.org/10.1112/blms/17.4.305), see\nalso the first counterexample by Enflo [Enflo 1987](https://doi.org/10.1007%2FBF02392260), submitted\nin 1981. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InvariantSubspaceProblem","statement":"∃ T, IsEmpty (InvariantSubspaceProblem.ClosedInvariantSubspace T)","subjects":["47"],"theorem":"InvariantSubspaceProblem.Invariant_subspace_problem_l1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.InvariantSubspaceProblem","statement":"∀ {H : Type u_2} [inst : TopologicalSpace H], ¬TopologicalSpace.SeparableSpace H → Nontrivial H","subjects":["47"],"theorem":"InvariantSubspaceProblem.TopologicalSpace.nontrivial_of_not_separableSpace"},{"answerKinds":[],"category":"research open","docstring":"Show that every bounded linear operator `T : H → H` on a separable Hilbert space `H` of dimension\nat least 2 has a non-trivial closed `T`-invariant subspace: a closed linear subspace `W` of `H`,\nwhich is different from `H` and from `{0}`, such that `T ( W ) ⊂ W`. One needs the assumption that\nthe dimension of `H` is at least 2 because otherwise any subspace would be either `H` or `{0}`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InvariantSubspaceProblem","statement":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℂ H] [TopologicalSpace.SeparableSpace H]\n  [CompleteSpace H],\n  2 ≤ Module.rank ℂ H → ∀ (T : H →L[ℂ] H), Nonempty (InvariantSubspaceProblem.ClosedInvariantSubspace T)","subjects":["47"],"theorem":"InvariantSubspaceProblem.Invariant_subspace_problem"},{"answerKinds":[],"category":"research solved","docstring":"Every (bounded) linear operator `T : H → H` on a finite-dimensional linear space `H` of dimension\nat least 2 has a non-trivial (closed) `T`-invariant subspace. This can be solved using the Jordan\nnormal form, which is\n[not yet in mathlib](https://leanprover-community.github.io/undergrad_todo.html). ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InvariantSubspaceProblem","statement":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : Module ℂ H],\n  FiniteDimensional ℂ H →\n    2 ≤ Module.rank ℂ H → ∀ (T : H →L[ℂ] H), Nonempty (InvariantSubspaceProblem.ClosedInvariantSubspace T)","subjects":["47"],"theorem":"InvariantSubspaceProblem.Invariant_subspace_problem_finite_dimensional"},{"answerKinds":[],"category":"research solved","docstring":"Every bounded linear operator `T : H → H` on a non-separable Hilbert space `H` has a\nnon-trivial closed `T`-invariant subspace. Such an invariant space is given by considering the\nclosure of the linear span of the orbit of any single non-zero vector. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.InvariantSubspaceProblem","statement":"∀ {H : Type u_1} [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace ℂ H] [CompleteSpace H],\n  ¬TopologicalSpace.SeparableSpace H → ∀ (T : H →L[ℂ] H), Nonempty (InvariantSubspaceProblem.ClosedInvariantSubspace T)","subjects":["47"],"theorem":"InvariantSubspaceProblem.Invariant_subspace_problem_non_separable"},{"answerKinds":[],"category":"research open","docstring":"Let $P = (m_1, \\dots, m_k)$ be a tuple of positive even integers. Let\n$\\pi_P(n)$ denote the number of primes $p\\leq n$ such that $(p, p + m_1, \\dots, p + m_k)$\nforms an admissible prime constellation. Let $w(q; m_1, \\dots, m_k)$ denote the\nnumber of distinct residues of $0, m_1, \\dots, m_k$ modulo $q$, and let\n$$\n  C_P = 2 ^ k\\prod_{\\substack{q\\ \\text{prime} \\\\ q\\geq 3}}\n    \\frac{1 - \\frac{w(q; m_1, \\dots, m_k)}{q}}{\\left(1 - \\frac{1}{q}\\right)^{k+1}}.\n$$\nThen\n$$\n  \\pi_P(n)\\sim C_P\\int_2^n\\frac{dt}{\\log^{k+1}t}.\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.HardyLittlewood","statement":"∀ {k : ℕ} (m : Fin k.succ → ℕ), HardyLittlewood.FirstHardyLittlewoodConjectureFor m","subjects":["11"],"theorem":"HardyLittlewood.first_hardy_littlewood_conjecture"},{"answerKinds":[],"category":"research open","docstring":"For integers $x, y \\geq 2$,\n$$\n  \\pi(x + y) \\leq \\pi(x) + \\pi(y),\n$$\nwhere $\\pi(z)$ denotes the prime-counting function, giving the number of primes up to\nand including $z$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.HardyLittlewood","statement":"∀ {x y : ℕ}, 2 ≤ x → 2 ≤ y → HardyLittlewood.SecondHardyLittlewoodConjectureFor x y","subjects":["11"],"theorem":"HardyLittlewood.second_hardy_littlewood_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"Richards [Ri74] showed that only one of the two Hardy-Littlewood conjectures can be true.\n\n[Ri74] Richards, Ian (1974). _On the Incompatibility of Two Conjectures Concerning Primes_. Bull. Amer. Math. Soc. 80: 419–438.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.HardyLittlewood","statement":"(∀ {k : ℕ} (m : Fin k.succ → ℕ), HardyLittlewood.FirstHardyLittlewoodConjectureFor m) →\n  ¬∀ {x y : ℕ}, 2 ≤ x → 2 ≤ y → HardyLittlewood.SecondHardyLittlewoodConjectureFor x y","subjects":["11"],"theorem":"HardyLittlewood.not_first_and_secondHardyLittlewoodConjecture"},{"answerKinds":[],"category":"research solved","docstring":"An upper bound of the maximal length of the longest snake in a box is given by\n$$\n1 + 2^{n-1}\\frac{6n}{6n + \\frac{1}{6\\sqrt{6}}n^{\\frac 1 2} - 7}.\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SnakeInTheBox","statement":"∀ (n : ℕ), ↑(SnakeInBox.LongestSnakeInTheBox n) ≤ 1 + 2 ^ (n - 1) * (6 * ↑n) / (6 * ↑n + 1 / (6 * √6) * √↑n)","subjects":["5"],"theorem":"SnakeInBox.snake_upper_bound"},{"answerKinds":[],"category":"test","docstring":"The longest snake in the $0$-dimensional cube, i.e. the cube consisting of one point, is zero,\nsince there only is one induced path and it is of length zero.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SnakeInTheBox","statement":"SnakeInBox.LongestSnakeInTheBox 0 = 0","subjects":["5"],"theorem":"SnakeInBox.snake_zero_zero"},{"answerKinds":[],"category":"research solved","docstring":"The best length found so far for dimension nine is 190.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SnakeInTheBox","statement":"190 ≤ SnakeInBox.LongestSnakeInTheBox 9","subjects":["5"],"theorem":"SnakeInBox.snake_dim_nine_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"The maximum length for the snake-in-the-box problem is known for dimensions zero through eight;\nit is $0, 1, 2, 4, 7, 13, 26, 50, 98$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SnakeInTheBox","statement":"List.map SnakeInBox.LongestSnakeInTheBox (List.range 9) = [0, 1, 2, 4, 7, 13, 26, 50, 98]","subjects":["5"],"theorem":"SnakeInBox.snake_small_dimensions"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"For dimension $9$, the length of the longest snake in the box is not known.\nThis is currently the smallest dimension where this question is open.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SnakeInTheBox","statement":"SnakeInBox.LongestSnakeInTheBox 9 = sorry","subjects":["5"],"theorem":"SnakeInBox.snake_dim_nine"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ {n : Type u_1} [inst : DecidableEq n] (S : Finset n), UnionClosed.IsUnionClosed S.powerset","subjects":["5"],"theorem":"UnionClosed.isUnionClosed_powerset"},{"answerKinds":[],"category":"research solved","docstring":"Yu [Yu23] showed that the union-closed sets conjecture holds with a constant of approximately\n0.38234 instead of 1/2.\n[Yu23] Yu, Lei (2023). \"Dimension-free bounds for the union-closed sets conjecture\". Entropy. 25 (5): 767.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ {n : Type u_1} [inst : DecidableEq n] {A : Finset (Finset n)} [Nonempty n],\n  A ≠ {∅} → UnionClosed.IsUnionClosed A → ∃ i, 0.38234 * ↑A.card ≤ ↑{x ∈ A | i ∈ x}.card","subjects":["5"],"theorem":"UnionClosed.union_closed.variants.yu"},{"answerKinds":[],"category":"research solved","docstring":"Vuckovic and Zivkovic [Vu17] showed that the union-closed sets conjecture holds for set families\nwhose universal set has cardinality at most 12.\n[Vu17] Vuckovic, Bojan; Zivkovic, Miodrag (2017). \"The 12-Element Case of Frankl's Conjecture\" (PDF). IPSI BGD Transactions on Internet Research. 13 (1): 65.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ {n : Type u_1} [inst : DecidableEq n] {A : Finset (Finset n)} [inst_1 : Fintype n] [Nonempty n],\n  A ≠ {∅} → UnionClosed.IsUnionClosed A → Fintype.card n ≤ 12 → ∃ i, 1 / 2 * ↑A.card ≤ ↑{x ∈ A | i ∈ x}.card","subjects":["5"],"theorem":"UnionClosed.union_closed.variants.univ_card"},{"answerKinds":[],"category":"research solved","docstring":"We can show the union-closed sets conjecture is true for the case where the set family contains\nsome singleton.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ {n : Type u_1} [inst : DecidableEq n] {A : Finset (Finset n)},\n  UnionClosed.IsUnionClosed A → ∀ (i : n), {i} ∈ A → ∃ i, 1 / 2 * ↑A.card ≤ ↑{x ∈ A | i ∈ x}.card","subjects":["5"],"theorem":"UnionClosed.union_closed.variants.singleton_mem"},{"answerKinds":[],"category":"research solved","docstring":"Roberts and Simpson [Ro10] showed that the union-closed sets conjecture holds for set families of\nsize at most 46.\nTheir method, however, combined with the result of [Vu17], further shows that it holds for `#A ≤ 50`\nas well.\n[Ro10] Roberts, Ian; Simpson, Jamie (2010). \"A note on the union-closed sets conjecture\" (PDF). Australas. J. Combin. 47: 265–267.\n[Vu17] Vuckovic, Bojan; Zivkovic, Miodrag (2017). \"The 12-Element Case of Frankl's Conjecture\" (PDF). IPSI BGD Transactions on Internet Research. 13 (1): 65.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ {n : Type u_1} [inst : DecidableEq n] {A : Finset (Finset n)} [Nonempty n],\n  A ≠ {∅} → UnionClosed.IsUnionClosed A → A.card ≤ 50 → ∃ i, 1 / 2 * ↑A.card ≤ ↑{x ∈ A | i ∈ x}.card","subjects":["5"],"theorem":"UnionClosed.union_closed.variants.family_card"},{"answerKinds":[],"category":"research open","docstring":"If the UC conjecture is tight for some family `A` then $|A| = 2^k$ for some $k$.\n\nReference: Conjecture 3 in https://www.nieuwarchief.nl/serie5/pdf/naw5-2023-24-4-225.pdf.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ {n : Type u_1} [inst : DecidableEq n] {A : Finset (Finset n)} [Nonempty n],\n  A ≠ {∅} ∧ A ≠ ∅ →\n    UnionClosed.IsUnionClosed A → (∀ (i : n), ↑{x ∈ A | i ∈ x}.card = 1 / 2 * ↑A.card) → ∃ k, A.card = 2 ^ k","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"UnionClosed.union_closed.variants.cardinality_even_of_union_closed_tight"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ (n : Type u_2) [inst : DecidableEq n] [inst_1 : Fintype n], UnionClosed.IsUnionClosed Finset.univ","subjects":["5"],"theorem":"UnionClosed.isUnionClosed_univ"},{"answerKinds":[],"category":"research solved","docstring":"The union-closed sets conjecture is sharp in the sense that if we replace the constant `1/2` with\nany larger constant, then the conjecture fails.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ {n : Type u_1} [inst : DecidableEq n] [Fintype n] (c : ℝ),\n  1 / 2 < c →\n    ¬∀ (A : Finset (Finset n)), A ≠ {∅} → UnionClosed.IsUnionClosed A → ∃ i, c * ↑A.card ≤ ↑{x ∈ A | i ∈ x}.card","subjects":["5"],"theorem":"UnionClosed.union_closed.variants.sharpness"},{"answerKinds":[],"category":"research solved","docstring":"We can show the union-closed sets conjecture is true for the case where the universal set has\ncardinality 2, by brute force.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ (A : Finset (Finset (Fin 2))), A ≠ {∅} → UnionClosed.IsUnionClosed A → ∃ i, 1 / 2 * ↑A.card ≤ ↑{x ∈ A | i ∈ x}.card","subjects":["5"],"theorem":"UnionClosed.union_closed.variants.univ_card_two"},{"answerKinds":[],"category":"research open","docstring":"For every finite union-closed family of sets, other than the family containing only the empty set,\nthere exists an element that belongs to at least half of the sets in the family.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.UnionClosed","statement":"∀ {n : Type u_1} [inst : DecidableEq n] {A : Finset (Finset n)} [Nonempty n],\n  A ≠ {∅} → UnionClosed.IsUnionClosed A → ∃ i, 1 / 2 * ↑A.card ≤ ↑{x ∈ A | i ∈ x}.card","subjects":["5"],"theorem":"UnionClosed.union_closed"},{"answerKinds":[],"category":"research solved","docstring":"If there exists a constant `c > 0` such that\n`(n + 1).nth Nat.Prime - n.nth Nat.Prime < (n.nth Nat.Prime) ^ (1 / 2 - c)` for all large `n`,\nthen Legendre's conjecture is asymptotically true.\n\nFormal proof linked here provided by AlphaProof.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mzhorvath1/formal-conjectures/blob/a4568d467b4f42884b6a4bd09c40d65f92113ee7/FormalConjectures/Wikipedia/LegendreConjecture.lean#L48"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LegendreConjecture","statement":"(∃ c > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ↑(Nat.nth Nat.Prime (n + 1)) - ↑(Nat.nth Nat.Prime n) < ↑(Nat.nth Nat.Prime n) ^ (1 / 2 - c)) →\n  ∀ᶠ (n : ℕ) in Filter.atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p","subjects":["11"],"theorem":"LegendreConjecture.bounded_gap_legendre"},{"answerKinds":[],"category":"research solved","docstring":"Ferreira proved that the conjecture is true for sufficiently large n.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LegendreConjecture","statement":"∀ᶠ (n : ℕ) in Filter.atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p","subjects":["11"],"theorem":"LegendreConjecture.legendre_conjecture.ferreira_large_n"},{"answerKinds":[],"category":"research open","docstring":"Does there always exist at least one prime between consecutive perfect squares?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LegendreConjecture","statement":"True ↔ ∀ n ≥ 1, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p","subjects":["11"],"theorem":"LegendreConjecture.legendre_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a $3 \\times 3$ semi-magic square whose entries are all distinct positive\ninteger cubes? A square is semi-magic if all rows and columns sum to the same total.\n\nMore precisely, we seek a $3 \\times 3$ matrix with entries $a_{ij}$ such that each\n$a_{ij} = n_{ij}^3$ for some positive integer $n_{ij}$, all nine cubes are distinct,\nand all row sums and column sums are equal.\n\n*Reference:*\n[Semi-Magic Square of Cubes](https://unsolvedproblems.org/index_files/SquareofCubes.htm)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MagicSquares","statement":"True ↔\n  ∃ m t,\n    Function.Injective2 m ∧\n      (∀ (i j : Fin 3), ∃ n, 0 < n ∧ m i j = n ^ 3) ∧ (∀ (i : Fin 3), ∑ j, m i j = t) ∧ ∀ (j : Fin 3), ∑ i, m i j = t","subjects":["5","11"],"theorem":"MagicSquares.exists_semi_magic_square_cubes"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a $3 \\times 3$ matrix such that every entry is a distinct square,\nand all rows, columns, and diagonals add up to the same value?\n\n0 is excluded, as a Magic Square of Squares with 0 and 8 distinct squares is know is knownn.\nSee [Magic Square of Squares](https://static.nsta.org/pdfs/QuantumV6N3.pdf)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MagicSquares","statement":"True ↔\n  ∃ m t,\n    Function.Injective2 m ∧\n      (∀ (i j : Fin 3), 0 < m i j ∧ IsSquare (m i j)) ∧\n        (∀ (i : Fin 3), ∑ j, m i j = t) ∧\n          (∀ (j : Fin 3), ∑ i, m i j = t) ∧ m 0 0 + m 1 1 + m 2 2 = t ∧ m 0 2 + m 1 1 + m 2 0 = t","subjects":["11"],"theorem":"MagicSquares.exists_magic_square_squares"},{"answerKinds":[],"category":"research open","docstring":"**Brocard's Conjecture**\nFor every `n ≥ 2`, between the squares of the `n`-th and `(n+1)`-th primes,\nthere are at least four prime numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BrocardConjecture","statement":"∀ (n : ℕ),\n  1 ≤ n → 4 ≤ (Finset.filter Nat.Prime (Finset.Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2))).card","subjects":["11"],"theorem":"Brocard.brocard_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"Ferreira proved that Brocard's conjecture is true for sufficiently large n.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BrocardConjecture","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  4 ≤ (Finset.filter Nat.Prime (Finset.Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2))).card","subjects":["11"],"theorem":"Brocard.brocard_conjecture.ferreira_large_n"},{"answerKinds":[],"category":"research solved","docstring":"The rank of the Elkies-Klagsbrun curve is at least 29. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"29 ≤ Module.finrank ℤ EllipticCurveRank.WeierstrassCurve.elkiesKlagsbrun29.Point","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.twentynine_le_rank_elkiesKlagsbrun29"},{"answerKinds":[],"category":"research solved","docstring":"The rank of an elliptic curve over a number field is always finite by the Mordell–Weil theorem.\nConsequently, the rank is always finite, so `finrank ℤ E⟮K⟯ = 0` really means that the group of\nrational points is torsion, not that it is of infinite rank. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ {K : Type u_1} [inst : Field K] [NumberField K] [inst_2 : DecidableEq K] (E : WeierstrassCurve.Affine K)\n  [WeierstrassCurve.IsElliptic E], Module.Finite ℤ E.Point","subjects":["11","14"],"theorem":"EllipticCurveRank.mordell_weil"},{"answerKinds":[],"category":"research open","docstring":"The rank of the Elkies-Klagsbrun curve is exactly 29. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"Module.finrank ℤ EllipticCurveRank.WeierstrassCurve.elkiesKlagsbrun29.Point = 29","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.rank_elkiesKlagsbrun29"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"WeierstrassCurve.IsElliptic EllipticCurveRank.WeierstrassCurve.ranksunbounded30","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.instIsEllipticRatRanksunbounded30"},{"answerKinds":[],"category":"research solved","docstring":"The rank of the Elkies curve is at least 28. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"28 ≤ Module.finrank ℤ EllipticCurveRank.WeierstrassCurve.elkies28.Point","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.twentyeight_le_rank_elkies28"},{"answerKinds":[],"category":"research solved","docstring":"Is there an elliptic curve over ℚ of rank at least 31?\n\nThe answer is yes: such a curve was found by Claude, Levent Alpöge and Ava Howell in 2026.\nSee https://elliptic-rank.icarm.cloud/curve/302 and `WeierstrassCurve.claudeAlpogeHowell31`\nbelow. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∃ E, 31 ≤ E.rank","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.exists_rank_ge_thirtyone"},{"answerKinds":[],"category":"API","docstring":"The naïve height of the Weierstrass model $E^d$ is $|d|^6$ times the naïve height of $E$.\nOrdering the twists of $E$ by $|d|$ is therefore the same as ordering them by naïve height. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (E : EllipticCurveRank.RatEllipticCurve) (d : ℤ),\n  max (4 * (d ^ 2 * E.A).natAbs ^ 3) (27 * (d ^ 3 * E.B).natAbs ^ 2) = d.natAbs ^ 6 * E.naiveHeight","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.naiveHeight_quadraticTwist"},{"answerKinds":[],"category":"research open","docstring":"From [PPVW2016], Section 3.1: \"from the mid-1960s to the present,\nit seems that most experts conjectured unboundedness.\" ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (n : ℕ), ∃ E, n ≤ E.rank","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.unbounded_rank_conjecture"},{"answerKinds":[],"category":"API","docstring":"The quadratic twist of an elliptic curve over $\\mathbb{Q}$ by a nonzero $d$ is again an\nelliptic curve. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (E : EllipticCurveRank.RatEllipticCurve) {d : ℤ}, d ≠ 0 → WeierstrassCurve.IsElliptic (E.quadraticTwist d)","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.isElliptic_quadraticTwist"},{"answerKinds":[],"category":"research open","docstring":"The rank of the Claude–Alpöge–Howell curve is exactly 31.\nIt has rank exactly 31 assuming the generalized Riemann hypothesis and the\nBirch and Swinnerton-Dyer conjecture.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"Module.finrank ℤ EllipticCurveRank.WeierstrassCurve.claudeAlpogeHowell31.Point = 31","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.rank_claudeAlpogeHowell31"},{"answerKinds":[],"category":"test","docstring":"See https://elliptic-rank.icarm.cloud/curve/302.\nUser profile: https://elliptic-rank.icarm.cloud/user/18\n        and   https://elliptic-rank.icarm.cloud/user/53\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"WeierstrassCurve.Δ EllipticCurveRank.WeierstrassCurve.claudeAlpogeHowell31 =\n  2 ^ 15 * 3 ^ 4 * 5 ^ 4 * 7 ^ 6 * 11 ^ 4 * 13 ^ 5 * 19 ^ 2 * 23 ^ 2 * 29 ^ 3 * 37 ^ 2 * 41 ^ 2 * 73 ^ 2 * 131 ^ 2 *\n                167 ^ 2 *\n              7547 *\n            632881 *\n          966509 *\n        18145679437533309132469 *\n      767028866604834801397681553 *\n    30580600452196904409276223329355584892025407195996968868775951126238056443210297","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.Δ_claudeAlpogeHowell31"},{"answerKinds":[],"category":"research open","docstring":"[PPVW2016] 8.2(b): for 1 ≤ r ≤ 20, the number of elliptic curves over ℚ with rank `r` and\nnaïve height at most `H` is asymptotically `H ^ ((21 - r) / 24 + o(1))`.\nNote: ℰ_H in 8.2(b) should be ℰ_{≤H}, see the statement of Theorem 7.3.3.\nWhen `r = 1`, the exponent is `20 / 24 = 5 / 6`, which agrees with the exponent in\n`card_heightLE_div_pow_five_div_six_tensto` and is consistent with\n`half_rank_zero_and_half_rank_one`.\nThe equality is asserted only for large `H`, matching the `o(1)` above: it cannot hold\nat small `H`, because `heightLE H` is empty for `H ≤ 3` (`naiveHeight E ≤ 3` forces\n`A = B = 0`, which both `Δ_ne_zero` and `reduced` exclude) and consists only of the two\nrank-zero curves `y² = x³ ± x` for `4 ≤ H ≤ 26`, while the right hand side is\n`Real.rpow` and hence strictly positive. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (r : ℕ),\n  1 ≤ r →\n    r ≤ 20 →\n      ∃ f,\n        Filter.Tendsto f Filter.atTop (nhds 0) ∧\n          ∀ᶠ (H : ℕ) in Filter.atTop,\n            ↑{E | E ∈ EllipticCurveRank.RatEllipticCurve.heightLE H ∧ r ≤ E.rank}.ncard = ↑H ^ ((21 - ↑r) / 24 + f H)","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.rank_height_count_asymptotic"},{"answerKinds":[],"category":"research solved","docstring":"The rank of the Claude–Alpöge–Howell curve is at least 31. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"31 ≤ Module.finrank ℤ EllipticCurveRank.WeierstrassCurve.claudeAlpogeHowell31.Point","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.thirtyone_le_rank_claudeAlpogeHowell31"},{"answerKinds":[],"category":"research open","docstring":"Goldfeld's conjecture with $d$ restricted to squarefree integers, which is how it is usually\nstated informally: 50% of the quadratic twists of $E$ have rank $0$ and 50% have rank $1$. Every\nquadratic twist of $E$ is isomorphic to $E^d$ for a unique squarefree $d$, so the squarefree $d$\nenumerate the distinct twists without repetition. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (E : EllipticCurveRank.RatEllipticCurve) (r : ℕ),\n  r = 0 ∨ r = 1 →\n    Filter.Tendsto\n      (fun H =>\n        ↑{d | d ∈ EllipticCurveRank.RatEllipticCurve.twistIndexLE H ∧ Squarefree d ∧ E.twistRank d = r}.ncard /\n          ↑{d | d ∈ EllipticCurveRank.RatEllipticCurve.twistIndexLE H ∧ Squarefree d}.ncard)\n      Filter.atTop (nhds (1 / 2))","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.goldfeld_conjecture.variants.squarefree"},{"answerKinds":[],"category":"research solved","docstring":"Theorem 3 of [BS2013]:\nwhen elliptic curves over ℚ are ordered by height, their average rank is < .885. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"Filter.limsup\n    (fun H =>\n      ↑(∑ᶠ (E : ↑(EllipticCurveRank.RatEllipticCurve.heightLE H)), (↑E).rank) /\n        ↑(EllipticCurveRank.RatEllipticCurve.heightLE H).ncard)\n    Filter.atTop <\n  0.885","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.avg_rank_lt_0885"},{"answerKinds":[],"category":"test","docstring":"See https://mathoverflow.net/a/478050. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"WeierstrassCurve.Δ EllipticCurveRank.WeierstrassCurve.elkies28 =\n  2 ^ 15 * 3 ^ 6 * 5 ^ 6 * 7 ^ 4 * 11 ^ 2 * 13 ^ 4 * 17 ^ 5 * 19 ^ 3 * 48463 * 20650099 *\n      315574902691581877528345013999136728634663121 *\n    376018840263193489397987439236873583997122096511452343225772113000611087671413","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.Δ_elkies28"},{"answerKinds":[],"category":"API","docstring":"The twist of $E$ by $d = 1$ is $E$ itself. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (E : EllipticCurveRank.RatEllipticCurve), E.quadraticTwist 1 = E.toWeierstrass","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.quadraticTwist_one"},{"answerKinds":[],"category":"textbook","docstring":"Formula (5.1.1) of [PPVW2016]: The number of elliptic curves over ℚ with naïve height at most\n`H` is asymptotically `2^(4/3)*3^(-3/2)/ζ(10) * H^(5/6)`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"Filter.Tendsto (fun H => ↑(EllipticCurveRank.RatEllipticCurve.heightLE H).ncard / ↑H ^ (5 / 6)) Filter.atTop\n  (nhds (2 ^ (4 / 3) * 3 ^ (-3 / 2) / (riemannZeta 10).re))","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.card_heightLE_div_pow_five_div_six_tensto"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"WeierstrassCurve.IsElliptic EllipticCurveRank.WeierstrassCurve.claudeAlpogeHowell31","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.instIsEllipticRatClaudeAlpogeHowell31"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"WeierstrassCurve.IsElliptic EllipticCurveRank.WeierstrassCurve.elkies28","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.elkies28IsElliptic"},{"answerKinds":[],"category":"research open","docstring":"Conjecture by Goldfeld and Katz–Sarnak: if elliptic curves over ℚ are ordered by their\nheights, then 50% of the curves have rank 0 and 50% have rank 1.\nSee p. 28 of https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Bhargava.pdf. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (r : ℕ),\n  r = 0 ∨ r = 1 →\n    Filter.Tendsto\n      (fun H =>\n        ↑{E | E ∈ EllipticCurveRank.RatEllipticCurve.heightLE H ∧ E.rank = r}.ncard /\n          ↑(EllipticCurveRank.RatEllipticCurve.heightLE H).ncard)\n      Filter.atTop (nhds (1 / 2))","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.half_rank_zero_and_half_rank_one"},{"answerKinds":[],"category":"research open","docstring":"From [PPVW2016], Section 8.2:\n\"Our heuristic predicts (a) All but finitely many E ∈ ℰ satisfy rk E(ℚ) ≤ 21\".\nIn other words, there are only finitely many elliptic curves over ℚ (up to isomorphism)\nwith rank greater than 21.\nNotice that this contradicts the previous conjecture. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"{E | 21 < E.rank}.Finite","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.finite_twentyone_lt_finrank"},{"answerKinds":[],"category":"research open","docstring":"Is there an elliptic curve over ℚ of rank at least 32?\nThe largest known rank of an elliptic curve over ℚ as of 2026 is at least 31 (and exactly\n31 assuming the generalized Riemann hypothesis and Birch and Swinnerton-Dyer conjecture).\nSee https://elliptic-rank.icarm.cloud/curve/302. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∃ E, 32 ≤ E.rank","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.exists_rank_ge_thirtytwo"},{"answerKinds":[],"category":"API","docstring":"There are $2H$ nonzero integers $d$ with $|d| \\leq H$. This is the normalisation used in\n[Smith2025]. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (H : ℕ), (EllipticCurveRank.RatEllipticCurve.twistIndexLE H).ncard = 2 * H","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.ncard_twistIndexLE"},{"answerKinds":[],"category":"test","docstring":"See https://mathoverflow.net/a/478050. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"WeierstrassCurve.Δ EllipticCurveRank.WeierstrassCurve.elkiesKlagsbrun29 =\n  -2 ^ 19 * 3 ^ 7 * 5 ^ 7 * 7 ^ 4 * 11 ^ 5 * 13 ^ 3 * 17 ^ 4 * 31 ^ 3 * 41 ^ 2 * 43 ^ 2 * 61 ^ 2 * 233 * 241 ^ 2 *\n          4139 *\n        678146849364709860535420504397393 *\n      159788990966780131363155786084695062643236502969 *\n    4402149008473369392540402625019227412319473055901","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.Δ_elkiesKlagsbrun29"},{"answerKinds":[],"category":"test","docstring":"See https://elliptic-rank.icarm.cloud/curve/273. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"WeierstrassCurve.Δ EllipticCurveRank.WeierstrassCurve.ranksunbounded30 =\n  -2 ^ 16 * 3 ^ 12 * 5 ^ 8 * 7 ^ 5 * 13 ^ 5 * 31 ^ 2 * 41 ^ 2 * 47 ^ 4 * 53 ^ 3 * 67 ^ 3 * 379 ^ 2 * 4349 *\n      25721454817 *\n    97018222656318846556561979214040553412450110580812087282349817173780902099339117104673990259247421230916714670243202937","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.Δ_ranksunbounded30"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"WeierstrassCurve.IsElliptic EllipticCurveRank.WeierstrassCurve.elkiesKlagsbrun29","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.elkiesKlagsbrun29IsElliptic"},{"answerKinds":[],"category":"research open","docstring":"The rank of the Elkies curve is exactly 28. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"Module.finrank ℤ EllipticCurveRank.WeierstrassCurve.elkies28.Point = 28","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.rank_elkies28"},{"answerKinds":[],"category":"research open","docstring":"[PPVW2016] 8.2(c): the number of elliptic curves over ℚ with rank ≥ 21 and naïve height\nat most `H` is asymptotically at most `H ^ o(1)`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∃ f,\n  Filter.Tendsto f Filter.atTop (nhds 0) ∧\n    ∀ (H : ℕ), 1 < H → ↑{E | E ∈ EllipticCurveRank.RatEllipticCurve.heightLE H ∧ 21 ≤ E.rank}.ncard ≤ ↑H ^ f H","subjects":["11","14"],"subsets":["FC100OpenSet1"],"theorem":"EllipticCurveRank.RatEllipticCurve.twentyone_le_rank_height_count_asymptotic"},{"answerKinds":[],"category":"research solved","docstring":"The rank of the ranksunbounded curve is at least 30. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"30 ≤ Module.finrank ℤ EllipticCurveRank.WeierstrassCurve.ranksunbounded30.Point","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.thirty_le_rank_ranksunbounded30"},{"answerKinds":[],"category":"research open","docstring":"**Goldfeld's conjecture** ([Goldfeld1979], Conjecture B), in the form stated in the\nintroduction of [Smith2025]: when the quadratic twists $E^d$ of an elliptic curve $E$ over\n$\\mathbb{Q}$ are ordered by $|d|$, 50% of them have rank $0$ and 50% have rank $1$. See\n`goldfeld_conjecture.variants.two_le` for the remaining case $2 \\leq r$.\n\nGoldfeld states the conjecture for the analytic rank of $E^d$, which the Birch and\nSwinnerton-Dyer conjecture predicts is equal to the Mordell–Weil rank used here. Corollary 1.2\nof [Smith2025] proves that the Birch and Swinnerton-Dyer conjecture for the quadratic twist\nfamily of $E$ implies Goldfeld's conjecture for $E$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (E : EllipticCurveRank.RatEllipticCurve) (r : ℕ),\n  r = 0 ∨ r = 1 →\n    Filter.Tendsto\n      (fun H =>\n        ↑{d | d ∈ EllipticCurveRank.RatEllipticCurve.twistIndexLE H ∧ E.twistRank d = r}.ncard /\n          ↑(EllipticCurveRank.RatEllipticCurve.twistIndexLE H).ncard)\n      Filter.atTop (nhds (1 / 2))","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.goldfeld_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**Goldfeld's conjecture** ([Goldfeld1979], Conjecture B), in the form stated in the\nintroduction of [Smith2025], in the case $2 \\leq r$: a density $0$ of the quadratic twists $E^d$\nof an elliptic curve $E$ over $\\mathbb{Q}$ have rank $r$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"∀ (E : EllipticCurveRank.RatEllipticCurve) (r : ℕ),\n  2 ≤ r →\n    Filter.Tendsto\n      (fun H =>\n        ↑{d | d ∈ EllipticCurveRank.RatEllipticCurve.twistIndexLE H ∧ E.twistRank d = r}.ncard /\n          ↑(EllipticCurveRank.RatEllipticCurve.twistIndexLE H).ncard)\n      Filter.atTop (nhds 0)","subjects":["11","14"],"theorem":"EllipticCurveRank.RatEllipticCurve.goldfeld_conjecture.variants.two_le"},{"answerKinds":[],"category":"research open","docstring":"The rank of the ranksunbounded curve is exactly 30. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EllipticCurveRank","statement":"Module.finrank ℤ EllipticCurveRank.WeierstrassCurve.ranksunbounded30.Point = 30","subjects":["11","14"],"theorem":"EllipticCurveRank.WeierstrassCurve.rank_ranksunbounded30"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a composite number $n > 1$ such that Euler’s totient function\n$\\varphi(n)$ divides $n - 1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LehmerTotient","statement":"True ↔ ∃ n > 1, ¬Prime n ∧ n.totient ∣ n - 1","subjects":["11"],"theorem":"LehmerTotient.lehmer_totient"},{"answerKinds":[],"category":"textbook","docstring":"Eleven unit squares can be packed into a square of side length < 3.877084.\n\nReference: [Wikipedia](https://en.wikipedia.org/wiki/Square_packing#In_a_square)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"Nonempty (SquarePacking.Packing 11 SquarePacking.UnitSquare (SquarePacking.Square 3.877084))","subjects":["51"],"theorem":"SquarePacking.eleven_square_packing_in_square_bound"},{"answerKinds":[],"category":"textbook","docstring":"Seventeen unit squares can be packed into a square of side length < 4.6756.\n\nReference: [Wikipedia](https://en.wikipedia.org/wiki/Square_packing#In_a_square)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"Nonempty (SquarePacking.Packing 17 SquarePacking.UnitSquare (SquarePacking.Square 4.6756))","subjects":["51"],"theorem":"SquarePacking.seventeen_square_packing_in_square_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the smallest square that can contain 21 unit circles?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"IsLeast {x | Nonempty (SquarePacking.Packing 21 SquarePacking.UnitCircle (SquarePacking.Square x))} sorry","subjects":["51"],"theorem":"SquarePacking.least_twenty_one_circle_packing_in_square"},{"answerKinds":[],"category":"textbook","docstring":"Twenty-one unit circles can be packed into a square of side length < 9.359.\n\nReference: [Visualizations](https://erich-friedman.github.io/packing/cirinsqu/)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"Nonempty (SquarePacking.Packing 21 SquarePacking.UnitCircle (SquarePacking.Square 9.359))","subjects":["51"],"theorem":"SquarePacking.twenty_one_circle_packing_in_square_bound"},{"answerKinds":[],"category":"test","docstring":"The degenerate circle is empty.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"SquarePacking.Circle 0 = ∅","subjects":["51"],"theorem":"SquarePacking.circle_zero"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the smallest square that can contain 11 unit squares?\n\nReference: [Wikipedia](https://en.wikipedia.org/wiki/Square_packing#In_a_square)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"IsLeast {x | Nonempty (SquarePacking.Packing 11 SquarePacking.UnitSquare (SquarePacking.Square x))} sorry","subjects":["51"],"theorem":"SquarePacking.least_eleven_square_packing_in_square"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the smallest circle that can contain 3 unit squares?\n\nReference: [Wikipedia](https://en.wikipedia.org/wiki/Square_packing#In_a_circle)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"IsLeast {r | Nonempty (SquarePacking.Packing 3 SquarePacking.UnitSquare (SquarePacking.Circle r))} sorry","subjects":["51"],"theorem":"SquarePacking.least_three_square_packing_in_circle"},{"answerKinds":[],"category":"textbook","docstring":"Three unit squares can be packed into a circle of radius $(5 \\sqrt{17}) / 16 \\approx 1.288$.\n\nReference: [Wikipedia](https://en.wikipedia.org/wiki/Square_packing#In_a_circle)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"Nonempty (SquarePacking.Packing 3 SquarePacking.UnitSquare (SquarePacking.Circle (5 * NNReal.sqrt 17 / 16)))","subjects":["51"],"theorem":"SquarePacking.three_square_packing_in_circle_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the smallest square that can contain 17 unit squares?\n\nReference: [Wikipedia](https://en.wikipedia.org/wiki/Square_packing#In_a_square)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"IsLeast {x | Nonempty (SquarePacking.Packing 17 SquarePacking.UnitSquare (SquarePacking.Square x))} sorry","subjects":["51"],"theorem":"SquarePacking.least_seventeen_square_packing_in_square"},{"answerKinds":[],"category":"textbook","docstring":"Fifteen unit circles can be packed into a circle of radius\n$1 + \\sqrt{6 + 2/\\sqrt{5} + 4 \\sqrt{1 + 2/\\sqrt{5}}} \\approx 4.521$.\n\nReference:\nGraham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ.\nDense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"Nonempty\n  (SquarePacking.Packing 15 SquarePacking.UnitCircle\n    (SquarePacking.Circle (1 + NNReal.sqrt (6 + 2 / NNReal.sqrt 5 + 4 * NNReal.sqrt (1 + 2 / NNReal.sqrt 5)))))","subjects":["51"],"theorem":"SquarePacking.fifteen_circle_packing_in_circle_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the smallest circle that can contain 15 unit circles?\n\nReference:\nGraham RL, Lubachevsky BD, Nurmela KJ, Ostergard PRJ.\nDense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.\n[Pirl (1969)](https://doi.org/10.1002/mana.19690400110) conjectured this configuration to be optimal.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SquarePacking","statement":"IsLeast {r | Nonempty (SquarePacking.Packing 15 SquarePacking.UnitCircle (SquarePacking.Circle r))} sorry","subjects":["51"],"theorem":"SquarePacking.least_fifteen_circle_packing_in_circle"},{"answerKinds":[],"category":"research solved","docstring":"The Modularity Theorem (formerly Shimura-Taniyama-Weil conjecture): every elliptic curve\nover $\\mathbb{Q}$ is modular.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ModularityConjecture","statement":"∀ (E : WeierstrassCurve ℚ) [inst : E.IsElliptic], ModularityConjecture.modularityConjecture E","subjects":["11"],"theorem":"ModularityConjecture.modularity_conjecture"},{"answerKinds":[],"category":"research open","docstring":"There are no distinct primes $p$ and $q$ such that $\\frac{q^p - 1}{q - 1}$ divides $\\frac{p^q - 1}{p - 1}$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.FeitThompsonPrimeConjecture","statement":"∀ (p q : ℕ), Nat.Prime p → Nat.Prime q → p < q → ¬(q ^ p - 1) / (q - 1) ∣ (p ^ q - 1) / (p - 1)","subjects":["11"],"theorem":"FeitThompsonPrimeConjecture.feit_thompson_primes"},{"answerKinds":[],"category":"research open","docstring":"**PSW conjecture** (Selfridge's test)\nLet $p$ be an odd number, with $p \\equiv \\pm 2 \\pmod{5}$, $2^{p-1} \\equiv 1 \\pmod{p}$\nand $F_{p+1} \\equiv 0 \\pmod{p}$, then $p$ is a prime number.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Selfridge","statement":"∀ (p : ℕ), Selfridge.IsSelfridge p → Nat.Prime p","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Selfridge.selfridge_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Selfridge conjectured that the number of prime factors of the `n`-th Fermat number does not grow\nmonotonically in $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Selfridge","statement":"¬Monotone Selfridge.fermatFactors","subjects":["11"],"theorem":"Selfridge.selfridge_seq_conjecture"},{"answerKinds":[],"category":"textbook","docstring":"Selfridge's test variant:\nLet $p$ be an odd number, with $p \\equiv \\pm 1 \\pmod{5}$, $2^{p-1} \\equiv 1 \\pmod{p}$\nand $F_{p-1} \\equiv 0 \\pmod{p}$, then $p$ is a prime number.\n\nThis test does not work.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Selfridge","statement":"∃ n, Selfridge.IsPseudoSelfridge n ∧ ¬Nat.Prime n","subjects":["11"],"theorem":"Selfridge.selfridge_conjecture.variants.exist_pseudo_counterexample"},{"answerKinds":[],"category":"research solved","docstring":"Selfridge conjectured that the number of prime factors of the `n`-th Fermat number does not grow\nmonotonically in $n$.\n\nA sufficient condition for this conjecture to hold is that there exists a Fermat prime larger than\n65537.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Selfridge","statement":"∀ (n : ℕ), Prime n.fermatNumber → n ≥ 5 → ¬Monotone Selfridge.fermatFactors","subjects":["11"],"theorem":"Selfridge.selfridge_seq_conjecture.variants.sufficient_condition"},{"answerKinds":[],"category":"textbook","docstring":"Selfridge's test variant:\nLet $p$ be an odd number, with $p \\equiv \\pm 1 \\pmod{5}$, $2^{p-1} \\equiv 1 \\pmod{p}$\nand $F_{p-1} \\equiv 0 \\pmod{p}$, then $p$ is a prime number.\n\nThe number $30889$ is a conterexample to this test satisfying $30889 ≡ - 1 \\mod 5$\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Selfridge","statement":"Selfridge.IsPseudoSelfridge 30889 ∧ ¬Nat.Prime 30889 ∧ 30889 ≡ 4 [MOD 5]","subjects":["11"],"theorem":"Selfridge.selfridge_conjecture.variants.pseudo_counterexample'"},{"answerKinds":[],"category":"textbook","docstring":"Selfridge's test variant:\nLet $p$ be an odd number, with $p \\equiv \\pm 1 \\pmod{5}$, $2^{p-1} \\equiv 1 \\pmod{p}$\nand $F_{p-1} \\equiv 0 \\pmod{p}$, then $p$ is a prime number.\n\nThe number $6601$ is a conterexample to this test satisfying $6601 ≡ 1 \\mod 5$\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Selfridge","statement":"Selfridge.IsPseudoSelfridge 6601 ∧ ¬Nat.Prime 6601 ∧ 6601 ≡ 1 [MOD 5]","subjects":["11"],"theorem":"Selfridge.selfridge_conjecture.variants.pseudo_counterexample"},{"answerKinds":[],"category":"test","docstring":"Sanity check: a system with no sets at all ($m = 0$) admits any colouring vacuously;\nin particular the all-ones colouring satisfies every discrepancy bound.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.BeckFialaConjecture","statement":"∀ (n t : ℕ) (S : Fin 0 → Finset (Fin n)),\n  (∀ (j : Fin n), {i | j ∈ S i}.card ≤ t) →\n    ∃ χ, (∀ (j : Fin n), χ j = 1 ∨ χ j = -1) ∧ ∀ (i : Fin 0), |∑ j ∈ S i, χ j| ≤ √↑t","subjects":["5"],"theorem":"BeckFiala.beck_fiala_conjecture.variants.no_sets"},{"answerKinds":[],"category":"research open","docstring":"**The Beck–Fiala conjecture**\n\nThere exists a universal constant $C > 0$ such that every set system\n$S_1, \\dots, S_m \\subseteq [n]$ of degree at most $t$ admits a colouring\n$\\chi \\colon [n] \\to \\{-1, +1\\}$ with\n$\\left|\\sum_{j \\in S_i} \\chi(j)\\right| \\le C \\sqrt{t}$ for every $i$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BeckFialaConjecture","statement":"∃ C,\n  0 < C ∧\n    ∀ (n m t : ℕ) (S : Fin m → Finset (Fin n)),\n      (∀ (j : Fin n), {i | j ∈ S i}.card ≤ t) →\n        ∃ χ, (∀ (j : Fin n), χ j = 1 ∨ χ j = -1) ∧ ∀ (i : Fin m), |∑ j ∈ S i, χ j| ≤ C * √↑t","subjects":["5"],"theorem":"BeckFiala.beck_fiala_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"**The Beck–Fiala theorem**\n\nIf $S_1, \\dots, S_m \\subseteq [n]$ is a set system of degree at most $t$, i.e. every\n$j \\in [n]$ lies in at most $t$ of the sets, and $t \\ge 1$, then there is a colouring\n$\\chi \\colon [n] \\to \\{-1, +1\\}$ with $\\left|\\sum_{j \\in S_i} \\chi(j)\\right| \\le 2t - 1$\nfor every $i$.\n\nThe hypothesis $t \\ge 1$ is necessary: a system of degree $0$ consists of empty sets\nonly, whose discrepancy is $0 > 2 \\cdot 0 - 1$.\n\n[J. Beck and T. Fiala, *\"Integer-making\" theorems*,\nDiscrete Applied Mathematics **3** (1981), 1–8.]\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BeckFialaConjecture","statement":"∀ (n m t : ℕ),\n  1 ≤ t →\n    ∀ (S : Fin m → Finset (Fin n)),\n      (∀ (j : Fin n), {i | j ∈ S i}.card ≤ t) →\n        ∃ χ, (∀ (j : Fin n), χ j = 1 ∨ χ j = -1) ∧ ∀ (i : Fin m), |∑ j ∈ S i, χ j| ≤ 2 * ↑t - 1","subjects":["5"],"theorem":"BeckFiala.beck_fiala_theorem"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Szekeres [ES35] proved that the number of powerful integers up to $x$ satisfies\n$$Q(x) = \\frac{\\zeta(3/2)}{\\zeta(3)} x^{1/2} + O(x^{1/3}).$$\nIn particular $Q(x) \\sim \\frac{\\zeta(3/2)}{\\zeta(3)} \\sqrt{x}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PowerfulNumbersDensity","statement":"(fun x => ↑(PowerfulNumbersDensity.Q x) - PowerfulNumbersDensity.A * x ^ (1 / 2)) =O[Filter.atTop] fun x => x ^ (1 / 3)","subjects":["11"],"theorem":"PowerfulNumbersDensity.asymptotic_erdos_szekeres"},{"answerKinds":[],"category":"research solved","docstring":"Bateman and Grosswald [BG58] proved the sharper asymptotic\n$$Q(x) = \\frac{\\zeta(3/2)}{\\zeta(3)} x^{1/2} + \\frac{\\zeta(2/3)}{\\zeta(2)} x^{1/3} + O(x^{1/6}).$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PowerfulNumbersDensity","statement":"(fun x =>\n    ↑(PowerfulNumbersDensity.Q x) - PowerfulNumbersDensity.A * x ^ (1 / 2) -\n      PowerfulNumbersDensity.B * x ^ (1 / 3)) =O[Filter.atTop]\n  fun x => x ^ (1 / 6)","subjects":["11"],"theorem":"PowerfulNumbersDensity.asymptotic_bateman_grosswald"},{"answerKinds":[],"category":"research open","docstring":"Can the exponent $1/6$ in the error term of the Bateman–Grosswald asymptotic be improved\nunconditionally? That is, is there $\\delta > 0$ such that\n$$Q(x) = \\frac{\\zeta(3/2)}{\\zeta(3)} x^{1/2} + \\frac{\\zeta(2/3)}{\\zeta(2)} x^{1/3} +\nO(x^{1/6 - \\delta})?$$\nImprovements are known under the Riemann Hypothesis.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PowerfulNumbersDensity","statement":"True ↔\n  ∃ δ > 0,\n    (fun x =>\n        ↑(PowerfulNumbersDensity.Q x) - PowerfulNumbersDensity.A * x ^ (1 / 2) -\n          PowerfulNumbersDensity.B * x ^ (1 / 3)) =O[Filter.atTop]\n      fun x => x ^ (1 / 6 - δ)","subjects":["11"],"theorem":"PowerfulNumbersDensity.error_term_improvement"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that there are infinitely many Wolstenholme primes.\n\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Wolstenholme_prime#Expected_number_of_Wolstenholme_primes)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WolstenholmePrime","statement":"{p | WolstenholmePrime.IsWolstenholmePrime p}.Infinite","subjects":["11"],"theorem":"WolstenholmePrime.wolstenholme_prime_infinite"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WolstenholmePrime","statement":"WolstenholmePrime.IsWolstenholmePrime 2124679","subjects":["11"],"theorem":"WolstenholmePrime.wolstenholme_prime_2124679"},{"answerKinds":[],"category":"textbook","docstring":"Wolstenholme's theorem states that any prime $p > 3$ satisfies $\\binom{2p-1}{p-1} \\equiv 1 (\\pmod{p^3})$.\n\nFormal proof linked here provided by AlphaProof.\n*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem)\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/d833ed31d82693f10bed7a4c9ac329545b556a03/FormalConjectures/Wikipedia/WolstenholmePrime.lean#L34"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WolstenholmePrime","statement":"∀ p > 3, Nat.Prime p → (2 * p - 1).choose (p - 1) ≡ 1 [MOD p ^ 3]","subjects":["11"],"theorem":"WolstenholmePrime.wolstenholme_theorem"},{"answerKinds":[],"category":"test","docstring":"Two known Wolstenholme primes: 16843 and 2124679.\n\nFormal proof linked here provided by AlphaProof\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WolstenholmePrime","statement":"WolstenholmePrime.IsWolstenholmePrime 16843","subjects":["11"],"theorem":"WolstenholmePrime.wolstenholme_prime_16483"},{"answerKinds":[],"category":"textbook","docstring":"Equivalently, a prime $p > 7$ is a Wolstenholme prime if it divides the numerator of the Bernoulli number $B_{p-3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WolstenholmePrime","statement":"∀ (p : ℕ), WolstenholmePrime.IsWolstenholmePrime p ↔ p > 7 ∧ Nat.Prime p ∧ ↑p ∣ (bernoulli' (p - 3)).num","subjects":["11"],"theorem":"WolstenholmePrime.wolstenholme_bernoulli"},{"answerKinds":[],"category":"textbook","docstring":"Another equivalent definition is that a prime $p > 7$ is a Wolstenholme prime\nif it $p^3$ divides the numerator of the harmonic number $H_{p-1}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WolstenholmePrime","statement":"∀ (p : ℕ), WolstenholmePrime.IsWolstenholmePrime p ↔ p > 7 ∧ Nat.Prime p ∧ ↑(p ^ 3) ∣ (harmonic (p - 1)).num","subjects":["11"],"theorem":"WolstenholmePrime.wolstenholme_harmonic"},{"answerKinds":[],"category":"research open","docstring":"**The Komlós conjecture**\n\nThere exists a universal constant $K > 0$ such that for all $n, m \\in \\mathbb{N}$ and\nall vectors $v\\_1, \\dots, v\\_n \\in \\mathbb{R}^m$ with $\\|v\\_i\\|\\_2 \\le 1$ (encoded here as\n$\\sum\\_j v\\_{ij}^2 \\le 1$), there exist signs $\\varepsilon\\_i \\in \\{-1, +1\\}$ such that\n$\\left\\|\\sum\\_i \\varepsilon\\_i v\\_i\\right\\|\\_\\infty \\le K$, i.e.\n$\\left|\\sum\\_i \\varepsilon\\_i v\\_{ij}\\right| \\le K$ for every coordinate $j$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.KomlosConjecture","statement":"∃ K,\n  0 < K ∧\n    ∀ (n m : ℕ) (v : Fin n → Fin m → ℝ),\n      (∀ (i : Fin n), ∑ j, v i j ^ 2 ≤ 1) →\n        ∃ ε, (∀ (i : Fin n), ε i = 1 ∨ ε i = -1) ∧ ∀ (j : Fin m), |∑ i, ε i * v i j| ≤ K","subjects":["5"],"theorem":"KomlosConjecture.komlos_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"**Banaszczyk's theorem**\n\nThere exists a constant $C > 0$ such that for all $n, m \\in \\mathbb{N}$ and all vectors\n$v\\_1, \\dots, v\\_n \\in \\mathbb{R}^m$ with $\\|v\\_i\\|\\_2 \\le 1$, there exist signs\n$\\varepsilon\\_i \\in \\{-1, +1\\}$ such that\n$\\left\\|\\sum\\_i \\varepsilon\\_i v\\_i\\right\\|\\_\\infty \\le C \\sqrt{\\log(n + 2)}$.\nThis is the best known bound towards the Komlós conjecture. (The shift $n + 2$ inside\nthe logarithm is a harmless normalization keeping it positive for $n \\in \\{0, 1\\}$.)\n\n[W. Banaszczyk, *Balancing vectors and Gaussian measures of n-dimensional convex bodies*,\nRandom Structures & Algorithms **12** (1998), 351–360.]\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.KomlosConjecture","statement":"∃ C,\n  0 < C ∧\n    ∀ (n m : ℕ) (v : Fin n → Fin m → ℝ),\n      (∀ (i : Fin n), ∑ j, v i j ^ 2 ≤ 1) →\n        ∃ ε, (∀ (i : Fin n), ε i = 1 ∨ ε i = -1) ∧ ∀ (j : Fin m), |∑ i, ε i * v i j| ≤ C * √(Real.log (↑n + 2))","subjects":["5"],"theorem":"KomlosConjecture.komlos_conjecture.variants.banaszczyk"},{"answerKinds":[],"category":"test","docstring":"Sanity check: with no vectors at all ($n = 0$), the empty signed sum is $0$ in every\ncoordinate, so any constant bound holds.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.KomlosConjecture","statement":"∀ (m : ℕ) (v : Fin 0 → Fin m → ℝ), ∃ ε, (∀ (i : Fin 0), ε i = 1 ∨ ε i = -1) ∧ ∀ (j : Fin m), |∑ i, ε i * v i j| ≤ 1","subjects":["5"],"theorem":"KomlosConjecture.komlos_conjecture.variants.zero_vectors"},{"answerKinds":[],"category":"research open","docstring":"**The infinitude of cousin primes**\nThere are infinitely many primes $p$ such that $p + 4$ is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Dickson","statement":"Infinite ↑{p | Prime p ∧ Prime (p + 4)}","subjects":["11"],"theorem":"Dickson.infinite_cousin_primes"},{"answerKinds":[],"category":"research open","docstring":"**The infinitude of Sophie Germain primes**\nThere are infinitely many primes $p$ such that $2p + 1$ is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Dickson","statement":"Infinite ↑{p | Prime p ∧ Prime (2 * p + 1)}","subjects":["11"],"theorem":"Dickson.infinite_safe_primes"},{"answerKinds":[],"category":"research open","docstring":"**Dickson's conjecture**\nIf a finite set of linear integer forms $f_i(n) = a_i n+b_i$ satisfies Schinzel condition,\nthere exist infinitely many natural numbers $m$ such that $f_i(m)$ are primes for all $i$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Dickson","statement":"∀ (fs : Finset (Polynomial ℤ)),\n  (∀ f ∈ fs, f.degree = 1 ∧ BunyakovskyCondition f) →\n    SchinzelCondition fs → Infinite ↑{n | ∀ f ∈ fs, Nat.Prime (Polynomial.eval (↑n) f).natAbs}","subjects":["11"],"theorem":"Dickson.dickson_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**The generalized twin-prime conjecture**\nFor any positive integer $k$ there are infinitely many primes $p$ such that $p + 2k$ is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Dickson","statement":"∀ (k : ℕ), 0 < k → Infinite ↑{p | Nat.Prime p ∧ Nat.Prime (p + 2 * k)}","subjects":["11"],"theorem":"Dickson.generalized_twin_primes"},{"answerKinds":[],"category":"research open","docstring":"**The infinitude of sexy primes**\nThere are infinitely many primes $p$ such that $p + 6$ is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Dickson","statement":"Infinite ↑{p | Prime p ∧ Prime (p + 6)}","subjects":["11"],"theorem":"Dickson.infinite_sexy_primes"},{"answerKinds":[],"category":"test","docstring":"$n = 0 ↔ φ(n) = 0$ ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.CarmichaelTotient","statement":"¬CarmichaelTotient.CarmichaelTotientFor 0","subjects":["11"],"theorem":"CarmichaelTotient.carchimichealTotientFor_zero"},{"answerKinds":[],"category":"research open","docstring":"*Carmichael's totient function conjecture*: For every positive natural number $n$,\nthere exists a natural number $m$ with $m ≠ n$, such that $φ(n) = φ(m)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CarmichaelTotient","statement":"∀ ⦃n : ℕ⦄, 0 < n → CarmichaelTotient.CarmichaelTotientFor n","subjects":["11"],"theorem":"CarmichaelTotient.charmichaelTotient"},{"answerKinds":[],"category":"research solved","docstring":"In Theorem 6 in [F1998], Kevin Ford proves that the smallest counterexample to\nCarmichael's totient function conjecture must be $≥ 10 ^ (10 ^ 10)$ ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CarmichaelTotient","statement":"∀ {n : ℕ}, 0 < n → n < 10 ^ 10 ^ 10 → CarmichaelTotient.CarmichaelTotientFor n","subjects":["11"],"theorem":"CarmichaelTotient.carchimaelTotient_bound"},{"answerKinds":[],"category":"textbook","docstring":"For every odd number $n$, $φ(2n) = φ(n)$ ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.CarmichaelTotient","statement":"∀ {n : ℕ}, Odd n → CarmichaelTotient.CarmichaelTotientFor n","subjects":["11"],"theorem":"CarmichaelTotient.carmichealTotientFor_odd"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M = DedekindNumber.M'","subjects":["5","6"],"theorem":"DedekindNumber.M_eq_M'"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M' 1 = 3","subjects":["6"],"theorem":"DedekindNumber.M'_one"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"∀ {n : ℕ} (f : (Fin n → Bool) → Bool), Monotone f → DedekindNumber.IsSperner (DedekindNumber.toSperner f)","subjects":["6"],"theorem":"DedekindNumber.toSperner_isSperner"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.kisielewiczFormula 0 = 2","subjects":["5","6"],"theorem":"DedekindNumber.kisielewiczFormula_zero"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M 3 = 20","subjects":["5"],"theorem":"DedekindNumber.M_three"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"∀ {n : ℕ} (v : Fin n → Bool), DedekindNumber.χ (DedekindNumber.supp v) = v","subjects":["5"],"theorem":"DedekindNumber.χ_supp"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.kisielewiczFormula 1 = 3","subjects":["5","6"],"theorem":"DedekindNumber.kisielewiczFormula_one"},{"answerKinds":[],"category":"research solved","docstring":"Kisielewicz (1988) proved the following arithmetic formula for the Dedekind numbers:\n$$\n  M(n) = \\sum_{k=0}^{2^{2^n}}\\prod_{j = 1}^{2 ^ n - 1}\\prod_{i = 0}^{j - 1} \\left(\n    1 - b_i^kb_j^k \\prod_{m = 0}^{\\log_2 i} (1 - b_m^i + b_m^ib_m^j)\\right),\n$$\nwhere $b_i^k$ is the $i$-th bit of $k$. However, this formula is not computationally\nefficient for large $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M = DedekindNumber.kisielewiczFormula","subjects":["5","6"],"theorem":"DedekindNumber.M_eq_kisielewiczFormula"},{"answerKinds":[],"category":"textbook","docstring":"Every true set of a monotone Boolean function contains a minimal true set. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"∀ {n : ℕ} {f : (Fin n → Bool) → Bool},\n  Monotone f →\n    ∀ {s : Finset (Fin n)},\n      f (DedekindNumber.χ s) = true →\n        ∃ t ⊆ s, f (DedekindNumber.χ t) = true ∧ ∀ u ⊆ t, f (DedekindNumber.χ u) = true → t ⊆ u","subjects":["5","6"],"theorem":"DedekindNumber.exists_minimal_true_subset"},{"answerKinds":[],"category":"textbook","docstring":"Converting a Sperner family to a monotone function and back yields the same family. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"∀ {n : ℕ} (A : Finset (Finset (Fin n))),\n  DedekindNumber.IsSperner A → DedekindNumber.toSperner (DedekindNumber.fromSperner A) = A","subjects":["5","6"],"theorem":"DedekindNumber.toSperner_fromSperner"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M' 2 = 6","subjects":["6"],"theorem":"DedekindNumber.M'_two"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M' 3 = 20","subjects":["6"],"theorem":"DedekindNumber.M'_three"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"∀ {n : ℕ} (s : Finset (Fin n)), DedekindNumber.supp (DedekindNumber.χ s) = s","subjects":["6"],"theorem":"DedekindNumber.supp_χ"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M 1 = 3","subjects":["5"],"theorem":"DedekindNumber.M_one"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M 2 = 6","subjects":["5"],"theorem":"DedekindNumber.M_two"},{"answerKinds":[],"category":"textbook","docstring":"Converting a monotone function to a Sperner family and back yields the same function. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"∀ {n : ℕ} (f : (Fin n → Bool) → Bool), Monotone f → DedekindNumber.fromSperner (DedekindNumber.toSperner f) = f","subjects":["5","6"],"theorem":"DedekindNumber.fromSperner_toSperner"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"∀ {n : ℕ} (s t : Finset (Fin n)), DedekindNumber.χ s ≤ DedekindNumber.χ t ↔ s ⊆ t","subjects":["6"],"theorem":"DedekindNumber.χ_le_iff"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"∀ {n : ℕ} (v : Fin n → Bool) (i : Fin n), i ∈ DedekindNumber.supp v ↔ v i = true","subjects":["6"],"theorem":"DedekindNumber.mem_supp_iff"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.kisielewiczFormula 2 = 6","subjects":["5","6"],"theorem":"DedekindNumber.kisielewiczFormula_two"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"No closed-form expression that allows efficient computation of Dedekind numbers is\ncurrently known.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M = sorry","subjects":["5","6"],"theorem":"DedekindNumber.M_eq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.kisielewiczFormula 3 = 20","subjects":["5","6"],"theorem":"DedekindNumber.kisielewiczFormula_three"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"∀ {n : ℕ} (A : Finset (Finset (Fin n))), DedekindNumber.IsSperner A → Monotone (DedekindNumber.fromSperner A)","subjects":["6"],"theorem":"DedekindNumber.fromSperner_monotone"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"In particular, the Dedekind number for `n = 10` is currently unknown.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M 10 = sorry","subjects":["5","6"],"theorem":"DedekindNumber.Dedekind_10"},{"answerKinds":[],"category":"test","docstring":"Values for small n ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M 0 = 2","subjects":["5"],"theorem":"DedekindNumber.M_zero"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DedekindNumber","statement":"DedekindNumber.M' 0 = 2","subjects":["6"],"theorem":"DedekindNumber.M'_zero"},{"answerKinds":[],"category":"research open","docstring":"The Elliott–Halberstam conjecture: for every $\\theta < 1$ and $A > 0$ there exists a\nconstant $C > 0$ such that\n$$\\sum_{1 \\le q \\le x^{\\theta}} E(x; q) \\le \\frac{C x}{\\log^A x}$$\nfor all $x > 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ElliottHalberstamConjecture","statement":"∀ θ < 1,\n  ∀ (A : ℝ),\n    0 < A →\n      ∃ C > 0,\n        ∀ (x : ℕ), 2 < x → ∑ q ∈ Finset.Icc 1 ⌊↑x ^ θ⌋₊, ElliottHalberstamConjecture.E x q ≤ C * ↑x / Real.log ↑x ^ A","subjects":["11"],"theorem":"ElliottHalberstamConjecture.elliott_halberstam"},{"answerKinds":[],"category":"research solved","docstring":"The Elliott–Halberstam conjecture fails at the endpoint $\\theta = 1$, as shown by\nFriedlander and Granville [FG89].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ElliottHalberstamConjecture","statement":"¬∀ (A : ℝ),\n    0 < A →\n      ∃ C > 0, ∀ (x : ℕ), 2 < x → ∑ q ∈ Finset.Icc 1 x, ElliottHalberstamConjecture.E x q ≤ C * ↑x / Real.log ↑x ^ A","subjects":["11"],"theorem":"ElliottHalberstamConjecture.elliott_halberstam.variants.friedlander_granville"},{"answerKinds":[],"category":"research solved","docstring":"The Bombieri–Vinogradov theorem: the Elliott–Halberstam conjecture holds for every\n$\\theta < 1/2$. This result may be regarded as an averaged form of the generalized\nRiemann hypothesis.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ElliottHalberstamConjecture","statement":"∀ θ < 1 / 2,\n  ∀ (A : ℝ),\n    0 < A →\n      ∃ C > 0,\n        ∀ (x : ℕ), 2 < x → ∑ q ∈ Finset.Icc 1 ⌊↑x ^ θ⌋₊, ElliottHalberstamConjecture.E x q ≤ C * ↑x / Real.log ↑x ^ A","subjects":["11"],"theorem":"ElliottHalberstamConjecture.elliott_halberstam.variants.bombieri_vinogradov"},{"answerKinds":[],"category":"research open","docstring":"**Inscribed square problem**\nDoes every Jordan curve admit an inscribed square?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InscribedSquare","statement":"True ↔\n  ∀ (γ : Circle → EuclideanSpace ℝ (Fin 2)),\n    Topology.IsEmbedding γ → ∃ t₁ t₂ t₃ t₄, InscribedSquare.IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) 1","subjects":["51"],"theorem":"InscribedSquare.inscribed_square_problem"},{"answerKinds":[],"category":"research solved","docstring":"It is known that every *smooth* Jordan curve admits inscribed rectangles of all aspect ratios.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InscribedSquare","statement":"∀ (γ : Circle → EuclideanSpace ℝ (Fin 2)),\n  Topology.IsEmbedding γ →\n    ContMDiff (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 1))) (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 2)))\n        (↑⊤) γ →\n      ∀ r > 0, ∃ t₁ t₂ t₃ t₄, InscribedSquare.IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) r","subjects":["51"],"theorem":"InscribedSquare.exists_inscribed_rectangle_of_smooth"},{"answerKinds":[],"category":"research solved","docstring":"It is known that every Jordan curve admits at least one inscribed rectangle.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InscribedSquare","statement":"∀ (γ : Circle → EuclideanSpace ℝ (Fin 2)),\n  Topology.IsEmbedding γ → ∃ t₁ t₂ t₃ t₄ r, InscribedSquare.IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) r","subjects":["51"],"theorem":"InscribedSquare.exists_inscribed_rectangle"},{"answerKinds":[],"category":"research solved","docstring":"It is also known that every $C^2$ Jordan curve admits an inscribed square.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InscribedSquare","statement":"∀ (γ : Circle → EuclideanSpace ℝ (Fin 2)),\n  Topology.IsEmbedding γ →\n    ContMDiff (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 1))) (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 2))) 2\n        γ →\n      ∃ t₁ t₂ t₃ t₄, InscribedSquare.IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) 1","subjects":["51"],"theorem":"InscribedSquare.exists_inscribed_square_of_C2"},{"answerKinds":[],"category":"research open","docstring":"**Inscribed rectangle problem**\nDoes every Jordan curve admit inscribed rectangles of any given aspect ratio?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.InscribedSquare","statement":"True ↔\n  ∀ (γ : Circle → EuclideanSpace ℝ (Fin 2)),\n    Topology.IsEmbedding γ → ∀ r > 0, ∃ t₁ t₂ t₃ t₄, InscribedSquare.IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) r","subjects":["51"],"theorem":"InscribedSquare.inscribed_rectangle_problem"},{"answerKinds":[],"category":"research solved","docstring":"**Six standard deviations suffice** (Spencer, 1985)\n\nFor every $n$ and every family of $n$ subsets $S_1, \\dots, S_n$ of $\\{1, \\dots, n\\}$,\nthere is a colouring $\\chi : \\{1, \\dots, n\\} \\to \\{-1, +1\\}$ such that\n$\\left|\\sum_{j \\in S_i} \\chi(j)\\right| \\le 6\\sqrt{n}$ for every $i$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SixStandardDeviations","statement":"∀ (n : ℕ) (S : Fin n → Finset (Fin n)),\n  ∃ χ, (∀ (j : Fin n), χ j = 1 ∨ χ j = -1) ∧ ∀ (i : Fin n), |∑ j ∈ S i, χ j| ≤ 6 * √↑n","subjects":["5"],"theorem":"SixStandardDeviations.six_standard_deviations"},{"answerKinds":[],"category":"test","docstring":"Sanity check: with no points and no sets ($n = 0$), the empty colouring works and\nthe bound $6\\sqrt{0} = 0$ holds vacuously.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SixStandardDeviations","statement":"∀ (S : Fin 0 → Finset (Fin 0)), ∃ χ, (∀ (j : Fin 0), χ j = 1 ∨ χ j = -1) ∧ ∀ (i : Fin 0), |∑ j ∈ S i, χ j| ≤ 6 * √0","subjects":["5"],"theorem":"SixStandardDeviations.six_standard_deviations.variants.zero_sets"},{"answerKinds":[],"category":"research solved","docstring":"**The random colouring bound**\n\nThere is a constant $K > 0$ such that for every $n$ and every family of $n$ subsets\n$S_1, \\dots, S_n$ of $\\{1, \\dots, n\\}$, there is a colouring\n$\\chi : \\{1, \\dots, n\\} \\to \\{-1, +1\\}$ with\n$\\left|\\sum_{j \\in S_i} \\chi(j)\\right| \\le K \\sqrt{n \\log(n + 2)}$ for every $i$.\nThis follows from a Chernoff bound applied to a uniformly random colouring, and is\nthe benchmark that Spencer's theorem improves upon by removing the logarithmic\nfactor. (The shift $n + 2$ inside the logarithm is a harmless normalization keeping\nit positive for $n \\in \\{0, 1\\}$.)\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Lemmy00/spencer-random-coloring-lean/blob/a691214a7263c1010fdd10560091e9f8cfce903a/SpencerRandomColoring/RandomBound.lean#L37"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SixStandardDeviations","statement":"∃ K,\n  0 < K ∧\n    ∀ (n : ℕ) (S : Fin n → Finset (Fin n)),\n      ∃ χ, (∀ (j : Fin n), χ j = 1 ∨ χ j = -1) ∧ ∀ (i : Fin n), |∑ j ∈ S i, χ j| ≤ K * √(↑n * Real.log (↑n + 2))","subjects":["5"],"theorem":"SixStandardDeviations.six_standard_deviations.variants.random_bound"},{"answerKinds":[],"category":"research open","docstring":"For any given integer $a > 0$, are there infinitely many primes $p$ such that\n$a^{p-1} \\equiv 1 \\pmod{p^2}$? The case $a = 1$ is trivial, since every prime qualifies\n(`isWieferichPrimeBase_one_iff`). So is the case $a = 0$ under our definition\n(`isWieferichPrimeBase_zero_iff`), which is why the source's restriction to $a > 0$ is dropped.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WieferichPrime","statement":"True ↔ ∀ (a : ℕ), {p | IsWieferichPrimeBase a p}.Infinite","subjects":["11"],"theorem":"WieferichPrime.infinite_isWieferichPrimeBase"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many Wieferich primes. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WieferichPrime","statement":"{p | IsWieferichPrime p}.Infinite","subjects":["11"],"theorem":"WieferichPrime.infinite_isWieferichPrime"},{"answerKinds":[],"category":"research open","docstring":"Are there any Wieferich primes to base $47$? None is currently known. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WieferichPrime","statement":"True ↔ ∃ p, IsWieferichPrimeBase 47 p","subjects":["11"],"theorem":"WieferichPrime.exists_isWieferichPrimeBase_47"},{"answerKinds":[],"category":"research open","docstring":"Are $1093$ and $3511$ the only Wieferich primes? They are the only known ones: PrimeGrid's search,\ncompleted in 2022, shows that any other Wieferich prime exceeds $2^{64}$. On the other hand, the\nheuristic count of $\\log \\log x$ Wieferich primes up to $x$ predicts that there are infinitely\nmany, see `infinite_isWieferichPrime`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WieferichPrime","statement":"True ↔ ∀ (p : ℕ), IsWieferichPrime p ↔ p = 1093 ∨ p = 3511","subjects":["11"],"theorem":"WieferichPrime.isWieferichPrime_iff"},{"answerKinds":[],"category":"test","docstring":"The prime $1093$ is a Wieferich prime: $2^{1092} \\equiv 1 \\pmod{1093^2}$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WieferichPrime","statement":"IsWieferichPrime 1093","subjects":["11"],"theorem":"WieferichPrime.isWieferichPrime_1093"},{"answerKinds":[],"category":"test","docstring":"The prime $3511$ is a Wieferich prime: $2^{3510} \\equiv 1 \\pmod{3511^2}$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WieferichPrime","statement":"IsWieferichPrime 3511","subjects":["11"],"theorem":"WieferichPrime.isWieferichPrime_3511"},{"answerKinds":[],"category":"test","docstring":"The prime $2$ is not a Wieferich prime, so no hypothesis excluding it is needed. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WieferichPrime","statement":"¬IsWieferichPrime 2","subjects":["11"],"theorem":"WieferichPrime.not_isWieferichPrime_two"},{"answerKinds":[],"category":"research open","docstring":"The **Agoh-Giuga Conjecture**, Agoh's formulation ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"AgohGiuga.AgohGiugaCongr","subjects":["11"],"theorem":"AgohGiuga.agoh_giuga"},{"answerKinds":[],"category":"research solved","docstring":"A composite number $n$ is weak Giuga if and only if\n$$\n\\sum_{p\\mid n} \\frac{1}{p} - \\frac{1}{n} \\in\\mathbb{N}.\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"∀ {n : ℕ}, n.Composite → (AgohGiuga.IsWeakGiuga n ↔ ∃ m, ∑ p ∈ n.primeFactors, 1 / ↑p - 1 / ↑n = ↑m)","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"AgohGiuga.isWeakGiuga_iff_sum_primeFactors"},{"answerKinds":[],"category":"research solved","docstring":"A composite number $n$ is weak Giuga if and only if $p \\mid (\\frac{n}{p} - 1)$ for all\nprime divisors $p$ of $n$.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/2663234a28260853790aa5752d8d4550ff0ab1ca/FormalConjectures/Wikipedia/AgohGiuga.lean#L97"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"∀ {n : ℕ}, n.Composite → (AgohGiuga.IsWeakGiuga n ↔ ∀ p ∈ n.primeFactors, p ∣ n / p - 1)","subjects":["11"],"theorem":"AgohGiuga.isWeakGiuga_iff_prime_dvd"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"IsCarmichael 561","subjects":["11"],"theorem":"AgohGiuga.isCarmichael_561"},{"answerKinds":[],"category":"research solved","docstring":"Giuga showed that a Giuga number has at least 9 prime factors.\nRef: G. Giuga, _Su una presumibile proprieta caratteristica dei numeri primi_\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"∀ (a : ℕ), AgohGiuga.IsStrongGiuga a → 9 ≤ a.primeFactors.card","subjects":["11"],"theorem":"AgohGiuga.agoh_giuga.variants.le_primeFactors_card_of_isStrongGiuga"},{"answerKinds":[],"category":"research solved","docstring":"The two statements of the conjecture are equivalent. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"AgohGiuga.AgohGiugaCongr ↔ AgohGiuga.AgohGiugaSum","subjects":["11"],"theorem":"AgohGiuga.agoh_giuga.variants.equivalence"},{"answerKinds":[],"category":"research open","docstring":"The **Agoh-Giuga Conjecture**, Giuga's formulation ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"AgohGiuga.AgohGiugaSum","subjects":["11"],"theorem":"AgohGiuga.agoh_giuga.variants.giuga"},{"answerKinds":[],"category":"textbook","docstring":"A composite number `a` is Carmichael if and only if it is squarefree\nand, for all prime `p` dividing `a`, we have `p - 1 ∣ a - 1`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"∀ (a : ℕ), a.Composite → (IsCarmichael a ↔ Squarefree a ∧ ∀ (p : ℕ), Nat.Prime p → p ∣ a → p - 1 ∣ a - 1)","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"AgohGiuga.korselts_criterion"},{"answerKinds":[],"category":"research solved","docstring":"Giuga showed that a number `n` is strong Giuga if and only if it is\nCarmichael and `∑_{p|n} 1/p - 1/n ∈ ℕ` (i.e., if and only if it is Carmichael\nand weak Giuga).\nRef: G. Giuga, _Su una presumibile proprieta caratteristica dei numeri primi_\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"∀ {a : ℕ}, a.Composite → (AgohGiuga.IsStrongGiuga a ↔ IsCarmichael a ∧ ∃ n, ∑ p ∈ a.primeFactors, 1 / ↑p - 1 / ↑a = ↑n)","subjects":["11"],"theorem":"AgohGiuga.isStrongGiuga_iff"},{"answerKinds":[],"category":"research solved","docstring":"Let `G(X)` denote the number of exceptions `n ≤ X` to Giuga’s conjecture.\nThen for `X` larger than an absolute constant which can be made\nexplicit, `G(X) ≪ X^{1/2} log X`.\nRef: Vicentiu Tipu, _A Note on Giuga’s Conjecture_\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"∀ (G : ℕ → ℕ),\n  (G = fun x => (Finset.filter AgohGiuga.IsStrongGiuga (Finset.Icc 1 x)).card) →\n    ∃ N O, ∀ n ≥ N, ↑(G n) ≤ O * √↑n * Real.log ↑n","subjects":["11"],"theorem":"AgohGiuga.agoh_giuga.variants.isStrongGiuga_growth"},{"answerKinds":[],"category":"textbook","docstring":"A composite Carmichael number is squarefree. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"∀ {a : ℕ}, a.Composite → IsCarmichael a → Squarefree a","subjects":["11"],"theorem":"AgohGiuga.squarefree_of_isCarmichael"},{"answerKinds":[],"category":"research solved","docstring":"Every strong Giuga number is a Carmichael number.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AgohGiuga","statement":"∀ (a : ℕ), AgohGiuga.IsStrongGiuga a → IsCarmichael a","subjects":["11"],"theorem":"AgohGiuga.agoh_giuga.variants.isStrongGiuga_implies_isCarmichael"},{"answerKinds":[],"category":"research solved","docstring":"**Mills' constant lower bound** (Caldwell–Cheng, 2005): assuming the Riemann hypothesis,\nMills' constant begins at $1.3063778838\\ldots$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mills","statement":"RiemannHypothesis → ∀ {A : ℝ}, Mills.IsMinMills A → A ∈ Set.Ioo 1.3063778838 1.3063778839","subjects":["11"],"theorem":"Mills.lower_bound_of_RH"},{"answerKinds":[],"category":"research solved","docstring":"**Mills' theorem** (Mills, 1947).\nThere is a real number $A > 1$ such that $\\lfloor A^{3^n}\\rfloor$ is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mills","statement":"∃ A > 1, Mills.IsMills A","subjects":["11"],"theorem":"Mills.exists'"},{"answerKinds":[],"category":"research solved","docstring":"**Mills' constant.**\nThere is a *least* Mills number.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mills","statement":"∃ A, Mills.IsMinMills A","subjects":["11"],"theorem":"Mills.exists_least"},{"answerKinds":[],"category":"research solved","docstring":"**Mills' constant is irrational** (Saito, 2024).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mills","statement":"∀ {A : ℝ}, Mills.IsMinMills A → Irrational A","subjects":["11"],"theorem":"Mills.irrational"},{"answerKinds":[],"category":"research open","docstring":"Kummer–Vandiver conjecture states that for every prime $p$, the class number of the maximal\nreal subfield of $\\mathbb{Q}(\\zeta_p)$ is not divisible by $p$.\n-","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.KummerVandiver","statement":"∀ (p : ℕ+), p.Prime → ¬↑p ∣ NumberField.classNumber ↥(NumberField.maximalRealSubfield (CyclotomicField ↑p ℚ))","subjects":["11"],"theorem":"KummerVandiver.kummer_vandiver"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many real quadratic fields `ℚ(√d)` with class number one,\nwhere `d > 1` is a squarefree integer.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ClassNumberProblem","statement":"{d | Squarefree d ∧ d > 1 ∧ ClassNumberProblem.IsClassNumberOne d}.Infinite","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"ClassNumberProblem.class_number_problem"},{"answerKinds":[],"category":"research solved","docstring":"**Stark–Heegner theorem** : For any squarefree integer `d < 0`, the class number of the imaginary\nquadratic field Q(√d) is one if and only if `d ∈ {-1, -2, -3, -7, -11, -19, -43, -67, -163}`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ClassNumberProblem","statement":"{d | Squarefree d ∧ d < 0 ∧ ClassNumberProblem.IsClassNumberOne d} = {-1, -2, -3, -7, -11, -19, -43, -67, -163}","subjects":["11"],"theorem":"ClassNumberProblem.class_number_problem.variants.imaginary"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Determine the value of the Busy Beaver function at n = 6.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BusyBeaver","statement":"BusyBeaver.BB 6 = sorry","subjects":["3"],"theorem":"BusyBeaver.BB_6"},{"answerKinds":[],"category":"textbook","docstring":"The value of the Busy Beaver function for 2 states is 6. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BusyBeaver","statement":"BusyBeaver.BB 2 = 6","subjects":["3"],"theorem":"BusyBeaver.BB_2"},{"answerKinds":[],"category":"API","docstring":"To compute `BB n`, we need only consider machines with states and symbols indexed in `Fin`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BusyBeaver","statement":"∀ (n : ℕ) [inst : NeZero n], ↑(BusyBeaver.BB n) = sSup {N | ∃ M, ∃ (_ : M.IsHalting), M.haltingNumber = N}","subjects":["3"],"subsets":["FC100SolvedSet1"],"theorem":"BusyBeaver.sanity_check"},{"answerKinds":[],"category":"textbook","docstring":"The value of the Busy Beaver function for 4 states is 107. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BusyBeaver","statement":"BusyBeaver.BB 4 = 107","subjects":["3"],"theorem":"BusyBeaver.BB_4"},{"answerKinds":[],"category":"test","docstring":"The value of the Busy Beaver function for 1 state is 1. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BusyBeaver","statement":"BusyBeaver.BB 1 = 1","subjects":["3"],"theorem":"BusyBeaver.BB_1"},{"answerKinds":[],"category":"research solved","docstring":"The value of the Busy Beaver function for 5 states is 47176870. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BusyBeaver","statement":"BusyBeaver.BB 5 = 47176870","subjects":["3"],"theorem":"BusyBeaver.BB_5"},{"answerKinds":[],"category":"textbook","docstring":"The value of the Busy Beaver function for 3 states is 21. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BusyBeaver","statement":"BusyBeaver.BB 3 = 21","subjects":["3"],"theorem":"BusyBeaver.BB_3"},{"answerKinds":[],"category":"textbook","docstring":"The primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31 are regular. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RegularPrimes","statement":"{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31} ⊆ RegularPrimes.regularPrimes","subjects":["11"],"theorem":"RegularPrimes.small_regular_primes"},{"answerKinds":[],"category":"textbook","docstring":"The prime 37 is not a regular prime. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RegularPrimes","statement":"¬RegularPrimes.IsRegularPrime 37","subjects":["11"],"theorem":"RegularPrimes.not_isRegularPrime_37_first"},{"answerKinds":[],"category":"research solved","docstring":"The set of irregular primes is infinite. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RegularPrimes","statement":"RegularPrimes.irregularPrimes.Infinite","subjects":["11"],"theorem":"RegularPrimes.infinitude_of_irregularprimes"},{"answerKinds":[],"category":"textbook","docstring":"An equivalent definition of a regular prime `p` is that it does not divide the numerator of the\nfirst `p-3` Bernoulli numbers. Not in Mathlib. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RegularPrimes","statement":"∀ (p : ℕ) [inst : Fact (Nat.Prime p)],\n  RegularPrimes.IsRegularPrime p ↔ ∀ k ∈ Finset.Icc 2 (p - 3), ¬↑p ∣ (bernoulli' k).num","subjects":["11"],"theorem":"RegularPrimes.isRegularPrime_iff_Bernoulli"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: The set of regular primes is infinite. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RegularPrimes","statement":"RegularPrimes.RegularPrimeConjecture","subjects":["11"],"theorem":"RegularPrimes.regularprime_conjecture"},{"answerKinds":[],"category":"textbook","docstring":"The prime 37 is not a regular prime. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RegularPrimes","statement":"¬RegularPrimes.IsRegularPrime 37","subjects":["11"],"theorem":"RegularPrimes.not_isRegularPrime_37_second"},{"answerKinds":[],"category":"research solved","docstring":"The first Cuboid conjecture\n\nThe DeepMind prover agent has found a formal proof of this statement.\n\nAn (independent) informal solution can be found here:\n*Reference:* [arxiv/2510.11768](https://arxiv.org/abs/2510.11768) **Irreducibility of the Cuboid Polynomial P_{a,u}(t) via a Rank-Zero Elliptic Curve** by *Valery Asiryan*\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/34c93bbad127a9a5354b9d53478d338eb65edb88/FormalConjectures/Wikipedia/EulerBrick.lean#L1804"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EulerBrick","statement":"EulerBrick.CuboidOne","subjects":["12"],"theorem":"EulerBrick.cuboidOne"},{"answerKinds":[],"category":"research open","docstring":"The third Cuboid conjecture ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EulerBrick","statement":"EulerBrick.CuboidThree","subjects":["12"],"subsets":["FC100OpenSet1"],"theorem":"EulerBrick.cuboidThree"},{"answerKinds":[],"category":"research open","docstring":"Is there an Euler brick in $4$-dimensional space?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EulerBrick","statement":"True ↔ ∃ sides, EulerBrick.IsEulerHyperBrick 4 sides","subjects":["11"],"theorem":"EulerBrick.four_dim_euler_brick_existence"},{"answerKinds":[],"category":"research open","docstring":"Is there an Euler brick in $n$-dimensional space for any $n > 3$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EulerBrick","statement":"True ↔ ∀ n > 3, ∃ sides, EulerBrick.IsEulerHyperBrick n sides","subjects":["11"],"theorem":"EulerBrick.n_dim_euler_brick_existence"},{"answerKinds":[],"category":"research open","docstring":"The second Cuboid conjecture ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EulerBrick","statement":"EulerBrick.CuboidTwo","subjects":["12"],"theorem":"EulerBrick.cuboidTwo"},{"answerKinds":[],"category":"research open","docstring":"Is there a perfect Euler brick?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EulerBrick","statement":"True ↔ ∃ a b c, EulerBrick.IsPerfectCuboid a b c","subjects":["11"],"theorem":"EulerBrick.perfect_euler_brick_existence"},{"answerKinds":[],"category":"research solved","docstring":"In [Sh12], Ruslan notes that a perfect Euler brick does not exist\nif all three Cuboid conjectures hold. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.EulerBrick","statement":"EulerBrick.CuboidOne → EulerBrick.CuboidTwo → EulerBrick.CuboidThree → ¬∃ a b c, EulerBrick.IsPerfectCuboid a b c","subjects":["12"],"theorem":"EulerBrick.cuboid_perfect_euler_brick"},{"answerKinds":[],"category":"research solved","docstring":"The two-dimensional case, proved by Davies [Da71].\n\n[Da71] Davies, R. O., _Some remarks on the Kakeya problem_. Math. Proc. Cambridge Philos. Soc. 69 (1971), no. 3, 417–421.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Kakeya","statement":"Kakeya.KakeyaSetConjectureDim 2","subjects":["42"],"theorem":"Kakeya.kakeya_2d"},{"answerKinds":[],"category":"research solved","docstring":"The finite field Kakeya conjecture asserts that any Kakeya set in `𝔽_qⁿ` has size at\nleast `c_n · qⁿ` for some constant `c_n` depending only on `n`.\nThis was first proved by Dvir [Dv08]. The best known bound to date, due to Bukh and Chao [BuCh21],\nestablishes that any Kakeya set in `𝔽_qⁿ` has size at least `qⁿ / (2 - 1/q)^(n - 1)`.\n\n[Dv08] Dvir, Z., _On the size of Kakeya sets in finite fields_. Journal of the American Mathematical Society 22 (2009), no. 4, 1093–1097.\n[BuCh21] Bukh, B. and Chao, T.-W., _Sharp density bounds on the finite field Kakeya problem_. Discrete Analysis 26 (2021), 9 pp.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Kakeya","statement":"∀ {F : Type u_1} [inst : Field F] [inst_1 : Fintype F] {n : ℕ} (K : Finset (Fin n → F)),\n  Kakeya.IsKakeyaFinite K → ↑(Fintype.card F) ^ n / (2 - 1 / ↑(Fintype.card F)) ^ (n - 1) ≤ ↑K.card","subjects":["52"],"theorem":"Kakeya.kakeya_finite"},{"answerKinds":[],"category":"research open","docstring":"The Kakeya set conjecture: Kakeya sets in $\\mathbb{R}^n$ have Hausdorff dimension $n$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Kakeya","statement":"∀ n > 0, Kakeya.KakeyaSetConjectureDim n","subjects":["42"],"theorem":"Kakeya.kakeya_set_conjecture"},{"answerKinds":[],"category":"test","docstring":"A trivial example: the closed ball of radius 1 in `ℝⁿ` is a Kakeya set.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Kakeya","statement":"∀ (n : ℕ), Kakeya.IsKakeya (Metric.closedBall 0 1)","subjects":["42"],"theorem":"Kakeya.isKakeya_closedBall"},{"answerKinds":[],"category":"research solved","docstring":"The three-dimensional case, proved by Wang, Zahl [WaZa25].\n\n[WaZa25] Wang, H. and Zahl, J., _Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions_. arXiv preprint, arXiv:2502.17655, 2025.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Kakeya","statement":"Kakeya.KakeyaSetConjectureDim 3","subjects":["42"],"theorem":"Kakeya.kakeya_3d"},{"answerKinds":[],"category":"research open","docstring":"**The Andrews-Curtis conjecture.**\n\nEvery normally generating `n`-tuple in the free group of rank `n` is\nAndrews-Curtis equivalent to the standard tuple of free generators.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AndrewsCurtis","statement":"∀ (n : ℕ) (r : AndrewsCurtis.RelatorTuple n),\n  AndrewsCurtis.NormallyGenerates r → AndrewsCurtis.Equivalent r (AndrewsCurtis.standardRelators n)","subjects":["20"],"theorem":"AndrewsCurtis.andrews_curtis_conjecture"},{"answerKinds":[],"category":"research open","docstring":"The **Köthe conjecture**: every left nil radical is contained in the Köthe radical. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Koethe","statement":"∀ {R : Type u_1} [inst : Ring R] {I : Ideal R}, Koethe.IsNil I → ↑I ⊆ ↑(Koethe.KotheRadical R)","subjects":["16"],"theorem":"Koethe.KotherConjecture.variants.le_KotherRadical"},{"answerKinds":[],"category":"research solved","docstring":"The **Amitsur Conjecture**: If `J` is a nil ideal in `R`, then `J[x]` is a nil ideal of the polynomial ring `R[x]`.\nThis is known to be false, see Agata Smoktunowicz, _Polynomial rings over nil rings need not be nil_.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Koethe","statement":"∀ {R : Type u_1} [inst : Ring R] (J : TwoSidedIdeal R), Koethe.IsNil J → Koethe.IsNil (TwoSidedIdeal.map Polynomial.C J)","subjects":["16"],"theorem":"Koethe.amitsur_conjecture"},{"answerKinds":[],"category":"research open","docstring":"The **Köthe conjecture**: for any positive integer `n`, the Köthe radical of `R` is the matrix ideal `M_2(Nil*(R))`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Koethe","statement":"∀ {R : Type u_1} [inst : Ring R] {I : TwoSidedIdeal R},\n  Koethe.IsNil I →\n    ∀ (n : Type u_2) [inst_1 : Fintype n],\n      TwoSidedIdeal.matrix n (Koethe.KotheRadical R) = Koethe.KotheRadical (Matrix n n R)","subjects":["16"],"subsets":["FC100OpenSet1"],"theorem":"Koethe.KotherConjecture.variants.matrixOver_KotherRadical"},{"answerKinds":[],"category":"research open","docstring":"The **Köthe conjecture**: In any ring, the sum of two nil left ideals is nil. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Koethe","statement":"∀ {R : Type u_1} [inst : Ring R] (I J : Ideal R), Koethe.IsNil I → Koethe.IsNil J → Koethe.IsNil (I + J)","subjects":["16"],"theorem":"Koethe.KotheConjecture"},{"answerKinds":[],"category":"research open","docstring":"The **Köthe conjecture**: for any nil ideal `I` of `R`, the matrix ideal `M_2(I)` is a nil ideal\nof the matrix ring `M_2(R)`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Koethe","statement":"∀ {R : Type u_1} [inst : Ring R] {I : TwoSidedIdeal R}, Koethe.IsNil I → Koethe.IsNil (TwoSidedIdeal.matrix (Fin 2) I)","subjects":["16"],"theorem":"Koethe.KotherConjecture.variants.two_by_two_matrix"},{"answerKinds":[],"category":"research open","docstring":"The **Köthe conjecture**: for any nil ideal `I` of `R`, the matrix ideal `M_n(I)` is a nil ideal\nof the matrix ring `M_n(R)`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Koethe","statement":"∀ {R : Type u_1} [inst : Ring R] {I : TwoSidedIdeal R},\n  Koethe.IsNil I → ∀ (n : Type u_2) [inst_1 : Fintype n], Koethe.IsNil (TwoSidedIdeal.matrix n I)","subjects":["16"],"theorem":"Koethe.KotherConjecture.variants.general_matrix"},{"answerKinds":[],"category":"research solved","docstring":"There is a convex set of area 0.270911861 that covers all worms.\n\n*Reference:*\nWang, Wei (2006), \"An improved upper bound for the worm problem\",\nActa Mathematica Sinica, 49 (4): 835–846, MR 2264090.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MoserWorm","statement":"∃ X,\n  MeasurableSet X ∧\n    Convex ℝ X ∧\n      MeasureTheory.volume X = 0.270911861 ∧\n        ∀ w ∈ MoserWorm.Worms, ∃ e v, LinearEquiv.det e.toLinearEquiv = 1 ∧ w ⊆ (fun x => e x + v) '' X","subjects":["52"],"theorem":"MoserWorm.convex_mosers_worm_problem_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"There is a set of area 0.260437 that covers all worms.\n\n*Reference:*\nNorwood, Rick; Poole, George (2003), \"An improved upper bound for Leo Moser's worm problem\",\nDiscrete and Computational Geometry, 29 (3): 409–417, doi:10.1007/s00454-002-0774-3, MR 1961007.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MoserWorm","statement":"∃ X ∈ MoserWorm.WormCovers, MeasureTheory.volume X = 0.260437","subjects":["52"],"theorem":"MoserWorm.mosers_worm_problem_upper_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"**Convex Moser's Worm Problem**\nWhat is the minimal area (or greatest lower bound on the area)\nof a *convex* shape that can cover every unit-length curve?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MoserWorm","statement":"IsGLB {v | ∃ X ∈ MoserWorm.WormCovers, Convex ℝ X ∧ MeasureTheory.volume X = v} sorry","subjects":["52"],"theorem":"MoserWorm.convex_mosers_worm_problem"},{"answerKinds":[],"category":"textbook","docstring":"A disc of radius 1 / 2 is a worm cover.\n\nThis follows by translating the center of the disc to the midpoint of the worm.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.MoserWorm","statement":"Metric.closedBall 0 0.5 ∈ MoserWorm.WormCovers","subjects":["52"],"theorem":"MoserWorm.disc_mem_worm_covers"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"**Moser's Worm Problem**\nWhat is the minimal area (or greatest lower bound on the area)\nof a shape that can cover every unit-length curve?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MoserWorm","statement":"IsGLB {v | ∃ X ∈ MoserWorm.WormCovers, MeasureTheory.volume X = v} sorry","subjects":["52"],"theorem":"MoserWorm.mosers_worm_problem"},{"answerKinds":[],"category":"research solved","docstring":"0.232239 is a lower bound on the area of a convex set that covers all worms.\n\n*Reference:*\nKhandhawit, Tirasan; Pagonakis, Dimitrios; Sriswasdi, Sira (2013),\n\"Lower Bound for Convex Hull Area and Universal Cover Problems\",\nInternational Journal of Computational Geometry & Applications,\n23 (3): 197–212, arXiv:1101.5638, doi:10.1142/S0218195913500076, MR 3158583, S2CID 207132316.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MoserWorm","statement":"0.232239 ∈ lowerBounds {v | ∃ X ∈ MoserWorm.WormCovers, Convex ℝ X ∧ MeasureTheory.volume X = v}","subjects":["52"],"theorem":"MoserWorm.convex_mosers_worm_problem_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"The minimal area of a convex shape that can cover every unit-length curve is attained.\nThis follows from the Blaschke selection theorem.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MoserWorm","statement":"∃ bound, IsLeast {v | ∃ X ∈ MoserWorm.WormCovers, Convex ℝ X ∧ MeasureTheory.volume X = v} bound","subjects":["52"],"theorem":"MoserWorm.convex_mosers_worm_problem_bound_attained"},{"answerKinds":[],"category":"textbook","docstring":"The only clique of size `n` in a complete graph on `n` vertices is the entire set of vertices. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Conway99Graph","statement":"∀ {V : Type} [inst : Fintype V], ⊤.cliqueSet (Fintype.card V) = {Set.univ.toFinset}","subjects":["5"],"theorem":"Conway99Graph.completeGraph_cliqueSet"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Conway99Graph","statement":"Conway99Graph.Conway9.LocallyLinear","subjects":["5"],"theorem":"Conway99Graph.conway9_locallyLinear"},{"answerKinds":[],"category":"test","docstring":"The triangle is an example with 3 vertices satisfying the condition.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Conway99Graph","statement":"(SimpleGraph.completeGraph (Fin 3)).LocallyLinear ∧\n  Conway99Graph.NonEdgesAreDiagonals (SimpleGraph.completeGraph (Fin 3))","subjects":["5"],"theorem":"Conway99Graph.triangle_locallyLinear_and_nonEdgesAreDiagonals"},{"answerKinds":[],"category":"research open","docstring":"Does there exist an undirected graph with 99 vertices, in which each two adjacent vertices have\nexactly one common neighbor, and in which each two non-adjacent vertices have exactly two common\nneighbors?\nEquivalently, every edge should be part of a unique triangle and every non-adjacent pair should be\none of the two diagonals of a unique 4-cycle.\nThe first condition is equivalent to being locally linear.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Conway99Graph","statement":"True ↔ ∃ G, G.LocallyLinear ∧ Conway99Graph.NonEdgesAreDiagonals G","subjects":["5"],"theorem":"Conway99Graph.conway99Graph"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Conway99Graph","statement":"Conway99Graph.NonEdgesAreDiagonals Conway99Graph.Conway9","subjects":["5"],"theorem":"Conway99Graph.conway9_nonEdgesAreDiagonals"},{"answerKinds":[],"category":"textbook","docstring":"A finset of vertices in a complete graph is always a clique. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Conway99Graph","statement":"∀ {V : Type} (s : Finset V), ⊤.IsClique ↑s","subjects":["5"],"theorem":"Conway99Graph.completeGraphIsClique"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Conway99Graph","statement":"(SimpleGraph.completeGraph (Fin 3) □ SimpleGraph.completeGraph (Fin 3)).cliqueSet 3 =\n  {x | ∃ q, {x | ∃ p, (p, q) = x} = x} ∪ {x | ∃ q, {x | ∃ p, (q, p) = x} = x}","subjects":["5"],"theorem":"Conway99Graph.completeGraph_boxProd_completeGraph_cliqueSet"},{"answerKinds":[],"category":"research open","docstring":"**Existence of an infinite club.**  A *club* is an abundancy equivalence class, i.e.\nthe set of all positive integers friendly with a given $n$.  It is unknown whether any club\nis infinite.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SolitaryNumber","statement":"True ↔ ∃ n, 0 < n ∧ {m | SolitaryNumber.Friendly m n}.Infinite","subjects":["11"],"theorem":"SolitaryNumber.infinite_club_exists"},{"answerKinds":[],"category":"research open","docstring":"**Is 10 a solitary number?**  The smallest positive integer whose solitary status is\ncurrently unresolved is $10$, with abundancy index $\\sigma(10) / 10 = 9/5$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SolitaryNumber","statement":"True ↔ SolitaryNumber.IsSolitary 10","subjects":["11"],"theorem":"SolitaryNumber.is_ten_solitary"},{"answerKinds":[],"category":"API","docstring":"Absolute normality implies normality in at least one base. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.AlgebraicNormality","statement":"∀ {x : ℝ}, NormalNumber.IsAbsolutelyNormal x → ∃ b, 2 ≤ b ∧ NormalNumber.IsNormalInBase b x","subjects":["11"],"theorem":"AlgebraicNormality.normal_in_some_base_of_absolutely_normal"},{"answerKinds":[],"category":"research open","docstring":"The weaker normality conjecture: every irrational algebraic real is normal in at least one\ninteger base `b ≥ 2`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AlgebraicNormality","statement":"True ↔ ∀ (x : ℝ), AlgebraicNormality.IsIrrationalAlgebraic x → ∃ b, 2 ≤ b ∧ NormalNumber.IsNormalInBase b x","subjects":["11","12","41"],"theorem":"AlgebraicNormality.irrational_algebraic_normal_in_some_base"},{"answerKinds":[],"category":"research open","docstring":"The strong normality conjecture: every irrational algebraic real is absolutely normal. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AlgebraicNormality","statement":"True ↔ ∀ (x : ℝ), AlgebraicNormality.IsIrrationalAlgebraic x → NormalNumber.IsAbsolutelyNormal x","subjects":["11","12","41"],"theorem":"AlgebraicNormality.irrational_algebraic_absolutely_normal"},{"answerKinds":[],"category":"research solved","docstring":"The inequality $p_{n+1}-p_n < (\\log p_n)^2-\\log p_n$ holds for all $n>4$.\nA consequence of Firuzbakht's conjecture.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Firoozbakht","statement":"∀ (n : ℕ),\n  3 < n →\n    (∀ (n : ℕ), Firoozbakht.firoozbakhtSeq (n + 1) < Firoozbakht.firoozbakhtSeq n) →\n      ↑(Nat.nth Prime (n + 1)) - ↑(Nat.nth Prime n) < Real.log ↑(Nat.nth Prime n) ^ 2 - Real.log ↑(Nat.nth Prime n)","subjects":["11"],"theorem":"Firoozbakht.firoozbakht_conjecture_consequence"},{"answerKinds":[],"category":"research open","docstring":"**Firoozbakht's conjecture**\nThe inequality $\\sqrt[n+1]{p_{n+1}} < \\sqrt[n]{p_n}$ holds for all prime numbers $p_n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Firoozbakht","statement":"∀ (n : ℕ), Firoozbakht.firoozbakhtSeq (n + 1) < Firoozbakht.firoozbakhtSeq n","subjects":["11"],"theorem":"Firoozbakht.firoozbakht_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**Non-Power-of-2 Almost Perfect Numbers Conjecture.**\nDoes there exist an almost perfect number that is not a power of 2?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.AlmostPerfectNumbers","statement":"True ↔ ∃ n, AlmostPerfectNumbers.AlmostPerfect n ∧ ¬∃ k, n = 2 ^ k","subjects":["11"],"theorem":"AlmostPerfectNumbers.exists_almost_perfect_not_power_of_two"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FortuneConjecture","statement":"FortuneConjecture.fortunateNumber 0 = 3","subjects":["11"],"theorem":"FortuneConjecture.fortunateNumber_zero"},{"answerKinds":[],"category":"research open","docstring":"**Fortune's Conjecture**: Every Fortunate number is prime. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.FortuneConjecture","statement":"True ↔ ∀ (n : ℕ), Nat.Prime (FortuneConjecture.fortunateNumber n)","subjects":["11"],"theorem":"FortuneConjecture.fortune_conjecture"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FortuneConjecture","statement":"FortuneConjecture.fortunateNumber 2 = 7","subjects":["11"],"theorem":"FortuneConjecture.fortunateNumber_two"},{"answerKinds":[],"category":"API","docstring":"For any natural number `N` there is some `m > 1` with `N + m` prime; an\nimmediate consequence of the infinitude of primes. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FortuneConjecture","statement":"∀ (N : ℕ), ∃ m, 1 < m ∧ Nat.Prime (N + m)","subjects":["11"],"theorem":"FortuneConjecture.exists_one_lt_prime_add"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FortuneConjecture","statement":"FortuneConjecture.fortunateNumber 1 = 5","subjects":["11"],"theorem":"FortuneConjecture.fortunateNumber_one"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FortuneConjecture","statement":"FortuneConjecture.fortunateNumber 3 = 13","subjects":["11"],"theorem":"FortuneConjecture.fortunateNumber_three"},{"answerKinds":[],"category":"API","docstring":"Minimality of `fortunateNumber n`: no smaller integer $m > 1$ makes\n`primorial (Nat.nth Nat.Prime n) + m` prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FortuneConjecture","statement":"∀ (n m : ℕ), 1 < m → Nat.Prime (primorial (Nat.nth Nat.Prime n) + m) → FortuneConjecture.fortunateNumber n ≤ m","subjects":["11"],"theorem":"FortuneConjecture.fortunateNumber_le"},{"answerKinds":[],"category":"API","docstring":"`fortunateNumber n` is greater than $1$, and adding it to the primorial of\nthe $(n+1)$-st prime yields a prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FortuneConjecture","statement":"∀ (n : ℕ),\n  1 < FortuneConjecture.fortunateNumber n ∧\n    Nat.Prime (primorial (Nat.nth Nat.Prime n) + FortuneConjecture.fortunateNumber n)","subjects":["11"],"theorem":"FortuneConjecture.fortunateNumber_spec"},{"answerKinds":[],"category":"research open","docstring":"**Andrica's conjecture**\nThe inequality $\\sqrt{p_{n+1}}-\\sqrt{p_n} < 1$ holds for all $n$, where $p_n$ is the $n$-th prime number.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Andrica","statement":"∀ (n : ℕ), √↑(Nat.nth Nat.Prime (n + 1)) - √↑(Nat.nth Nat.Prime n) < 1","subjects":["11"],"theorem":"Andrica.andrica_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"Ferreira proved that Andrica's conjecture is true for sufficiently large n.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Andrica","statement":"∀ᶠ (n : ℕ) in Filter.atTop, √↑(Nat.nth Nat.Prime (n + 1)) - √↑(Nat.nth Nat.Prime n) < 1","subjects":["11"],"theorem":"Andrica.andrica_conjecture.ferreira_large_n"},{"answerKinds":[],"category":"textbook","docstring":"If Mahler's conjecture is true, i.e. there are no Z-numbers, then `Ω(3/2)` exceeds `1/2`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mahler32","statement":"(∀ (x : ℝ), IsZNumber x → False) → 1 / 2 < Mahler32.Ω (3 / 2)","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Mahler32.mahler_conjecture.variants.consequence"},{"answerKinds":[],"category":"research solved","docstring":"It is known that for all rational `p/q > 1` in lowest terms, we have `Ω(p/q) > 1/p`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mahler32","statement":"∀ (p q : ℕ), 1 < q → p.Coprime q → q < p → 1 / ↑p < Mahler32.Ω (↑p / ↑q)","subjects":["11"],"theorem":"Mahler32.mahler_conjecture.variants.flatto_lagarias_pollington"},{"answerKinds":[],"category":"research open","docstring":"The **Mahler Conjecture** states that there are no Z-numbers. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mahler32","statement":"∀ (x : ℝ), IsZNumber x → False","subjects":["11"],"theorem":"Mahler32.mahler_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Taxicab number for $k=5$, $m=2$, and $n=2$ is not known.\nWhether such a number exists is also not known. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Taxicab","statement":"True ↔ ∃ x, Taxicab.IsTaxicabFor 5 2 2 x","subjects":["11"],"theorem":"Taxicab.taxicab_for_5_2_2"},{"answerKinds":[],"category":"test","docstring":"Using Aristotle (Harmonic) we get a compact proof that\n4 is the taxicab number for $k=1, m=2, n=2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Taxicab","statement":"Taxicab.IsTaxicabFor 1 2 2 4","subjects":["11"],"theorem":"Taxicab.taxicab_4"},{"answerKinds":[],"category":"research open","docstring":"Taxicab number for $k=5$ and $m=2$ is not-known for any $n ≥ 2$.\nWhether such a number exists is also not known. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Taxicab","statement":"True ↔ ∃ n ≥ 2, ∃ x, Taxicab.IsTaxicabFor 5 2 n x","subjects":["11"],"theorem":"Taxicab.taxicab_for_5_2_n"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Taxicab","statement":"Taxicab.IsTaxicabFor' 1 2 2 4","subjects":["11"],"theorem":"Taxicab.taxicab_4'"},{"answerKinds":[],"category":"test","docstring":"$1729$ is a possible taxicab number for $k=3, m=2, n=2$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Taxicab","statement":"Taxicab.IsTaxicabFor' 3 2 2 1729","subjects":["11"],"theorem":"Taxicab.taxicab_1729"},{"answerKinds":[],"category":"research solved","docstring":"The value of $N(r)$ can be expressed as\n$$\n  N(r) = 1 + 4\\sum_{i=0}^{\\infty}\\left(\\left\\lfloor\\frac{r^2}{4i+1}\\right\\rfloor -\n    \\left\\lfloor\\frac{r^2}{4i + 3}\\right\\rfloor\\right).\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.GaussCircleProblem","statement":"∀ (r : ℝ), 0 ≤ r → ↑(GaussCircleProblem.N r) = 1 + 4 * ∑' (i : ℕ), (⌊r ^ 2 / (4 * ↑i + 1)⌋ - ⌊r ^ 2 / (4 * ↑i + 3)⌋)","subjects":["11"],"theorem":"GaussCircleProblem.exact_form_floor"},{"answerKinds":[],"category":"research solved","docstring":"Hardy and Laundau independently found a lower bound by showing that\n$$\n  |E(r)| \\neq o\\left(r^{1/2}(\\log r)^{1/4}\\right)\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.GaussCircleProblem","statement":"¬GaussCircleProblem.E =o[Filter.atTop] fun r => √r * √√(Real.log r)","subjects":["11"],"theorem":"GaussCircleProblem.error_not_isLittleO"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that the correct bound is\n$$\n  |E(r)| = O\\left(r^{1/2 + o(1)}\\right)\n$$\n\n[Ha59]  Hardy, G. H. (1959). _Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work_(3rd ed.). New York: Chelsea Publishing Company. p. 67\n\nSee also https://arxiv.org/abs/2305.03549\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.GaussCircleProblem","statement":"∃ o, ∃ (_ : Filter.Tendsto o Filter.atTop (nhds 0)), GaussCircleProblem.E =O[Filter.atTop] fun r => r ^ (1 / 2 + o r)","subjects":["11"],"theorem":"GaussCircleProblem.error_isBigO"},{"answerKinds":[],"category":"research solved","docstring":"Gauss proved that\n$$\n  |E(r)|\\leq 2\\sqrt{2}\\pi r,\n$$\nfor sufficiently large $r$.\n\n[Ha59]  Hardy, G. H. (1959). _Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work_(3rd ed.). New York: Chelsea Publishing Company. p. 67\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.GaussCircleProblem","statement":"∀ᶠ (r : ℝ) in Filter.atTop, |GaussCircleProblem.E r| ≤ 2 * √2 * Real.pi * r","subjects":["11"],"theorem":"GaussCircleProblem.error_le"},{"answerKinds":[],"category":"research open","docstring":"Brennan's conjecture, part 2: $B_b(-2) = 1$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Brennanconjecture","statement":"BrennanConjecture.universalSpectrumBounded (-2) = 1","subjects":["30"],"theorem":"BrennanConjecture.brennan_universalSpectrumBounded"},{"answerKinds":[],"category":"API","docstring":"Brennan's conjecture: $B(-2) = B_b(-2) = 1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Brennanconjecture","statement":"BrennanConjecture.universalSpectrum (-2) = 1 ∧ BrennanConjecture.universalSpectrumBounded (-2) = 1","subjects":["30"],"theorem":"BrennanConjecture.brennan"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Brennanconjecture","statement":"∀ (τ : ℝ), BrennanConjecture.universalSpectrumBounded τ ≤ BrennanConjecture.universalSpectrum τ","subjects":["30"],"theorem":"BrennanConjecture.universalSpectrumBounded_le"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Brennanconjecture","statement":"∀ (τ : ℝ), BrennanConjecture.integralMeansSpectrum id τ = 0","subjects":["30"],"theorem":"BrennanConjecture.integralMeansSpectrum_id"},{"answerKinds":[],"category":"research open","docstring":"Brennan's conjecture, part 1: $B(-2) = 1$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Brennanconjecture","statement":"BrennanConjecture.universalSpectrum (-2) = 1","subjects":["30"],"theorem":"BrennanConjecture.brennan_universalSpectrum"},{"answerKinds":[],"category":"API","docstring":"Brennan's conjecture, part 3: $B(-2) = B_b(-2)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Brennanconjecture","statement":"BrennanConjecture.universalSpectrum (-2) = BrennanConjecture.universalSpectrumBounded (-2)","subjects":["30"],"theorem":"BrennanConjecture.brennan_spectra_eq"},{"answerKinds":[],"category":"research open","docstring":"There exists a Hadamard matrix for all $n = 4k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Hadamard","statement":"∀ (k : ℕ), ∃ M, Hadamard.IsHadamard M","subjects":["15"],"theorem":"Hadamard.HadamardConjecture"},{"answerKinds":[],"category":"test","docstring":"Both definitions are equivalent.\n\nTODO(firsching): complete and golf the proof\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Hadamard","statement":"∀ (n : ℕ) (M : Matrix (Fin n) (Fin n) ℝ), Hadamard.IsHadamard' M ↔ Hadamard.IsHadamard M","subjects":["15"],"theorem":"Hadamard.isHadamard_equiv_isHadamard'"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Hadamard","statement":"∃ M, Hadamard.IsHadamard M","subjects":["15"],"theorem":"Hadamard.exists_hadamard_zero"},{"answerKinds":[],"category":"research open","docstring":"The smallest order for which no Hadamard matrix is presently known is $668 = 4 * 167$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Hadamard","statement":"∃ M, Hadamard.IsHadamard M","subjects":["15"],"theorem":"Hadamard.HadamardConjecture.variants.«167»"},{"answerKinds":[],"category":"test","docstring":"which satisfies the condition.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Hadamard","statement":"Hadamard.IsHadamard Hadamard.H12","subjects":["15"],"theorem":"Hadamard.isHadamard_H12"},{"answerKinds":[],"category":"research solved","docstring":"For all $k ≤ 166$, it is known there that there is a Hadamard matrix of size $4 * k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Hadamard","statement":"∀ k ≤ 166, ∃ M, Hadamard.IsHadamard M","subjects":["15"],"subsets":["FC100SolvedSet1"],"theorem":"Hadamard.HadamardConjecture.variants.first_cases"},{"answerKinds":[],"category":"research solved","docstring":"**There are infinitely many Steiner systems with $t = 5$.**\n\nKeevash (2014) proved that for any fixed $t$ and $k$, a Steiner system $S(t, k, n)$\nexists for all sufficiently large $n$ satisfying the necessary divisibility conditions.\nThis settles the long-standing open problem of whether infinitely many $S(5, k, n)$\nsystems exist. The proof is nonconstructive.\n\nOnly two explicit examples are known: $S(5, 6, 12)$ and $S(5, 8, 24)$, both Witt\ndesigns related to the Mathieu groups $M_{12}$ and $M_{24}$ respectively.\nNo Steiner system with $t \\geq 6$ has been explicitly constructed, though Keevash's\nresult guarantees their existence nonconstructively as well.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SteinerSystem","statement":"∃ S, S.Infinite","subjects":["5"],"theorem":"SteinerSystems.infinitely_many_steiner_t5"},{"answerKinds":[],"category":"research solved","docstring":"**Existence of $S(5, 8, 24)$**: The large Witt design.\n\nThere exists a unique Steiner system $S(5, 8, 24)$, known as the large Witt design.\nIt was constructed by Witt (1938) and is closely related to the Mathieu group $M_{24}$.\nThis is one of only two known Steiner systems with $t = 5$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SteinerSystem","statement":"Nonempty (SteinerSystems.SteinerSystem 5 8 24)","subjects":["5"],"theorem":"SteinerSystems.steiner_system_5_8_24"},{"answerKinds":[],"category":"research solved","docstring":"**There are infinitely many Steiner systems with $t = 4$.**\n\nKeevash (2014) proved that for any fixed $t$ and $k$, a Steiner system $S(t, k, n)$\nexists for all sufficiently large $n$ satisfying the necessary divisibility conditions.\nSince there are infinitely many such admissible $n$, this implies infinitely many\n$S(4, k, n)$ systems exist (for any fixed $k > 4$). The proof is nonconstructive.\n\nExplicit examples include $S(4, 5, 11)$ (the unique system, related to the Mathieu\ngroup $M_{11}$) and $S(4, 7, 23)$ (related to the Mathieu group $M_{23}$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SteinerSystem","statement":"∃ S, S.Infinite","subjects":["5"],"theorem":"SteinerSystems.infinitely_many_steiner_t4"},{"answerKinds":[],"category":"research open","docstring":"Construct an $S(t, k, n)$-Steiner system with $n > k > t > 5$, $t < 10$, and $n < 200$.\n\nNo example of a Steiner system with $t > 5$ is known, despite a 2014 existence theorem\nby Keevash showing that such systems must exist for sufficiently large $n$.\n\n*Reference:* [Large Steiner Systems](https://epoch.ai/frontiermath/open-problems/large-steiner-systems)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SteinerSystem","statement":"Nonempty SteinerSystems.LargeSteinerSystemWitness","subjects":["5"],"theorem":"SteinerSystems.large_steiner_systems"},{"answerKinds":[],"category":"research solved","docstring":"**Existence of $S(5, 6, 12)$**: The small Witt design.\n\nThere exists a unique Steiner system $S(5, 6, 12)$, known as the small Witt design.\nIt was constructed by Witt (1938) and is closely related to the Mathieu group $M_{12}$.\nThis is one of only two known Steiner systems with $t = 5$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SteinerSystem","statement":"Nonempty (SteinerSystems.SteinerSystem 5 6 12)","subjects":["5"],"theorem":"SteinerSystems.steiner_system_5_6_12"},{"answerKinds":[],"category":"research open","docstring":"The pebbling number conjecture:\nthe pebbling number of a Cartesian product of connected graphs is at most equal to the product\nof the pebbling numbers of the factors. See\n[Asplund, Hurlbert, and Kenter](https://arxiv.org/abs/1801.07808).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.PebblingNumberConjecture","statement":"∀ {V : Type} [inst : DecidableEq V] {W : Type} [inst_1 : Fintype V] [inst_2 : Fintype W] [inst_3 : DecidableEq W]\n  (G : SimpleGraph V) (H : SimpleGraph W),\n  G.Connected →\n    H.Connected →\n      PebblingNumberConjecture.PebblingNumber (G □ H) ≤\n        PebblingNumberConjecture.PebblingNumber G * PebblingNumberConjecture.PebblingNumber H","subjects":["5"],"theorem":"PebblingNumberConjecture.pebbling_number_conjecture"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.PebblingNumberConjecture","statement":"∀ {V : Type} [inst : DecidableEq V] (G : SimpleGraph V) (A : PebblingNumberConjecture.PebbleDistribution V),\n  PebblingNumberConjecture.IsReachable G A A","subjects":["5"],"theorem":"PebblingNumberConjecture.IsReachable.refl"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.PebblingNumberConjecture","statement":"∀ {V : Type} [inst : DecidableEq V] (G : SimpleGraph V) (A : PebblingNumberConjecture.PebbleDistribution V) {v w : V},\n  2 ≤ A v →\n    G.Adj v w →\n      PebblingNumberConjecture.IsPebblingMove G A fun u => if u = w then A u + 1 else if u = v then A u - 2 else A u","subjects":["5"],"theorem":"PebblingNumberConjecture.IsPebblingMove.refl"},{"answerKinds":[],"category":"API","docstring":"The pebbling number of the complete graph on `n` vertices is `n`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.PebblingNumberConjecture","statement":"∀ {V : Type} [inst : DecidableEq V] [inst_1 : Fintype V],\n  PebblingNumberConjecture.PebblingNumber (SimpleGraph.completeGraph V) = Fintype.card V","subjects":["5"],"theorem":"PebblingNumberConjecture.PebblingNumber_completeGraph"},{"answerKinds":[],"category":"textbook","docstring":"The discriminant of `ℚ[√d]` for `d ≥ 2` squarefree not congruent to 1 mod 4 is `4 * d`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∀ {d : ℤ} [inst : Fact (Squarefree d)] [inst_1 : Fact (d ≠ 1)],\n  ¬d ≡ 1 [ZMOD 4] → NumberField.discr (QuadraticAlgebra ℚ (↑d) 0) = 4 * d","subjects":["11"],"theorem":"QuadraticAlgebra.discr_rat_of_not_modEq_one"},{"answerKinds":[],"category":"textbook","docstring":"An integer `D` is a fundamental discriminant iff it is the discriminant of the explicit\nquadratic field `QuadraticAlgebra ℚ d 0` for some squarefree `d : ℤ` not equal to 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∀ {D : ℤ},\n  NumberField.IsFundamentalDiscr D ↔\n    ∃ d, ∃ (x : Fact (d ≠ 1)) (x_1 : Fact (Squarefree d)), NumberField.discr (QuadraticAlgebra ℚ (↑d) 0) = D","subjects":["11"],"theorem":"NumberField.isFundamentalDiscr_iff_exists_discr_quadraticAlgebra"},{"answerKinds":[],"category":"textbook","docstring":"An integer `D` is a fundamental discriminant iff it is the discriminant of some number field. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∀ {D : ℤ},\n  NumberField.IsFundamentalDiscr D ↔\n    ∃ K x, ∃ (x_1 : NumberField K), Algebra.IsQuadraticExtension ℚ K ∧ NumberField.discr K = D","subjects":["11"],"theorem":"NumberField.isFundamentalDiscr_iff_exists_discr_numberField"},{"answerKinds":[],"category":"textbook","docstring":"An algebra `L` is quadratic over a field `K` iff it is isomorphic to the explicit quadratic\nalgebra `QuadraticAlgebra K a b` for some `a b : K`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L],\n  Algebra.IsQuadraticExtension K L ↔ ∃ a b, Nonempty (L ≃ₐ[K] QuadraticAlgebra K a b)","subjects":["11"],"theorem":"Algebra.isQuadraticExtension_iff_exists_quadraticAlgebra"},{"answerKinds":[],"category":"textbook","docstring":"A number field `K` is quadratic iff it is isomorphic to the explicit quadratic field\n`QuadraticAlgebra ℚ d 0` for some squarefree `d : ℤ` not equal to 1. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K],\n  Algebra.IsQuadraticExtension ℚ K ↔ ∃ d, d ≠ 1 ∧ Squarefree d ∧ Nonempty (K ≃+* QuadraticAlgebra ℚ (↑d) 0)","subjects":["11"],"theorem":"NumberField.isQuadraticExtension_iff_exists_quadraticAlgebra"},{"answerKinds":[],"category":"textbook","docstring":"A quadratic algebra `L` over a field `K` is isomorphic to the explicit quadratic algebra\n`QuadraticAlgebra K a b` for some `a b : K`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∀ (K : Type u_1) (L : Type u_2) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L]\n  [Algebra.IsQuadraticExtension K L], ∃ a b, Nonempty (L ≃ₐ[K] QuadraticAlgebra K a b)","subjects":["11"],"theorem":"Algebra.exists_quadraticAlgebra_of_isQuadraticExtension"},{"answerKinds":[],"category":"textbook","docstring":"A quadratic number field `K` is isomorphic to the explicit quadratic field\n`QuadraticAlgebra ℚ d 0` for some squarefree `d : ℤ` not equal to 1. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] [Algebra.IsQuadraticExtension ℚ K],\n  ∃ d, d ≠ 1 ∧ Squarefree d ∧ Nonempty (K ≃+* QuadraticAlgebra ℚ (↑d) 0)","subjects":["11"],"theorem":"NumberField.exists_quadraticAlgebra_of_isQuadraticExtension"},{"answerKinds":[],"category":"research open","docstring":"A Lucas–Wieferich prime associated with $(a,b)$ is an odd prime $p$, not dividing $a^2 - 4b$, such\nthat $U_{p-\\varepsilon}(a,b) \\equiv 0 \\pmod{p^2}$ where $U(a,b)$ is the Lucas sequence of the first\nkind and $\\varepsilon$ is the Legendre symbol $\\left({\\tfrac {a^2-4b}{p}}\\right)$.\nThe discriminant of this number is the quantity $a^2 - 4b$. It is conjectured that there are\ninfinitely many Lucas–Wieferich primes of any given non-one fundamental discriminant.\n\nTODO: Source this conjecture\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∀ {D : ℤ},\n  NumberField.IsFundamentalDiscr D → D ≠ 1 → {p | ∃ a b, a ^ 2 - 4 * b = D ∧ IsLucasWieferichPrime a b p}.Infinite","subjects":["11"],"theorem":"WallSunSun.infinite_isWallSunSunPrime_of_disc_eq"},{"answerKinds":[],"category":"research open","docstring":"A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \\equiv 1 \\pmod{p^2}$, where $L_p$ is the\n$p$-th Lucas number. It is conjectured that there are infinitely many Wall-Sun-Sun primes.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"{p | IsWallSunSunPrime p}.Infinite","subjects":["11"],"theorem":"WallSunSun.infinite_isWallSunSunPrime"},{"answerKinds":[],"category":"textbook","docstring":"The discriminant of `ℚ[√d]` for `d ≥ 2` squarefree congruent to 1 mod 4 is `d`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∀ {d : ℤ} [inst : Fact (Squarefree d)] [inst_1 : Fact (d ≠ 1)],\n  d ≡ 1 [ZMOD 4] → NumberField.discr (QuadraticAlgebra ℚ (↑d) 0) = d","subjects":["11"],"theorem":"QuadraticAlgebra.discr_rat_of_modEq_one"},{"answerKinds":[],"category":"research open","docstring":"A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \\equiv 1 \\pmod{p^2}$, where $L_p$ is the\n$p$-th Lucas number. It is conjectured that there is at least one Wall–Sun–Sun prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.WallSunSun","statement":"∃ p, IsWallSunSunPrime p","subjects":["11"],"theorem":"WallSunSun.exists_isWallSunSunPrime"},{"answerKinds":[],"category":"research open","docstring":"**Fuglede's conjecture** in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Fuglede","statement":"True ↔ Fuglede.FugledeConjectureFor 2","subjects":["42","46","47"],"theorem":"Fuglede.FugledeConjecture.variants.dim_2"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Fuglede's conjecture** in three or higher dimensions has been disproven.\n(Note that counterexamples in lower dimensions would also disprove the conjecture in higher dimensions.)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Fuglede","statement":"False ↔ ∀ (n : ℕ), 3 ≤ n → Fuglede.FugledeConjectureFor n","subjects":["42","46","47"],"theorem":"Fuglede.FugledeConjecture.variants.dim_3_or_higher"},{"answerKinds":[],"category":"research open","docstring":"**Fuglede's conjecture** in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Fuglede","statement":"True ↔ Fuglede.FugledeConjectureFor 1","subjects":["42","46","47"],"theorem":"Fuglede.FugledeConjecture.variants.dim_1"},{"answerKinds":[],"category":"research open","docstring":"**Vizing's conjecture (1968).**\n\nFor all finite simple graphs $G$ and $H$, the domination number of the Cartesian (box)\nproduct satisfies $\\gamma(G \\,\\square\\, H) \\ge \\gamma(G)\\,\\gamma(H)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.VizingConjecture","statement":"True ↔\n  ∀ {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] (G : SimpleGraph α) (H : SimpleGraph β),\n    G.dominationNumber * H.dominationNumber ≤ (G □ H).dominationNumber","subjects":["5"],"theorem":"VizingConjecture.vizing_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"**The Suen–Tarr inequality (2012).**\n\nFor all finite simple graphs $G$ and $H$,\n$\\gamma(G \\,\\square\\, H) \\ge \\tfrac12\\,\\gamma(G)\\,\\gamma(H) +\n\\tfrac12\\min\\{\\gamma(G), \\gamma(H)\\}$, improving the Clark–Suen bound.\n\n*Reference:* [SuTa12].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.VizingConjecture","statement":"∀ {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] (G : SimpleGraph α) (H : SimpleGraph β),\n  G.dominationNumber * H.dominationNumber + min G.dominationNumber H.dominationNumber ≤ 2 * (G □ H).dominationNumber","subjects":["5"],"theorem":"VizingConjecture.vizing_conjecture.variants.suen_tarr"},{"answerKinds":[],"category":"research solved","docstring":"**Vizing's conjecture when `γ(H) = 1`.**\n\nIf `H` has a dominating vertex, so that $\\gamma(H) = 1$, then Vizing's inequality reduces to\n$\\gamma(G) \\le \\gamma(G \\,\\square\\, H)$, which is the projection bound\n`dominationNumber_le_dominationNumber_boxProd`. This is the simplest of the known cases of the\nconjecture (see [BDGHHKR12]).\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.VizingConjecture","statement":"∀ {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] (G : SimpleGraph α) (H : SimpleGraph β),\n  H.dominationNumber = 1 → G.dominationNumber * H.dominationNumber ≤ (G □ H).dominationNumber","subjects":["5"],"theorem":"VizingConjecture.vizing_conjecture.variants.dominationNumber_eq_one"},{"answerKinds":[],"category":"research solved","docstring":"**The Clark–Suen inequality (2000).**\n\nFor all finite simple graphs $G$ and $H$,\n$\\gamma(G \\,\\square\\, H) \\ge \\tfrac12\\,\\gamma(G)\\,\\gamma(H)$; that is, Vizing's conjecture holds\nup to a factor of $2$.\n\n*Reference:* [ClSu00].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.VizingConjecture","statement":"∀ {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] (G : SimpleGraph α) (H : SimpleGraph β),\n  G.dominationNumber * H.dominationNumber ≤ 2 * (G □ H).dominationNumber","subjects":["5"],"theorem":"VizingConjecture.vizing_conjecture.variants.clark_suen"},{"answerKinds":[],"category":"research open","docstring":"$\\zeta(7)$ is irrational.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RiemannZetaValues","statement":"∃ x, Irrational x ∧ riemannZeta 7 = ↑x","subjects":["11","33"],"theorem":"RiemannZetaValues.irrational_seven"},{"answerKinds":[],"category":"research open","docstring":"$\\zeta(2n + 1)$ is irrational for any $n\\in\\mathbb{N}^{+}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RiemannZetaValues","statement":"∀ (n : ℕ), 0 < n → ∃ x, Irrational x ∧ riemannZeta (2 * ↑n + 1) = ↑x","subjects":["11","33"],"theorem":"RiemannZetaValues.irrational_odd"},{"answerKinds":[],"category":"research solved","docstring":"There are infinitely many $\\zeta(2n + 1)$, $n \\in \\mathbb{N}$, that are irrational.\n\n[Ri00] Rivoal, T. (2000). _La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs_. Comptes Rendus de l'Académie des Sciences, Série I. 331 (4): 267–270.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RiemannZetaValues","statement":"{n | ∃ x, Irrational x ∧ riemannZeta (2 * ↑n + 1) = ↑x}.Infinite","subjects":["11","33"],"subsets":["FC100SolvedSet1"],"theorem":"RiemannZetaValues.infinite_irrational_at_odd"},{"answerKinds":[],"category":"research open","docstring":"$\\zeta(11)$ is irrational.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RiemannZetaValues","statement":"∃ x, Irrational x ∧ riemannZeta 11 = ↑x","subjects":["11","33"],"theorem":"RiemannZetaValues.irrational_eleven"},{"answerKinds":[],"category":"research solved","docstring":"At least one of $\\zeta(5), \\zeta(7), \\zeta(9)$ or $\\zeta(11)$ is irrational.\n\n[Zu01]  W. Zudilin (2001). _One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational_. Russ. Math. Surv. 56 (4): 774–776.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RiemannZetaValues","statement":"({5, 7, 9, 11} ∩ {a | ∃ x, Irrational x ∧ riemannZeta a = ↑x}).Nonempty","subjects":["11","33"],"theorem":"RiemannZetaValues.exists_irrational_of_five_seven_nine_eleven"},{"answerKinds":[],"category":"research solved","docstring":"$\\zeta(3)$ is irrational.\n\n[Ap79] Apéry, R. (1979). _Irrationalité de ζ(2) et ζ(3)_. Astérisque. 61: 11–13.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RiemannZetaValues","statement":"∃ x, Irrational x ∧ riemannZeta 3 = ↑x","subjects":["11","33"],"theorem":"RiemannZetaValues.irrational_three"},{"answerKinds":[],"category":"research open","docstring":"$\\zeta(5)$ is irrational.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RiemannZetaValues","statement":"∃ x, Irrational x ∧ riemannZeta 5 = ↑x","subjects":["11","33"],"subsets":["FC100OpenSet1"],"theorem":"RiemannZetaValues.irrational_five"},{"answerKinds":[],"category":"research open","docstring":"$\\zeta(9)$ is irrational.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RiemannZetaValues","statement":"∃ x, Irrational x ∧ riemannZeta 9 = ↑x","subjects":["11","33"],"theorem":"RiemannZetaValues.irrational_nine"},{"answerKinds":[],"category":"research open","docstring":"Given any set of $n$ complex numbers $\\{z_1, ..., z_n\\}$ that are linearly independent over\n$\\mathbb{Q}$, the field extension $\\mathbb{Q}(z_1, ..., z_n, e^{z_1}, ..., e^{z_n})$\nhas transcendence degree at least $n$ over $\\mathbb{Q}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Schanuel","statement":"∀ (n : ℕ) (z : Fin n → ℂ),\n  LinearIndependent ℚ z → ↑n ≤ Algebra.trdeg ℚ ↥(IntermediateField.adjoin ℚ (Set.range z ∪ Set.range (Complex.exp ∘ z)))","subjects":["11","33"],"theorem":"Schanuel.schanuel_conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FactorialPrime","statement":"FactorialPrime.IsFactorialPrime 23","subjects":["11"],"theorem":"FactorialPrime.twentyThree_isFactorialPrime"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many factorial primes. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.FactorialPrime","statement":"{p | FactorialPrime.IsFactorialPrime p}.Infinite","subjects":["11"],"theorem":"FactorialPrime.infinitely_many_factorial_primes"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.FactorialPrime","statement":"FactorialPrime.IsFactorialPrime 7","subjects":["11"],"theorem":"FactorialPrime.seven_isFactorialPrime"},{"answerKinds":[],"category":"research open","docstring":"Singmaster's conjecture: the number of times any number $t > 1$ appears in\nPascal's triangle is bounded.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Singmaster","statement":"∃ C, ∀ t > 1, (Singmaster.solutions t).Finite ∧ (Singmaster.solutions t).ncard ≤ C","subjects":["11"],"theorem":"Singmaster.singmaster"},{"answerKinds":[],"category":"research open","docstring":"$\\pi$ is normal in base 10.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.NormalityOfPi","statement":"NormalNumber.IsNormalInBase 10 Real.pi","subjects":["11"],"theorem":"NormalNumber.pi_normal_base_ten"},{"answerKinds":[],"category":"research open","docstring":"**Grimm's Conjecture**\nIf $n, n+1, \\dots, n+k-1$ are all composite numbers, then there are $k$ distinct primes $p_i$\nsuch that $p_i$ divides $n + i$ for all $0 \\le i \\le k-1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Grimm","statement":"∀ (n k : ℕ), 1 ≤ n → 1 ≤ k → (∀ (i : Fin k), (n + ↑i).Composite) → ∃ ps, ∀ (i : Fin k), Nat.Prime (ps i) ∧ ps i ∣ n + ↑i","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Grimm.grimm_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**Grimm's Conjecture, weaker version**\nIf $n, n+1, \\dots, n+k-1$ are all composite numbers, then their product\nhas at least $k$ distinct prime divisors.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Grimm","statement":"∀ (n k : ℕ),\n  1 ≤ n → 1 ≤ k → (∀ (i : Fin k), (n + ↑i).Composite) → ∃ ps, ∀ (i : Fin k), Nat.Prime (ps i) ∧ ∃ j, ps i ∣ n + ↑j","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Grimm.grimm_conjecture_weak"},{"answerKinds":[],"category":"research open","docstring":"$e + \\pi$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.exp 1 + Real.pi)","subjects":["11","33"],"theorem":"Transcendental.exp_add_pi_transcendental"},{"answerKinds":[],"category":"research solved","docstring":"At least one of Catalan constant and the Gompertz constant is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ catalanConstant ∨ Transcendental ℚ gompertzConstant","subjects":["11","33"],"theorem":"Transcendental.transcendental_catalanConstant_or_gompertzConstant"},{"answerKinds":[],"category":"research open","docstring":"The Gompertz constant $\\delta$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ gompertzConstant","subjects":["33"],"theorem":"Transcendental.transcendental_gompertzConstant"},{"answerKinds":[],"category":"research solved","docstring":"$\\Gamma(1/4)$ is transcendental.\n\n[Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.Gamma (1 / 4))","subjects":["33"],"theorem":"Transcendental.transcendental_gamma_one_div_four"},{"answerKinds":[],"category":"research open","docstring":"$\\pi^{\\pi^{\\pi^\\pi}}$ is not an integer.\n\nThis would follow from $\\pi^{\\pi^{\\pi^\\pi}}$ being transcendental,\nbut this formulation is of interest in its own right,\nas it could in principle be proven by direct computation.\n\n*Reference:* [YouTube](https://www.youtube.com/watch?v=BdHFLfv-ThQ)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"¬∃ n, Real.pi ^ Real.pi ^ Real.pi ^ Real.pi = ↑n","subjects":["11","33"],"theorem":"Transcendental.pi_pow_pi_pow_pi_pow_pi_not_integer"},{"answerKinds":[],"category":"research open","docstring":"$\\sin(e)$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.sin (Real.exp 1))","subjects":["11","33"],"theorem":"Transcendental.sin_exp_transcendental"},{"answerKinds":[],"category":"research open","docstring":"$\\log(\\pi)$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.log Real.pi)","subjects":["11","33"],"theorem":"Transcendental.rlog_pi_transcendental"},{"answerKinds":[],"category":"research open","docstring":"$e^{\\pi^2}$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.exp (Real.pi ^ 2))","subjects":["11","33"],"theorem":"Transcendental.exp_pow_pi_sq_transcendental"},{"answerKinds":[],"category":"research open","docstring":"$\\pi^e$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.pi ^ Real.exp 1)","subjects":["11","33"],"theorem":"Transcendental.pi_pow_exp_transcendental"},{"answerKinds":[],"category":"research open","docstring":"$\\pi^{\\pi^{\\pi^\\pi}}$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.pi ^ Real.pi ^ Real.pi ^ Real.pi)","subjects":["11","33"],"theorem":"Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental"},{"answerKinds":[],"category":"research open","docstring":"The Catalan constant $G$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ catalanConstant","subjects":["11","33"],"theorem":"Transcendental.transcendental_catalanConstant"},{"answerKinds":[],"category":"research solved","docstring":"$\\Gamma(1/6)$ is transcendental.\n\n[Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.Gamma (1 / 6))","subjects":["33"],"theorem":"Transcendental.transcendental_gamma_one_div_six"},{"answerKinds":[],"category":"research open","docstring":"$\\log(\\log(2))$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.log (Real.log 2))","subjects":["11","33"],"theorem":"Transcendental.rlog_rlog_two_transcendental"},{"answerKinds":[],"category":"research open","docstring":"$e^e$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.exp (Real.exp 1))","subjects":["11","33"],"theorem":"Transcendental.exp_exp_transcendental"},{"answerKinds":[],"category":"research open","docstring":"$\\pi^{\\sqrt{2}}$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.pi ^ √2)","subjects":["11","33"],"subsets":["FC100OpenSet1"],"theorem":"Transcendental.pi_pow_sqrt_two_transcendental"},{"answerKinds":[],"category":"research solved","docstring":"$\\Gamma(1/3)$ is transcendental.\n\n[Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.Gamma (1 / 3))","subjects":["33"],"theorem":"Transcendental.transcendental_gamma_one_div_three"},{"answerKinds":[],"category":"textbook","docstring":"At least one of $\\pi + e$ and $\\pi e$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.pi + Real.exp 1) ∨ Transcendental ℚ (Real.pi * Real.exp 1)","subjects":["11"],"theorem":"Transcendental.exp_add_pi_or_exp_add_mul_transcendental"},{"answerKinds":[],"category":"research open","docstring":"$\\Gamma(1/n)$ for `n ≥ 2` is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"∀ (n : ℕ), 2 ≤ n → Transcendental ℚ (Real.Gamma (1 / ↑n))","subjects":["33"],"theorem":"Transcendental.transcendental_gamma_one_div"},{"answerKinds":[],"category":"research open","docstring":"$e\\pi$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.exp 1 * Real.pi)","subjects":["11","33"],"theorem":"Transcendental.exp_mul_pi_transcendental"},{"answerKinds":[],"category":"research open","docstring":"$\\pi^{\\pi}$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.pi ^ Real.pi)","subjects":["11","33"],"theorem":"Transcendental.pi_pow_pi_transcendental"},{"answerKinds":[],"category":"research open","docstring":"$\\pi^{\\pi^{\\pi}}$ is transcendental.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.pi ^ Real.pi ^ Real.pi)","subjects":["11","33"],"subsets":["FC100OpenSet1"],"theorem":"Transcendental.pi_pow_pi_pow_pi_transcendental"},{"answerKinds":[],"category":"research solved","docstring":"$\\Gamma(1/2)$ is transcendental.\n\n[Ch84] Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Transcendental","statement":"Transcendental ℚ (Real.Gamma (1 / 2))","subjects":["33"],"theorem":"Transcendental.transcendental_gamma_one_div_two"},{"answerKinds":[],"category":"test","docstring":"The pair $(1,3)$ is the trivial solution of the Erdős–Moser equation. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.ErdosMoser","statement":"ErdosMoser.powerSum 1 3 = 3 ^ 1","subjects":["11"],"theorem":"ErdosMoser.powerSum_one_three"},{"answerKinds":[],"category":"test","docstring":"For $k=2$ and $m=3$, the left side is $1^2+2^2=5$, not $3^2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.ErdosMoser","statement":"ErdosMoser.powerSum 2 3 = 5 ∧ ErdosMoser.powerSum 2 3 ≠ 3 ^ 2","subjects":["11"],"theorem":"ErdosMoser.powerSum_two_three"},{"answerKinds":[],"category":"test","docstring":"Zero is a solution for every positive exponent, so the conjecture must require $m>0$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.ErdosMoser","statement":"∀ (k : ℕ), 0 < k → ErdosMoser.powerSum k 0 = 0 ^ k","subjects":["11"],"theorem":"ErdosMoser.powerSum_zero"},{"answerKinds":[],"category":"research open","docstring":"The only positive solution of $S_k(m)=m^k$ is $(k,m)=(1,3)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ErdosMoser","statement":"∀ (k m : ℕ), 0 < k → 0 < m → ErdosMoser.powerSum k m = m ^ k → k = 1 ∧ m = 3","subjects":["11"],"theorem":"ErdosMoser.erdos_moser_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"**The Lovász–Plummer conjecture (1970s), proved by Esperet, Kardoš, King, Král' and Norine\n(2011).**\n\nEvery bridgeless cubic graph on $n$ vertices has exponentially many perfect matchings: there is\na constant $c > 0$ such that the number of perfect matchings is at least $2^{cn}$.\n[EKKKN11] prove this with $2^{n/3656}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LovaszPlummerConjecture","statement":"∃ c,\n  0 < c ∧\n    ∀ {V : Type} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n      (∀ (v : V), G.degree v = 3) →\n        G.IsBridgeless → 2 ^ (c * ↑(Fintype.card V)) ≤ ↑(LovaszPlummerConjecture.perfectMatchingCount G)","subjects":["5"],"theorem":"LovaszPlummerConjecture.lovasz_plummer_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**Sheehan's conjecture (1977).**\n\nEvery $4$-regular graph with a Hamiltonian cycle has a second Hamiltonian cycle (one with a\ndifferent edge set). Sheehan's conjecture would settle the last open case of the question,\nraised by Smith's theorem for cubic graphs, of which regular Hamiltonian graphs have a second\nHamiltonian cycle: Thomassen [Th98] proved it for all $r$-regular graphs with $r \\ge 300$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LovaszPlummerConjecture","statement":"∀ {V : Type} [inst : Fintype V] [inst_1 : DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  (∀ (v : V), G.degree v = 4) →\n    ∀ (v : V) (c : G.Walk v v),\n      LovaszPlummerConjecture.IsHamiltonianCycle G c →\n        ∃ w c', LovaszPlummerConjecture.IsHamiltonianCycle G c' ∧ c'.edges.toFinset ≠ c.edges.toFinset","subjects":["5"],"theorem":"LovaszPlummerConjecture.sheehan_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"**The explicit bound of Esperet–Kardoš–King–Král'–Norine (2011).**\n\nEvery bridgeless cubic graph on $n$ vertices has at least $2^{n/3656}$ perfect matchings.\n\n*Reference:* [EKKKN11].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LovaszPlummerConjecture","statement":"∀ {V : Type} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  (∀ (v : V), G.degree v = 3) →\n    G.IsBridgeless → 2 ^ (↑(Fintype.card V) / 3656) ≤ ↑(LovaszPlummerConjecture.perfectMatchingCount G)","subjects":["5"],"theorem":"LovaszPlummerConjecture.lovasz_plummer_conjecture.variants.explicit"},{"answerKinds":[],"category":"research solved","docstring":"**Thomassen (1998): regular graphs of large degree.**\n\nEvery $r$-regular Hamiltonian graph with $r \\ge 300$ has a second Hamiltonian cycle.\n\n*Reference:* [Th98].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LovaszPlummerConjecture","statement":"∀ {V : Type} [inst : Fintype V] [inst_1 : DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj] (r : ℕ),\n  300 ≤ r →\n    (∀ (v : V), G.degree v = r) →\n      ∀ (v : V) (c : G.Walk v v),\n        LovaszPlummerConjecture.IsHamiltonianCycle G c →\n          ∃ w c', LovaszPlummerConjecture.IsHamiltonianCycle G c' ∧ c'.edges.toFinset ≠ c.edges.toFinset","subjects":["5"],"theorem":"LovaszPlummerConjecture.sheehan_conjecture.variants.thomassen"},{"answerKinds":[],"category":"test","docstring":"For example, `0` is part of an attracting `2`-cycle of `z ↦ z ^ 2 - 1`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"Mandelbrot.IsAttractingCycle (fun z => z ^ 2 - 1) 2 0","subjects":["37"],"theorem":"Mandelbrot.isAttractingCycle_z_squared_minus_one"},{"answerKinds":[],"category":"API","docstring":"The `multibrotSet n` is equivalently the set of all parameters `c` for which the orbit of `0`\nunder `z ↦ z ^ n + c` does not leave the closed disk of radius `2 ^ (n - 1)⁻¹` around the origin. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"∀ {n : ℕ}, 2 ≤ n → Mandelbrot.multibrotSet n = {c | ∀ (k : ℕ), ‖(fun z => z ^ n + c)^[k] 0‖ ≤ 2 ^ (↑n - 1)⁻¹}","subjects":["37"],"theorem":"Mandelbrot.multibrotSet_eq"},{"answerKinds":[],"category":"test","docstring":"On the other hand, while `2` is part of a `1`-cycle of `z ↦ z ^ 2 - 2`, that cycle is not\nattracting. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"¬Mandelbrot.IsAttractingCycle (fun z => z ^ 2 - 2) 1 2","subjects":["37"],"theorem":"Mandelbrot.not_isAttractingCycle_z_squared_minus_two"},{"answerKinds":[],"category":"research open","docstring":"The MLC conjecture, stating that the mandelbrot set is locally connected. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"LocallyConnectedSpace ↑Mandelbrot.mandelbrotSet","subjects":["37"],"subsets":["FC100OpenSet1"],"theorem":"Mandelbrot.MLC"},{"answerKinds":[],"category":"research open","docstring":"The boundary of any Multibrot set is conjectured to have zero area.\nNote that we don't need to exclude the trivial cases `n = 0` and `n = 1` because the conjecture\nholds for them. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"∀ {n : ℕ}, MeasureTheory.volume (frontier (Mandelbrot.multibrotSet n)) = 0","subjects":["37"],"theorem":"Mandelbrot.volume_frontier_multibrotSet_eq_zero"},{"answerKinds":[],"category":"test","docstring":"The boundary of any Multibrot set is measurable because it is closed, so it makes sense to\nask about its area. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"∀ {n : ℕ}, MeasurableSet (frontier (Mandelbrot.multibrotSet n))","subjects":["37"],"theorem":"Mandelbrot.multibrotSet_frontier_measurable"},{"answerKinds":[],"category":"research open","docstring":"The boundary of the Mandelbrot set is conjectured to have zero area. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"MeasureTheory.volume (frontier Mandelbrot.mandelbrotSet) = 0","subjects":["37"],"subsets":["FC100OpenSet1"],"theorem":"Mandelbrot.volume_frontier_mandelbrotSet_eq_zero"},{"answerKinds":[],"category":"research open","docstring":"The density of hyperbolicity conjecture for Multibrot sets, stating that the set of all\nparameters `c` for which `fun z ↦ z ^ n + c` has an attracting cycle is dense in `multibrotSet n`.\nNote that we need to require `2 ≤ n` because the conjecture is trivially false for `n = 1`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"∀ {n : ℕ},\n  2 ≤ n → Mandelbrot.multibrotSet n ⊆ closure {c | ∃ m z, Mandelbrot.IsAttractingCycle (fun z => z ^ n + c) m z}","subjects":["37"],"theorem":"Mandelbrot.density_of_hyperbolicity_general_exponent"},{"answerKinds":[],"category":"test","docstring":"No function has an attracting cycle of period `0`. This is important in that it means we don't\nneed to require `0 < n` in the conjectures below. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"∀ (f : ℂ → ℂ) (z : ℂ), ¬Mandelbrot.IsAttractingCycle f 0 z","subjects":["37"],"theorem":"Mandelbrot.no_attractingCycle_period_zero"},{"answerKinds":[],"category":"API","docstring":"The mandelbrot set is equivalently the set of all parameters `c` for which the orbit of `0`\nunder `z ↦ z ^ 2 + c` does not leave the closed disk of radius two around the origin. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"Mandelbrot.mandelbrotSet = {c | ∀ (k : ℕ), ‖(fun z => z ^ 2 + c)^[k] 0‖ ≤ 2}","subjects":["37"],"theorem":"Mandelbrot.mandelbrotSet_eq"},{"answerKinds":[],"category":"research open","docstring":"A stronger version of the MLC conjecture, stating that all multibrots are locally connected.\nNote that we don't need to require `2 ≤ n` because the conjecture holds in the trivial cases `n = 0`\nand `n = 1` too. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"∀ (n : ℕ), LocallyConnectedSpace ↑(Mandelbrot.multibrotSet n)","subjects":["37"],"theorem":"Mandelbrot.MLC_general_exponent"},{"answerKinds":[],"category":"research open","docstring":"The density of hyperbolicity conjecture, stating that the set of all parameters `c` for which\n`fun z ↦ z ^ 2 + c` has an attracting cycle is dense in the Mandelbrot set. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Mandelbrot","statement":"Mandelbrot.mandelbrotSet ⊆ closure {c | ∃ m z, Mandelbrot.IsAttractingCycle (fun z => z ^ 2 + c) m z}","subjects":["37"],"theorem":"Mandelbrot.density_of_hyperbolicity"},{"answerKinds":[],"category":"test","docstring":"The case $M = 0$ fails: the hypothesis is vacuous, so $a = 0$ cannot be concluded. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Buchi","statement":"¬Buchi.IsBuchi 0","subjects":["11"],"theorem":"Buchi.buchi_false_M0"},{"answerKinds":[],"category":"research open","docstring":"**Büchi's problem (first open case, $M = 5$)**:\nFor all integers $x$ and $a$, if $(x+n)^2 + a$ is a perfect square for $n = 0, 1, 2, 3, 4$,\nthen $a = 0$.\n\nNon-trivial sequences of length 3 and 4 are known to exist, so $M = 5$ is the first open case.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Buchi","statement":"Buchi.IsBuchi 5","subjects":["11"],"theorem":"Buchi.buchi_problem_M5"},{"answerKinds":[],"category":"test","docstring":"The case $M = 2$ fails: $(7+n)^2 - 48$ is a perfect square for $n = 0, 1$\n($1^2$ and $4^2$), but $a = -48 \\neq 0$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Buchi","statement":"¬Buchi.IsBuchi 2","subjects":["11"],"theorem":"Buchi.buchi_false_M2"},{"answerKinds":[],"category":"research open","docstring":"**Büchi's problem**\nThere exists a positive integer $M$ such that, for all integers $x$ and $a$,\nif $(x+n)^2 + a$ is a square for $M$ consecutive values of $n$, then $a = 0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Buchi","statement":"True ↔ ∃ M, 1 ≤ M ∧ Buchi.IsBuchi M","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Buchi.buchi_problem"},{"answerKinds":[],"category":"test","docstring":"The case $M = 1$ fails: $0^2 + 4 = 4 = 2^2$ is a square but $a = 4 \\neq 0$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Buchi","statement":"¬Buchi.IsBuchi 1","subjects":["11"],"theorem":"Buchi.buchi_false_M1"},{"answerKinds":[],"category":"test","docstring":"The case $M = 4$ fails: $(246+n)^2 - 60480$ is a perfect square for $n = 0, 1, 2, 3$\n($6^2$, $23^2$, $32^2$, and $39^2$), but $a = -60480 \\neq 0$.\nThe sequence $(6, 23, 32, 39)$ is a nontrivial Büchi sequence of length 4 (Hensley 1983). ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Buchi","statement":"¬Buchi.IsBuchi 4","subjects":["11"],"theorem":"Buchi.buchi_false_M4"},{"answerKinds":[],"category":"test","docstring":"The case $M = 3$ fails: $(24+n)^2 - 576$ is a perfect square for $n = 0, 1, 2$\n($0^2$, $7^2$, and $10^2$), but $a = -576 \\neq 0$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Buchi","statement":"¬Buchi.IsBuchi 3","subjects":["11"],"theorem":"Buchi.buchi_false_M3"},{"answerKinds":[],"category":"test","docstring":"A known rational Diophantine 6-tuple. [Gi99]\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DiophantineTuple","statement":"DiophantineTuple.IsRationalDiophantineTuple {11 / 192, 35 / 192, 155 / 27, 512 / 27, 1235 / 48, 180873 / 16}","subjects":["11"],"theorem":"DiophantineTuple.gibbs_6_tuple"},{"answerKinds":[],"category":"textbook","docstring":"If every integral Diophantine triple has a unique extension by a larger element, then there\nis no integral Diophantine 5-tuple. [Du]\n\nProof: suppose a 5-tuple $a < b < c < d_1 < d_2$ exists; then $d_1 = d_2$ by unique extension.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DiophantineTuple","statement":"DiophantineTuple.HasUniqueExtensionOfForall → DiophantineTuple.NoIntegralDiophantineFiveTuple","subjects":["11"],"theorem":"DiophantineTuple.noIntegralDiophantineFiveTuple_of_hasUniqueExtensionOfForall"},{"answerKinds":[],"category":"test","docstring":"There exists a Diophantine 4-tuple; this example is due to Fermat.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.DiophantineTuple","statement":"DiophantineTuple.IsIntegralDiophantineTuple {1, 3, 8, 120}","subjects":["11"],"theorem":"DiophantineTuple.fermat_4_tuple"},{"answerKinds":[],"category":"research open","docstring":"The \"strong Diophantine 5-tuple conjecture\", so-called because it implies the Diophantine\n5-tuple theorem (see `noIntegralDiophantineFiveTuple_of_hasUniqueExtensionOfForall`). [Du]\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DiophantineTuple","statement":"DiophantineTuple.HasUniqueExtensionOfForall","subjects":["11"],"theorem":"DiophantineTuple.hasUniqueExtension_of_forall"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a rational Diophantine 7-tuple? [Du16]\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DiophantineTuple","statement":"True ↔ ∃ t, DiophantineTuple.IsRationalDiophantineTuple t ∧ t.card = 7","subjects":["11"],"theorem":"DiophantineTuple.rational_7_tuple"},{"answerKinds":[],"category":"textbook","docstring":"The property of being a Diophantine tuple is closed under subsets. [Du16]\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DiophantineTuple","statement":"∀ {R : Type u_1} [inst : Semiring R] (s t : Finset R),\n  DiophantineTuple.IsDiophantineTuple t → s ⊆ t → DiophantineTuple.IsDiophantineTuple s","subjects":["11"],"theorem":"DiophantineTuple.isDiophantineTuple_of_subset"},{"answerKinds":[],"category":"research solved","docstring":"The Diophantine 5-tuple theorem: there does not exist an integral Diophantine 5-tuple. [HTZ19]\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DiophantineTuple","statement":"DiophantineTuple.NoIntegralDiophantineFiveTuple","subjects":["11"],"theorem":"DiophantineTuple.noIntegralDiophantineFiveTuple"},{"answerKinds":[],"category":"research solved","docstring":"`HasUniqueExtension` is known to hold for certain triples, including $\\{1, 3, 8\\}$: this is\nessentially the Baker–Davenport theorem [BD69], which states that $120$ is the only integer\n$d$ such that $\\{1, 3, 8, d\\}$ is a Diophantine tuple. Together with a finite check ruling out\n$d < 8$, this shows that no integral Diophantine tuple properly extends $\\{1, 3, 8, 120\\}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.DiophantineTuple","statement":"DiophantineTuple.HasUniqueExtension 1 3 8","subjects":["11"],"theorem":"DiophantineTuple.hasUniqueExtension_of_1_3_8"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RamanujanTau","statement":"RamanujanTau.τ 2 = -24","subjects":["11"],"theorem":"RamanujanTau.τ_two"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.RamanujanTau","statement":"RamanujanTau.τ 0 = 0","subjects":["11"],"theorem":"RamanujanTau.τ_zero"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RamanujanTau","statement":"Multipliable fun n => (1 - PowerSeries.X ^ ↑n) ^ 24","subjects":["11"],"theorem":"RamanujanTau.multipliable"},{"answerKinds":[],"category":"research solved","docstring":"The Ramanujan-Petersson conjecture: $|\\tau(p)| \\le 2 p^{11/2}$ for primes $p$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RamanujanTau","statement":"∀ (p : ℕ), Prime p → ↑|RamanujanTau.τ p| ≤ 2 * ↑p ^ (11 / 2)","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"RamanujanTau.ramanujan_petersson"},{"answerKinds":[],"category":"research open","docstring":"Lehmer's conjecture: $\\tau(n) \\ne 0$ for all $n > 0$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.RamanujanTau","statement":"∀ n > 0, RamanujanTau.τ n ≠ 0","subjects":["11"],"theorem":"RamanujanTau.lehmer_ramanujan_tau"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.RamanujanTau","statement":"RamanujanTau.τ 1 = 1","subjects":["11"],"theorem":"RamanujanTau.τ_one"},{"answerKinds":[],"category":"research open","docstring":"Weak form of Hall's conjecture: relax the exponent from $1/2$ to $1/2 - \\varepsilon$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Hall","statement":"∀ ε > 0, Hall.HallConjectureExp (2⁻¹ - ε)","subjects":["11"],"theorem":"Hall.weak_hall_conjecture"},{"answerKinds":[],"category":"test","docstring":"Elkies' example $(x, y) = (5853886516781223, 447884928428402042307918)$ shows that such $C$ must be\nless than $0.0215$. Note that simple `linarith` does not work here.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Hall","statement":"∀ (C : ℝ), Hall.HallIneq C 2⁻¹ → C < 215e-4","subjects":["11"],"theorem":"Hall.elkies_bound"},{"answerKinds":[],"category":"research solved","docstring":"Danilov proved that one cannot replace the exponent $1/2$ with larger number.\nIn other words, for any $\\delta > 0$, there is no positive constant $C$ such that\n$|y^2 - x^3| > C |x| ^ {1/2 + \\delta}$ for all integers $x, y$ with $y^2 \\ne x^3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Hall","statement":"∀ δ > 0, ¬Hall.HallConjectureExp (2⁻¹ + δ)","subjects":["11"],"theorem":"Hall.danilov"},{"answerKinds":[],"category":"research open","docstring":"Original Hall's conjecture with exponent $1/2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Hall","statement":"Hall.HallConjectureExp 2⁻¹","subjects":["11"],"theorem":"Hall.hall_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Tunnell's theorem (sufficient condition assuming BSD) for even squarefree congruent numbers. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CongruentNumber","statement":"∀ (n : ℕ),\n  Squarefree n →\n    Even n → 2 * (CongruentNumber.C n).ncard = (CongruentNumber.D n).ncard → CongruentNumber.congruentNumber n","subjects":["11"],"theorem":"CongruentNumber.Tunnell_even_converse"},{"answerKinds":[],"category":"research solved","docstring":"Tunnell's theorem (necessary condition) for even squarefree congruent numbers. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CongruentNumber","statement":"∀ (n : ℕ),\n  Squarefree n →\n    Even n → CongruentNumber.congruentNumber n → 2 * (CongruentNumber.C n).ncard = (CongruentNumber.D n).ncard","subjects":["11"],"theorem":"CongruentNumber.Tunnell_even"},{"answerKinds":[],"category":"textbook","docstring":"1 is not a congruent number, as proved by Fermat via infinite descent. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CongruentNumber","statement":"¬CongruentNumber.congruentNumber 1","subjects":["11"],"theorem":"CongruentNumber.not_congruentNumber_1"},{"answerKinds":[],"category":"test","docstring":"Zagier's rational right triangle witnesses that $157$ is a congruent number.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.CongruentNumber","statement":"CongruentNumber.congruentNumber 157","subjects":["11"],"theorem":"CongruentNumber.congruentNumber_157_zagier"},{"answerKinds":[],"category":"test","docstring":"The rational right triangle with side lengths $\\frac{3}{2}$, $\\frac{20}{3}$, and\n$\\frac{41}{6}$ witnesses that $5$ is a congruent number.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.CongruentNumber","statement":"CongruentNumber.congruentNumber 5","subjects":["11"],"theorem":"CongruentNumber.congruentNumber_5"},{"answerKinds":[],"category":"research solved","docstring":"Tunnell's theorem (necessary condition) for odd squarefree congruent numbers. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CongruentNumber","statement":"∀ (n : ℕ),\n  Squarefree n →\n    Odd n → CongruentNumber.congruentNumber n → 2 * (CongruentNumber.A n).ncard = (CongruentNumber.B n).ncard","subjects":["11"],"theorem":"CongruentNumber.Tunnell_odd"},{"answerKinds":[],"category":"test","docstring":"The rational right triangle with side lengths $\\frac{35}{12}$, $\\frac{24}{5}$, and\n$\\frac{337}{60}$ witnesses that $7$ is a congruent number.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.CongruentNumber","statement":"CongruentNumber.congruentNumber 7","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"CongruentNumber.congruentNumber_7"},{"answerKinds":[],"category":"test","docstring":"The $3$-$4$-$5$ right triangle witnesses that $6$ is a congruent number.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.CongruentNumber","statement":"CongruentNumber.congruentNumber 6","subjects":["11"],"theorem":"CongruentNumber.congruentNumber_6"},{"answerKinds":[],"category":"research open","docstring":"Tunnell's theorem (sufficient condition assuming BSD) for odd squarefree congruent numbers. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CongruentNumber","statement":"∀ (n : ℕ),\n  Squarefree n →\n    Odd n → 2 * (CongruentNumber.A n).ncard = (CongruentNumber.B n).ncard → CongruentNumber.congruentNumber n","subjects":["11"],"theorem":"CongruentNumber.Tunnell_odd_converse"},{"answerKinds":[],"category":"research solved","docstring":"The following tighter bound depending on the degree $d$ of the polynomial $p$,\nin the case of $p$ all roots having the same norm has been shown by Tischler.\n$|p(z) - p(c)|/|z-c| \\le (d-1)/d \\cdot |p'(z)|$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MeanValueProblem","statement":"∀ (p : Polynomial ℂ),\n  2 ≤ p.natDegree →\n    (∀ (x y : ℂ), p.IsRoot x ∧ p.IsRoot y → ‖x‖ = ‖y‖) →\n      ∀ (z : ℂ) (K : ℝ),\n        ∃ c,\n          Polynomial.eval c (Polynomial.derivative p) = 0 ∧\n            ‖Polynomial.eval z p - Polynomial.eval c p‖ / ‖z - c‖ ≤\n              (↑p.natDegree - 1) / ↑p.natDegree * ‖Polynomial.eval z (Polynomial.derivative p)‖","subjects":["12"],"theorem":"MeanValueProblem.mean_value_problem_of_roots_same_norm"},{"answerKinds":[],"category":"research open","docstring":"Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$\nthere is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MeanValueProblem","statement":"∀ (p : Polynomial ℂ),\n  2 ≤ p.degree →\n    ∀ (z : ℂ) (K : ℝ),\n      ∃ c,\n        Polynomial.eval c (Polynomial.derivative p) = 0 ∧\n          ‖Polynomial.eval z p - Polynomial.eval c p‖ / ‖z - c‖ ≤ ‖Polynomial.eval z (Polynomial.derivative p)‖","subjects":["12"],"theorem":"MeanValueProblem.mean_value_problem"},{"answerKinds":[],"category":"research solved","docstring":"The following weaker version of the mean value problem has been proven.\nGiven a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$\nthere a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ 4|p'(z)|$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MeanValueProblem","statement":"∀ (p : Polynomial ℂ),\n  2 ≤ p.degree →\n    ∀ (z : ℂ) (K : ℝ),\n      ∃ c,\n        Polynomial.eval c (Polynomial.derivative p) = 0 ∧\n          ‖Polynomial.eval z p - Polynomial.eval c p‖ / ‖z - c‖ ≤ 4 * ‖Polynomial.eval z (Polynomial.derivative p)‖","subjects":["12"],"theorem":"MeanValueProblem.mean_value_problem_leq_4"},{"answerKinds":[],"category":"research solved","docstring":"The following tighter bound depending on the degree $d$ of the polynomial $p$,\nin the case of $p$ only having real roots has been shown by Tischler.\n$|p(z)-p(c)|/|z-c| \\le (d-1)/d \\cdot |p'(z)|$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MeanValueProblem","statement":"∀ (p : Polynomial ℂ),\n  2 ≤ p.natDegree →\n    (∀ (x : ℂ), p.IsRoot x → x.im = 0) →\n      ∀ (z : ℂ) (K : ℝ),\n        ∃ c,\n          Polynomial.eval c (Polynomial.derivative p) = 0 ∧\n            ‖Polynomial.eval z p - Polynomial.eval c p‖ / ‖z - c‖ ≤\n              (↑p.natDegree - 1) / ↑p.natDegree * ‖Polynomial.eval z (Polynomial.derivative p)‖","subjects":["12"],"theorem":"MeanValueProblem.mean_value_problem_of_real_roots"},{"answerKinds":[],"category":"test","docstring":"Sanity check: digit reversal of `120` is `21`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.LychrelNumbers","statement":"LychrelNumbers.rev10 120 = 21","subjects":["11"],"theorem":"LychrelNumbers.rev10_120"},{"answerKinds":[],"category":"research open","docstring":"**Lychrel conjecture (base 10):** conjecturally, there are no Lychrel numbers in base 10.\n\nEquivalently, every positive integer eventually becomes a palindrome under the Lychrel iteration.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LychrelNumbers","statement":"True ↔ ∀ (n : ℕ), 0 < n → ¬LychrelNumbers.IsLychrel10 n","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"LychrelNumbers.no_lychrel_numbers_base10"},{"answerKinds":[],"category":"test","docstring":"Sanity check: `121` is a base-10 palindrome. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.LychrelNumbers","statement":"LychrelNumbers.IsPalindrome10 121","subjects":["11"],"theorem":"LychrelNumbers.palindrome_121"},{"answerKinds":[],"category":"test","docstring":"Sanity check: `56 → 121` in one Lychrel step. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.LychrelNumbers","statement":"LychrelNumbers.lychrelStep^[1] 56 = 121","subjects":["11"],"theorem":"LychrelNumbers.lychrelIter_56_one"},{"answerKinds":[],"category":"test","docstring":"Sanity check: the Lychrel iteration at `56` reaches a palindrome. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.LychrelNumbers","statement":"∃ k, LychrelNumbers.IsPalindrome10 (LychrelNumbers.lychrelStep^[k] 56)","subjects":["11"],"theorem":"LychrelNumbers.eventually_palindrome_56"},{"answerKinds":[],"category":"research open","docstring":"The first widely studied open case: whether `196` is a base-10 Lychrel number.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.LychrelNumbers","statement":"True ↔ LychrelNumbers.IsLychrel10 196","subjects":["11"],"theorem":"LychrelNumbers.isLychrel10_196"},{"answerKinds":[],"category":"API","docstring":"An equivalent formulation of `no_lychrel_numbers_base10`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.LychrelNumbers","statement":"(∀ (n : ℕ), 0 < n → ∃ k, LychrelNumbers.IsPalindrome10 (LychrelNumbers.lychrelStep^[k] n)) ↔\n  ∀ (n : ℕ), 0 < n → ¬LychrelNumbers.IsLychrel10 n","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"LychrelNumbers.eventually_palindrome_base10"},{"answerKinds":[],"category":"research solved","docstring":"**The Sato–Tate conjecture**: for a non-CM elliptic curve $E$ over $\\mathbb{Q}$ and\n$-1 \\le a \\le b \\le 1$, the proportion of primes $p < N$ whose normalized coefficient\nbelongs to $[a,b]$ tends to\n$$\n\\frac{2}{\\pi}\\int_a^b \\sqrt{1-x^2}\\,dx\n$$\nas $N \\to \\infty$.\n\nThe finitely many primes where the supplied equation fails to have integral\ncoefficients and nonsingular reduction do not affect this limit.\n\nEstablished through the work of Clozel–Harris–Taylor [CHT08], Taylor [Tay08],\nHarris–Shepherd-Barron–Taylor [HST10], and\nBarnet-Lamb–Geraghty–Harris–Taylor [BGHT11]. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.SatoTateConjecture","statement":"∀ (E : WeierstrassCurve ℚ) [inst : E.IsElliptic],\n  ¬SatoTateConjecture.HasCM E →\n    ∀ (a b : ℝ),\n      -1 ≤ a →\n        a ≤ b →\n          b ≤ 1 →\n            Filter.Tendsto (fun N => ↑(SatoTateConjecture.primeCountInInterval E a b N) / ↑N.primesBelow.card)\n              Filter.atTop (nhds (SatoTateConjecture.satoTateMeasure a b))","subjects":["11","14"],"theorem":"SatoTateConjecture.satoTate_conjecture"},{"answerKinds":[],"category":"test","docstring":"Sanity check: for $-1 \\le a \\le b \\le 1$, the closed-form interval mass `satoTateMeasure`\nagrees with the integral form `satoTateIntegral`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.SatoTateConjecture","statement":"∀ (a b : ℝ), -1 ≤ a → a ≤ b → b ≤ 1 → SatoTateConjecture.satoTateMeasure a b = SatoTateConjecture.satoTateIntegral a b","subjects":["11","14"],"theorem":"SatoTateConjecture.satoTateMeasure_eq_satoTateIntegral"},{"answerKinds":[],"category":"research open","docstring":"The Bing-Borsuk Conjecture: every $n$-dimensional homogeneous absolute neighborhood retract\nis a topological $n$-manifold. A topological space $X$ is an $n$-dimensional manifold\nwhen `T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X)`. The hypothesis `[MetrizableSpace X]`\nimplies `T2Space X` so this does not appear in the conclusion.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BingBorsuk","statement":"∀ (n : ℕ) (X : Type) [inst : TopologicalSpace X] [TopologicalSpace.MetrizableSpace X] [HomogeneousSpace X]\n  [IsAbsoluteNeighborhoodRetract X], HasLebesgueCoveringDimensionEq X n → Nonempty (ChartedSpace (Fin n → ℝ) X)","subjects":["54","57"],"theorem":"BingBorsuk.bing_borsuk_conjecture"},{"answerKinds":[],"category":"research open","docstring":"For all odd integers $n ≥ 9$ there are odd prime numbers $p,q,r,s$ and natural numbers $a,b$\nsuch that $p+2q = n$, $2+pq = 2^a+r$, $2p+q = 2^b+s$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Lemoine","statement":"∀ (n : ℕ),\n  8 < n →\n    Odd n →\n      ∃ p q r s a b,\n        Lemoine.OddPrime p ∧\n          Lemoine.OddPrime q ∧\n            Lemoine.OddPrime r ∧ Lemoine.OddPrime s ∧ p + 2 * q = n ∧ 2 + p * q = 2 ^ a + r ∧ 2 * p + q = 2 ^ b + s","subjects":["11"],"theorem":"Lemoine.lemoine_conjecture_extension"},{"answerKinds":[],"category":"research open","docstring":"For all odd integers $n ≥ 7$ there are prime numbers $p,q$ such that $n = p+2q$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Lemoine","statement":"∀ (n : ℕ), 6 < n → Odd n → ∃ p q, Nat.Prime p ∧ Nat.Prime q ∧ p + 2 * q = n","subjects":["11"],"theorem":"Lemoine.lemoine_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**Sendov's conjecture** states that for a polynomial\n$$f(z)=(z-r_{1})\\cdots (z-r_{n}),\\qquad (n\\geq 2)$$\nwith all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a\ndistance no more than $1$ from at least one critical point. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Sendov","statement":"∀ (n : ℕ), 2 ≤ n → Sendov.Nat.SatisfiesSendovConjecture n","subjects":["12","30","52"],"theorem":"Sendov.sendov_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"**Sendov's conjecture** states that for a polynomial\n$$f(z)=(z-r_{1})\\cdots (z-r_{n}),\\qquad (n\\geq 2)$$\nwith all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a\ndistance no more than $1$ from at least one critical point.\n\nIt has been shown that Sendov's conjecture holds when the degree of $n$ is at most $9$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Sendov","statement":"∀ n ∈ Set.Icc 2 9, Sendov.Nat.SatisfiesSendovConjecture n","subjects":["12","30","52"],"theorem":"Sendov.sendov_conjecture.variants.le_nine"},{"answerKinds":[],"category":"research solved","docstring":"**Sendov's conjecture** states that for a polynomial\n$$f(z)=(z-r_{1})\\cdots (z-r_{n}),\\qquad (n\\geq 2)$$\nwith all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a\ndistance no more than $1$ from at least one critical point.\n\nIt has been shown that Sendov's conjecture holds for polynomials of sufficiently large degree.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Sendov","statement":"∀ᶠ (n : ℕ) in Filter.atTop, Sendov.Nat.SatisfiesSendovConjecture n","subjects":["12","30","52"],"theorem":"Sendov.sendov_conjecture.variants.eventually_true"},{"answerKinds":[],"category":"research solved","docstring":"**Shitov's bound (2019)**: Every synchronizing DFA with $n$ states admits a synchronizing\nword of length at most $\\left(\\frac{7}{48} + \\frac{2 \\cdot 15625}{1597536}\\right) n^3 + o(n^3)$,\nwhere the $o(n^3)$ term is uniform over all alphabets. This is the best known upper bound\ntowards the Černý conjecture.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CernyConjecture","statement":"∃ f,\n  (f =o[Filter.atTop] fun n => n ^ 3) ∧\n    ∀ {α : Type u_1} {σ : Type u_2} [inst : Fintype σ] (M : DFA α σ),\n      M.IsSynchronizing →\n        ∃ w,\n          M.IsSynchronizingWord w ∧\n            ↑w.length ≤ (7 / 48 + 2 * 15625 / 1597536) * ↑(Fintype.card σ) ^ 3 + f ↑(Fintype.card σ)","subjects":["68"],"theorem":"CernyConjecture.shitov_upper_bound"},{"answerKinds":[],"category":"research open","docstring":"**Černý Conjecture**: Every synchronizing DFA with $n$ states admits a\nsynchronizing word of length at most $(n - 1)^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.CernyConjecture","statement":"True ↔\n  ∀ {α : Type u_1} {σ : Type u_2} [inst : Fintype σ] (M : DFA α σ),\n    M.IsSynchronizing → ∃ w, M.IsSynchronizingWord w ∧ w.length ≤ (Fintype.card σ - 1) ^ 2","subjects":["68"],"theorem":"CernyConjecture.cerny_conjecture"},{"answerKinds":[],"category":"research open","docstring":"**The zero-divisor conjecture**\n\nIf `G` is torsion-free, then the group algebra `K[G]` has no non-trivial zero divisors.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Kaplansky","statement":"∀ (K : Type u_1) [inst : Field K] (G : Type u_2) [inst_1 : Group G],\n  IsMulTorsionFree G → NoZeroDivisors (MonoidAlgebra K G)","subjects":["16","20"],"theorem":"Kaplansky.zero_divisor_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"If $P$ is the Promislow group, then the group ring $\\mathbb{C}[P]$ has a non-trivial unit.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Kaplansky","statement":"∃ u, ¬Kaplansky.IsTrivialUnit ↑u","subjects":["16","20"],"subsets":["FC100SolvedSet1"],"theorem":"Kaplansky.UnitConjecture.counterexamples.ii"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Kaplansky","statement":"∀ (K : Type u_1) [inst : Field K] (G : Type u_2) [inst_1 : Group G] {u : MonoidAlgebra K G},\n  Kaplansky.IsTrivialUnit u → IsUnit u","subjects":["16","20"],"theorem":"Kaplansky.IsTrivialUnit.isUnit"},{"answerKinds":[],"category":"API","docstring":"The Promislow group is torsion-free.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Kaplansky","statement":"IsMulTorsionFree Kaplansky.PromislowGroup","subjects":["20"],"theorem":"Kaplansky.promislow_group_is_torsionfree"},{"answerKinds":[],"category":"research solved","docstring":"There is a counterexample to **Unit Conjecture** in any characteristic.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Kaplansky","statement":"∀ (p : ℕ),\n  p = 0 ∨ Nat.Prime p → ∃ G x, ∃ (_ : IsMulTorsionFree G), ∃ K x_2, ∃ (_ : CharP K p), ∃ u, ¬Kaplansky.IsTrivialUnit ↑u","subjects":["16","20"],"theorem":"Kaplansky.counter_unit_conjecture_weak"},{"answerKinds":[],"category":"research open","docstring":"**The idempotent conjecture**\n\nIf `G` is torsion-free, then `K[G]` has no non-trivial idempotents.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Kaplansky","statement":"∀ (K : Type u_1) [inst : Field K] (G : Type u_2) [inst_1 : Group G],\n  IsMulTorsionFree G → ∀ (a : MonoidAlgebra K G), IsIdempotentElem a → a = 0 ∨ a = 1","subjects":["16","20"],"theorem":"Kaplansky.idempotent_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"If $P$ is the Promislow group, then the group ring $\\mathbb{F}_p[P]$ has a non-trivial unit.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.Kaplansky","statement":"∀ (p : ℕ) [hp : Fact (Nat.Prime p)], ∃ u, ¬Kaplansky.IsTrivialUnit ↑u","subjects":["16","20"],"theorem":"Kaplansky.UnitConjecture.counterexamples.i"},{"answerKinds":[],"category":"research solved","docstring":"The **Unit Conjecture** is false.\n\nAt least there is a counterexample for any prime and zero characteristic:\n[Mu21] Murray, A. (2021). More Counterexamples to the Unit Conjecture for Group Rings.\n[Pa21] Passman, D. (2021). On the counterexamples to the unit conjecture for group rings.\n[Ga24] Gardam, G. (2024). Non-trivial units of complex group rings.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.Kaplansky","statement":"∃ G x,\n  ∃ (_ : IsMulTorsionFree G),\n    ∀ (p : ℕ), p = 0 ∨ Nat.Prime p → ∃ K x_3, ∃ (_ : CharP K p), ∃ u, ¬Kaplansky.IsTrivialUnit ↑u","subjects":["16","20"],"theorem":"Kaplansky.counter_unit_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ be a finitely generated group, and assume there exists $n$ such that for every $g$ in $G$,\n$g^n = 1$. Is $G$ necessarily finite?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.BoundedBurnsideProblem","statement":"True ↔ ∀ (G : Type) [inst : Group G], Group.FG G → ∀ n > 0, (∀ (g : G), g ^ n = 1) → Finite G","subjects":["20"],"theorem":"BoundedBurnsideProblem.bounded_burnside_problem"},{"answerKinds":[],"category":"research open","docstring":"**Artin's Conjecture on Primitive Roots**, second half, power version\nIf $a = b^m$ is a perfect power of a number $b$ whose squarefree part $b_0\\equiv 1 \\pmod{4}$,\nthen the density of the set $S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$\nis given by\n$$C \\left(\\prod_{p \\mid m} \\frac{p(p-2)}{(p ^ 2 - p - 1)}\\right)\n\\left(1 - \\prod_{p \\mid \\gcd(b_0, m)} \\frac{1}{2 - p}\n\\prod_{p \\mid b_0, p\\nmid m} \\frac{1}{(1 + p - p ^ 2)}\\right),$$\nwhere $C$ is Artin's constant.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","statement":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          b.squarefreePart ≡ 1 [MOD 4] →\n            (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n              (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m *\n                ArtinPrimitiveRootsConjecture.entanglementFactor b m)\n              {p | Nat.Prime p}","subjects":["11"],"theorem":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one"},{"answerKinds":[],"category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, second half, conditional on GRH.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","statement":"∀ (a a_0 b : ℤ),\n  a = a_0 * b ^ 2 →\n    (∀ (n : ℤ) (m : ℕ), m ≠ 1 → a ≠ n ^ m) →\n      Squarefree a_0 →\n        ¬a_0 ≡ 1 [ZMOD 4] →\n          (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n              χ.IsPrimitive →\n                ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n            (ArtinPrimitiveRootsConjecture.S a).HasDensity ArtinPrimitiveRootsConjecture.ArtinConstant {p | Nat.Prime p}","subjects":["11"],"theorem":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, second half, power version, conditional on GRH\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","statement":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          ¬b.squarefreePart ≡ 1 [MOD 4] →\n            (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n                χ.IsPrimitive →\n                  ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n              (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n                (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m)\n                {p | Nat.Prime p}","subjects":["11"],"theorem":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one"},{"answerKinds":[],"category":"research open","docstring":"**Artin's Conjecture on Primitive Roots**, second half, power version\nIf $a = b^m$ is a perfect odd power of a number $b$ whose squarefree part\n$b_0\\not\\equiv 1 \\pmod{4}$, then the density of the set $S(a)$ of primes $p$ such that\n$a$ is a primitive root modulo $p$ is given by\n$$C\\prod_{p \\mid m} \\frac{p(p - 2)}{p^2 - p - 1}$$,\nwhere $C$ is Artin's constant.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","statement":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          ¬b.squarefreePart ≡ 1 [MOD 4] →\n            (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n              (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m)\n              {p | Nat.Prime p}","subjects":["11"],"theorem":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one"},{"answerKinds":[],"category":"research open","docstring":"**Artin's Conjecture on Primitive Roots**, first half.\nLet $a$ be an integer that is not a square number and not $−1$. Then the set $S(a)$\nof primes $p$ such that $a$ is a primitive root modulo $p$ has a positive asymptotic\ndensity inside the set of primes. In particular, $S(a)$ is infinite.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","statement":"∀ (a : ℤ), ¬IsSquare a → a ≠ -1 → ∃ x > 0, (ArtinPrimitiveRootsConjecture.S a).HasDensity x {p | Nat.Prime p}","subjects":["11"],"theorem":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, second half, different residue version\nIf $a$ is a square number or $a = −1$, then the density of the set $S(a)$ of primes\n$p$ such that $a$ is a primitive root modulo $p$ is $0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","statement":"∀ (a : ℤ), IsSquare a ∨ a = -1 → (ArtinPrimitiveRootsConjecture.S a).HasDensity 0 {p | Nat.Prime p}","subjects":["11"],"theorem":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_square_or_minus_one"},{"answerKinds":[],"category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, second half, power version, conditional on GRH.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","statement":"∀ (a m b : ℕ),\n  a = b ^ m →\n    (∀ (u v : ℕ), 1 < u → b ≠ v ^ u) →\n      1 < m →\n        Odd m →\n          b.squarefreePart ≡ 1 [MOD 4] →\n            (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n                χ.IsPrimitive →\n                  ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n              (ArtinPrimitiveRootsConjecture.S ↑a).HasDensity\n                (ArtinPrimitiveRootsConjecture.ArtinConstant * ArtinPrimitiveRootsConjecture.powCorrectionFactor m *\n                  ArtinPrimitiveRootsConjecture.entanglementFactor b m)\n                {p | Nat.Prime p}","subjects":["11"],"theorem":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one"},{"answerKinds":[],"category":"research solved","docstring":"**Artin's Conjecture on Primitive Roots**, first half, conditional on GRH.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","statement":"∀ (a : ℤ),\n  ¬IsSquare a →\n    a ≠ -1 →\n      (∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n          χ.IsPrimitive →\n            ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2) →\n        ∃ x > 0, (ArtinPrimitiveRootsConjecture.S a).HasDensity x {p | Nat.Prime p}","subjects":["11"],"theorem":"ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.i"},{"answerKinds":[],"category":"research open","docstring":"**Artin's Conjecture on Primitive Roots**, second half.\nWrite $a = a_0 b^2$ where $a_0$ is squarefree. Under the conditions that $a$ is not a perfect\npower and $a_0\\not\\equiv 1\\pmod{4}$ (sequence A85397 in the OEIS), the density of the set\n$S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$ is independent of $a$ and\nequals Artin's constant.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture","statement":"∀ (a a_0 b : ℤ),\n  a = a_0 * b ^ 2 →\n    (∀ (n : ℤ) (m : ℕ), m ≠ 1 → a ≠ n ^ m) →\n      Squarefree a_0 →\n        ¬a_0 ≡ 1 [ZMOD 4] →\n          (ArtinPrimitiveRootsConjecture.S a).HasDensity ArtinPrimitiveRootsConjecture.ArtinConstant {p | Nat.Prime p}","subjects":["11"],"theorem":"ArtinPrimitiveRootsConjecture.artin_primitive_roots.parts.ii"},{"answerKinds":[],"category":"textbook","docstring":"There exist unique constants $A$, $B$, $\\varphi$, and $\\theta$ satisfying the spec. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MovingSofa","statement":"∃! ABφθ, MovingSofa.GerversSofa.ABφθSpec ABφθ.1 ABφθ.2.1 ABφθ.2.2.1 ABφθ.2.2.2","subjects":["49"],"theorem":"MovingSofa.GerversSofa.ABφθSpec.existsUnique"},{"answerKinds":["non-Prop"],"category":"research solved","docstring":"What is the sofa constant? ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MovingSofa","statement":"MovingSofa.sofaConstant = MeasureTheory.volume MovingSofa.gerversSofa","subjects":["49"],"theorem":"MovingSofa.sofaConstant_eq"},{"answerKinds":[],"category":"research solved","docstring":"Gerver's sofa attains the sofa constant, conjectured by [Ge92] and claimed by [Ba24]. ","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MovingSofa","statement":"MovingSofa.sofaConstant = MeasureTheory.volume MovingSofa.gerversSofa","subjects":["49"],"theorem":"MovingSofa.sofaConstant_eq_volume_gerversSofa"},{"answerKinds":[],"category":"test","docstring":"The unit square $[0,1]^2$ is a valid moving sofa (with the identity motion).\nIt sits in the corner where both hallways overlap, so the stationary motion works.\nThis is a sanity check that the `IsMovingSofa` definition is not vacuous.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.MovingSofa","statement":"∃ m, MovingSofa.IsMovingSofa MovingSofa.unitSquare m","subjects":["49"],"theorem":"MovingSofa.isMovingSofa_unitSquare"},{"answerKinds":[],"category":"research open","docstring":"Gerver's sofa is the unique sofa that attains the sofa constant, up to a rigid motion.\n\nThe motion is needed: `horizontalHallway` is $(-\\infty, 1] \\times [0, 1]$, so a leftward\ntranslate of any moving sofa is again one, obtained by sliding right and then following the\noriginal motion. It has the same area, so uniqueness cannot hold on the nose.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.MovingSofa","statement":"∀ (s : Set (EuclideanSpace ℝ (Fin 2))),\n  (∃ m, MovingSofa.IsMovingSofa s m) →\n    (MeasureTheory.volume s = MovingSofa.sofaConstant ↔ ∃ g, s = ⇑g '' MovingSofa.gerversSofa)","subjects":["49"],"theorem":"MovingSofa.volume_eq_sofaConstant_iff_congruent_gerversSofa"},{"answerKinds":[],"category":"test","docstring":"The sofa constant is at least 1, as witnessed by the unit square. ","hasSorryFreeProof":true,"module":"FormalConjectures.Wikipedia.MovingSofa","statement":"1 ≤ MovingSofa.sofaConstant","subjects":["49"],"theorem":"MovingSofa.one_le_sofaConstant"},{"answerKinds":[],"category":"research open","docstring":"The **Fermat–Catalan conjecture** states that the equation\n$a^m + b^n = c^k$ has only finitely many solutions $(a,b,c,m,n,k)$ with distinct triplets of values\n$(a^m, b^n, c^k)$ where $a, b, c$ are positive coprime integers and $m, n, k$ are positive integers satisfying\n$\\frac 1 m + \\frac 1 n + \\frac 1 k < 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.FermatCatalanConjecture","statement":"FermatCatalanConjecture.fermatCatalanConjecture","subjects":["11"],"theorem":"FermatCatalanConjecture.fermat_catalan"},{"answerKinds":[],"category":"research solved","docstring":"By the **Darmon-Granville** theorem,\nfor any fixed choice of positive integers m, n and k satisfying $\\frac 1 m + \\frac 1 n + \\frac 1 k < 1$,\nonly finitely many coprime triples $(a, b, c)$ solving $a^m + b^n = c^k$ exist.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Wikipedia.FermatCatalanConjecture","statement":"∀ (m n k : ℕ),\n  0 < m →\n    0 < n →\n      0 < k →\n        1 / ↑m + 1 / ↑n + 1 / ↑k < 1 →\n          {(a, b, c) | 0 < a ∧ 0 < b ∧ 0 < c ∧ a ^ m + b ^ n = c ^ k ∧ {a, b, c}.Pairwise Nat.Coprime}.Finite","subjects":["11"],"theorem":"FermatCatalanConjecture.fermat_catalan.variants.darmon_granville"},{"answerKinds":[],"category":"research solved","docstring":"The exceptional set for Problem 10.8 has Hausdorff dimension zero (Einsiedler\nand Kleinbock [EK07].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_8","statement":"∀ (p : ℕ),\n  Nat.Prime p →\n    dimH {ξ | Filter.liminf (fun q => ↑q * ↑(padicNorm p ↑q) * distToNearestInt (↑q * ξ)) Filter.atTop ≠ 0} = 0","subjects":["11"],"theorem":"Bugeaud08.problem_10_8.variants.exceptional_set_dimH_zero"},{"answerKinds":[],"category":"research solved","docstring":"The quadratic case of Problem 10.8. de Mathan and Teulié [dMT04] solved the\n$p$-adic Littlewood conjecture for quadratic real numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_8","statement":"∀ (ξ : ℝ) (p : ℕ),\n  Nat.Prime p →\n    (minpoly ℚ ξ).natDegree = 2 → sInf {x | ∃ q, 1 ≤ q ∧ x = ↑q * ↑(padicNorm p ↑q) * distToNearestInt (↑q * ξ)} = 0","subjects":["11"],"theorem":"Bugeaud08.problem_10_8.variants.quadratic"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.8 ($p$-adic Littlewood conjecture). For every real number $\\xi$ and\nevery prime number $p$,\n$$\\inf_{q \\ge 1} q \\cdot \\lVert q \\xi \\rVert \\cdot |q|_p = 0,$$\nwhere $\\lVert \\cdot \\rVert$ denotes the distance to the nearest integer and\n$|\\cdot|_p$ denotes the $p$-adic absolute value. Posed by de Mathan and\nTeulié [dMT04].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_8","statement":"∀ (ξ : ℝ) (p : ℕ), Nat.Prime p → sInf {x | ∃ q, 1 ≤ q ∧ x = ↑q * ↑(padicNorm p ↑q) * distToNearestInt (↑q * ξ)} = 0","subjects":["11"],"theorem":"Bugeaud08.problem_10_8"},{"answerKinds":[],"category":"test","docstring":"`mSeq` has intermediate (subexponential but super-polynomial) growth: for every\n`0 < α < 1` its terms eventually dominate $\\exp(n^\\alpha)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_6","statement":"∀ (α : ℝ), 0 < α → α < 1 → Bugeaud06.HasIntermediateGrowth α Bugeaud06.mSeq","subjects":["11"],"theorem":"Bugeaud06.example_hasIntermediateGrowth"},{"answerKinds":[],"category":"research solved","docstring":"The **Pollington–de Mathan theorem** [Pol79][Mat80]. For every lacunary sequence\n$(m_n)_{n \\ge 1}$ of positive integers, the set of real numbers $\\xi$ for which\n$(\\{\\xi m_n\\})_{n \\ge 1}$ is *not* dense modulo one has full Hausdorff dimension. ","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_6","statement":"∀ (m : ℕ → ℕ), (∀ (n : ℕ), 0 < m n) → IsLacunary m → dimH {ξ | ¬Dense (Set.range fun n => ↑(ξ * ↑(m n)))} = 1","subjects":["11"],"theorem":"Bugeaud06.pollington_de_mathan"},{"answerKinds":[],"category":"research solved","docstring":"**Boshernitzan's theorem** [Bos94]. Given a real sublacunary sequence $r$, the set of\nreal numbers $\\xi$ for which $(\\{\\xi r_n\\})_{n \\ge 1}$ is *not* dense modulo one has\nHausdorff dimension zero. ","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_6","statement":"∀ (r : ℕ → ℝ),\n  (∀ (n : ℕ), 0 < r n) →\n    ¬BddAbove (Set.range r) →\n      Filter.Tendsto (fun n => r (n + 1) / r n) Filter.atTop (nhds 1) →\n        dimH {ξ | ¬Dense (Set.range fun n => ↑(ξ * r n))} = 0","subjects":["11"],"theorem":"Bugeaud06.boshernitzan"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.6. Find a very rapidly increasing sequence $(m_n)_{n \\ge 1}$ of positive\nintegers such that $(\\{\\xi m_n\\})_{n \\ge 1}$ is dense modulo one for every irrational\nnumber $\\xi$. Note: Furstenberg's $2^m3^n$ is sublacunary but requires two parameters. ","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_6","statement":"∃ m,\n  StrictMono m ∧ Bugeaud06.IsGenuinelySublacunary m ∧ ∀ (ξ : ℝ), Irrational ξ → Dense (Set.range fun n => ↑(ξ * ↑(m n)))","subjects":["11"],"theorem":"Bugeaud06.problem_10_6_variant_1"},{"answerKinds":[],"category":"test","docstring":"The sequence `mSeq`, given by $m_{n+1} = \\lceil m_n (1 + 1/\\log n) \\rceil$, is\ngenuinely sublacunary: taking $c = 1$, we have $m_{n+1}/m_n \\ge 1 + 1/\\log n$ because\n$\\lceil m_n (1 + 1/\\log n) \\rceil \\ge m_n (1 + 1/\\log n)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_6","statement":"Bugeaud06.IsGenuinelySublacunary Bugeaud06.mSeq","subjects":["11"],"theorem":"Bugeaud06.example_isGenuineSublacunary"},{"answerKinds":[],"category":"test","docstring":"The Pollington–de Mathan theorem implies that a lacunary sequence cannot answer\nProblem 10.6. ","hasSorryFreeProof":true,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_6","statement":"(∀ (m : ℕ → ℕ), (∀ (n : ℕ), 0 < m n) → IsLacunary m → dimH {ξ | ¬Dense (Set.range fun n => ↑(ξ * ↑(m n)))} = 1) →\n  ∃ m, (∀ (n : ℕ), 0 < m n) ∧ IsLacunary m ∧ ¬∀ (ξ : ℝ), Irrational ξ → Dense (Set.range fun n => ↑(ξ * ↑(m n)))","subjects":["11"],"theorem":"Bugeaud06.problem_lacunary_not_dense_of_pollington_de_mathan"},{"answerKinds":[],"category":"research solved","docstring":"**Furstenberg's theorem** [Fur67] (the $\\times 2, \\times 3$ case). For every irrational\nnumber $\\xi$, the two-parameter family $(\\{\\xi \\, 2^m 3^n\\})_{m, n \\ge 1}$ is dense modulo\none. ","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_6","statement":"∀ (ξ : ℝ), Irrational ξ → Dense {x | ∃ m n, 0 < m ∧ 0 < n ∧ x = ↑(ξ * ↑(2 ^ m * 3 ^ n))}","subjects":["11"],"theorem":"Bugeaud06.furstenberg_two_three"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.6, intermediate-growth variant. ","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_6","statement":"∃ m,\n  StrictMono m ∧\n    (∃ α, 0 < α ∧ α < 1 ∧ Bugeaud06.HasIntermediateGrowth α m) ∧\n      ∀ (ξ : ℝ), Irrational ξ → Dense (Set.range fun n => ↑(ξ * ↑(m n)))","subjects":["11"],"theorem":"Bugeaud06.problem_10_6_variant_2"},{"answerKinds":[],"category":"research solved","docstring":"Mahler [Mah68] proved that the set of Z-numbers has Lebesgue measure zero.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_9","statement":"MeasureTheory.volume {ξ | IsZNumber ξ} = 0","subjects":["11"],"theorem":"Bugeaud09.problem_10_9.variants.measure_zero"},{"answerKinds":[],"category":"test","docstring":"Sanity check: `1` is not a Z-number, since for `n = 1` we have\n$\\{1 \\cdot (3/2)^1\\} = \\{3/2\\} = 1/2 \\not< 1/2$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_9","statement":"¬IsZNumber 1","subjects":["11"],"theorem":"Bugeaud09.not_isZNumber_one"},{"answerKinds":[],"category":"research solved","docstring":"Flatto, Lagarias, and Pollington [FLP95] proved that the set of Z-numbers has\nHausdorff dimension strictly less than $1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_9","statement":"dimH {ξ | IsZNumber ξ} < 1","subjects":["11"],"theorem":"Bugeaud09.problem_10_9.variants.dimH_lt_one"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.9. There are no real numbers $\\xi$ such that $0 \\le \\{\\xi (3/2)^n\\} < 1/2$\nfor every positive integer $n$, i.e. no Z-number exists. Posed by Mahler [Mah68].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_9","statement":"∀ (x : ℝ), IsZNumber x → False","subjects":["11"],"theorem":"Bugeaud09.problem_10_9"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.5 (first part). Let $\\mathbb{K}$ be a real number field. Then, for any\n$\\varepsilon > 0$, there exists a lacunary sequence $(t_n)_{n \\ge 1}$ of positive numbers\nin $\\mathbb{K}$ such that\n$$\\limsup_{n \\to \\infty} \\{\\xi t_n\\} \\ge 1 - \\varepsilon,$$\nfor any real number $\\xi$ not in $\\mathbb{K}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_5","statement":"∀ (K : IntermediateField ℚ ℝ) [FiniteDimensional ℚ ↥K] {ε : ℝ},\n  0 < ε →\n    ∃ t,\n      (∀ (n : ℕ), 0 < ↑(t n)) ∧\n        (IsLacunaryReal fun k => ↑(t k)) ∧ ∀ ξ ∉ K, 1 - ε ≤ Filter.limsup (fun n => Int.fract (ξ * ↑(t n))) Filter.atTop","subjects":["11"],"theorem":"Bugeaud05.problem_10_5"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.5 (\"moreover\" clause). With the same hypotheses as `problem_10_5`, the\nsequence $(t_n)$ can be chosen so that, for any real $\\xi$ not in $\\mathbb{K}$, each\nsubinterval of $[0, 1]$ of length $\\varepsilon$ contains a limit point of the sequence\n$(\\{\\xi t_n\\})_{n \\ge 1}$. This is strictly stronger than `problem_10_5`: the limsup\nbound is the special case at the subinterval $[1 - \\varepsilon, 1]$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_5","statement":"∀ (K : IntermediateField ℚ ℝ) [FiniteDimensional ℚ ↥K] {ε : ℝ},\n  0 < ε →\n    ∃ t,\n      (∀ (n : ℕ), 0 < ↑(t n)) ∧\n        (IsLacunaryReal fun k => ↑(t k)) ∧\n          ∀ ξ ∉ K,\n            ∀ a ∈ Set.Icc 0 (1 - ε),\n              ∃ y ∈ Set.Icc a (a + ε), MapClusterPt y Filter.atTop fun n => Int.fract (ξ * ↑(t n))","subjects":["11"],"theorem":"Bugeaud05.problem_10_5_moreover"},{"answerKinds":[],"category":"test","docstring":"The \"moreover\" form of Problem 10.5 implies the first part: applying the cluster-point\ndensity to the subinterval $[1 - \\varepsilon, 1]$ yields the required lower bound on the\nlimsup. ","hasSorryFreeProof":true,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_5","statement":"(∀ (K : IntermediateField ℚ ℝ) [FiniteDimensional ℚ ↥K] {ε : ℝ},\n    0 < ε →\n      ∃ t,\n        (∀ (n : ℕ), 0 < ↑(t n)) ∧\n          (IsLacunaryReal fun k => ↑(t k)) ∧\n            ∀ ξ ∉ K,\n              ∀ a ∈ Set.Icc 0 (1 - ε),\n                ∃ y ∈ Set.Icc a (a + ε), MapClusterPt y Filter.atTop fun n => Int.fract (ξ * ↑(t n))) →\n  ∀ (K : IntermediateField ℚ ℝ) [FiniteDimensional ℚ ↥K] {ε : ℝ},\n    0 < ε →\n      ∃ t,\n        (∀ (n : ℕ), 0 < ↑(t n)) ∧\n          (IsLacunaryReal fun k => ↑(t k)) ∧\n            ∀ ξ ∉ K, 1 - ε ≤ Filter.limsup (fun n => Int.fract (ξ * ↑(t n))) Filter.atTop","subjects":["11"],"theorem":"Bugeaud05.problem_10_5_of_moreover"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.4. Let $\\xi$ be a non-zero real number and $\\alpha > 1$ be a real\nnumber. The spectrum of the sequence $(\\xi \\alpha^n)_{n \\ge 1}$ is at most\ncountable. Posed by Mendès France [Men73].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_4","statement":"∀ (ξ : ℝ), ξ ≠ 0 → ∀ (α : ℝ), 1 < α → (Bugeaud04.Spectrum fun n => ξ * α ^ n).Countable","subjects":["11"],"theorem":"Bugeaud04.spectrum_xi_alpha_pow_countable"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.1. Are there a transcendental number $\\alpha$ and a positive real\nnumber $\\xi$ such that $\\lVert \\xi \\alpha^n \\rVert$ tends to~$0$ as~$n$ tends to infinity? [Har19]\n(Trivial for $|\\alpha| < 1$)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.IntDistanceDistribution","statement":"True ↔\n  ∃ α ξ,\n    1 < |α| ∧ Transcendental ℚ α ∧ 0 < ξ ∧ Filter.Tendsto (fun n => distToNearestInt (ξ * α ^ n)) Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Bugeaud01.problem_10_1"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.2. To prove that $\\lVert e^n \\rVert$ does not tend to 0 as n tends to\ninfinity.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.IntDistanceDistribution","statement":"¬Filter.Tendsto (fun n => distToNearestInt (Real.exp ↑n)) Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Bugeaud01.problem_10_2"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.3. To prove that there exists a positive real number~$c$ such\nthat $\\lVert e^n \\rVert > e^{−cn}$, for every~$n \\ge 1$. Posed by Mahler [Mah53].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.IntDistanceDistribution","statement":"∃ c, 0 < c ∧ ∀ (n : ℕ), 1 ≤ n → Real.exp (-c * ↑n) < distToNearestInt (Real.exp ↑n)","subjects":["11"],"theorem":"Bugeaud01.problem_10_3"},{"answerKinds":[],"category":"test","docstring":"Waldschmidt's conjecture is stronger than Mahler's: since $\\log n \\le n$ for $n \\ge 1$,\nthe polynomial lower bound $n^{-c}$ dominates the exponential lower bound $e^{-cn}$.\nFor the $n = 1$ case (not covered by Waldschmidt's $n \\ge 2$), we choose a larger constant\nusing the numerical bound $\\lVert e \\rVert = 3 - e > 0$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.IntDistanceDistribution","statement":"(∃ c, 0 < c ∧ ∀ (n : ℕ), 2 ≤ n → ↑n ^ (-c) < distToNearestInt (Real.exp ↑n)) →\n  ∃ c, 0 < c ∧ ∀ (n : ℕ), 1 ≤ n → Real.exp (-c * ↑n) < distToNearestInt (Real.exp ↑n)","subjects":["11"],"theorem":"Bugeaud01.problem_10_3_of_waldschmidt"},{"answerKinds":[],"category":"research open","docstring":"Waldschmidt [Wal03] conjectured that a stronger result holds, namely\nthat there exists a positive real number~$c$ such that $\\lVert e^n \\rVert > n^{-c}$ for\nevery~$n \\ge 2$. This is supported by metrical results [Kok45].\n\nNote: the bound $n^{-c}$ equals $1$ when $n = 1$ for all $c$, while the distance to the nearest\ninteger is always at most $1/2$, so the conjecture must start at $n \\ge 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.IntDistanceDistribution","statement":"∃ c, 0 < c ∧ ∀ (n : ℕ), 2 ≤ n → ↑n ^ (-c) < distToNearestInt (Real.exp ↑n)","subjects":["11"],"theorem":"Bugeaud01.waldschmidt"},{"answerKinds":[],"category":"research open","docstring":"Problem 10.7. Let $\\varepsilon$ be a positive real number. Are there arbitrarily\nlarge real numbers $\\alpha$ such that $\\alpha$ is not a Pisot number and all the\nfractional parts $\\{\\alpha^n\\}$, $n \\ge 1$, are lying in an interval of length\n$\\varepsilon / \\alpha$? [Bug12b]\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_7","statement":"True ↔\n  ∀ (ε : ℝ),\n    0 < ε → ∀ (M : ℝ), ∃ α, M < α ∧ ¬IsPisot α ∧ ∃ c, ∀ (n : ℕ), 1 ≤ n → Int.fract (α ^ n) ∈ Set.Icc c (c + ε / α)","subjects":["11"],"theorem":"Bugeaud07.problem_10_7"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does the series\n$$\n  \\sum_{n=1}^{\\infty} \\frac{\\left(\\frac{2}{3} + \\frac{1}{3}\\sin n\\right)^n}{n}\n$$\nconverge?\n\nAfter computing approximately $10^7$ terms, the partial sums approximate $2.163$.\n\nSee https://arxiv.org/abs/2007.11017 for a proof of the convergence,\nrelying on an irrationality measure for pi.\n\nAlso see\nhttps://github.com/AxiomMath/gdm-formal-conjectures/blob/main/docs/BorweinSineSeries.md\nfor a partial formalization of the conjecture,\nconditional on such an irrationality measure of pi (cf https://arxiv.org/abs/1912.06345).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/AxiomMath/gdm-formal-conjectures/blob/main/BorweinSineSeries/solution.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Books.BorweinSineSeries","statement":"True ↔ Summable fun n => (2 / 3 + 1 / 3 * Real.sin ↑↑n) ^ ↑n / ↑↑n","subjects":["26","40"],"theorem":"BorweinSineSeries.borwein_sine_series"},{"answerKinds":[],"category":"research open","docstring":"The sequence `(3/2)^n` is equidistributed modulo `1`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.UniformDistributionOfSequences.Equidistribution","statement":"IsEquidistributedModuloOne fun n => (3 / 2) ^ n","subjects":["11"],"theorem":"Equidistribution.isEquidistributedModuloOne_three_halves_pow"},{"answerKinds":[],"category":"research open","docstring":"For any transcendental number `x`, the sequence `x * (3 / 2) ^ n` is\nequidistributed modulo 1. ","hasSorryFreeProof":false,"module":"FormalConjectures.Books.UniformDistributionOfSequences.Equidistribution","statement":"∀ (x : ℝ), Transcendental ℚ x → IsEquidistributedModuloOne fun n => x * (3 / 2) ^ n","subjects":["11"],"theorem":"Equidistribution.isEquidistributedModuloOne_transcendental_three_halves_pow"},{"answerKinds":[],"category":"textbook","docstring":"If a point `x` is an accumulation point of a sequence `s_0, s_1, ...` then\nthere is a subsequence of `s` that tends to `x`\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.UniformDistributionOfSequences.Equidistribution","statement":"∀ (x : ℝ) (s : ℕ → ℝ),\n  Equidistribution.IsAccumulationPoint x s → ∃ u, StrictMono u ∧ Filter.Tendsto (s ∘ u) Filter.atTop (nhds x)","subjects":["11","54"],"theorem":"Equidistribution.isAccumulationPoint_exists_subsequence_tendsto"},{"answerKinds":[],"category":"research solved","docstring":"The sequence `(3/2)^n` has infinitely many accumulation points modulo `1`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.UniformDistributionOfSequences.Equidistribution","statement":"{x | Equidistribution.IsAccumulationPoint x fun n => Int.fract ((3 / 2) ^ n)}.Infinite","subjects":["11"],"theorem":"Equidistribution.isAccumulationPoint_three_halves_pow_infinite"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Find an accumulation point of the sequence `(3/2)^n` modulo `1`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.UniformDistributionOfSequences.Equidistribution","statement":"Equidistribution.IsAccumulationPoint sorry fun n => Int.fract ((3 / 2) ^ n)","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Equidistribution.isAccumulationPoint_three_halves_pow"},{"answerKinds":[],"category":"test","docstring":"There is an accumulation point of the sequence `(3/2)^n` modulo `1`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Books.UniformDistributionOfSequences.Equidistribution","statement":"∃ p, Equidistribution.IsAccumulationPoint p fun n => Int.fract ((3 / 2) ^ n)","subjects":["11"],"theorem":"Equidistribution.isAccumulationPoint_three_halves_pow_exists"},{"answerKinds":[],"category":"test","docstring":"The rational number $1/2$ is fusible. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.FusibleNumber","statement":"FusibleNumber.IsFusible (1 / 2)","subjects":["5"],"theorem":"FusibleNumber.isFusible_one_half"},{"answerKinds":[],"category":"research open","docstring":"If `x` is a fusible number and `y` is its successor, then the interval `[x + 1, y + 1)` can be\ndivided into intervals `[ℓₙ, ℓₙ₊₁)`, such that the fusible numbers in `[ℓₙ, ℓₙ₊₁)` are obtained by\nfusing the `n + 1`st successor of `x` with a fusible number.\nThis formalization differs from Conjecture 7.1 in the paper in four ways:\n(1) it is obtained from Conjecture 7.1 by plugging in `n + 1` into `n`, which simplifies the expressions\n  and removes the need to assume `n ≥ 1`;\n(2) the `n + 1`st successor `s^(n+1)(x)` is replaced by the explicit value `x + (2 - 1 / 2 ^ n) * m`;\n(3) instead of defining `y` to be the successor of `x`, we assert that there is no fusible number\n  strictly between `x` and `y`;\n(4) instead of using `∃ z, IsFusible z ∧ q = s^(n+1)(x) ~ z` we use the value of `z` determined by the equality,\n  namely `z = 2 * q - 1 - s^(n+1)(x)`, and it is easy to see `z ∈ [x + 1 - m / 2 ^ n, x + 1)` as required. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.FusibleNumber","statement":"∀ (x y q : ℚ) (n : ℕ),\n  FusibleNumber.IsFusible x →\n    FusibleNumber.IsFusible y →\n      x < y →\n        (∀ (z : ℚ), FusibleNumber.IsFusible z → z ∉ Set.Ioo x y) →\n          have m := y - x;\n          have ℓ := fun n => y + 1 - m / 2 ^ n;\n          FusibleNumber.IsFusible q →\n            q ∈ Set.Ico (ℓ n) (ℓ (n + 1)) → FusibleNumber.IsFusible (2 * q - 1 - x - (2 - 1 / 2 ^ n) * m)","subjects":["5"],"theorem":"FusibleNumber.conj_7_1"},{"answerKinds":[],"category":"test","docstring":"The rational number $1$ is fusible. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.FusibleNumber","statement":"FusibleNumber.IsFusible 1","subjects":["5"],"theorem":"FusibleNumber.isFusible_one"},{"answerKinds":[],"category":"research open","docstring":"For a graph $G$, we define $\\Delta(G)$ to be the maximum degree, $\\omega(G)$ to be the size of the\nlargest clique subgraph, and $\\chi(G)$ to be the chromatic number. Reed's omega, delta, and chi\nconjecture states that $$\\chi(G) \\leq \\lceil \\frac{1}{2}(\\omega(G) + \\Delta(G) + 1) \\rceil.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ReedOmegaDeltaChi","statement":"∀ {V : Type} (G : SimpleGraph V),\n  have χ := G.chromaticNumber;\n  have ω := G.ecliqueNum;\n  have Δ := G.emaxDegree;\n  2 * χ ≤ ω + Δ + 2","subjects":["5"],"theorem":"ReedOmegaDeltaChi.reed_omega_delta_chi_conjecture"},{"answerKinds":[],"category":"research open","docstring":"For a finite graph $G$, we define $\\Delta(G)$ to be the maximum degree, $\\omega(G)$ to be the\nsize of the largest clique subgraph, and $\\chi(G)$ to be the chromatic number. Reed's omega,\ndelta, and chi conjecture states that $$\\chi(G) \\leq \\lceil \\frac{1}{2}(\\omega(G) + \\Delta(G) + 1) \\rceil.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ReedOmegaDeltaChi","statement":"∀ {V : Type} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  have χ := G.chromaticNumber;\n  have ω := G.cliqueNum;\n  have Δ := G.maxDegree;\n  2 * χ ≤ ↑ω + ↑Δ + 2","subjects":["5"],"theorem":"ReedOmegaDeltaChi.reed_omega_delta_chi_conjecture_for_finite_graphs"},{"answerKinds":[],"category":"research open","docstring":"The simplest open case is when $\\Delta(G) = 6$ and $\\omega(G) = 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ReedOmegaDeltaChi","statement":"∀ {V : Type} (G : SimpleGraph V), G.emaxDegree = 6 ∧ G.cliqueNum = 2 → G.chromaticNumber ≤ 5","subjects":["5"],"theorem":"ReedOmegaDeltaChi.reed_conjecture_Δ_6_ω_2"},{"answerKinds":[],"category":"research open","docstring":"De Giorgi's conjecture holds in dimension $n = 8$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DeGiorgi","statement":"DeGiorgi.DeGiorgi_conclusion 8","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_eight"},{"answerKinds":[],"category":"research open","docstring":"De Giorgi's conjecture holds in dimension $n = 5$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DeGiorgi","statement":"DeGiorgi.DeGiorgi_conclusion 5","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_five"},{"answerKinds":[],"category":"research open","docstring":"De Giorgi's conjecture holds in dimension $n ≤ 8$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DeGiorgi","statement":"∀ {n : ℕ} [inst : NeZero n], n ≤ 8 → DeGiorgi.DeGiorgi_conclusion n","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_le_eight"},{"answerKinds":[],"category":"research solved","docstring":"De Giorgi's conjecture holds in dimension $n = 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DeGiorgi","statement":"DeGiorgi.DeGiorgi_conclusion 2","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_two"},{"answerKinds":[],"category":"research open","docstring":"De Giorgi's conjecture holds in dimension $n = 6$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DeGiorgi","statement":"DeGiorgi.DeGiorgi_conclusion 6","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_six"},{"answerKinds":[],"category":"research solved","docstring":"De Giorgi's conjecture trivially holds in dimension $n = 1$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.DeGiorgi","statement":"DeGiorgi.DeGiorgi_conclusion 1","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_one"},{"answerKinds":[],"category":"research open","docstring":"De Giorgi's conjecture holds in dimension $n = 7$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DeGiorgi","statement":"DeGiorgi.DeGiorgi_conclusion 7","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_seven"},{"answerKinds":[],"category":"research solved","docstring":"De Giorgi's conjecture holds in dimension $n = 3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DeGiorgi","statement":"DeGiorgi.DeGiorgi_conclusion 3","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_three"},{"answerKinds":[],"category":"research open","docstring":"De Giorgi's conjecture holds in dimension $n = 4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DeGiorgi","statement":"DeGiorgi.DeGiorgi_conclusion 4","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_four"},{"answerKinds":[],"category":"research solved","docstring":"In dimension $n ≥ 9$, the conclusion of De Giorgi's conjecture does not hold.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DeGiorgi","statement":"∀ {n : ℕ} [inst : NeZero n], n ≥ 9 → ¬DeGiorgi.DeGiorgi_conclusion n","subjects":["35"],"theorem":"DeGiorgi.DeGiorgi_ge_nine"},{"answerKinds":[],"category":"research open","docstring":"For any `k ≥ 2`, let `a₁,...,aₖ` and `b₁,...,bₖ` be integers with `aᵢ > 0`. Suppose that for\nevery prime `p` there exists an integer `n` such that `p ∤ ∏ i, (aᵢ n + bᵢ)`. Then there exist\ninfinitely many `n` such that `aᵢ n + bᵢ` is prime for all `i`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.PrimeTuples","statement":"∀ {k : ℕ},\n  2 ≤ k →\n    ∀ (a : Fin k → ℕ+) (b : Fin k → ℕ),\n      (∀ (p : ℕ), Nat.Prime p → ∃ n, ¬p ∣ ∏ i, (↑(a i) * n + b i)) →\n        {n | ∀ (i : Fin k), Nat.Prime (↑(a i) * n + b i)}.Infinite","subjects":["11"],"theorem":"PrimeTuplesConjecture.prime_tuples_conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Dubner","statement":"¬DubnerConjecture.IsTwinPrime 100","subjects":["11"],"theorem":"DubnerConjecture.t5"},{"answerKinds":[],"category":"research open","docstring":"Every even number greater than 4208 is the sum of two twin primes.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Dubner","statement":"∀ (n : ℕ), 4208 < n → Even n → ∃ p q, DubnerConjecture.IsTwinPrime p ∧ DubnerConjecture.IsTwinPrime q ∧ p + q = n","subjects":["11"],"theorem":"DubnerConjecture.dubner_conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Dubner","statement":"DubnerConjecture.IsTwinPrime 101","subjects":["11"],"theorem":"DubnerConjecture.t4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Dubner","statement":"DubnerConjecture.IsTwinPrime 3","subjects":["11"],"theorem":"DubnerConjecture.t2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Dubner","statement":"DubnerConjecture.IsTwinPrime 5","subjects":["11"],"theorem":"DubnerConjecture.t3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Dubner","statement":"¬DubnerConjecture.IsTwinPrime 2","subjects":["11"],"theorem":"DubnerConjecture.t1"},{"answerKinds":[],"category":"research open","docstring":"**Problem 4.2.** Let $\\Omega \\subset \\mathbb{R}$ be a finite union of three or more\nintervals. If $\\Omega$ weakly tiles its complement, must it also tile its complement\nproperly? ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.WeakTiling","statement":"True ↔\n  ∀ (n : ℕ),\n    3 ≤ n →\n      ∀ (Ω : Set ℝ),\n        WeakTiling.IsUnionOfNIntervals n Ω →\n          ∀ (ν : MeasureTheory.Measure ℝ), WeakTiling.IsWeakTilingMeasure Ω ν → ∃ T, WeakTiling.IsProperTiling Ω T","subjects":["42","46"],"theorem":"WeakTiling.problem_4_2"},{"answerKinds":[],"category":"research open","docstring":"**Problem 4.1.** Let $\\Omega \\subset \\mathbb{R}$ be a finite union of intervals and $\\nu$\na weak tiling measure for $\\Omega$. Must $\\mathrm{supp}(\\nu)$ have bounded density? ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.WeakTiling","statement":"True ↔\n  ∀ (Ω : Set ℝ),\n    WeakTiling.IsFiniteUnionOfIntervals Ω →\n      ∀ (ν : MeasureTheory.Measure ℝ), WeakTiling.IsWeakTilingMeasure Ω ν → WeakTiling.HasBoundedDensity ν.support","subjects":["42","46"],"theorem":"WeakTiling.problem_4_1"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Problem 4.3.** Let $\\Omega \\subset \\mathbb{R}$ be a finite union of intervals and $\\nu$\n    a weak tiling measure for $\\Omega$. Must $\\nu$ be expressible as a convex combination of\n    proper tiling measures?\n\nThe answer is negative.\n\nAn AI-assisted investigation by Kenta Kitamura produced a counterexample for\n$\\Omega = (0,1) \\cup (2,3) \\cup (12,13) \\cup (30,31)$.\nA finite computation shows that every convex combination of proper tiling measures\nfor this $\\Omega$ assigns equal mass to $\\{7\\}$ and $\\{15\\}$.\nHowever, the constructed weak tiling measure $\\nu$ satisfies\n$\\nu(\\{7\\}) = \\frac{1}{2}$ and $\\nu(\\{15\\}) = 0$, so $\\nu$ cannot be expressed\nas a convex combination of proper tiling measures.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/weak-tiling-counterexample/blob/5bf93234cc51f02fd7681407d77dcebde592f3ac/formal-conjectures-v4.27.0/WeakTilingCounterexample.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Paper.WeakTiling","statement":"False ↔\n  ∀ (Ω : Set ℝ),\n    WeakTiling.IsFiniteUnionOfIntervals Ω →\n      ∀ (ν : MeasureTheory.Measure ℝ),\n        WeakTiling.IsWeakTilingMeasure Ω ν →\n          ∃ T c,\n            (∀ (i : ℕ), WeakTiling.IsProperTiling Ω (T i)) ∧\n              ∑' (i : ℕ), c i = 1 ∧\n                ν =\n                  MeasureTheory.Measure.sum fun i =>\n                    ↑(c i) • MeasureTheory.Measure.sum fun t => MeasureTheory.Measure.dirac ↑t","subjects":["42","46"],"theorem":"WeakTiling.problem_4_3"},{"answerKinds":[],"category":"research solved","docstring":"The smallest number of vertices of a triangle-free graph with chromatic number 4 and f=3 is at most 19. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"SimpleGraph.F 4 ≤ 19","subjects":["5"],"theorem":"SimpleGraph.F_four_le"},{"answerKinds":[],"category":"API","docstring":"**Lemma 4.** Let `G` be a triangle-free graph with `n` vertices and let `v` be a vertex of `G`.\nThere exists a triangle-free graph `H` containing `G` as an induced subgraph such that:\n(i) the degree of `v` in `H` is one more than its degree in `G`;\n(ii) for every vertex `w` of `G` other than `v` the degree of `w` in `H` is the same as its degree in `G`;\n(iii) if `J` is the subgraph of `H` induced by the vertices not in `G`, then `f(J)=3` and `δ(J) ≥ 2n`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] (G : SimpleGraph α) [inst_2 : DecidableRel G.Adj],\n  G.Connected →\n    G.CliqueFree 3 →\n      ∀ (v : α),\n        ∃ β x H x_1 i,\n          H.CliqueFree 3 ∧\n            H.degree (i v) = G.degree v + 1 ∧\n              (∀ (w : α), w ≠ v → H.degree (i w) = G.degree w) ∧\n                let J := SimpleGraph.induce (Set.range ⇑i).compl H;\n                J.degreeSequenceMultiplicity = 3 ∧ J.minDegree ≥ 2 * Fintype.card α","subjects":["5"],"theorem":"SimpleGraph.lemma4"},{"answerKinds":[],"category":"API","docstring":"**Lemma 2 (c)**\nInequality involving sums of terms of a nondecreasing sequence with no three terms equal. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ (d : ℕ → ℕ) (n : ℕ),\n  Monotone d →\n    (∀ (k : ℕ), 0 < d k) →\n      (∀ (i : ℕ), d (i + 2) ≠ d i) →\n        2 * n * n + 2 * n ≤ ∑ i ∈ Finset.Icc (2 * n + 2) (4 * n + 2), d i - ∑ i ∈ Finset.Icc 1 (2 * n + 1), d i","subjects":["5"],"theorem":"DegreeSequencesTriangleFree.lemma2_c"},{"answerKinds":[],"category":"API","docstring":"**Lemma 3.** For every `n` there exists a bipartite graph with\n`8 n` vertices, minimum degree `n + 1`, and `f = 3`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ (n : ℕ), 0 < n → ∃ G x, G.IsBipartite ∧ G.minDegree = n + 1 ∧ G.degreeSequenceMultiplicity = 3","subjects":["5"],"theorem":"SimpleGraph.lemma3"},{"answerKinds":[],"category":"research solved","docstring":"The smallest number of vertices of a triangle-free graph with chromatic number 3 and f=3 is 7. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"SimpleGraph.F 3 = 7","subjects":["5"],"theorem":"SimpleGraph.F_three"},{"answerKinds":[],"category":"API","docstring":"**Lemma 2 (d)**\nInequality involving sums of terms of a nondecreasing sequence with no three terms equal. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ (d : ℕ → ℕ) (n : ℕ),\n  Monotone d →\n    (∀ (k : ℕ), 0 < d k) →\n      (∀ (i : ℕ), d (i + 2) ≠ d i) →\n        2 * n * n + 4 * n + 2 ≤ ∑ i ∈ Finset.Icc (2 * n + 2) (4 * n + 3), d i - ∑ i ∈ Finset.Icc 1 (2 * n + 1), d i","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"DegreeSequencesTriangleFree.lemma2_d"},{"answerKinds":[],"category":"research solved","docstring":"**Theorem 1.** If a triangle-free graph has `f = 2`,\nthen it is bipartite, has minimum degree `1`, and\nits degree sequence is compact. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] (G : SimpleGraph α),\n  G.Connected →\n    ∀ [inst_2 : DecidableRel G.Adj],\n      G.CliqueFree 3 → G.degreeSequenceMultiplicity = 2 → G.IsBipartite ∧ G.minDegree = 1 ∧ G.HasCompactdegreeSequence","subjects":["5"],"theorem":"SimpleGraph.theorem1"},{"answerKinds":[],"category":"API","docstring":"**Lemma 1 (a)**\nIf a sequence `d` is nondecreasing and no three terms are equal, then terms at distance 2 differ by at least 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ (d : ℕ → ℕ) (k : ℕ), Monotone d → (∀ (k : ℕ), d (k + 2) ≠ d k) → 1 ≤ d (k + 2) - d k","subjects":["5"],"theorem":"DegreeSequencesTriangleFree.lemma1_a"},{"answerKinds":[],"category":"research solved","docstring":"**Theorem 2.** Every triangle-free graph is an induced subgraph of one\nwith `f = 3`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ {α : Type u_1} [Fintype α] [DecidableEq α] (G : SimpleGraph α) [DecidableRel G.Adj],\n  G.Connected → G.CliqueFree 3 → ∃ β x H x_1 i, H.CliqueFree 3 ∧ H.degreeSequenceMultiplicity = 3","subjects":["5"],"theorem":"SimpleGraph.theorem2"},{"answerKinds":[],"category":"API","docstring":"**Lemma 1 (b)**\nIf a sequence `d` is nondecreasing and no three terms are equal, then terms at distance `2 * r` differ by at least `r`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ (d : ℕ → ℕ) (k r : ℕ), Monotone d → (∀ (i : ℕ), d (i + 2) ≠ d i) → r ≤ d (k + 2 * r) - d k","subjects":["5"],"theorem":"DegreeSequencesTriangleFree.lemma1_b"},{"answerKinds":[],"category":"API","docstring":"**Lemma 2 (b)**\nInequality involving sums of terms of a nondecreasing sequence with no three terms equal. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ (d : ℕ → ℕ) (n : ℕ),\n  Monotone d →\n    (∀ (k : ℕ), 0 < d k) →\n      (∀ (i : ℕ), d (i + 2) ≠ d i) →\n        2 * n * n + 2 * n + 1 ≤ ∑ i ∈ Finset.Icc (2 * n + 1) (4 * n + 1), d i - ∑ i ∈ Finset.Icc 1 (2 * n), d i","subjects":["5"],"theorem":"DegreeSequencesTriangleFree.lemma2_b"},{"answerKinds":[],"category":"API","docstring":"**Lemma 2 (a)**\nInequality involving sums of terms of a nondecreasing sequence with no three terms equal. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.DegreeSequencesTriangleFree","statement":"∀ (d : ℕ → ℕ) (n : ℕ),\n  Monotone d →\n    (∀ (k : ℕ), 0 < d k) →\n      (∀ (i : ℕ), d (i + 2) ≠ d i) →\n        2 * n * n ≤ ∑ i ∈ Finset.Icc (2 * n + 1) (4 * n), d i - ∑ i ∈ Finset.Icc 1 (2 * n), d i","subjects":["5"],"theorem":"DegreeSequencesTriangleFree.lemma2_a"},{"answerKinds":[],"category":"research open","docstring":"For any tree $T$ with $n$ edges, the complete graph $K_{2n+1}$ decomposes into\n$2n+1$ edge-disjoint copies of $T$.\n\nA \"copy\" of $T$ is the image $T.\\text{map}(f_i)$ of $T$ under a vertex embedding\n$f_i : V \\hookrightarrow \\text{Fin}(2n+1)$; the copies are pairwise edge-disjoint\nand together cover every edge of $K_{2n+1}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.RingelConjecture","statement":"∀ {V : Type} [Finite V] (T : SimpleGraph V),\n  T.IsTree →\n    ∀ (n : ℕ),\n      T.edgeSet.ncard = n →\n        ∃ f,\n          (Pairwise fun i j => Disjoint (SimpleGraph.map (⇑(f i)) T).edgeSet (SimpleGraph.map (⇑(f j)) T).edgeSet) ∧\n            ⨆ i, SimpleGraph.map (⇑(f i)) T = ⊤","subjects":["5"],"theorem":"RingelConjecture.ringel_conjecture"},{"answerKinds":[],"category":"research open","docstring":"This statement can be reduced to the prime case only.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Kurepa","statement":"∀ (p : ℕ), 2 < p → Nat.Prime p → Kurepa.left_factorial p % p ≠ 0","subjects":["11"],"theorem":"Kurepa.kurepa_conjecture.variants.prime"},{"answerKinds":[],"category":"research open","docstring":"## Kurepa's conjecture\n\nFor all $n$, $$!n\\not\\equiv 0 \\mod n$$\n\nThis appears as B44 \"Sums of factorials.\"\nin [Unsolved Problems in Number Theory](https://doi.org/10.1007/978-0-387-26677-0)\nby *Richard K. Guy*\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Kurepa","statement":"∀ (n : ℕ), 2 < n → Kurepa.left_factorial n % n ≠ 0","subjects":["11"],"theorem":"Kurepa.kurepa_conjecture"},{"answerKinds":[],"category":"test","docstring":"Sanity check: for small values we can just compute that the conjecture is true.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Kurepa","statement":"∀ (n : ℕ), 2 < n → n < 50 → n.factorial.gcd (Kurepa.left_factorial n) = 2","subjects":["11"],"theorem":"Kurepa.kurepa_conjecture.variants.gcd.first_cases"},{"answerKinds":[],"category":"textbook","docstring":"Kurepa's conjecture for all integers greater than 2 is equivalent to the statement that $\\gcd(n!, !n) = 2$ for all integers greater than 2.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Kurepa","statement":"(∀ (n : ℕ), 2 < n → Kurepa.left_factorial n % n ≠ 0) ↔ ∀ (n : ℕ), 2 < n → n.factorial.gcd (Kurepa.left_factorial n) = 2","subjects":["11"],"theorem":"Kurepa.kurepa_conjecture.gcd_reduction"},{"answerKinds":[],"category":"test","docstring":"Sanity check: for small values we can just compute that the conjecture is true\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Kurepa","statement":"∀ (n : ℕ), 2 < n → n < 50 → Kurepa.left_factorial n % n ≠ 0","subjects":["11"],"theorem":"Kurepa.kurepa_conjecture.variants.first_cases"},{"answerKinds":[],"category":"research open","docstring":"An equivalent formulation in terms of the gcd of $n!$ and $!n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Kurepa","statement":"∀ (n : ℕ), 2 < n → n.factorial.gcd (Kurepa.left_factorial n) = 2","subjects":["11"],"theorem":"Kurepa.kurepa_conjecture.variants.gcd"},{"answerKinds":[],"category":"textbook","docstring":"Kurepa's conjecture for all integers greater than 2 is equivalent to the conjecture restricted to primes greater than 2.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Kurepa","statement":"(∀ (n : ℕ), 2 < n → Kurepa.left_factorial n % n ≠ 0) ↔ ∀ (p : ℕ), 2 < p → Nat.Prime p → Kurepa.left_factorial p % p ≠ 0","subjects":["11"],"theorem":"Kurepa.kurepa_conjecture.prime_reduction"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Variant of the Bézier-Bernstein Voronovskaja problem which treats \"sufficiently smooth\" as an\neventual condition in the smoothness order $m$: for all sufficiently large finite $m$, every\n$C^m$ function on $[0,1]$ should have the asserted asymptotic formula.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.VoronovskajaTypeFormula","statement":"∀ (α : ℝ),\n  0 < α →\n    α ≠ 1 →\n      have limitFormula := sorry;\n      ∀ᶠ (m : ℕ) in Filter.atTop,\n        ∀ (f : ℝ → ℝ),\n          ∀ x ∈ unitInterval,\n            ContDiffOn ℝ (↑m) f unitInterval →\n              Filter.Tendsto (fun n => √↑n * (VoronovskajaTypeFormula.bezierBernstein n α f x - f x)) Filter.atTop\n                (nhds (limitFormula f x))","subjects":["26","40","47"],"theorem":"VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth"},{"answerKinds":[],"category":"research open","docstring":"Existence-only version of the eventual-smoothness variant. This separates the first part of the\nsource problem, proving that the scaled sequence has some limit, from the stronger task of finding\nan explicit expression for that limit.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.VoronovskajaTypeFormula","statement":"∀ (α : ℝ),\n  0 < α →\n    α ≠ 1 →\n      ∀ᶠ (m : ℕ) in Filter.atTop,\n        ∀ (f : ℝ → ℝ),\n          ∀ x ∈ unitInterval,\n            ContDiffOn ℝ (↑m) f unitInterval →\n              ∃ L,\n                Filter.Tendsto (fun n => √↑n * (VoronovskajaTypeFormula.bezierBernstein n α f x - f x)) Filter.atTop\n                  (nhds L)","subjects":["26","40","47"],"theorem":"VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth.limit_exists"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Conjecture: Voronovskaja-type formula for Bézier-Bernstein operators\nwith shape parameter $\\alpha > 0$, $\\alpha \\neq 1$.\n\nThe source asks for sufficiently smooth functions. This concrete version uses\n`ContDiffOn ℝ 2 f I` as a readable baseline regularity assumption; since the\ndomain is the compact interval $[0,1]$, this also explains why no separate\nboundedness assumption is included here. The variants below record the unknown\nsmoothness threshold more explicitly.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.VoronovskajaTypeFormula","statement":"∀ (α : ℝ),\n  0 < α →\n    α ≠ 1 →\n      ∀ (f : ℝ → ℝ),\n        ∀ x ∈ unitInterval,\n          ContDiffOn ℝ 2 f unitInterval →\n            Filter.Tendsto (fun n => √↑n * (VoronovskajaTypeFormula.bezierBernstein n α f x - f x)) Filter.atTop\n              (nhds sorry)","subjects":["26","40","47"],"theorem":"VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Variant of the Bézier-Bernstein Voronovskaja problem with the required smoothness order itself\nleft as an answer. Replacing `(answer(sorry) : ℕ × ((ℝ → ℝ) → ℝ → ℝ))` by a concrete value lets one\nstate the conjecture for a chosen regularity threshold.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.VoronovskajaTypeFormula","statement":"∀ (α : ℝ),\n  0 < α →\n    α ≠ 1 →\n      have p := sorry;\n      have m := p.1;\n      have limitFormula := p.2;\n      ∀ (f : ℝ → ℝ),\n        ∀ x ∈ unitInterval,\n          ContDiffOn ℝ (↑m) f unitInterval →\n            Filter.Tendsto (fun n => √↑n * (VoronovskajaTypeFormula.bezierBernstein n α f x - f x)) Filter.atTop\n              (nhds (limitFormula f x))","subjects":["26","40","47"],"theorem":"VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators.variants.answer_smoothness"},{"answerKinds":[],"category":"research solved","docstring":"Classical Voronovskaja theorem (α = 1).\n\nFor functions $f$ that are $C^2$ on $[0,1]$, the limit:\n$$\nn\\bigl( B_n f(x) - f(x) \\bigr)\n\\;\\longrightarrow\\;\n\\frac{1}{2}\\, x(1 - x)\\, f''(x)\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.VoronovskajaTypeFormula","statement":"∀ (f : ℝ → ℝ),\n  ∀ x ∈ unitInterval,\n    ContDiffOn ℝ 2 f unitInterval →\n      have f'' := iteratedDerivWithin 2 f unitInterval x;\n      Filter.Tendsto (fun n => ↑n * (VoronovskajaTypeFormula.bezierBernstein n 1 f x - f x)) Filter.atTop\n        (nhds (1 / 2 * x * (1 - x) * f''))","subjects":["26","40","47"],"theorem":"VoronovskajaTypeFormula.voronovskaja_theorem.bernstein_operators"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ClaudesCycles","statement":"∀ {m : ℕ} [inst : NeZero m] {b b' : Fin 3}, b ≠ b' → ∀ (v : ClaudesCycles.Vertex m), ClaudesCycles.bumpAt b v b' = v b'","subjects":["5"],"theorem":"ClaudesCycles.bumpAt_apply_of_ne"},{"answerKinds":[],"category":"research solved","docstring":"The case `m = 2` is impossible: the cube digraph on `(ZMod 2)³` does not have a\nHamiltonian arc decomposition [Aub82]. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ClaudesCycles","statement":"¬ClaudesCycles.HasHamiltonianArcDecomposition 2","subjects":["5"],"theorem":"ClaudesCycles.cube_hamiltonian_arc_decomposition_impossible_m2"},{"answerKinds":[],"category":"research solved","docstring":"For odd `m > 1`, the cube digraph on `(ZMod m)³` has a Hamiltonian arc decomposition\ninto three directed cycles [Knu26]. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/kim-em/KnuthClaudeLean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ClaudesCycles","statement":"∀ {m : ℕ} [inst : NeZero m], Odd m → 1 < m → ClaudesCycles.HasHamiltonianArcDecomposition m","subjects":["5"],"theorem":"ClaudesCycles.cube_hamiltonian_arc_decomposition"},{"answerKinds":[],"category":"test","docstring":"The hypothesis `1 < m` on `cube_hamiltonian_arc_decomposition` is load-bearing. At `m = 1`\nthe vertex type has one element, so the only permutation of it is the identity, which is not a\ncycle. Note that `Odd 1` holds, so without `1 < m` the odd statement would be false. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ClaudesCycles","statement":"¬ClaudesCycles.HasHamiltonianArcDecomposition 1","subjects":["5"],"theorem":"ClaudesCycles.not_hasHamiltonianArcDecomposition_one"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ClaudesCycles","statement":"∀ {m : ℕ} [inst : NeZero m] (b : Fin 3) (v : ClaudesCycles.Vertex m), ClaudesCycles.bumpAt b v b = v b + 1","subjects":["5"],"theorem":"ClaudesCycles.bumpAt_apply_self"},{"answerKinds":[],"category":"test","docstring":"The three arcs leaving a vertex are distinct, which is what makes the `∃!` in\n`HasHamiltonianArcDecomposition` a condition about arcs rather than about heads that might\ncoincide. It needs `1 < m`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ClaudesCycles","statement":"∀ {m : ℕ} [inst : NeZero m],\n  1 < m → ∀ (v : ClaudesCycles.Vertex m), Function.Injective fun b => ClaudesCycles.bumpAt b v","subjects":["5"],"theorem":"ClaudesCycles.bumpAt_injective"},{"answerKinds":["Prop"],"category":"research solved","docstring":"For even `m > 2`, the cube digraph on `(ZMod m)³` has a Hamiltonian arc decomposition.\nKnuth records this as settled in the final section of [Knu26], by [Ho26] with the proof in\n[GPT26] for even `m ≥ 8`, and by [AM26] for the even case generally. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ClaudesCycles","statement":"True ↔ ∀ (m : ℕ) (x : NeZero m), Even m → 2 < m → ClaudesCycles.HasHamiltonianArcDecomposition m","subjects":["5"],"theorem":"ClaudesCycles.cube_hamiltonian_arc_decomposition_even"},{"answerKinds":[],"category":"research open","docstring":"If F is a decreasing family of sets of some finite type α, then there is some element\nx of α such that the family consisting of all members of F containing x is an intersecting\nsubfamily of F with maximal cardinality.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Chvatal","statement":"∀ {α : Type} [Fintype α] [inst : DecidableEq α] [Nonempty α] (F : Finset (Finset α)),\n  Chvatal.Decreasing F → ∃ x, ∀ G ⊆ F, Chvatal.Intersecting G → G.card ≤ {A ∈ F | x ∈ A}.card","subjects":["5"],"theorem":"Chvatal.exists_maximal_star"},{"answerKinds":[],"category":"test","docstring":"Every axiom that is non-negative is reachable. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.BranchingVAS","statement":"∀ {d : ℕ} (b : BranchingVAS.Bvas d) {v : Fin d → ℤ}, v ∈ b.axioms → 0 ≤ v → b.Reachable v","subjects":["3","68"],"theorem":"BranchingVAS.reachable_of_mem_axioms"},{"answerKinds":[],"category":"test","docstring":"The vector [0, 0, 12] is reachable in the first example BVAS.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.BranchingVAS","statement":"BranchingVAS.exampleBvas.Reachable ![0, 0, 12]","subjects":["3","68"],"theorem":"BranchingVAS.reachable_example"},{"answerKinds":[],"category":"research solved","docstring":"The reachability problem for branching vector addition systems is decidable.\n\nThat is, is the predicate taking a branching vector addition system `b` together\nwith a target vector `t` and returning whether `t` is reachable in `b` is a\ncomputable predicate.\n\nAs of August 2026, the solution is quite recently announced and is not\nyet peer-reviewed.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.BranchingVAS","statement":"∀ {d : ℕ}, ComputablePred fun p => p.1.Reachable p.2","subjects":["3","68"],"theorem":"BranchingVAS.reachability_decidable"},{"answerKinds":[],"category":"test","docstring":"Reachable vectors are always non-negative. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.BranchingVAS","statement":"∀ {d : ℕ} (v : Fin d → ℤ) (b : BranchingVAS.Bvas d), b.Reachable v → 0 ≤ v","subjects":["3","68"],"theorem":"BranchingVAS.reachable_imp_pos"},{"answerKinds":[],"category":"test","docstring":"The vector [0, 0] is not reachable in the second example BVAS.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.BranchingVAS","statement":"¬BranchingVAS.exampleBvas2.Reachable ![0, 0]","subjects":["3","68"],"theorem":"BranchingVAS.not_reachable_example"},{"answerKinds":[],"category":"research solved","docstring":"The Gourevitch series identity:\nThe following identity holds:\n$\\sum_{n=0}^{\\infty} \\frac{1 + 14 n + 76 n^2 + 168 n^3}{2^{20 n}} \\binom{2n}{n}^7 = \\frac{32}{\\pi^3}.$\nThis was originally conjectured in [G2003] by Guillera and proven in [A2025] by Au.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Gourevitch","statement":"∑' (n : ℕ), (1 + 14 * ↑n + 76 * ↑n ^ 2 + 168 * ↑n ^ 3) / 2 ^ (20 * n) * ↑n.centralBinom ^ 7 = 32 / Real.pi ^ 3","subjects":["11","33"],"subsets":["FC100SolvedSet1"],"theorem":"Gourevitch.gourevitch_series_identity"},{"answerKinds":[],"category":"research open","docstring":"For $N = 16$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{C}$? ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.MonochromaticQuantumGraph","statement":"True ↔ ¬∃ W, MonochromaticQuantumGraph.EqSystemN 16 3 W","subjects":["5","14","81"],"subsets":["FC100OpenSet1"],"theorem":"MonochromaticQuantumGraph.eqSystem16_no_solution_d3"},{"answerKinds":["Prop"],"category":"research solved","docstring":"For $N = 4$ and $D = 4$, does there exist no solution to the monochromatic quantum\ngraph equation system over $\\mathbb{C}$?\n\nThis is the $D = N$ case, proved using `eqSystem_no_solution_even_ge4_d_eq_n_explicit`. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/af88acbf9da0f26e3e934743a819e986e02f6875/FormalConjectures/Paper/MonochromaticQuantumGraph.lean#L1021"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Paper.MonochromaticQuantumGraph","statement":"True ↔ ¬∃ W, MonochromaticQuantumGraph.EqSystemN 4 4 W","subjects":["5","14","81"],"theorem":"MonochromaticQuantumGraph.eqSystem4_no_solution_d4"},{"answerKinds":["Prop"],"category":"research solved","docstring":"For all even $N \\geq 4$ and $D = N$, does there exist no solution to the monochromatic quantum\ngraph equation system over $\\mathbb{C}$?\n\nThe DeepMind prover agent has found a formal proof for this statement.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/af88acbf9da0f26e3e934743a819e986e02f6875/FormalConjectures/Paper/MonochromaticQuantumGraph.lean#L1006"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Paper.MonochromaticQuantumGraph","statement":"True ↔ ∀ N ≥ 4, Even N → ¬∃ W, MonochromaticQuantumGraph.EqSystemN N N W","subjects":["5","14","81"],"theorem":"MonochromaticQuantumGraph.eqSystem_no_solution_even_ge4_d_eq_n_explicit"},{"answerKinds":[],"category":"research open","docstring":"For all even $N \\geq 6$ and $D \\geq 3$, does there exist no solution to the monochromatic\nquantum graph equation system over $\\mathbb{Z}$? ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.MonochromaticQuantumGraph","statement":"True ↔ ∀ (N D : ℕ), N ≥ 6 → Even N → D ≥ 3 → ¬∃ W, MonochromaticQuantumGraph.EqSystemN N D W","subjects":["5","14","81"],"theorem":"MonochromaticQuantumGraph.eqSystem_no_solution_ge6_ge3_int"},{"answerKinds":[],"category":"research open","docstring":"For $N = 8$ and $D = 3$, does there exist no solution to the monochromatic quantum graph\nequation system over $\\mathbb{Z}$ with weights in $\\{-1, 0, 1\\}$? 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","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.MonochromaticQuantumGraph","statement":"True ↔ ¬∃ W, MonochromaticQuantumGraph.EqSystemN 10 6 W","subjects":["5","14","81"],"theorem":"MonochromaticQuantumGraph.eqSystem10_no_solution_d6"},{"answerKinds":[],"category":"research open","docstring":"The Latin Tableau Conjecture: If G is the simple graph\nof a Young diagram, then G is CDS-colorable. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinTableau","statement":"∀ (μ : YoungDiagram), μ.toSimpleGraph.CDSColorable","subjects":["5"],"theorem":"LatinTableau.SimpleGraph.LatinTableauConjecture"},{"answerKinds":[],"category":"test","docstring":"The number of transversals of the Cayley table of $\\mathbb{Z}_n$ for odd $n$ forms\n[OEIS A006717](https://oeis.org/A006717), starting with\n$z(1) = 1, z(3) = 3, z(5) = 15, z(7) = 133$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.LatinSquare","statement":"[LatinSquare.z 1, LatinSquare.z 3, LatinSquare.z 5, LatinSquare.z 7] = [1, 3, 15, 133]","subjects":["5"],"theorem":"LatinSquare.z_odd_values"},{"answerKinds":[],"category":"test","docstring":"The $0 \\times 0$ Cayley table has exactly $1$ transversal (vacuously). ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.LatinSquare","statement":"LatinSquare.z 0 = 1","subjects":["5"],"theorem":"LatinSquare.z_zero"},{"answerKinds":[],"category":"research open","docstring":"The smallest unresolved case of the MOLS existence problem: whether there are `11` mutually\northogonal latin squares of order `12`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinSquare","statement":"True ↔ LatinSquare.HasCompleteMOLS 12","subjects":["5"],"theorem":"LatinSquare.molsOrder12"},{"answerKinds":[],"category":"research solved","docstring":"Theorem 7.2 in [Wa2011]:\nFor all $n \\geq 5$,\n$$\n15^{n/5} \\leq T(n) \\leq c^n \\sqrt{n} \\cdot n!\n$$\nwhere $c = \\sqrt{\\frac{3 - \\sqrt{3}}{6}} \\cdot e^{\\sqrt{3}/6}$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinSquare","statement":"have c := √((3 - √3) / 6) * Real.exp (√3 / 6);\n∀ n ≥ 5, ↑(LatinSquare.T n) ∈ Set.Icc (15 ^ (↑n / 5)) (c ^ n * √↑n * ↑n.factorial)","subjects":["5"],"theorem":"LatinSquare.maxTransversalsBound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"MOLS existence problem: determine exactly which orders `n` admit a complete set of `n - 1`\nmutually orthogonal latin squares.\n\nEquivalently, this asks for which orders affine planes of order `n` exist. Complete sets are known\nfor prime-power orders; the smallest currently unresolved order is `12`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinSquare","statement":"sorry = {n | LatinSquare.HasCompleteMOLS n}","subjects":["5"],"theorem":"LatinSquare.molsExistenceProblem"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 5.1 in [Wa2011]:\nEvery latin square has a near-transversal\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinSquare","statement":"True ↔ ∀ (n : ℕ) (L : LatinSquare n), ∃ ρ σ, IsNearTransversal L ρ σ","subjects":["5"],"theorem":"LatinSquare.latinSquareNearTransversal"},{"answerKinds":[],"category":"research open","docstring":"The smallest odd number for which this conjecture is not known is 11.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinSquare","statement":"True ↔ ∀ (L : LatinSquare 11), ∃ σ, IsTransversal L σ","subjects":["5"],"theorem":"LatinSquare.latinSquareOrder11Transversal"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 6.7 in [Wa2011]:\nThere exist real constants $0 < c_1 < c_2 < 1$ such that\n$$\nc_1^n n! \\leq z_n \\leq c_2^n n!\n$$\nfor all odd $n \\geq 3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinSquare","statement":"True ↔\n  ∃ c₁ > 0,\n    ∃ c₂ < 1,\n      ∃ (_ : c₁ < c₂), ∀ n ≥ 3, Odd n → ↑(LatinSquare.z n) ∈ Set.Icc (c₁ ^ n * ↑n.factorial) (c₂ ^ n * ↑n.factorial)","subjects":["5"],"theorem":"LatinSquare.numTransversalsZn"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The conjecture is known to be true for $n \\leq 9$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinSquare","statement":"True ↔ ∀ n ≤ 9, Odd n → ∀ (L : LatinSquare n), ∃ σ, IsTransversal L σ","subjects":["5"],"theorem":"LatinSquare.oddOrderLeq9LatinSquareTransversal"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 3.2 in [Wa2011]:\nEach Latin square of odd order has at least one transversal.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinSquare","statement":"True ↔ ∀ (n : ℕ), Odd n → ∀ (L : LatinSquare n), ∃ σ, IsTransversal L σ","subjects":["5"],"theorem":"LatinSquare.oddOrderLatinSquareTransversal"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 6.9 in [Wa2011]:\n$$\n\\lim_{\\substack{n \\to \\infty \\\\ n \\text{ odd}}} \\frac{1}{n} \\log(z_n / n!) = -1\n$$\nIt is not even known if this limit exists. Note that $z_n = 0$ for even $n$ (see `z_even`), so the\nlimit must be restricted to odd $n$; here we parametrise odd $n$ as $2k + 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.LatinSquare","statement":"True ↔\n  Filter.Tendsto (fun k => 1 / (2 * ↑k + 1) * Real.log (↑(LatinSquare.z (2 * k + 1)) / ↑(2 * k + 1).factorial))\n    Filter.atTop (nhds (-1))","subjects":["5"],"theorem":"LatinSquare.growthRateZn"},{"answerKinds":[],"category":"textbook","docstring":"The Cayley table of $\\mathbb{Z}_n$ for positive even $n$ has no transversals. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.LatinSquare","statement":"∀ (n : ℕ), LatinSquare.z (2 * (n + 1)) = 0","subjects":["5"],"theorem":"LatinSquare.z_even"},{"answerKinds":[],"category":"research open","docstring":"Problem 13 in [Ar2013]:\nIs it true that every infinite homogeneous compact hausdorff\nspace contains a non-trivial convergent sequence? ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Homogenous","statement":"True ↔\n  ∀ (X : Type) (x : TopologicalSpace X),\n    ¬Finite X →\n      T2Space X →\n        CompactSpace X →\n          Homogeneous.HomogeneousSpace X → ∃ s, Function.Injective s ∧ ∃ a, Filter.Tendsto s Filter.atTop (nhds a)","subjects":["54"],"theorem":"Homogeneous.homogeneousSpace_exists_inj_tendsto"},{"answerKinds":[],"category":"research open","docstring":"Problem 17 in [Ar2013]:\nIs it true that every nonempty ω-monolithic compact hausdorff space contains a point with a\nfirst countable neighborhood basis?\n\nNote: `Nonempty X` is required since the conclusion asserts the existence of a point.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Homogenous","statement":"True ↔\n  ∀ (X : Type) (x : TopologicalSpace X),\n    T2Space X →\n      CompactSpace X → Nonempty X → Homogeneous.CountablyMonolithicSpace X → ∃ x_1, (nhds x_1).IsCountablyGenerated","subjects":["54"],"theorem":"Homogeneous.countablyMonolithicSpace_exists_nhds_generated_countable"},{"answerKinds":[],"category":"research open","docstring":"Problem 15 in [Ar2013]:\nIs every homogeneous ω-monolithic compact hausdorff space first countable? ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Homogenous","statement":"True ↔\n  ∀ (X : Type) (x : TopologicalSpace X),\n    T2Space X →\n      CompactSpace X →\n        Homogeneous.HomogeneousSpace X → Homogeneous.CountablyMonolithicSpace X → FirstCountableTopology X","subjects":["54"],"theorem":"Homogeneous.firstCountableTopology_of_countablyMonolithicSpace"},{"answerKinds":[],"category":"test","docstring":"Every discrete space is homogeneous. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Homogenous","statement":"∀ (X : Type u_1) [inst : TopologicalSpace X] [DiscreteTopology X], Homogeneous.HomogeneousSpace X","subjects":["54"],"theorem":"Homogeneous.DiscreteTopology.toHomogeneousSpace"},{"answerKinds":[],"category":"test","docstring":"Every Metrizable space is ω-monolithic. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.Homogenous","statement":"∀ (X : Type u_1) [inst : TopologicalSpace X] [TopologicalSpace.MetrizableSpace X],\n  Homogeneous.CountablyMonolithicSpace X","subjects":["54"],"theorem":"Homogeneous.MetrizableSpace.countablyMonolithicSpace"},{"answerKinds":[],"category":"research open","docstring":"Problem 16 in [Ar2013]:\nIs the cardinality of every homogeneous ω-monolithic compact hausdorff space not greater than 𝔠? ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Homogenous","statement":"True ↔\n  ∀ (X : Type) (x : TopologicalSpace X),\n    T2Space X →\n      CompactSpace X →\n        Homogeneous.HomogeneousSpace X → Homogeneous.CountablyMonolithicSpace X → Cardinal.mk X ≤ Cardinal.continuum","subjects":["54"],"theorem":"Homogeneous.countablyMonolithicSpace_card_lt"},{"answerKinds":[],"category":"research open","docstring":"Problem 14 in [Ar2013]:\nIs it possible to represent an arbitrary compact hausdorff space as an image\nof a homogeneous compact space under a continuous mapping? ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Homogenous","statement":"True ↔\n  ∀ (X : Type) (x : TopologicalSpace X),\n    T2Space X →\n      CompactSpace X →\n        ∃ Y x_1, T2Space Y ∧ CompactSpace Y ∧ Homogeneous.HomogeneousSpace Y ∧ ∃ f, Continuous f ∧ Function.Surjective f","subjects":["54"],"theorem":"Homogeneous.homogeneousSpace_exists_surjective"},{"answerKinds":[],"category":"research open","docstring":"There are no indecomposable vector bundles of rank 2 on $\\mathbb{P}^n$ for $n \\ge 7$.\nThis is Conjecture 6.3 in [Har1974].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.HartshorneConjecture","statement":"∀ (n : ℕ),\n  7 ≤ n →\n    ∀ (𝓕 : (ProjectiveSpace (Fin (n + 1)) (AlgebraicGeometry.Spec (CommRingCat.of ℂ))).VectorBundles),\n      𝓕.rank = 2 → Nonempty (𝓕.Splitting (Fin 2))","subjects":["14"],"theorem":"AlgebraicGeometry.Scheme.harthshorne_conjecture"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.Paper.HartshorneConjecture","statement":"∀ (S : AlgebraicGeometry.Scheme), CategoryTheory.Limits.HasFiniteCoproducts S.VectorBundles","subjects":["14"],"theorem":"AlgebraicGeometry.Scheme.hasFiniteCoproductsVectorBundles"},{"answerKinds":[],"category":"research open","docstring":"For any tree $T$ with $n$ edges, the complete graph $K_{2n+1}$ decomposes into\n$2n+1$ edge-disjoint copies of $T$ via cyclic shifts of a single embedding.\n\nThe $2n+1$ copies are $f_0, f_1, \\dots, f_{2n}$ where $f_i(v) = f_0(v) + i$ for all vertices\n$v$ — each copy is obtained by adding $i \\pmod{2n+1}$ to every vertex of the base copy.\nThis is strictly stronger than `RingelConjecture.ringel_conjecture`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.KotzigConjecture","statement":"∀ {V : Type} [Finite V] (T : SimpleGraph V),\n  T.IsTree →\n    ∀ (n : ℕ),\n      T.edgeSet.ncard = n →\n        ∃ f,\n          (∀ (i : Fin (2 * n + 1)) (v : V), (f i) v = (f 0) v + i) ∧\n            (Pairwise fun i j => Disjoint (SimpleGraph.map (⇑(f i)) T).edgeSet (SimpleGraph.map (⇑(f j)) T).edgeSet) ∧\n              ⨆ i, SimpleGraph.map (⇑(f i)) T = ⊤","subjects":["5"],"theorem":"KotzigConjecture.kotzig_conjecture"},{"answerKinds":[],"category":"test","docstring":"$\\mathcal{Z}_0 = \\mathbb{Q}$, so $\\dim_\\mathbb{Q}(\\mathcal{Z}_0) = 1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ZagierMZV","statement":"Module.finrank ℚ ↥(ZagierMZV.mzvSpanOfWeight 0) = 1","subjects":["11"],"theorem":"ZagierMZV.dim_mzv_weight_zero"},{"answerKinds":[],"category":"test","docstring":"Euler's identity for $\\zeta(4) = \\pi^4/90$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ZagierMZV","statement":"ZagierMZV.multiZeta [4] = Real.pi ^ 4 / 90","subjects":["11"],"theorem":"ZagierMZV.multiZeta_four"},{"answerKinds":[],"category":"test","docstring":"Euler's identity: $\\zeta(2) = \\pi^2/6$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ZagierMZV","statement":"ZagierMZV.multiZeta [2] = Real.pi ^ 2 / 6","subjects":["11"],"theorem":"ZagierMZV.multiZeta_two"},{"answerKinds":[],"category":"test","docstring":"There is no admissible index of weight 1 (since $s_1 \\geq 2$ is required). ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ZagierMZV","statement":"∀ (s : List ℕ), ZagierMZV.AdmissibleIndex s → ZagierMZV.weight s ≠ 1","subjects":["11"],"theorem":"ZagierMZV.no_admissible_weight_one"},{"answerKinds":[],"category":"test","docstring":"$\\mathcal{Z}_1 = \\emptyset$, so $\\dim_\\mathbb{Q}(\\mathcal{Z}_1) = 0$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ZagierMZV","statement":"Module.finrank ℚ ↥(ZagierMZV.mzvSpanOfWeight 1) = 0","subjects":["11"],"theorem":"ZagierMZV.dim_mzv_weight_one"},{"answerKinds":[],"category":"test","docstring":"The first few values of $d_n$ are $1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, \\ldots$ ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ZagierMZV","statement":"[ZagierMZV.zagierDim 0, ZagierMZV.zagierDim 1, ZagierMZV.zagierDim 2, ZagierMZV.zagierDim 3, ZagierMZV.zagierDim 4,\n    ZagierMZV.zagierDim 5, ZagierMZV.zagierDim 6, ZagierMZV.zagierDim 7, ZagierMZV.zagierDim 8, ZagierMZV.zagierDim 9] =\n  [1, 0, 1, 1, 1, 2, 2, 3, 4, 5]","subjects":["11"],"theorem":"ZagierMZV.zagierDim_first_values"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ZagierMZV","statement":"ZagierMZV.multiZeta [] = 1","subjects":["11"],"theorem":"ZagierMZV.multiZeta_empty"},{"answerKinds":[],"category":"research solved","docstring":"**Upper bound** [Te02, DG05]\n\nThe dimension of the $\\mathbb{Q}$-vector space of MZVs of weight $n$ is at most $d_n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ZagierMZV","statement":"∀ (n : ℕ), Module.finrank ℚ ↥(ZagierMZV.mzvSpanOfWeight n) ≤ ZagierMZV.zagierDim n","subjects":["11"],"theorem":"ZagierMZV.zagier_upper_bound"},{"answerKinds":[],"category":"research open","docstring":"**Zagier's conjecture**\n\nThe $\\mathbb{Q}$-dimension of the vector space spanned by all multiple zeta values\nof weight $n$ equals $d_n$, where $d_n$ is the Zagier dimension sequence\nsatisfying $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, and $d_n = d_{n-2} + d_{n-3}$ for $n \\geq 3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ZagierMZV","statement":"True ↔ ∀ (n : ℕ), Module.finrank ℚ ↥(ZagierMZV.mzvSpanOfWeight n) = ZagierMZV.zagierDim n","subjects":["11"],"theorem":"ZagierMZV.zagier_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Is there a Lindelöf space with singletons as Gδ sets with cardinality greater than the continuum?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.CardinalityLindelof","statement":"∃ X x, HasGδSingletons X ∧ LindelofSpace X ∧ Cardinal.continuum < Cardinal.mk X","subjects":["54"],"theorem":"CardinalityLindelof.HasGδSingletons.lindelof_card"},{"answerKinds":[],"category":"test","docstring":"Assuming `n` is a positive multiple of 4, the sensitivity of `nisanExample`\nis $n/2 + 2$. It is achieved by any $x$ with Hamming weight $n/2 + 2$: flipping\nany of its $n/2 + 2$ one-bits moves the input into the accepting layer of weight $n/2 + 1$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.StrongSensitivityConjecture","statement":"∀ (n : ℕ),\n  4 ∣ n → 0 < n → StrongSensitivityConjecture.sensitivity (StrongSensitivityConjecture.nisanExample n) = n / 2 + 2","subjects":["68"],"theorem":"StrongSensitivityConjecture.nisanExample_sensitivity"},{"answerKinds":[],"category":"test","docstring":"Assuming `n` is a multiple of 4 and $n \\ge 8$, the block sensitivity of\n`nisanExample` is $3n/4$, achieved by any $x$ with Hamming weight $n/2$.\nAn optimal block configuration uses all $n/2$ one-bits as singleton blocks\nand forms $n/4$ disjoint size-2 blocks from the zero-bits. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.StrongSensitivityConjecture","statement":"∀ (n : ℕ),\n  4 ∣ n → 8 ≤ n → StrongSensitivityConjecture.blockSensitivity (StrongSensitivityConjecture.nisanExample n) = 3 * n / 4","subjects":["68"],"theorem":"StrongSensitivityConjecture.nisanExample_blockSensitivity"},{"answerKinds":[],"category":"research open","docstring":"Strong Sensitivity Conjecture,\nfor every Boolean function `f : {0,1}^n → {0,1}`,\n`bs(f) ≤ s(f)^2`.\n\nWe call this the *strong* sensitivity conjecture because the original sensitivity\nconjecture only asked for a polynomial bound in terms of `s(f)`. Huang's\ncelebrated result (often called the sensitivity theorem) gives a quartic bound,\n`bs(f) ≤ s(f)^4`, thereby settling the original conjecture. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.StrongSensitivityConjecture","statement":"∀ {n : ℕ} (f : (Fin n → Bool) → Bool),\n  StrongSensitivityConjecture.blockSensitivity f ≤ StrongSensitivityConjecture.sensitivity f ^ 2","subjects":["68"],"theorem":"StrongSensitivityConjecture.strong_sensitivity_conjecture"},{"answerKinds":[],"category":"test","docstring":"Every first countable space is weakly first countable,\nsimply take $N x$ as a countable neighborhood basis of $x$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.WeaklyFirstCountable","statement":"∀ (X : Type u_1) [inst : TopologicalSpace X] [FirstCountableTopology X],\n  WeaklyFirstCountable.WeaklyFirstCountableTopology X","subjects":["54"],"theorem":"WeaklyFirstCountable.FirstCountableTopology.weaklyFirstCountableTopology"},{"answerKinds":[],"category":"research open","docstring":"Problem 4 in [Ar2013]: If a Tychonoff weakly first-countable space has countable\nSouslin number, then does its cardinality not exceed the continuum? ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.WeaklyFirstCountable","statement":"True ↔\n  ∀ (X : Type) (x : TopologicalSpace X),\n    T35Space X →\n      WeaklyFirstCountable.WeaklyFirstCountableTopology X →\n        WeaklyFirstCountable.HasCountableSouslinNumber X → Cardinal.mk X ≤ Cardinal.continuum","subjects":["54"],"theorem":"WeaklyFirstCountable.cardinalMk_le_continuum_of_weaklyFirstCountable_of_countableSouslinNumber"},{"answerKinds":[],"category":"test","docstring":"Every separable space has countable Souslin number. ","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.WeaklyFirstCountable","statement":"∀ (X : Type u) [inst : TopologicalSpace X] [TopologicalSpace.SeparableSpace X],\n  WeaklyFirstCountable.HasCountableSouslinNumber X","subjects":["54"],"theorem":"WeaklyFirstCountable.hasCountableSouslinNumber_of_separable"},{"answerKinds":[],"category":"research open","docstring":"Problem 3 in [Ar2013]: Give an example in ZFC of a weakly first-\ncountable compact Hausdorff space which is not first countable.\n\nNote: [Ar2013] uses a blanket convention that all spaces are\nTychonoff and \"compact\" means compact Hausdorff. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.WeaklyFirstCountable","statement":"∃ X x, WeaklyFirstCountable.WeaklyFirstCountableTopology X ∧ CompactSpace X ∧ T2Space X ∧ ¬FirstCountableTopology X","subjects":["54"],"theorem":"WeaklyFirstCountable.existsWeaklyFirstCountableCompactNotFirstCountable"},{"answerKinds":[],"category":"research open","docstring":"Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first-\ncountable compact Hausdorff space X such that $𝔠 < |X|$.\n\nNote: [Ar2013] uses a blanket convention that all spaces are\nTychonoff and \"compact\" means compact Hausdorff. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.WeaklyFirstCountable","statement":"True ↔\n  ∃ X x,\n    WeaklyFirstCountable.WeaklyFirstCountableTopology X ∧\n      CompactSpace X ∧ T2Space X ∧ Cardinal.continuum < Cardinal.mk X","subjects":["54"],"theorem":"WeaklyFirstCountable.existsWeaklyFirstCountableCompactBig"},{"answerKinds":[],"category":"textbook","docstring":"There are weakly first countable spaces which are not first countable,\nfor example the [Arens Space](https://topology.pi-base.org/spaces/S000156). ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.WeaklyFirstCountable","statement":"∃ X x, WeaklyFirstCountable.WeaklyFirstCountableTopology X ∧ ¬FirstCountableTopology X","subjects":["54"],"theorem":"WeaklyFirstCountable.exists_weakly_first_countable_not_first_countable"},{"answerKinds":[],"category":"research solved","docstring":"Under CH, such a space (for Problem 3 in [Ar2013]) exists as\nconstructed in [Ya1976] by Yakovlev. ","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.WeaklyFirstCountable","statement":"∀ [Fact (Cardinal.aleph 1 = Cardinal.continuum)],\n  ∃ X x, WeaklyFirstCountable.WeaklyFirstCountableTopology X ∧ CompactSpace X ∧ T2Space X ∧ ¬FirstCountableTopology X","subjects":["54"],"theorem":"WeaklyFirstCountable.CH.existsWeaklyFirstCountableCompactNotFirstCountable"},{"answerKinds":[],"category":"research open","docstring":"**Serre's uniformity question over $\\mathbb{Q}$** [Ser72, Lem17]: is there a bound\n$C$ such that every non-CM elliptic curve over $\\mathbb{Q}$ has surjective mod-$p$\nGalois representation for every prime $p > C$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.SerreUniformity","statement":"True ↔\n  ∃ C,\n    ∀ (E : WeierstrassCurve ℚ) [inst : E.IsElliptic],\n      E.j ∉ SerreUniformity.cmJInvariants → ∀ (p : ℕ), Nat.Prime p → C < p → SerreUniformity.HasFullTorsionAction E p","subjects":["11","14"],"theorem":"SerreUniformity.serre_uniformity"},{"answerKinds":[],"category":"research open","docstring":"**Serre's uniformity conjecture over $\\mathbb{Q}$, explicit form**: the bound $C = 37$\nworks, i.e. every non-CM elliptic curve over $\\mathbb{Q}$ has surjective mod-$p$ Galois\nrepresentation for every prime $p > 37$. From the introduction of [Lem17]: \"This conjecture\nremains open today, but, over the last forty years, there has been a lot of progress towards\na proof for $K = \\mathbb{Q}$ — it is believed that, in this case, $p_K = 37$.\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.SerreUniformity","statement":"True ↔\n  ∀ (E : WeierstrassCurve ℚ) [inst : E.IsElliptic],\n    E.j ∉ SerreUniformity.cmJInvariants → ∀ (p : ℕ), Nat.Prime p → 37 < p → SerreUniformity.HasFullTorsionAction E p","subjects":["11","14"],"theorem":"SerreUniformity.serre_uniformity.variants.bound_37"},{"answerKinds":[],"category":"research solved","docstring":"The $S_3$-conjecture holds for solvable groups: every nontrivial *solvable* finite ah-group is\nisomorphic to $S_3$. This was proved independently by Zhang (1994) and by\nKnörr–Lempken–Thielcke (1995).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ConjugacyClassSizes","statement":"∀ (G : Type) [inst : Group G] [inst_1 : Fintype G] [Group.IsSolvable G] [Nontrivial G],\n  ConjugacyClassSizes.HasDistinctConjClassSizes G → Nonempty (G ≃* Equiv.Perm (Fin 3))","subjects":["20"],"theorem":"ConjugacyClassSizes.conjClassSizes_iff_sym_three_solvable"},{"answerKinds":[],"category":"test","docstring":"The symmetric group $S_3$ is anti-homogeneous, since its three conjugacy classes have sizes $1$, $2$ and $3$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ConjugacyClassSizes","statement":"ConjugacyClassSizes.HasDistinctConjClassSizes (Equiv.Perm (Fin 3))","subjects":["20"],"theorem":"ConjugacyClassSizes.hasDistinctConjClassSizes_perm_fin_three"},{"answerKinds":[],"category":"test","docstring":"An anti-homogeneous group has trivial center, since each central element is in its own conjugacy class.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ConjugacyClassSizes","statement":"∀ {G : Type u_1} [inst : Group G] [inst_1 : Fintype G],\n  ConjugacyClassSizes.HasDistinctConjClassSizes G → Subgroup.center G = ⊥","subjects":["20"],"theorem":"ConjugacyClassSizes.trivial_center_of_hasDistinctConjClassSizes"},{"answerKinds":[],"category":"research open","docstring":"**Markel's $S_3$-conjecture** (1973): any nontrivial finite ah-group is isomorphic to $S_3$.\n\nThe conjecture is open in general; it is known to be true for solvable groups.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.ConjugacyClassSizes","statement":"∀ (G : Type) [inst : Group G] [inst_1 : Fintype G] [Nontrivial G],\n  ConjugacyClassSizes.HasDistinctConjClassSizes G → Nonempty (G ≃* Equiv.Perm (Fin 3))","subjects":["20"],"theorem":"ConjugacyClassSizes.conjClassSizes_iff_sym_three"},{"answerKinds":[],"category":"test","docstring":"The trivial group is anti-homogeneous, since it has a single conjugacy class.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ConjugacyClassSizes","statement":"∀ {G : Type u_1} [inst : Group G] [inst_1 : Fintype G] [Subsingleton G], ConjugacyClassSizes.HasDistinctConjClassSizes G","subjects":["20"],"theorem":"ConjugacyClassSizes.hasDistinctConjClassSizes_of_subsingleton"},{"answerKinds":[],"category":"API","docstring":"Equivalently, two conjugacy classes with the same cardinality must coincide.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.ConjugacyClassSizes","statement":"∀ {G : Type u_1} [inst : Group G] [inst_1 : Fintype G],\n  ConjugacyClassSizes.HasDistinctConjClassSizes G ↔\n    ∀ (a b : ConjClasses G), ConjugacyClassSizes.conjClassCard a = ConjugacyClassSizes.conjClassCard b → a = b","subjects":["20"],"theorem":"ConjugacyClassSizes.hasDistinctConjClassSizes_iff"},{"answerKinds":[],"category":"research open","docstring":"Let $T_N = \\sum_{k=1}^{N} k = \\frac{N(N+1)}{2}$.\nIf $T_N$ is even (equivalently $N \\equiv 0 \\pmod 4$ or $N \\equiv 3 \\pmod 4$),\nthen under optimal play the game `Catch-Up($\\{1, \\ldots, N\\}$)` ends in a draw.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.CatchUpConjecture","statement":"∀ (N : ℕ), Even (N * (N + 1) / 2) → CatchUp.value (Finset.Icc 1 N) = CatchUp.Outcome.draw","subjects":["11","91"],"theorem":"CatchUp.value_of_even_mul_succ_self_div_two"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.CasasAlvero","statement":"∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : Field L] {f : K →+* L} {P : Polynomial K},\n  CasasAlvero.HasCasasAlveroProp (Polynomial.map f P) ↔ CasasAlvero.HasCasasAlveroProp P","subjects":["12"],"theorem":"CasasAlvero.HasCasasAlveroProp.map_iff"},{"answerKinds":[],"category":"research solved","docstring":"The Casas-Alvero conjecture fails in positive characteristic `p` for polynomials of degree `p + 1`.\nThis was shown by Graf von Bothmer, Labs, Schicho, and van de Woestijne.\n\nReference: [The Casas-Alvero conjecture for infinitely many degrees](https://arxiv.org/pdf/math/0605090)\n\nFormal proof linked here provided by AlphaProof.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mzhorvath1/formal-conjectures/blob/4f2343508f2c157f35abb7be4814bd550280ce81/FormalConjectures/Paper/CasasAlvero.lean#163"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Paper.CasasAlvero","statement":"∀ {p : ℕ},\n  Nat.Prime p →\n    ∃ K x,\n      ∃ (_ : CharP K p),\n        have P := Polynomial.X ^ (p + 1) - Polynomial.X ^ p;\n        P.Monic ∧ CasasAlvero.HasCasasAlveroProp P ∧ ¬∃ α, P = (Polynomial.X - Polynomial.C α) ^ P.natDegree","subjects":["12"],"theorem":"CasasAlvero.casas_alvero.positive_char_counterexample"},{"answerKinds":[],"category":"research open","docstring":"The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial `P`\nhas the Casas-Alvero property, then `P = (X - α)ᵈ` for some `α`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.CasasAlvero","statement":"∀ {K : Type u_1} [inst : Field K] [CharZero K] (P : Polynomial K),\n  P.Monic → CasasAlvero.HasCasasAlveroProp P → ∃ α, P = (Polynomial.X - Polynomial.C α) ^ P.natDegree","subjects":["12"],"theorem":"CasasAlvero.casas_alvero_conjecture"},{"answerKinds":[],"category":"API","docstring":"Note that whether we use `HasCasasAlveroProp` or `HasCasasAlveroPropᵣ` to state the Casas-Alvero conjecture,\nwe obtain the following equivalent statements.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Paper.CasasAlvero","statement":"(∀ {K : Type u} [inst : Field K] [CharZero K] (P : Polynomial K),\n    P.Monic → CasasAlvero.HasCasasAlveroProp P → ∃ α, P = (Polynomial.X - Polynomial.C α) ^ P.natDegree) ↔\n  ∀ {K : Type u} [inst : Field K] [CharZero K] (P : Polynomial K),\n    P.Monic → CasasAlvero.HasCasasAlveroPropᵣ P → ∃ α, P = (Polynomial.X - Polynomial.C α) ^ P.natDegree","subjects":["12"],"theorem":"CasasAlvero.casas_alvero_iffᵣ"},{"answerKinds":[],"category":"research solved","docstring":"The Casas-Alvero conjecture holds for polynomials of prime power degree.\nThis was proved by Graf von Bothmer, Labs, Schicho, and van de Woestijne.\n\nReference: [The Casas-Alvero conjecture for infinitely many degrees](https://arxiv.org/pdf/math/0605090)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.CasasAlvero","statement":"∀ {K : Type u_1} [inst : Field K] [CharZero K] (P : Polynomial K),\n  P.Monic →\n    ∀ (p k : ℕ),\n      Nat.Prime p →\n        P.natDegree = p ^ k → CasasAlvero.HasCasasAlveroProp P → ∃ α, P = (Polynomial.X - Polynomial.C α) ^ P.natDegree","subjects":["12"],"theorem":"CasasAlvero.casas_alvero.prime_power"},{"answerKinds":[],"category":"research solved","docstring":"The Casas-Alvero conjecture holds for polynomials of degree `2p^k` where `p` is prime.\nThis was proved by Graf von Bothmer, Labs, Schicho, and van de Woestijne.\n\nReference: [The Casas-Alvero conjecture for infinitely many degrees](https://arxiv.org/pdf/math/0605090)\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.CasasAlvero","statement":"∀ {K : Type u_1} [inst : Field K] [CharZero K] (P : Polynomial K),\n  P.Monic →\n    ∀ (p k : ℕ),\n      Nat.Prime p →\n        P.natDegree = 2 * p ^ k →\n          CasasAlvero.HasCasasAlveroProp P → ∃ α, P = (Polynomial.X - Polynomial.C α) ^ P.natDegree","subjects":["12"],"theorem":"CasasAlvero.casas_alvero.double_prime_power"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.CasasAlvero","statement":"∀ {K : Type u_1} [inst : Field K] {P : Polynomial K} [IsAlgClosed K],\n  CasasAlvero.HasCasasAlveroProp P ↔ CasasAlvero.HasCasasAlveroPropᵣ P","subjects":["12"],"theorem":"CasasAlvero.hasCasasAlveroProp_iffᵣ"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Paper.CasasAlvero","statement":"∀ {K : Type u_1} [inst : Field K] {P : Polynomial K},\n  CasasAlvero.HasCasasAlveroPropᵣ P → CasasAlvero.HasCasasAlveroProp P","subjects":["12"],"theorem":"CasasAlvero.HasCasasAlveroPropᵣ.hasCasasAlveroProp"},{"answerKinds":["Prop"],"category":"research solved","docstring":"There exists a convex polyhedron with nonempty interior for which the Rupert property does\nnot hold.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/jcreedcmu/Noperthedron"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Paper.Rupert","statement":"False ↔ ∀ (vertices : Finset (Fin 3 → ℝ)), (interior ((convexHull ℝ) ↑vertices)).Nonempty → Rupert.IsRupert vertices","subjects":["52"],"theorem":"Rupert.is_every_convex_polyhedron_rupert"},{"answerKinds":[],"category":"research open","docstring":"**The Tu-Deng conjecture.** For $k \\ge 2$ and a nonzero residue $t$ modulo $2^k - 1$, there\nare at most $2^{k-1}$ pairs of residues $(a, b)$ with $a + b = t$ whose binary weights\n(of their representatives in $0, \\dots, 2^k - 2$) sum to at most $k - 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Paper.TuDengConjecture","statement":"∀ (k : ℕ),\n  2 ≤ k →\n    ∀ (t : ZMod (2 ^ k - 1)),\n      t ≠ 0 →\n        {p |\n              p.1 + p.2 = t ∧\n                TuDengConjecture.binaryWeight p.1.val + TuDengConjecture.binaryWeight p.2.val ≤ k - 1}.ncard ≤\n          2 ^ (k - 1)","subjects":["5","11","94"],"theorem":"TuDengConjecture.tu_deng_conjecture"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $A \\subset \\mathbb{Z}$ be a set of size $n$. For how many $\\theta \\in \\mathbb{R}/\\mathbb{Z}$\nmust we have $\\sum_{a \\in A} \\cos(2\\pi a\\theta) = 0$? The answer is the function `minZeros`. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«82»","statement":"sorry = Green82.minZeros","subjects":["11","42"],"theorem":"Green82.green_82"},{"answerKinds":[],"category":"research solved","docstring":"Upper bound: there exists $A$ with at most $C(n \\log n)^{2/3}$ zeros. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«82»","statement":"∃ C > 0,\n  ∀ n ≥ 1,\n    ∃ A,\n      A.card = n ∧\n        ↑{θ | θ ∈ Set.Ico 0 1 ∧ ∑ a ∈ A, Real.cos (2 * Real.pi * ↑a * θ) = 0}.ncard ≤\n          ↑⌊C * (↑n * Real.log ↑n) ^ (2 / 3)⌋₊","subjects":["11","42"],"theorem":"Green82.green_82_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Lower bound: every set has at least $(\\log \\log n)^{1-o(1)}$ zeros. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«82»","statement":"∃ o,\n  (o =o[Filter.atTop] fun x => 1) ∧\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (A : Finset ℤ),\n        A.card = n →\n          ↑⌊Real.log (Real.log ↑n) ^ (1 - |o n|)⌋₊ ≤\n            ↑{θ | θ ∈ Set.Ico 0 1 ∧ ∑ a ∈ A, Real.cos (2 * Real.pi * ↑a * θ) = 0}.ncard","subjects":["11","42"],"theorem":"Green82.green_82_lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Let `A ⊂ R` be a set of positive measure. Does $A$ contain an affine copy of `{1, 1/2, 1/4, . . . }`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«94»","statement":"True ↔ ∀ (A : Set ℝ), MeasurableSet A ∧ MeasureTheory.volume A > 0 → ∃ a b, a ≠ 0 ∧ ∀ (n : ℕ), a * (1 / 2 ^ n) + b ∈ A","subjects":["28"],"theorem":"Green94.green_94"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let `A ⊂ R` be a set of positive outer measure. Does $A$ contain an affine copy of `{1, 1/2, 1/4, . . . }`?\n\nThe answer is \"no\".\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/153d79d6c82c76fe1bee860742af800840c974d9/FormalConjectures/GreensOpenProblems/94.lean#L174"}],"hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«94»","statement":"False ↔ ∀ (A : Set ℝ), MeasureTheory.volume A > 0 → ∃ a b, a ≠ 0 ∧ ∀ (n : ℕ), a * (1 / 2 ^ n) + b ∈ A","subjects":["28"],"theorem":"Green94.green_94_outer_measure"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Which finite groups have the smallest biggest product-free sets?\n\nWe formalise this as: determine the supremum of exponents $\\alpha$ such that every nontrivial\nfinite group of order $n$ contains a product-free set of size $\\geq c n^{\\alpha}$ for some\nabsolute constant $c > 0$. (The trivial group is excluded since its only product-free subset\nis empty.) Kedlaya [Ke97] showed that $\\alpha = 11/14$ is admissible, and Green suggests this\nexponent may well be sharp; the candidate extremal family is the Ree groups ${}^2G_2(q)$,\n$q = 3^{2m+1}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«5»","statement":"IsLUB\n  {α |\n    ∃ c > 0,\n      ∀ (G : Type) [inst : Group G] [inst_1 : Fintype G],\n        Nontrivial G → ∃ S, IsProductFree ↑S ∧ c * ↑(Fintype.card G) ^ α ≤ ↑S.card}\n  sorry","subjects":["5","20"],"theorem":"Green5.green_5"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"A good model problem would be to determine the largest product-free subsets of\n$\\mathrm{SL}_2(\\mathbb{F}_p)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«5»","statement":"∀ (p : ℕ) [Fact (Nat.Prime p)],\n  have S := sorry;\n  MaximalFor IsProductFree Set.ncard ↑S","subjects":["5","20"],"theorem":"Green5.green_5.variants.sl_two"},{"answerKinds":[],"category":"research solved","docstring":"Kedlaya [Ke97] observed, refining some work of Babai and Sós [BaSo85], that it follows from the\nclassification of finite simple groups that every finite group $G$ of order $n$ has a\nproduct-free subset of size $\\gg n^{11/14}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«5»","statement":"∃ c > 0,\n  ∀ (G : Type) [inst : Group G] [inst_1 : Fintype G],\n    Nontrivial G → ∃ S, IsProductFree ↑S ∧ c * ↑(Fintype.card G) ^ (11 / 14) ≤ ↑S.card","subjects":["5","20"],"theorem":"Green5.green_5.variants.kedlaya"},{"answerKinds":[],"category":"research solved","docstring":"For the largest product-free subsets of $\\mathrm{SL}_2(\\mathbb{F}_p)$ of order $n$, the\nbest-known upper bound is $O(n^{8/9})$, due to Gowers [Go08].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«5»","statement":"∃ C > 0,\n  ∀ (p : ℕ) [inst : Fact (Nat.Prime p)] (S : Finset (Matrix.SpecialLinearGroup (Fin 2) (ZMod p))),\n    IsProductFree ↑S → ↑S.card ≤ C * ↑(Fintype.card (Matrix.SpecialLinearGroup (Fin 2) (ZMod p))) ^ (8 / 9)","subjects":["5","20"],"theorem":"Green5.green_5.variants.gowers_sl_two"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $G$ is a finite group, and let $A \\subset G \\times G$ be a subset of density $\\alpha$.\nIs it true that there are $\\gg_\\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$\nall lie in $A$?\n\nNote: A is taken as $\\alpha$-dense, i.e. $|A| \\ge \\alpha |G|^2$ [Au16, Question 2]\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«18»","statement":"True ↔\n  ∀ α > 0,\n    ∃ c > 0,\n      ∃ m₀,\n        ∀ (G : Type u_1) [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (A : Finset (G × G)),\n          Fintype.card G ≥ m₀ →\n            ↑A.card ≥ α * ↑(Fintype.card G) ^ 2 → ↑(Green18.numNaiveCorners A) ≥ c * ↑(Fintype.card G) ^ 3","subjects":["5","11","20"],"theorem":"Green18.green_18"},{"answerKinds":[],"category":"research solved","docstring":"[So13] proved this is true for \"BMZ corners\". Follows from the proof of Theorem 2.1, p.1456-1457.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«18»","statement":"∀ α > 0,\n  ∃ c > 0,\n    ∃ m₀,\n      ∀ (G : Type u_1) [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (A : Finset (G × G)),\n        Fintype.card G ≥ m₀ →\n          ↑A.card ≥ α * ↑(Fintype.card G) ^ 2 → ↑(Green18.numBmzCorners A) ≥ c * ↑(Fintype.card G) ^ 3","subjects":["5","11","20"],"theorem":"Green18.green_18.bmz_corners"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"For which values of $k$ is the following true: whenever we partition $[N] = A_1 \\cup \\dots \\cup A_k$,\n$\\left|\\bigcup^k_{i=1} (A_i \\hat{+} A_i)\\right| \\geq \\frac{1}{10} N$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«25»","statement":"{k | ∀ᶠ (N : ℕ) in Filter.atTop, Green25.Property25 (k N) N} = sorry","subjects":["5","11"],"theorem":"Green25.green_25"},{"answerKinds":[],"category":"research solved","docstring":"For $k \\ll \\log \\log N$, this is true [Gr24].\n\nNOTE: [ESS89] derive a precise constant for which this is true, but $\\ll$ would imply that it works\nfor any constant. We thus prefer a little-o statement.\n ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«25»","statement":"∀ (k : ℕ → ℕ),\n  (∀ᶠ (N : ℕ) in Filter.atTop, 1 ≤ k N ∧ k N ≤ N) ∧ (fun N => ↑(k N)) =o[Filter.atTop] Green25.bestLower →\n    ∀ᶠ (N : ℕ) in Filter.atTop, Green25.Property25 (k N) N","subjects":["5","11"],"theorem":"Green25.green_25.variants.lower_ess89"},{"answerKinds":[],"category":"research solved","docstring":"For $k \\gg N / \\log N$, it need not be true [Gr24].\n\nIn this version, cases like $k(N) = N$ or $N-1$ can produce trivial counter examples.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«25»","statement":"∃ k,\n  (∀ᶠ (N : ℕ) in Filter.atTop, 1 ≤ k N ∧ k N ≤ N) ∧\n    (Green25.bestUpper =O[Filter.atTop] fun N => ↑(k N)) ∧ ¬∀ᶠ (N : ℕ) in Filter.atTop, Green25.Property25 (k N) N","subjects":["5","11"],"theorem":"Green25.green_25.variants.upper_ess89_trivial"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"We conjecture that the best-known upper bound can be lowered. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«25»","statement":"have ans := sorry;\n(∀ᶠ (N : ℕ) in Filter.atTop, 1 ≤ ans N ∧ ans N ≤ N) ∧\n  (fun N => ↑(ans N)) =o[Filter.atTop] Green25.bestUpper ∧ ¬∀ᶠ (N : ℕ) in Filter.atTop, Green25.Property25 (ans N) N","subjects":["5","11"],"theorem":"Green25.green_25.upper"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"We conjecture that the best-known lower bound can be raised. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«25»","statement":"have ans := sorry;\nGreen25.bestLower =o[Filter.atTop] ans ∧\n  (∀ᶠ (N : ℕ) in Filter.atTop, 1 ≤ ans N ∧ ans N ≤ ↑N) ∧\n    ∀ (k : ℕ → ℕ),\n      (∀ᶠ (N : ℕ) in Filter.atTop, 1 ≤ k N ∧ k N ≤ N) ∧ (fun N => ↑(k N)) =O[Filter.atTop] ans →\n        ∀ᶠ (N : ℕ) in Filter.atTop, Green25.Property25 (k N) N","subjects":["5","11"],"theorem":"Green25.green_25.lower"},{"answerKinds":[],"category":"research solved","docstring":"For $k \\gg N / \\log N$, it need not be true [Gr24].\n\nWe formalize by enforcing a $N / \\log N$ growth rate on $k(N)$. This avoids cases like $k(N) = N$\nor $N-1$, which produce trivial counter examples with singletons and thus empty restricted sumsets.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«25»","statement":"∃ k,\n  (∀ᶠ (N : ℕ) in Filter.atTop, 1 ≤ k N ∧ k N ≤ N) ∧\n    (fun N => ↑(k N)) =Θ[Filter.atTop] Green25.bestUpper ∧ ¬∀ᶠ (N : ℕ) in Filter.atTop, Green25.Property25 (k N) N","subjects":["5","11"],"theorem":"Green25.green_25.variants.upper_ess89"},{"answerKinds":[],"category":"research open","docstring":"Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«42»","statement":"True ↔ Green42.CohnElkiesOptimal 2 (√3 / 6)","subjects":["51","52"],"theorem":"Green42.green_42"},{"answerKinds":[],"category":"research solved","docstring":"[CoEl03] proved this when $d = 1$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«42»","statement":"Green42.CohnElkiesOptimal 1 (1 / 2)","subjects":["51","52"],"theorem":"Green42.green_42.variants.dimension_one"},{"answerKinds":[],"category":"research solved","docstring":"In [CKM17], [Vi17] was adapted to $d = 24$. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/math-inc/Sphere-Packing-Lean/blob/main/SpherePacking/Dim24/MainTheorem.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«42»","statement":"Green42.CohnElkiesOptimal 24 1","subjects":["51","52"],"theorem":"Green42.green_42.variants.dimension_twenty_four"},{"answerKinds":[],"category":"research solved","docstring":"[Vi17] established the case $d = 8$. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/math-inc/Sphere-Packing-Lean/blob/main/SpherePacking/Dim8/MainTheorem.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«42»","statement":"Green42.CohnElkiesOptimal 8 (1 / 16)","subjects":["51","52"],"theorem":"Green42.green_42.variants.dimension_eight"},{"answerKinds":[],"category":"research solved","docstring":"The best-known upper bound for a Sidon subset of $\\{0, 1\\}^n$ ($N = 2^n$) is $N^{0.5753}$ [CLZ01].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"∃ C, ∀ (n : ℕ) (S : Finset (𝔽₂ n)), Green31.IsBinarySidon ↑S → ↑S.card ≤ C * (2 ^ n) ^ 0.5753","subjects":["5","11"],"theorem":"Green31.green_31.variants.sidon_01n_clz01"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], for all sufficiently\nlarge $N$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"have ans := sorry;\n(∀ᶠ (N : ℕ) in Filter.atTop, Green31.F N ≤ ans N) ∧\n  ∃ c < 0.98183, ∃ C, ∀ᶠ (N : ℕ) in Filter.atTop, ans N - √↑N ≤ c * ↑N ^ 4⁻¹ + C","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Green31.green_31.variants.upper_eventually"},{"answerKinds":[],"category":"research open","docstring":"Another very nice old problem is whether there is a Sidon subset of $\\{0, 1\\}^n$ of size $N^{0.51}$,\nwhere $N = 2^n$ [Gr24].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"True ↔ ∃ S, (∀ (n : ℕ), Green31.IsBinarySidon ↑(S n)) ∧ ∀ᶠ (n : ℕ) in Filter.atTop, (2 ^ n) ^ 0.51 ≤ ↑(S n).card","subjects":["5","11"],"theorem":"Green31.green_31.variants.sidon_01n"},{"answerKinds":[],"category":"research solved","docstring":"[BFR23] obtained a small improvement, getting an upper bound of $F(N) \\le N^{1/2} + 0.998 N^{1/4}$\nfor large $N$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"∀ᶠ (N : ℕ) in Filter.atTop, Green31.F N ≤ √↑N + 0.998 * ↑N ^ 4⁻¹","subjects":["5","11"],"theorem":"Green31.green_31.variants.upper_bfr23"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Can we improve the lower bound $N^{1/2} + O(1)$, at least for infinitely many $N$? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"have ans := sorry;\nFilter.Tendsto (fun N => ans N - √↑N) Filter.atTop Filter.atTop ∧ ∃ᶠ (N : ℕ) in Filter.atTop, ans N ≤ Green31.F N","subjects":["5","11"],"theorem":"Green31.green_31.lower"},{"answerKinds":[],"category":"research solved","docstring":"[Li69] proved $F(n) \\le n^{1/2} + n^{1/4} + O(1)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"∃ C, ∀ᶠ (N : ℕ) in Filter.atTop, Green31.F N ≤ √↑N + ↑N ^ 4⁻¹ + C","subjects":["5","11"],"theorem":"Green31.green_31.variants.upper_li69"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Can we improve the lower bound $N^{1/2} + O(1)$, for all sufficiently large $N$? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"have ans := sorry;\nFilter.Tendsto (fun N => ans N - √↑N) Filter.atTop Filter.atTop ∧ ∀ᶠ (N : ℕ) in Filter.atTop, ans N ≤ Green31.F N","subjects":["5","11"],"theorem":"Green31.green_31.variants.lower_eventually"},{"answerKinds":[],"category":"research open","docstring":"It is not known whether, if $G$ is an abelian group of size $n$, there always exists a Sidon subset\nof $G$ of size $0.01\\sqrt{n}$ [Gr24].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"True ↔ ∀ (G : Type) [inst : AddCommGroup G] [inst_1 : Fintype G], ∃ S, IsSidon ↑S ∧ 1e-2 * √↑(Fintype.card G) ≤ ↑S.card","subjects":["5","11"],"theorem":"Green31.green_31.variants.abelian"},{"answerKinds":[],"category":"research open","docstring":"It is not known whether or not there exists a Sidon subset of $\\mathbb{Z}/p\\mathbb{Z}$ of size\n$(1 + o(1))\\sqrt{p}$, for all $p$ [Gr24].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"True ↔\n  ∃ S o,\n    (o =o[Filter.atTop] fun x => 1) ∧\n      (∀ (p : ℕ), Nat.Prime p → IsSidon ↑(S p)) ∧ ∀ (p : ℕ), Nat.Prime p → ↑(S p).card = (1 + o p) * √↑p","subjects":["5","11"],"theorem":"Green31.green_31.variants.zmod_p"},{"answerKinds":[],"category":"research solved","docstring":"The upper bound was further improved to $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"∃ C, ∀ᶠ (N : ℕ) in Filter.atTop, Green31.F N ≤ √↑N + 0.98183 * ↑N ^ 4⁻¹ + C","subjects":["5","11"],"theorem":"Green31.green_31.variants.upper_cho25"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], at least for infinitely\nmany $N$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«31»","statement":"have ans := sorry;\n(∃ᶠ (N : ℕ) in Filter.atTop, Green31.F N ≤ ans N) ∧\n  ∃ c < 0.98183, ∃ C, ∀ᶠ (N : ℕ) in Filter.atTop, ans N - √↑N ≤ c * ↑N ^ 4⁻¹ + C","subjects":["5","11"],"theorem":"Green31.green_31.upper"},{"answerKinds":[],"category":"research open","docstring":"Is $\\varepsilon^{-C}$ rotations enough? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«41»","statement":"True ↔ ∃ C, ∃ ε₀ > 0, ∀ ε ∈ Set.Ioc 0 ε₀, ↑(Green41.minCopies ε) ≤ ε ^ (-C)","subjects":["51","52"],"theorem":"Green41.green_41.variants.polynomial_bound"},{"answerKinds":[],"category":"research solved","docstring":"[Ma15] proved that for any $\\varepsilon > 0$, finitely many rotations of the pyjama set of width\n$\\varepsilon$ cover the plane. This implies that the set we are taking the infimum over in `minCopies`\nis non-empty.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«41»","statement":"∀ (ε : ℝ), 0 < ε → (Green41.coveringCopies ε).Nonempty","subjects":["51","52"],"theorem":"Green41.minCopies_set_nonempty"},{"answerKinds":[],"category":"research solved","docstring":"[KrLe25] have established the first quantitative bound, showing via an analysis of [Ma15]'s method\nthat $\\exp\\exp\\exp(\\varepsilon^{-C})$ rotations suffice.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«41»","statement":"∃ C, ∃ ε₀ > 0, ∀ ε ∈ Set.Ioc 0 ε₀, ↑(Green41.minCopies ε) ≤ Real.exp (Real.exp (Real.exp (ε ^ (-C))))","subjects":["51","52"],"theorem":"Green41.green_41.variants.kravitz_leng"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"How many rotated (about the origin) copies of the 'pyjama set'\n$\\\\{(x, y) \\in \\mathbb{R}^2 : \\text{dist}(x, \\mathbb{Z}) \\leq \\varepsilon\\\\}$ are needed to cover\n$\\mathbb{R}^2$?\n\nIn particular, can one find a better bound than the best-known bound from [KrLe25]?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«41»","statement":"∃ C > 0,\n  ∃ ε₀ > 0,\n    ∀ ε ∈ Set.Ioc 0 ε₀,\n      have ans := sorry;\n      ↑(Green41.minCopies ε) ≤ ans ∧ ans < Real.exp (Real.exp (Real.exp (ε ^ (-C))))","subjects":["51","52"],"theorem":"Green41.green_41"},{"answerKinds":[],"category":"research open","docstring":"Is there a better bound than the best-known bound from [KrLe25]?\nThis is an existential version of the main problem that does not require providing the bound explicitly.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«41»","statement":"True ↔\n  ∃ C > 0,\n    ∃ ε₀ > 0, ∀ ε ∈ Set.Ioc 0 ε₀, ∃ ans, ↑(Green41.minCopies ε) ≤ ans ∧ ans < Real.exp (Real.exp (Real.exp (ε ^ (-C))))","subjects":["51","52"],"theorem":"Green41.green_41.variants.exists_better_bound"},{"answerKinds":[],"category":"research solved","docstring":"Problem 9 (i): is $r_3(N) \\ll N(\\log N)^{-10}$?\n\nSolved in [BlSi20].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«9»","statement":"(fun N => ↑(Green9.r 3 N)) =O[Filter.atTop] fun N => ↑N * Real.log ↑N ^ (-10)","subjects":["5"],"theorem":"Green9.green_9_i"},{"answerKinds":[],"category":"research open","docstring":"Problem 9 (ii): is $r_5(N) \\ll N(\\log N)^{-c}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«9»","statement":"True ↔ ∃ c > 0, (fun N => ↑(Green9.r 5 N)) =O[Filter.atTop] fun N => ↑N * Real.log ↑N ^ (-c)","subjects":["5"],"theorem":"Green9.green_9_ii"},{"answerKinds":[],"category":"research open","docstring":"Problem 9 (iii): is $r_4(\\mathbf{F}_5^n) \\ll N^{1-c}$, where $N=5^n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«9»","statement":"True ↔ ∃ c > 0, (fun n => ↑(Finset.maxAPFreeCard 4 Finset.univ)) =O[Filter.atTop] fun n => (5 ^ n) ^ (1 - c)","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"Green9.green_9_iii"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the largest product-free set in the alternating group $A_n$? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«4»","statement":"∀ (n : ℕ),\n  have S := sorry;\n  MaximalFor Green4.ProdFree Set.ncard (S n)","subjects":["20"],"theorem":"Green4.green_4"},{"answerKinds":[],"category":"research solved","docstring":"In the case of large n, the problem was solved in\n[On the largest product-free subsets of the alternating groups](https://arxiv.org/pdf/2205.15191).\nSpecifically, this theorem formalizes the statement of theorem 1.1 in the mentioned paper\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«4»","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ (S : Set ↥(alternatingGroup (Fin n))),\n    MaximalFor Green4.ProdFree Set.ncard S →\n      ∃ x I, S = Green4.extremalFamily x I ∨ S = (fun x => x⁻¹) '' Green4.extremalFamily x I","subjects":["20"],"theorem":"Green4.large_green_4"},{"answerKinds":[],"category":"test","docstring":"Any set has a gap of length 0 (vacuously true). ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"∀ {p : ℕ} (A : Finset (ZMod p)), Green32.HasGap A 0","subjects":["5","11"],"theorem":"Green32.hasGap_zero"},{"answerKinds":[],"category":"research solved","docstring":"In the regime $\\omega(p) \\sim c p$, this is Szemerédi's theorem [Gr24]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"∀ (c : ℝ),\n  0 < c ∧ c < 1 → ∀ (ω : ℕ → ℝ), (Asymptotics.IsEquivalent Filter.atTop ω fun p => c * ↑p) → Green32.HasLargeGapDilate ω","subjects":["5","11"],"theorem":"Green32.green_32.variants.szemeredi_regime"},{"answerKinds":[],"category":"test","docstring":"The empty set has a gap of any length. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"∀ {p : ℕ} (L : ℕ), Green32.HasGap ∅ L","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Green32.hasGap_empty"},{"answerKinds":[],"category":"research solved","docstring":"[Sh20] has used the polynomial method to show that this is true with 100 replaced by 2 [Gr24].\n\nNote: More precisely [Sh20, Theorem 1] implies a gap of at least $\\lfloor 2p/|A| - 2 \\rfloor$.\nFor a set $A$ of size $\\lfloor \\sqrt{p} \\rfloor$, this guarantees a gap of at least\n$\\lfloor 2\\sqrt{p} \\rfloor - 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"∀ᶠ (p : ℕ) in Filter.atTop,\n  Nat.Prime p → ∀ (A : Finset (ZMod p)), A.card = ⌊√↑p⌋₊ → ∃ c, Green32.HasGap (c • A) (⌊2 * √↑p⌋₊ - 2)","subjects":["5","11"],"theorem":"Green32.green_32.variants.sh20_sqrt"},{"answerKinds":[],"category":"test","docstring":"Concrete: $\\{0\\}$ in $\\mathbb{Z}/5\\mathbb{Z}$ has a gap of length 4 starting at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"Green32.HasGap {0} 4","subjects":["5","11"],"theorem":"Green32.hasGap_concrete"},{"answerKinds":[],"category":"research solved","docstring":"[Sh20, Theorem 1] implies a gap of at least $\\lfloor 2p/|A| - 2 \\rfloor$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"∀ (p : ℕ), Nat.Prime p → ∀ (A : Finset (ZMod p)), 1 < A.card → ∃ c, Green32.HasGap (c • A) ⌊2 * ↑p / ↑A.card - 2⌋₊","subjects":["5","11"],"theorem":"Green32.green_32.variants.sh20_general"},{"answerKinds":[],"category":"research open","docstring":"Even what happens in the regime $\\omega(p) \\sim 10 \\log p$ is unclear [Gr24]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"True ↔ ∀ (ω : ℕ → ℝ), (Asymptotics.IsEquivalent Filter.atTop ω fun p => 10 * Real.log ↑p) → Green32.HasLargeGapDilate ω","subjects":["5","11"],"theorem":"Green32.green_32.variants.log_regime"},{"answerKinds":[],"category":"research open","docstring":"Let $p$ be a prime and let $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ be a set of size $\\lfloor \\sqrt{p} \\rfloor$.\nIs there a dilate of $A$ containing a gap of length $100\\sqrt{p}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"True ↔ Green32.HasLargeGapDilate fun p => √↑p","subjects":["5","11"],"theorem":"Green32.green_32"},{"answerKinds":[],"category":"research solved","docstring":"In the regime $\\omega(p) \\le c \\log p$, this is basically Dirichlet's lower bound for the size of\nBohr sets [Gr24].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"∃ c > 0, ∀ (ω : ℕ → ℝ), (∀ᶠ (p : ℕ) in Filter.atTop, 100 < ω p ∧ ω p ≤ c * Real.log ↑p) → Green32.HasLargeGapDilate ω","subjects":["5","11"],"theorem":"Green32.green_32.variants.dirichlet_regime"},{"answerKinds":[],"category":"research solved","docstring":"Tom Sanders' finite field variant [Gr24].\nIf $N = 2^n$ and $A$ is a subset of size $\\lfloor \\sqrt{N} \\rfloor$, then $A^c$ contains a coset of\nsize at least $100\\sqrt{N}$ for sufficiently large $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"∀ᶠ (n : ℕ) in Filter.atTop, ∀ (A : Finset (𝔽₂ n)), A.card = ⌊√(2 ^ n)⌋₊ → Green32.HasCosetHole A ⌊100 * √(2 ^ n)⌋₊","subjects":["5","11"],"theorem":"Green32.green_32.variants.finite_field"},{"answerKinds":[],"category":"test","docstring":"The empty set in $\\mathbb{F}_2^n$ has a coset hole (using the trivial subspace). ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"∀ (n : ℕ), Green32.HasCosetHole ∅ 0","subjects":["5","11"],"theorem":"Green32.hasCosetHole_empty"},{"answerKinds":[],"category":"test","docstring":"The full set in $\\mathbb{Z}/p\\mathbb{Z}$ has no gap of positive length. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«32»","statement":"∀ {p : ℕ} [inst : NeZero p], ¬Green32.HasGap Finset.univ 1","subjects":["5","11"],"theorem":"Green32.not_hasGap_univ"},{"answerKinds":[],"category":"research solved","docstring":"Lower bound for $c(2)$ from Green's first paper ([Gr01]); the constant is `sqrt(4/7)` (about 0.7559). ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«35»","statement":"ENNReal.ofReal √(4 / 7) ≤ Green35.c 2","subjects":["26","28","42"],"theorem":"Green35.variants.c_2_lower"},{"answerKinds":[],"category":"research solved","docstring":"Best-known upper bound for $c(\\infty)$ due to Matolcsi and Vinuesa ([MV10]). ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«35»","statement":"Green35.c ⊤ ≤ 0.7505","subjects":["26","28","42"],"theorem":"Green35.variants.c_inf_upper"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Upper bound for $c(p)$ for $1 < p \\le \\infty$, improving the best-known value at $p = \\infty$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«35»","statement":"have ub := sorry;\n(∀ (p : ENNReal), 1 < p → Green35.c p ≤ ub p) ∧ ub ⊤ < 0.7505","subjects":["26","28","42"],"theorem":"Green35.green_35.upper"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Lower bound for $c(p)$ for $1 < p \\le \\infty$, improving the known value at $p = 2$ or $p = \\infty$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«35»","statement":"have lb := sorry;\n(∀ (p : ENNReal), 1 < p → lb p ≤ Green35.c p) ∧ (ENNReal.ofReal √(4 / 7) < Green35.c 2 ∨ 0.64 < Green35.c ⊤)","subjects":["26","28","42"],"theorem":"Green35.green_35.lower"},{"answerKinds":[],"category":"research solved","docstring":"Best-known lower bound for $c(\\infty)$ due to Cloninger and Steinerberger ([CS17]). ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«35»","statement":"0.64 ≤ Green35.c ⊤","subjects":["26","28","42"],"theorem":"Green35.variants.c_inf_lower"},{"answerKinds":[],"category":"textbook","docstring":"A comparison bound from Young's inequality. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«35»","statement":"Green35.c 2 ^ 2 ≤ Green35.c ⊤","subjects":["26","28","42"],"theorem":"Green35.variants.c_inf_lower_young"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $A + A$ contains the first $n$ squares. Is $|A| \\geq n^{1 - o(1)}$?\n\nIt is known that necessarily $|A| \\geq n^{2/3 - o(1)}$, whilst in the other direction there do\nexist such $A$ with $|A| \\ll_C n / \\log^C n$ for any $C$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«61»","statement":"True ↔\n  ∃ f,\n    Filter.Tendsto f Filter.atTop (nhds 0) ∧\n      ∀ n ≥ 1, ∀ (A : Finset ℕ), Finset.image (fun x => x ^ 2) (Finset.Icc 1 n) ⊆ A + A → ↑n ^ (1 - f n) ≤ ↑A.card","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Green61.green_61"},{"answerKinds":[],"category":"research open","docstring":"Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$.\nIs every $x \\in \\{1, \\ldots, p-1\\}$ congruent to some product $a_1 a_2$ where $a_1, a_2 \\in A$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«62»","statement":"True ↔\n  ∀ᶠ (p : ℕ) in Filter.atTop,\n    Nat.Prime p →\n      have A := Finset.filter Nat.Prime (Finset.range p);\n      ∀ (x : ℕ), 1 ≤ x ∧ x < p → ∃ a₁ ∈ A, ∃ a₂ ∈ A, ↑x = ↑a₁ * ↑a₂","subjects":["11"],"theorem":"Green62.green_62"},{"answerKinds":["Prop"],"category":"research solved","docstring":"For $G = \\mathbb{Z}/3\\mathbb{Z}$ and the functional $a(0) = -1$, $a(1) = -3$, $a(2) = 3$,\ndoes the support function of $\\Phi$ at $a$ strictly exceed that of $\\Phi'$?\n\nNumerical evidence suggests the answer is **yes**:\n$$\\sup_{\\varphi \\in \\Phi'(\\mathbb{Z}/3\\mathbb{Z})} \\operatorname{Re}\\langle a, \\varphi \\rangle\n  \\;<\\; \\sup_{\\varphi \\in \\Phi(\\mathbb{Z}/3\\mathbb{Z})} \\operatorname{Re}\\langle a, \\varphi \\rangle.$$\n\nThe DeepMind prover agent provided a formal proof, showing that $\\frac{183095}{30000}$ separates the\nsupport functions.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/f5afe85e1e02611f63c32ae041b33c67b7938cba/FormalConjectures/GreensOpenProblems/57.lean#L1071"}],"hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«57»","statement":"have a := ![-1, -3, 3];\nTrue ↔ Green57.supportFn (ZMod 3) a (Green57.Φ' (ZMod 3)) < Green57.supportFn (ZMod 3) a (Green57.Φ (ZMod 3))","subjects":["5","11"],"theorem":"Green57.green_57.variants.z3_functional"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Do $\\Phi(\\mathbb{Z}/3\\mathbb{Z})$ and $\\Phi'(\\mathbb{Z}/3\\mathbb{Z})$ coincide?\n\nNumerical evidence suggests the answer is **no**: the integer functional $a = (-1, -3, 3)$\nseparates the two spaces. A branch-and-bound verification over the phase variables shows\n$\\max_{\\Phi'} \\operatorname{Re}\\langle a, \\varphi \\rangle < 6.112 < 6.115 \\approx\n\\max_{\\Phi} \\operatorname{Re}\\langle a, \\varphi \\rangle$.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/f5afe85e1e02611f63c32ae041b33c67b7938cba/FormalConjectures/GreensOpenProblems/57.lean#L1100"}],"hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«57»","statement":"False ↔ Green57.Φ (ZMod 3) = Green57.Φ' (ZMod 3)","subjects":["5","11"],"theorem":"Green57.green_57.variants.z3"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that for every finite abelian group $G$ the spaces $\\Phi(G)$ and $\\Phi'(G)$,\nobtained from kernels $\\phi(g) = \\mathbb{E}_{x_1 + x_2 + x_3 = g} f_1(x_2, x_3)\n      f_2(x_1, x_3) f_3(x_1, x_2)$ with $\\lVert f_i \\rVert_\\infty \\le 1$\n(where $f_i : G \\times G \\to \\mathbb{C}$), still coincide when the\nthird kernel is required to depend only on $x_1 + x_2$?\n\nGreen guesses that the answer is probably 'no'.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/f5afe85e1e02611f63c32ae041b33c67b7938cba/FormalConjectures/GreensOpenProblems/57.lean#L1120"}],"hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«57»","statement":"False ↔ ∀ (G : Type) [inst : AddCommGroup G] [inst_1 : Fintype G] [DecidableEq G], Green57.Φ G = Green57.Φ' G","subjects":["5","11"],"theorem":"Green57.green_57"},{"answerKinds":[],"category":"research open","docstring":"The **no-k-in-line problem**:\nFor $N \\geq k$ and $k > 2$, the AllowedSetSize is $(k - 1) N$, i. e. on an $N \\times N$ subset,\nthere is a set of $(k - 1) N$ points for which no $k$ lie on a line (and not such a set of bigger size).\n\nNote the range. [GK2025] proves this for $k > 10^{37}$, which is `no_k_in_line_big` below. At\n$k = 3$ it is the claim Green expects to fail for large $N$, so this statement is not a\nconjecture anyone has made across the whole range $k > 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«72»","statement":"∀ {k N : ℕ}, 2 < k → k ≤ N → Green72.NoKInLineFor k N","subjects":["5","52"],"theorem":"Green72.NoKInLine"},{"answerKinds":["Prop"],"category":"research open","docstring":"Is $2N$ attained for all sufficiently large $N$?\n\nThis is not the negation of `green_72`. Negating that one gives $\\exists^f N$ where this asks\n$\\forall^f N$, so both can be answered `False` if the behaviour oscillates. Green asks his\nquestion in the `green_72` form. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«72»","statement":"sorry ↔ ∀ᶠ (N : ℕ) in Filter.atTop, Green72.NoKInLineFor 3 N","subjects":["5","52"],"theorem":"Green72.green_72.variants.eventually"},{"answerKinds":[],"category":"research solved","docstring":"In [GK2025] Grebennikov and Kwan prove the no-k-in-line conjecture for $k > 10 ^ 37$\nand $N \\geq k$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«72»","statement":"∀ {k : ℕ} (N : ℕ), 10 ^ 37 < k → k ≤ N → Green72.NoKInLineFor k N","subjects":["5","52"],"theorem":"Green72.no_k_in_line_big"},{"answerKinds":[],"category":"research solved","docstring":"For $N \\leq 60$, this has been verified with computers. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«72»","statement":"∀ {N : ℕ}, 3 ≤ N → N ≤ 60 → Green72.NoKInLineFor 3 N","subjects":["5","52"],"theorem":"Green72.no_three_in_line_le"},{"answerKinds":["Prop"],"category":"research open","docstring":"**Green's Open Problem 72 / No-three-in-line problem**:\nFor $N$ sufficiently large, is it impossible to have $2N$ points in $[N]^2$ with no three in a\nline? Green suspects the answer is yes, and that $(3/2 + o(1))N$ is optimal. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«72»","statement":"sorry ↔ ∀ᶠ (N : ℕ) in Filter.atTop, ¬Green72.NoKInLineFor 3 N","subjects":["5","52"],"theorem":"Green72.green_72"},{"answerKinds":[],"category":"textbook","docstring":"By the pigeon hole principle, the size of a subset of an $N \\times N$ grid such that no $k$\npoints lie on a line is bounded by $\\leq (k - 1) * N$. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«72»","statement":"∀ {k N : ℕ}, Green72.AllowedSetSize k N ≤ (k - 1) * N","subjects":["5","52"],"theorem":"Green72.allowedSetSize_le"},{"answerKinds":[],"category":"research open","docstring":"Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set\nof size at least $\\frac n 3 + Ω(n)$, where $Ω(n) → ∞$ as $n → ∞$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«1»","statement":"True ↔\n  ∃ Ω,\n    Filter.Tendsto Ω Filter.atTop Filter.atTop ∧\n      ∀ (n : ℕ) (A : Finset ℕ), (∀ a ∈ A, 0 < a) → A.card = n → ∃ S ⊆ A, IsSumFree ↑S ∧ ↑n / 3 + Ω n ≤ ↑S.card","subjects":["5","11"],"theorem":"Green1.green_1"},{"answerKinds":[],"category":"research solved","docstring":"Upper bound: $m(p) \\ll (\\log p)^2$ [Be23, Theorem 5]. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/mkwatson/unique-sums-notes/blob/d4383f93a52e3a916ccce78d5fb7739c3bf3dabc/Green27Upper/Theorem.lean#L26"}],"hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«27»","statement":"Green27.m =O[Green27.primesAtTop] Green27.upperBest","subjects":["5","11"],"theorem":"Green27.green_27.variants.upper_be23"},{"answerKinds":[],"category":"research solved","docstring":"We have $m(p) \\geq \\omega(p) \\log p$ for some function $\\omega(p)$ tending to infinity [Be23, Theorem 3].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«27»","statement":"∃ ω,\n  Filter.Tendsto ω Green27.primesAtTop Filter.atTop ∧ ∀ᶠ (p : ℕ) in Green27.primesAtTop, ω p * Real.log ↑p ≤ Green27.m p","subjects":["5","11"],"theorem":"Green27.green_27.variants.lower_be23"},{"answerKinds":[],"category":"research solved","docstring":"Previous best-known upper bound $m(p) \\ll \\sqrt{p}$ from [Be23]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«27»","statement":"Green27.m =O[Green27.primesAtTop] fun p => √↑p","subjects":["5","11"],"theorem":"Green27.green_27.variants.previous_upper"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Propose a better upper bound along primes. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«27»","statement":"have ans := sorry;\nans =o[Green27.primesAtTop] Green27.upperBest ∧ Green27.m =O[Green27.primesAtTop] ans","subjects":["5","11"],"theorem":"Green27.green_27.upper"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the size of the smallest set $A \\subset \\mathbb{Z} / p\\mathbb{Z}$ (with at least two elements)\nfor which no element in the sumset $A + A$ has a unique representation?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«27»","statement":"Asymptotics.IsEquivalent Green27.primesAtTop sorry Green27.m","subjects":["5","11"],"theorem":"Green27.green_27.equivalent"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Propose a better lower bound along primes. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«27»","statement":"have ans := sorry;\nGreen27.lowerBest =o[Green27.primesAtTop] ans ∧ ans =O[Green27.primesAtTop] Green27.m","subjects":["5","11"],"theorem":"Green27.green_27.lower"},{"answerKinds":[],"category":"research solved","docstring":"Previous best-known lower bound $\\log p \\ll m(p)$ from [St76]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«27»","statement":"(fun p => Real.log ↑p) =O[Green27.primesAtTop] Green27.m","subjects":["5","11"],"theorem":"Green27.green_27.variants.previous_lower"},{"answerKinds":[],"category":"research open","docstring":"Let $A \\subset \\mathbb{F}_2^n$ be a set of density $\\alpha > 0$. Does $10A$ contain a coset\nof some subspace of dimension at least $n - O(\\log(1/\\alpha))$?\n\nMore precisely: does there exist an absolute constant $C > 0$ such that for all $n \\geq 1$ and all\nnonempty $A \\subseteq \\mathbb{F}_2^n$ with density $\\alpha > 0$, the sumset $10A$ contains a coset\nof some subspace of dimension at least $n - C \\log_2(1/\\alpha)$?\n\nThe sumset $10A$ is defined as $\\{a_1 + a_2 + \\cdots + a_{10} : a_i \\in A\\}$, using the pointwise\nscalar multiplication notation `10 • A` where `•` denotes the iterated addition of a set.\n\nNote: We model $\\mathbb{F}_2^n$ as `Fin n → ZMod 2`, which is an $n$-dimensional vector space\nover $\\mathbb{F}_2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«50»","statement":"True ↔\n  ∃ C > 0,\n    ∀ (n : ℕ) (A : Finset (𝔽₂ n)),\n      A.Nonempty →\n        have α := ↑A.dens;\n        ∃ W v, v +ᵥ ↑W ⊆ ↑(10 • A) ∧ ↑n - C * Real.logb 2 (1 / α) ≤ ↑(Module.finrank (ZMod 2) ↥W)","subjects":["5","11"],"theorem":"Green50.green_50"},{"answerKinds":["Prop"],"category":"research solved","docstring":"[BLR08] proved $W(3, r) \\gg r^{2 - 1/\\log \\log r}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ (fun r => ↑r ^ (2 - 1 / Real.log (Real.log ↑r))) =O[Filter.atTop] fun r => ↑(Green14.W 3 r)","subjects":["5","11"],"theorem":"Green14.green_14_lower_bound_brown_landman_robertson"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 3) = 9$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 3 = 9","subjects":["5","11"],"theorem":"Green14.W_3_3"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 38) > 1378$ from [AKS14, Table 3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 38 > 1378","subjects":["5","11"],"theorem":"Green14.W_3_38_lower"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"It remains an interesting open problem to actually write down a colouring showing (say)\n$W(3, r) \\ge 2r^2$ for some $r$. [Gr24]\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"have ans := sorry;\nlet r := ans.fst;\nhave c := ans.snd;\n3 ≤ r ∧\n  ¬((∃ s, {x | ∃ s' ∈ s, ↑s' = x}.IsAPOfLength 3 ∧ ∀ x ∈ s, c x = 0) ∨\n      ∃ s, {x | ∃ s' ∈ s, ↑s' = x}.IsAPOfLength ↑r ∧ ∀ x ∈ s, c x = 1)","subjects":["5","11"],"theorem":"Green14.green_14_variant_2r2"},{"answerKinds":["Prop"],"category":"research solved","docstring":"[Gr21] proved a lower bound of shape $W(3, r) \\gg \\exp(c(\\log r)^{4/3-o(1)})$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔\n  ∃ c o,\n    ∃ (_ : Filter.Tendsto o Filter.atTop (nhds 0)),\n      (fun r => Real.exp (c * Real.log ↑r ^ (4 / 3 - o r))) =O[Filter.atTop] fun r => ↑(Green14.W 3 r)","subjects":["5","11"],"theorem":"Green14.green_14_lower_bound_green"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 30) \\ge 903$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 30 ≥ 903","subjects":["5","11"],"theorem":"Green14.W_3_30_lower"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 6) = 32$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 6 = 32","subjects":["5","11"],"theorem":"Green14.W_3_6"},{"answerKinds":["Prop"],"category":"research solved","docstring":"[Sc20] proves the upper bound $W(3, r) < \\exp(r^{1-c})$ for some $c > 0$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ ∃ c, 0 < c ∧ (fun r => ↑(Green14.W 3 r)) =O[Filter.atTop] fun r => Real.exp (↑r ^ (1 - c))","subjects":["5","11"],"theorem":"Green14.green_14_upper_bound_schoen"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 26) \\ge 727$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 26 ≥ 727","subjects":["5","11"],"theorem":"Green14.W_3_26_lower"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 23) \\ge 516$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 23 ≥ 516","subjects":["5","11"],"theorem":"Green14.W_3_23_lower"},{"answerKinds":["Prop"],"category":"research solved","docstring":"[KeMe23] gives a corresponding upper bound $W(3, r) \\ll \\exp(C(\\log r)^C)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ ∃ C, (fun r => ↑(Green14.W 3 r)) =O[Filter.atTop] fun r => Real.exp (C * Real.log ↑r ^ C)","subjects":["5","11"],"theorem":"Green14.green_14_upper_bound_kelley_meka"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 36) > 1257$ from [AKS14, Table 3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 36 > 1257","subjects":["5","11"],"theorem":"Green14.W_3_36_lower"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 34) > 1143$ from [AKS14, Table 3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 34 > 1143","subjects":["5","11"],"theorem":"Green14.W_3_34_lower"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 10) = 97$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 10 = 97","subjects":["5","11"],"theorem":"Green14.W_3_10"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 17) = 279$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 17 = 279","subjects":["5","11"],"theorem":"Green14.W_3_17"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 4) = 18$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 4 = 18","subjects":["5","11"],"theorem":"Green14.W_3_4"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 29) \\ge 868$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 29 ≥ 868","subjects":["5","11"],"theorem":"Green14.W_3_29_lower"},{"answerKinds":["Prop"],"category":"research solved","docstring":"[Hu22] improved this to $W(3, r) \\gg \\exp(c(\\log r)^{2-o(1)})$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔\n  ∃ c o,\n    ∃ (_ : Filter.Tendsto o Filter.atTop (nhds 0)),\n      (fun r => Real.exp (c * Real.log ↑r ^ (2 - o r))) =O[Filter.atTop] fun r => ↑(Green14.W 3 r)","subjects":["5","11"],"theorem":"Green14.green_14_lower_bound_hunter"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 16) = 238$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 16 = 238","subjects":["5","11"],"theorem":"Green14.W_3_16"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 21) \\ge 416$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 21 ≥ 416","subjects":["5","11"],"theorem":"Green14.W_3_21_lower"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 5) = 22$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 5 = 22","subjects":["5","11"],"theorem":"Green14.W_3_5"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 35) > 1204$ from [AKS14, Table 3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 35 > 1204","subjects":["5","11"],"theorem":"Green14.W_3_35_lower"},{"answerKinds":[],"category":"research solved","docstring":"We know $W(3, r)$ does not have polynomial growth in $r$ [Gr21, p.3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"¬∃ d, (fun r => ↑(Green14.W 3 r)) =O[Filter.atTop] fun r => ↑r ^ d","subjects":["5","11"],"theorem":"Green14.green_14_polynomial_k_eq_3"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 13) = 160$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 13 = 160","subjects":["5","11"],"theorem":"Green14.W_3_13"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 7) = 46$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 7 = 46","subjects":["5","11"],"theorem":"Green14.W_3_7"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 33) > 1063$ from [AKS14, Table 3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 33 > 1063","subjects":["5","11"],"theorem":"Green14.W_3_33_lower"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 14) = 186$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 14 = 186","subjects":["5","11"],"theorem":"Green14.W_3_14"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 37) > 1338$ from [AKS14, Table 3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 37 > 1338","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Green14.W_3_37_lower"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 22) \\ge 464$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 22 ≥ 464","subjects":["5","11"],"theorem":"Green14.W_3_22_lower"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is $W(3, r) \\ll r^2$?\n\n[Gr21] proves a superpolynomial lower bound $W(3, r) \\gg \\exp(c(\\log r)^{4/3-o(1)})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"False ↔ (fun r => ↑(Green14.W 3 r)) =O[Filter.atTop] fun r => ↑r ^ 2","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Green14.green_14_quadratic"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 9) = 77$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 9 = 77","subjects":["5","11"],"theorem":"Green14.W_3_9"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 20) \\ge 389$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 20 ≥ 389","subjects":["5","11"],"theorem":"Green14.W_3_20_lower"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 18) = 312$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 18 = 312","subjects":["5","11"],"theorem":"Green14.W_3_18"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 24) \\ge 593$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 24 ≥ 593","subjects":["5","11"],"theorem":"Green14.W_3_24_lower"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 39) > 1418$ from [AKS14, Table 3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 39 > 1418","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Green14.W_3_39_lower"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 15) = 218$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 15 = 218","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Green14.W_3_15"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 19) = 349$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 19 = 349","subjects":["5","11"],"theorem":"Green14.W_3_19"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 28) \\ge 827$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 28 ≥ 827","subjects":["5","11"],"theorem":"Green14.W_3_28_lower"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 32) > 1006$ from [AKS14, Table 3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 32 > 1006","subjects":["5","11"],"theorem":"Green14.W_3_32_lower"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 27) \\ge 770$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 27 ≥ 770","subjects":["5","11"],"theorem":"Green14.W_3_27_lower"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 12) = 135$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 12 = 135","subjects":["5","11"],"theorem":"Green14.W_3_12"},{"answerKinds":["Prop"],"category":"research solved","docstring":"[LiSh10] proved $W(3, r) \\gg (r / \\log r)^2$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ (fun r => (↑r / Real.log ↑r) ^ 2) =O[Filter.atTop] fun r => ↑(Green14.W 3 r)","subjects":["5","11"],"theorem":"Green14.green_14_lower_bound_li_shu"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 11) = 114$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 11 = 114","subjects":["5","11"],"theorem":"Green14.W_3_11"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 31) > 930$ from [AKS14, Table 3]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 31 > 930","subjects":["5","11"],"theorem":"Green14.W_3_31_lower"},{"answerKinds":[],"category":"research open","docstring":"Is $W(k, r)$ a polynomial in $r$, for fixed $k$?\n\nWe formulate this as asking if $W(k, r)$ has polynomial growth in $r$.\nWe know it is not the case for $k = 3$ [Gr21, p.3].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ ∀ k ≥ 4, ∃ d, (fun r => ↑(Green14.W k r)) =O[Filter.atTop] fun r => ↑r ^ d","subjects":["5","11"],"theorem":"Green14.green_14_polynomial"},{"answerKinds":[],"category":"research solved","docstring":"$W(3, 8) = 58$ from [AKS14]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"Green14.W 3 8 = 58","subjects":["5","11"],"theorem":"Green14.W_3_8"},{"answerKinds":[],"category":"research open","docstring":"$W(3, 25) \\ge 656$ from [AKS14, Table 2]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«14»","statement":"True ↔ Green14.W 3 25 ≥ 656","subjects":["5","11"],"theorem":"Green14.W_3_25_lower"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \\subset A$,\n$|S| \\gg K^{-O(1)} |A|$, with $S^8 \\subset A^4$? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«29»","statement":"True ↔\n  ∃ C c,\n    0 < C ∧\n      0 < c ∧\n        ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] (K : ℝ) (A : Finset G),\n          1 ≤ K → IsApproximateSubgroup K ↑A → ∃ S ⊆ A, C * K ^ (-c) * ↑A.card ≤ ↑S.card ∧ S ^ 8 ⊆ A ^ 4","subjects":["20"],"theorem":"Green29.green_29"},{"answerKinds":[],"category":"research solved","docstring":"Such a conclusion is known with $|S| \\gg_K |A|$ [Br13 Problem 6.5, CrSi10, Sa10]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«29»","statement":"∀ (K : ℝ),\n  1 ≤ K →\n    ∃ c,\n      0 < c ∧\n        ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] (A : Finset G),\n          IsApproximateSubgroup K ↑A → ∃ S, c * ↑A.card ≤ ↑S.card ∧ S ^ 8 ⊆ A ^ 4","subjects":["20"],"subsets":["FC100SolvedSet1"],"theorem":"Green29.green_29.variant"},{"answerKinds":[],"category":"research open","docstring":"Suppose $A, B ⊆ \\{1, \\dots, N\\}$ both have size at least $N^{0.49}$. Must the sumset $A + B$\ncontain a composite number?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«58»","statement":"True ↔\n  ∀ᶠ (N : ℕ) in Filter.atTop,\n    ∀ A ⊆ Finset.Icc 1 N, ∀ B ⊆ Finset.Icc 1 N, ↑N ^ 0.49 ≤ ↑A.card → ↑N ^ 0.49 ≤ ↑B.card → ∃ m ∈ A + B, m.Composite","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Green58.green_58"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"Green39.proportionCoverable 7 4 2 = 3 / 5","subjects":["5","60"],"theorem":"Green39.proportionCoverable_7_4_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"Green39.proportionCoverable 2 1 2 = 1","subjects":["5","60"],"theorem":"Green39.proportionCoverable_2_1_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"Green39.proportionCoverable 3 4 2 = 0","subjects":["5","60"],"theorem":"Green39.proportionCoverable_a_gt_p"},{"answerKinds":[],"category":"research open","docstring":"Similar questions are interesting with $\\sqrt{p}$ replaced by $p^\\theta$ for any $\\theta \\le 1/2$. [Gr24]\n\nNOTE: using $C p^\\theta$ translates as stated makes the conjecture trivially false by the pigeonhole\nprinciple. Indeed for a set of size $p^\\theta$, we cover at most $C p^{2\\theta}$ elements, which is\nstrictly less than $p$ for $\\theta < 1/2$. We interpret the question as asking whether\n$O(p^{1-\\theta})$ translates suffice. This generalizes the main conjecture where\n$\\sqrt{p} = p^{1-1/2}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"True ↔\n  ∀ (θ : ℝ),\n    0 < θ →\n      θ ≤ 1 / 2 →\n        ∃ C > 1,\n          Filter.Tendsto\n            (fun p =>\n              have k := ⌊↑↑p ^ θ⌋₊;\n              have c := ⌊C * ↑↑p ^ (1 - θ)⌋₊;\n              ↑(Green39.proportionCoverable (↑p) k c))\n            Filter.atTop (nhds 1)","subjects":["5","60"],"theorem":"Green39.green_39.variant_theta"},{"answerKinds":[],"category":"research open","docstring":"\"I do not know how to answer this even with 100 replaced by 1.01.\" [Gr24]\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"True ↔\n  Filter.Tendsto\n    (fun p =>\n      have k := (↑p).sqrt;\n      have c := ⌊1.01 * ↑k⌋₊;\n      ↑(Green39.proportionCoverable (↑p) k c))\n    Filter.atTop (nhds 1)","subjects":["5","60"],"theorem":"Green39.green_39.variant_101"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"Green39.proportionCoverable 5 2 0 = 0","subjects":["5","60"],"theorem":"Green39.proportionCoverable_t_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"Green39.proportionCoverable 11 4 3 = 1 / 6","subjects":["5","60"],"theorem":"Green39.proportionCoverable_11_4_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"Green39.proportionCoverable 3 3 1 = 1","subjects":["5","60"],"theorem":"Green39.proportionCoverable_p_p_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"Green39.proportionCoverable 3 1 2 = 0","subjects":["5","60"],"theorem":"Green39.proportionCoverable_3_1_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"Green39.proportionCoverable 11 3 4 = 1 / 3","subjects":["5","60"],"theorem":"Green39.proportionCoverable_11_3_4"},{"answerKinds":[],"category":"research open","docstring":"If $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ is random, $|A| = \\sqrt{p}$, can we almost surely cover\n$\\mathbb{Z}/p\\mathbb{Z}$ with $100\\sqrt{p}$ translates of $A$? [Gr24]\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«39»","statement":"True ↔\n  Filter.Tendsto\n    (fun p =>\n      have k := (↑p).sqrt;\n      have c := 100 * k;\n      ↑(Green39.proportionCoverable (↑p) k c))\n    Filter.atTop (nhds 1)","subjects":["5","60"],"theorem":"Green39.green_39"},{"answerKinds":[],"category":"research open","docstring":"Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes\n$2 \\leqslant p_1 < p_2 < \\dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most\n$\\frac{1}{10} N$?\n\nWe interpret \"half the residue classes\" as $\\lfloor p_i / 2 \\rfloor$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«44»","statement":"True ↔\n  ∀ (N : ℕ) (p : Fin 1000 → ℕ) (A : (i : Fin 1000) → Finset (ZMod (p i))),\n    have remaining := {x ∈ Finset.Icc 1 N | ∀ (i : Fin 1000), ↑x ∉ A i};\n    (∀ (i : Fin 1000), Nat.Prime (p i)) →\n      StrictMono p → p 999 ^ 10 < N ^ 9 → (∀ (i : Fin 1000), (A i).card = p i / 2) → 10 * remaining.card ≤ N","subjects":["11"],"theorem":"Green44.green_44"},{"answerKinds":[],"category":"research solved","docstring":"The answer is affirmative if the primes are all less than $N^{1/2}$, by the large sieve. [Gr24] ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«44»","statement":"∀ (N : ℕ) (p : Fin 1000 → ℕ) (A : (i : Fin 1000) → Finset (ZMod (p i))),\n  have remaining := {x ∈ Finset.Icc 1 N | ∀ (i : Fin 1000), ↑x ∉ A i};\n  (∀ (i : Fin 1000), Nat.Prime (p i)) →\n    StrictMono p → p 999 ^ 2 < N → (∀ (i : Fin 1000), (A i).card = p i / 2) → 10 * remaining.card ≤ N","subjects":["11"],"theorem":"Green44.green_44.variants.less_than_sqrt"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values,\nand such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly\ndistributed on their ranges?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«28»","statement":"True ↔\n  ∀ (X Y : PMF ℤ),\n    X.support.Finite ∧ Y.support.Finite ∧ Green28.IsUniformOnSupport (Green28.indepSum X Y) →\n      Green28.IsUniformOnSupport X ∧ Green28.IsUniformOnSupport Y","subjects":["60"],"theorem":"Green28.green_28"},{"answerKinds":[],"category":"research open","docstring":"Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$\ndetermined by them?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«77»","statement":"True ↔ ∃ o, Filter.Tendsto o Filter.atTop (nhds 0) ∧ Erdos507.α =O[Filter.atTop] fun n => ↑n ^ (-2 + o n)","subjects":["5","52"],"theorem":"Green77.green_77"},{"answerKinds":[],"category":"research solved","docstring":"Green notes that a well-known, almost trivial argument gives an $O(X^{1/4})$ bound on the left.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«66»","statement":"∃ C > 0, ∀ᶠ (X : ℝ) in Filter.atTop, ∃ n, Green66.IsSumOfTwoSquares n ∧ ↑n ∈ Set.Icc (X - C * X ^ (1 / 4)) X","subjects":["11"],"theorem":"Green66.green_66.variants.trivial_bound"},{"answerKinds":[],"category":"research open","docstring":"Is there always a sum of two squares between $X - \\frac{1}{10}X^{1/4}$ and $X$?\nWe formalize this as an eventual statement for sufficiently large real $X$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«66»","statement":"True ↔ ∀ᶠ (X : ℝ) in Filter.atTop, ∃ n, Green66.IsSumOfTwoSquares n ∧ ↑n ∈ Set.Icc (X - 1 / 10 * X ^ (1 / 4)) X","subjects":["11"],"theorem":"Green66.green_66"},{"answerKinds":[],"category":"research solved","docstring":"From [Aa19] p.577: the trivial upper bound is $n^2$ (non asymptotic). ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«24»","statement":"∀ {n : ℕ}, Green24.max013AffineTranslates n ≤ n ^ 2","subjects":["5","11"],"theorem":"Green24.variants.upper_trivial"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set\n$\\lbrace 0,1,3 \\rbrace$ that $A$ can contain?\n\nConjectured in [Aa19] p.579: $\\left(\\frac{1}{3} + o(1)\\right) n^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«24»","statement":"∀ (n : ℕ), Green24.max013AffineTranslates n = sorry","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Green24.green_24"},{"answerKinds":[],"category":"research solved","docstring":"Asymptotic upper bound (1.2) in [Aa19]. Named after Hardy and Littlewood [HaL28]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«24»","statement":"Green24.variants.gamma ≤ 3 / 4","subjects":["5","11"],"theorem":"Green24.variants.upper_HL"},{"answerKinds":[],"category":"research solved","docstring":"Asymptotic lower bound (1.2) in [Aa19]. Named after Hardy and Littlewood [HaL28]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«24»","statement":"Green24.variants.gamma ≥ 1 / 12","subjects":["5","11"],"theorem":"Green24.variants.lower_HL"},{"answerKinds":[],"category":"research open","docstring":"Conjecture p.579 in [Aa19]: $\\left(\\frac{1}{3} + o(1)\\right) n^2$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«24»","statement":"Green24.variants.gamma = 1 / 3","subjects":["5","11"],"theorem":"Green24.variants.conjecture"},{"answerKinds":[],"category":"research open","docstring":"Is there an absolute constant $c > 0$ such that, whenever $A ⊆ \\mathbb{N}$ is a set of squares\nwith $|A| ≥ 2$, the sumset $A + A$ satisfies $|A + A| ≥ |A|^{1 + c}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«60»","statement":"True ↔ ∃ c > 0, ∀ (A : Finset ℕ), (∀ a ∈ A, IsSquare a) → 2 ≤ A.card → ↑(A + A).card ≥ ↑A.card ^ (1 + c)","subjects":["11"],"theorem":"Green60.green_60"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Determine the asymptotic equivalence class (theta) of `m(N, k)`. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«37»","statement":"∀ (k : ℕ), (fun N => ↑(Green37.m N k)) =Θ[Filter.atTop] sorry","subjects":["5","11"],"theorem":"Green37.green_37_theta"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Determine an upper bound (big O) for `m(N, k)`. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«37»","statement":"∀ (k : ℕ), (fun N => ↑(Green37.m N k)) =O[Filter.atTop] sorry","subjects":["5","11"],"theorem":"Green37.green_37_bigO"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Determine a strict upper bound (little o) for `m(N, k)`. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«37»","statement":"∀ (k : ℕ), (fun N => ↑(Green37.m N k)) =o[Filter.atTop] sorry","subjects":["5","11"],"theorem":"Green37.green_37_littleO"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Given a natural number `N`, what is the smallest size of a subset of `ℕ` that contains, for each `d = 1, …, N`,\nan arithmetic progression of length `k` with common difference `d`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«37»","statement":"∀ (N k : ℕ), IsLeast {m | ∃ A, A.card = m ∧ Green37.IsAPCover (↑A) N k} sorry","subjects":["5","11"],"theorem":"Green37.green_37"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Asymptotic version: determine the asymptotic behavior of `m(N, k)` as `N` grows.\nThe solver should determine what function `f : ℕ → ℝ` eventually equals `(fun N ↦ (m N k : ℝ))`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«37»","statement":"∀ (k : ℕ), ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Green37.m N k) = sorry N","subjects":["5","11"],"theorem":"Green37.green_37_asymptotic"},{"answerKinds":[],"category":"research open","docstring":"Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?\n\nThe following very particular instance is probably the simplest [Gr24]:\nSuppose that $A \\subset \\mathbb{N}$ is a set with the property that\n$|A \\pmod p| \\leqslant \\frac{1}{2}(p + 1)$ for all sufficiently large $p$.\nIs it true that either $|A \\cap [X]| \\ll X^{1/2} / \\log^{100} X$, or $A$ is contained in the\nimage of $\\mathbb{Z}$ under a quadratic map $\\phi : \\mathbb{Q} \\to \\mathbb{Q}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«47»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    (∀ᶠ (p : ℕ) in Filter.atTop, Nat.Prime p → ((fun a => ↑a) '' A).ncard ≤ (p + 1) / 2) →\n      ((fun X => ↑(A ∩ Set.Iic X).ncard) =O[Filter.atTop] fun X => √↑X / Real.log ↑X ^ 100) ∨\n        ∃ P, P.degree = 2 ∧ ∀ a ∈ A, ∃ z, ↑a = Polynomial.eval (↑z) P","subjects":["11"],"theorem":"Green47.green_47"},{"answerKinds":[],"category":"research solved","docstring":"[GrSa25, Theorem 1.1] found a permissible upper bound.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«22»","statement":"∀ᶠ (r : ℕ) in Filter.atTop, ↑(Green22.N₀ r) ≤ Green22.GreenSawhneyBound r","subjects":["5","11"],"theorem":"Green22.green_22.variants.green_sawhney_bound"},{"answerKinds":[],"category":"research solved","docstring":"[Mo17, Corollary 1.5] For any finite coloring of $\\mathbb{N}$ there exist (infinitely many)\n$x, y \\in \\mathbb{N}$ such that $\\{xy, x + y\\}$ is monochromatic.\n\nThis guarantees that $N_0(r)$ is well-defined.\n\nNote: [Mo17] also establishes that $x$ is of the same colour.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«22»","statement":"∀ (r : ℕ) (coloring : ℕ → Fin r),\n  {p | 0 < p.1 ∧ 0 < p.2 ∧ coloring p.1 = coloring (p.1 * p.2) ∧ coloring p.1 = coloring (p.1 + p.2)}.Infinite","subjects":["5","11"],"theorem":"Green22.green_22.variants.moreira_infinite"},{"answerKinds":[],"category":"research solved","docstring":"Since $x, y \\geq 3$, we must have $xy \\geq 9$, so $N_0(r) \\geq 9$\n(assuming $N_0(r)$ is well-defined, which follows from [Mo17]).\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«22»","statement":"∀ (r : ℕ), r ≠ 0 → 9 ≤ Green22.N₀ r","subjects":["5","11"],"theorem":"Green22.green_22.variants.lower_nine"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"If $\\{1, \\ldots, N\\}$ is $r$-coloured then, for $N \\geqslant N_0(r)$, there are integers\n$x, y \\geqslant 3$ such that $x + y, xy$ have the same colour.\n\nFind reasonable bounds for $N_0(r)$. The goal is to improve upon the Green-Sawhney bound.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«22»","statement":"have ans := sorry;\n∀ᶠ (r : ℕ) in Filter.atTop, ↑(Green22.N₀ r) ≤ ans r ∧ ans =o[Filter.atTop] Green22.GreenSawhneyBound","subjects":["5","11"],"theorem":"Green22.green_22"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $A \\subset \\mathbb{F}_2^n$ is a set with an additive complement of size $K$.\nDoes $2A$ contain a coset of codimension $O_K(1)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«52»","statement":"True ↔\n  ∃ c,\n    ∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)),\n      S.card = K → A + ↑S = Set.univ → ∃ V, ↑V ⊆ A + A ∧ n ≤ Module.finrank (ZMod 2) ↥V.direction + c K","subjects":["5","11"],"theorem":"Green52.green_52"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Could $2A$ even contain a coset of codimension $O(\\log K)$?\n\nFrom [Green's 2025 update](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.52):\n\n> Update 2025. Kaave Hosseini and Ryan Alweiss have independently pointed out\n> that the second question is far too optimistic. To see this, let $A$ be a Hamming ball\n> of radius $n/2 - \\sqrt{n}$. There is a set $S$ of size $O(n)$ such that\n> $A + S = \\mathbb{F}_2^n$; a random choice of $S$ will work. However, every subspace\n> contained in $A - A$ has codimension $\\gg \\sqrt{n}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/green-52-log-counterexample/blob/16cb5d0/lean/Green52LogCounterexampleFC.lean#L661-L668"}],"hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«52»","statement":"False ↔\n  ∃ C D,\n    ∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)),\n      0 < K →\n        S.card = K →\n          A + ↑S = Set.univ → ∃ V, ↑V ⊆ A + A ∧ ↑n ≤ ↑(Module.finrank (ZMod 2) ↥V.direction) + C * Real.log ↑K + D","subjects":["5","11"],"theorem":"Green52.green_52_log"},{"answerKinds":[],"category":"research solved","docstring":"$f(r) \\le r^r / r! \\sim e^r$ [Gr24]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"∀ (r : ℕ), Green40.f r ≤ ↑r ^ r / ↑r.factorial","subjects":["5","94"],"theorem":"Green40.green_40.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $\\tilde{f}(2) = 1$ [St94]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"Green40.f_tilde 2 = 1","subjects":["5","94"],"theorem":"Green40.green_40.variants.arbitrary_subsets_sanity_f_tilde_two"},{"answerKinds":[],"category":"research solved","docstring":"We evidently have $\\tilde{f}(r) \\le f(r)$ [Gr24]. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"∀ (r : ℕ), Green40.f_tilde r ≤ Green40.f r","subjects":["5","94"],"theorem":"Green40.green_40.f_tilde_le_f"},{"answerKinds":[],"category":"research solved","docstring":"The only value known is $f(1) = 1$, which follows from the existence of the Hamming code [Gr24]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"Green40.f 1 = 1","subjects":["5","94"],"theorem":"Green40.green_40.sanity_f_one"},{"answerKinds":[],"category":"research open","docstring":"Does $\\tilde{f}(r) \\to \\infty$? [Gr24] ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"True ↔ Filter.Tendsto Green40.f_tilde Filter.atTop (nhds ⊤)","subjects":["5","94"],"theorem":"Green40.green_40.variants.arbitrary_subsets"},{"answerKinds":[],"category":"research solved","docstring":"The best-known upper bound for $f(2)$ is $1.4238$ [CHL97]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"Green40.f 2 ≤ 1.4238","subjects":["5","94"],"theorem":"Green40.green_40.upper_bound_f_two"},{"answerKinds":[],"category":"research open","docstring":"Does $f(r) \\to \\infty$? [Gr24]","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"True ↔ Filter.Tendsto Green40.f Filter.atTop (nhds ⊤)","subjects":["5","94"],"theorem":"Green40.green_40"},{"answerKinds":[],"category":"research open","docstring":"Does $f_{\\text{all}}(r) \\to \\infty$? [Gr24]\n\nThe target filter is `𝓝 ⊤`, as in `green_40` and `green_40.variants.arbitrary_subsets`. On\n`ℝ≥0∞`, `atTop` is the principal ultrafilter at `⊤`, so `Tendsto f_all atTop atTop` would say\nthat `f_all r = ⊤` for all large `r` rather than that `f_all r → ∞`. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"True ↔ Filter.Tendsto Green40.f_all Filter.atTop (nhds ⊤)","subjects":["5","94"],"theorem":"Green40.green_40.variants.all_n"},{"answerKinds":[],"category":"research open","docstring":"The possibility that f(r) = 1 for all r has not been ruled out [Gr24] ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"True ↔ ∀ (r : ℕ), Green40.f r = 1","subjects":["5","94"],"theorem":"Green40.green_40.f_eq_one_for_all"},{"answerKinds":[],"category":"research open","docstring":"It is not known whether f(2) = 1 [Gr24] ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«40»","statement":"True ↔ Green40.f 2 = 1","subjects":["5","94"],"theorem":"Green40.green_40.f_two_eq_one"},{"answerKinds":[],"category":"test","docstring":"Trivial lower bound: if $A + A = \\mathbb{Z}/q\\mathbb{Z}$, then $|A|^2 \\geq q$,\nsince the sumset $A + A$ has at most $|A|^2$ elements. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«33»","statement":"∀ (q : ℕ+) (A : Finset (ZMod ↑q)), A + A = Finset.univ → ↑q ≤ A.card ^ 2","subjects":["5","11"],"theorem":"Green33.green_33.sanity_sq_bound"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many $q$ for which there is a set $A \\subset \\mathbb{Z}/q\\mathbb{Z}$,\n$|A| = (\\sqrt{2} + o(1))q^{1/2}$, with $A + A = \\mathbb{Z}/q\\mathbb{Z}$? [Gr24]\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«33»","statement":"True ↔ ∀ (ε : ℝ), 0 < ε → ∃ᶠ (q : ℕ+) in Filter.atTop, ∃ A, A + A = Finset.univ ∧ |↑A.card / √↑↑q - √2| < ε","subjects":["5","11"],"theorem":"Green33.green_33"},{"answerKinds":[],"category":"research open","docstring":"Let $A \\subset \\mathbf{Z}$ be a set of $n$ integers. Is there a set $S \\subset A$ of size\n$(\\log n)^{100}$ such that the restricted sumset$S \\hat{+} S$ is disjoint from $A$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«2»","statement":"True ↔\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∀ (A : Finset ℤ), A.card = n → ↑(Green2.maxRestrictedSumAvoidingSubsetSize A) ≥ Real.log ↑n ^ 100","subjects":["11"],"theorem":"Green2.green_2"},{"answerKinds":[],"category":"research solved","docstring":"From [Ch71] it is known that $M(A) \\le |A|^{2/5 + o(1)}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«2»","statement":"∃ o,\n  ∃ (_ : Filter.Tendsto o Filter.atTop (nhds 0)),\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∃ A, A.card = n ∧ ↑(Green2.maxRestrictedSumAvoidingSubsetSize A) ≤ ↑A.card ^ (2 / 5 + o A.card)","subjects":["11"],"theorem":"Green2.green_2_upper_bound_choi"},{"answerKinds":[],"category":"research solved","docstring":"From [Er65] it is known that $M(A) \\le \\frac{1}{3}|A| + O(1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«2»","statement":"∃ C, ∀ᶠ (n : ℕ) in Filter.atTop, ∃ A, A.card = n ∧ ↑(Green2.maxRestrictedSumAvoidingSubsetSize A) ≤ ↑A.card / 3 + C","subjects":["11"],"theorem":"Green2.green_2_upper_bound_erdos"},{"answerKinds":[],"category":"research solved","docstring":"From [Ru05] the best-known upper bound is $|S| \\lt e^{C \\sqrt{\\log |A|}}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«2»","statement":"∃ C > 0,\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∃ A, A.card = n ∧ ↑(Green2.maxRestrictedSumAvoidingSubsetSize A) < Real.exp (C * √(Real.log ↑A.card))","subjects":["11"],"theorem":"Green2.green_2_upper_bound_ruzsa"},{"answerKinds":[],"category":"research solved","docstring":"From [Sa21] it is known that there is always such an S with $|S| \\gt (\\log |A|)^{1+c}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«2»","statement":"∃ c > 0,\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∀ (A : Finset ℤ), A.card = n → ↑(Green2.maxRestrictedSumAvoidingSubsetSize A) ≥ Real.log ↑n ^ (1 + c)","subjects":["11"],"theorem":"Green2.green_2_lower_bound_sanders"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $A \\subset [0,1]$ is open and has measure greater than $\\frac{1}{3}$. Is there a solution to $xy = z$ with $x, y, z \\in A$? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«3»","statement":"True ↔\n  ∀ (A : Set ℝ),\n    IsOpen A → A ⊆ Set.Icc 0 1 → MeasureTheory.volume A > 1 / 3 → ∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x * y = z","subjects":["11"],"theorem":"Green3.green_3"},{"answerKinds":[],"category":"research open","docstring":"From [Schoen and Sisask] $f(N) \\ll N \\cdot e^{-c(\\log N)^{1/7}}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«16»","statement":"∃ c > 0, (fun N => ↑(Green16.f N)) =O[Filter.atTop] fun N => ↑N * Real.exp (-c * Real.log ↑N ^ (1 / 7))","subjects":["5","11"],"theorem":"Green16.green_16_upper_bound"},{"answerKinds":[],"category":"research open","docstring":"From [Ruzsa] $f(N) \\gg N^{1/2}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«16»","statement":"(fun N => ↑N ^ (1 / 2)) =O[Filter.atTop] fun N => ↑(Green16.f N)","subjects":["5","11"],"theorem":"Green16.green_16_lower_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«16»","statement":"∀ (N : ℕ),\n  ∃ A ⊆ Finset.Icc 1 N,\n    Green16.SolutionFree A ∧\n      A.card = sorry ∧ MaximalFor (fun B => B ⊆ Finset.Icc 1 N ∧ Green16.SolutionFree B) Finset.card A","subjects":["5","11"],"theorem":"Green16.green_16"},{"answerKinds":[],"category":"research open","docstring":"$f(N) \\gg N \\cdot e^{-c(\\log N)^{1/7}}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«16»","statement":"∃ c > 0, (fun N => ↑N * Real.exp (-c * Real.log ↑N ^ (1 / 7))) =O[Filter.atTop] fun N => ↑(Green16.f N)","subjects":["5","11"],"theorem":"Green16.green_16_conjectured_lower_bound"},{"answerKinds":[],"category":"research open","docstring":"From [Yufei Zhao]: Is there a subset of $\\{1, \\ldots, N\\}$ of size\n$N^{1/3 - o(1)}$ with no nontrivial solutions to $x + 2y + 3z = x' + 2y' + 3z'$? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«16»","statement":"True ↔ ∃ h, Filter.Tendsto h Filter.atTop (nhds 0) ∧ ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Green16.g N) ≥ ↑N ^ (1 / 3 - h N)","subjects":["5","11"],"theorem":"Green16.zhao_question"},{"answerKinds":[],"category":"research open","docstring":"The analogous problem in $\\mathbb{F}^n_p$ remains open. [Gr24] ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«26»","statement":"True ↔\n  ∀ (p : ℕ) [inst : Fact (Nat.Prime p)],\n    ∃ C, ∀ (n : ℕ) (A : Fin C → Set (𝔽 p n)), (∀ (i : Fin C), Green26.IsCube (A i)) → ∑ i, A i = Set.univ","subjects":["5","11","15"],"theorem":"Green26.green_26.variants.open"},{"answerKinds":[],"category":"research solved","docstring":"[Yu25] has solved the original problem (with 100 replaced by 4) ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«26»","statement":"∀ (n : ℕ) (A : Fin 4 → Set (𝔽₃ n)), (∀ (i : Fin 4), Green26.IsCube (A i)) → ∑ i, A i = Set.univ","subjects":["5","11","15"],"theorem":"Green26.green_26.variants.yu25"},{"answerKinds":[],"category":"research solved","docstring":"Let $A_1, \\dots, A_{100}$ be \"cubes\" in $\\mathbb{F}^n_3$.\nIs it true that $A_1 + \\dots + A_{100} = \\mathbb{F}^n_3$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«26»","statement":"∀ (n : ℕ) (A : Fin 100 → Set (𝔽₃ n)), (∀ (i : Fin 100), Green26.IsCube (A i)) → ∑ i, A i = Set.univ","subjects":["5","11","15"],"theorem":"Green26.green_26"},{"answerKinds":[],"category":"research solved","docstring":"[ALM91] showed that if 100 is replaced by $\\leq c(p) \\log n$ then the result is true for\n$\\mathbb{F}^n_p$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«26»","statement":"∀ (p : ℕ) [inst : Fact (Nat.Prime p)],\n  ∃ k,\n    ((fun n => ↑(k n)) =O[Filter.atTop] fun n => Real.log ↑n) ∧\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ (A : Fin (k n) → Set (𝔽 p n)), (∀ (i : Fin (k n)), Green26.IsCube (A i)) → ∑ i, A i = Set.univ","subjects":["5","11","15"],"theorem":"Green26.green_26.variants.alm91"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Can we improve the lower bound? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«38»","statement":"have ans := sorry;\nans ≤ᶠ[Filter.atTop] Green38.LargestAdmissibleCardinality ∧ ∃ c > Green38.C₁, (fun n => c ^ n) =O[Filter.atTop] ans","subjects":["5","11"],"theorem":"Green38.green_38.lower"},{"answerKinds":[],"category":"research solved","docstring":"The current best upper bound is $|A| \\leqslant (C_2 + o(1))^n$ where\n$C_2 = \\frac{7 \\cos(\\pi/7)}{1 + \\cos(\\pi/7)} \\approx 3.3177$ [La79, Corollary 5]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«38»","statement":"∀ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, Green38.LargestAdmissibleCardinality n ≤ (Green38.C₂ + ε) ^ n","subjects":["5","11"],"theorem":"Green38.green_38.variants.best_upper"},{"answerKinds":[],"category":"research solved","docstring":"The current best lower bound is $(C_1 - o(1))^n \\leqslant |A|$ where\n$C_1 = 367^{1/5} \\approx 3.2578$ [Po20, Section 9.1]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«38»","statement":"∀ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, (Green38.C₁ - ε) ^ n ≤ Green38.LargestAdmissibleCardinality n","subjects":["5","11"],"theorem":"Green38.green_38.variants.best_lower"},{"answerKinds":[],"category":"test","docstring":"The set of valid cardinalities we take the supremum over is bounded above. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«38»","statement":"∀ {n : ℕ}, BddAbove (Green38.ValidCardinalities n)","subjects":["5","11"],"theorem":"Green38.green_38.test_bound_above"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Can we improve the best upper bound? ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«38»","statement":"have ans := sorry;\nGreen38.LargestAdmissibleCardinality ≤ᶠ[Filter.atTop] ans ∧ ∃ c < Green38.C₂, ans =O[Filter.atTop] fun n => c ^ n","subjects":["5","11"],"theorem":"Green38.green_38.upper"},{"answerKinds":[],"category":"test","docstring":"{0, 2, 4} is a valid independent set in C_7, giving cardinality 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«38»","statement":"3 ∈ Green38.ValidCardinalities 1","subjects":["5"],"theorem":"Green38.green_38.test_n1_lower"},{"answerKinds":[],"category":"test","docstring":"0 is a valid cardinality, since the empty set vacuously satisfies the condition. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«38»","statement":"∀ {n : ℕ}, 0 ∈ Green38.ValidCardinalities n","subjects":["5","11"],"theorem":"Green38.green_38.test_zero_mem_validCardinalities"},{"answerKinds":[],"category":"research open","docstring":"Do there exist infinitely many primes $p$ for which $p - 2$ has an odd number of prime factors,\ncounted with multiplicity (i.e. $\\Omega(p - 2)$ is odd)?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«64»","statement":"True ↔ {p | Nat.Prime p ∧ Odd (ArithmeticFunction.cardFactors (p - 2))}.Infinite","subjects":["11"],"theorem":"Green64.green_64"},{"answerKinds":[],"category":"research open","docstring":"The same question as `green_64` but with $p - 1$ instead of $p - 2$.\nGreen notes this is \"probably more natural\".\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«64»","statement":"True ↔ {p | Nat.Prime p ∧ Odd (ArithmeticFunction.cardFactors (p - 1))}.Infinite","subjects":["11"],"theorem":"Green64.green_64.variants.p_sub_one"},{"answerKinds":[],"category":"test","docstring":"$5$ satisfies the condition: $5$ is prime and $5 - 2 = 3$ is prime, so $\\Omega(3) = 1$ is odd. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«64»","statement":"5 ∈ {p | Nat.Prime p ∧ Odd (ArithmeticFunction.cardFactors (p - 2))}","subjects":["11"],"theorem":"Green64.green_64_mem_five"},{"answerKinds":[],"category":"test","docstring":"$11$ does *not* satisfy the condition: although $11$ is prime, $11 - 2 = 9 = 3 ^ 2$ has\n$\\Omega(9) = 2$ prime factors, which is even. This shows the condition is non-trivial. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«64»","statement":"11 ∉ {p | Nat.Prime p ∧ Odd (ArithmeticFunction.cardFactors (p - 2))}","subjects":["11"],"theorem":"Green64.green_64_not_mem_eleven"},{"answerKinds":[],"category":"test","docstring":"$7$ satisfies the condition: $7$ is prime and $7 - 2 = 5$ is prime, so $\\Omega(5) = 1$ is odd. ","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«64»","statement":"7 ∈ {p | Nat.Prime p ∧ Odd (ArithmeticFunction.cardFactors (p - 2))}","subjects":["11"],"theorem":"Green64.green_64_mem_seven"},{"answerKinds":[],"category":"research open","docstring":"Variant using the exact simultaneous double product property from [CKS05, 4.1]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«36»","statement":"True ↔\n  ∀ ε > 0,\n    ∃ᶠ (n : ℕ) in Filter.atTop,\n      ∃ H x,\n        ∃ (_ : Finite H),\n          ∃ A B,\n            ↑n ^ (2 - ε) ≤ ↑(Nat.card H) ∧\n              ↑(Nat.card H) ≤ ↑n ^ (2 + ε) ∧\n                (∀ (i : Fin n), ↑n ^ (2 - ε) ≤ ↑(A i).card * ↑(B i).card) ∧ Green36.SimultaneousDoubleProduct A B","subjects":["5","20"],"theorem":"Green36.green_36.variants.cks05"},{"answerKinds":[],"category":"research open","docstring":"Do the following exist, for arbitrarily large $n$? An abelian group $H$ with $|H| = n^{2+o(1)}$,\ntogether with subsets $A_1, ..., A_n, B_1, ..., B_n$ satisfying $|A_i||B_i| \\ge n^{2-o(1)}$ and\n$|A_i + B_i| = |A_i||B_i|$, such that the sets $A_i + B_i$ are disjoint from the sets $A_j + B_k$\n($j \\neq k$)?\n\nNOTE: according to [CKS05, 4.1], the conditions should be $A_i + B_j$ disjoint from $A_j + B_k$ for\n$i \\neq k$. See `green_36.variants.cks05`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«36»","statement":"True ↔\n  ∀ ε > 0,\n    ∃ᶠ (n : ℕ) in Filter.atTop,\n      ∃ H x,\n        ∃ (_ : Finite H),\n          ∃ A B,\n            ↑n ^ (2 - ε) ≤ ↑(Nat.card H) ∧\n              ↑(Nat.card H) ≤ ↑n ^ (2 + ε) ∧\n                (∀ (i : Fin n), ↑n ^ (2 - ε) ≤ ↑(A i).card * ↑(B i).card) ∧ Green36.Green36Property A B","subjects":["5","20"],"theorem":"Green36.green_36"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $\\mathbb{F}_2^n$ is partitioned in to sets $A_1, ..., A_K$.\nDoes $2A_i$ contain a coset of codimension $O_K(1)$ for some $i$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«53»","statement":"True ↔\n  ∃ c,\n    ∀ (n K : ℕ) (A : Fin K → Set (𝔽₂ n)),\n      ⋃ i, A i = Set.univ →\n        Pairwise (Function.onFun Disjoint A) → ∃ i S, ↑S ⊆ A i + A i ∧ n ≤ Module.finrank (ZMod 2) ↥S.direction + c K","subjects":["5","11"],"theorem":"Green53.green_53"},{"answerKinds":[],"category":"research open","docstring":"Let $K \\subset \\mathbb{R}^n$ be a balanced compact set (that is, $\\lambda K \\subseteq K$ whenever\n$|\\lambda| \\leq 1$) and suppose that the normalised Gaussian measure $\\gamma_n(K) \\geq 0.99$.\nDoes $10K$ contain a compact convex set $C$ with $\\gamma_n(C) \\geq 0.01$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«54»","statement":"True ↔\n  ∀ (K : Set (ℕ → ℝ)),\n    IsCompact K →\n      Balanced ℝ K →\n        0.99 ≤ Green54.gaussianMeasureInf K →\n          ∃ C, IsCompact C ∧ Convex ℝ C ∧ C ⊆ 10 • K ∧ 1e-2 ≤ Green54.gaussianMeasureInf C","subjects":["46","52","60"],"theorem":"Green54.green_54"},{"answerKinds":[],"category":"research solved","docstring":"The same statement is known to be false for 2K instead of 10K.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«54»","statement":"¬∀ (K : Set (ℕ → ℝ)),\n    IsCompact K →\n      Balanced ℝ K →\n        0.99 ≤ Green54.gaussianMeasureInf K →\n          ∃ C, IsCompact C ∧ Convex ℝ C ∧ C ⊆ 2 • K ∧ 1e-2 ≤ Green54.gaussianMeasureInf C","subjects":["46","52","60"],"theorem":"Green54.green_54_known_case"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Suppose that $\\mathbb{N}$ is finitely coloured. Are there $x,y$ of the same colour such that $x^2 +\ny^2$ is a square?\n\nSolved in [FrKlMo25].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«23»","statement":"True ↔ ∀ (k : ℕ) (c : ℕ → Fin k), ∃ x y, 0 < x ∧ 0 < y ∧ c x = c y ∧ IsSquare (x ^ 2 + y ^ 2)","subjects":["5","11"],"theorem":"Green23.green_23"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ be an abelian group of size $N$, and suppose that $A \\subset G$ has density $\\alpha$.\nAre there at least $\\alpha^{15} N^{10}$ tuples $(x_1, \\dots, x_5, y_1, \\dots, y_5) \\in G^{10}$\nsuch that $x_i + y_j \\in A$ whenever $j \\in \\{i, i+1, i+2\\}$?\n\nNote: We interpret indices modulo 5.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«12»","statement":"True ↔\n  ∀ {G : Type u_1} [inst : AddCommGroup G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (A : Finset G),\n    have N := Fintype.card G;\n    have α := ↑A.card / ↑N;\n    have valid_tuples := {t | ∀ (i j : Fin 5), j ∈ {i, i + 1, i + 2} → t.1 i + t.2 j ∈ A};\n    ↑valid_tuples.card ≥ α ^ 15 * ↑N ^ 10","subjects":["5","11"],"theorem":"Green12.green_12"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $A + A$ must contain a coset of dimension $\\gg_\\alpha n$ [Gr13]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«51»","statement":"∀ (α : ℝ), 0 < α → α ≤ 1 → ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, c * ↑n ≤ ↑(Green51.guaranteedMaxCosetDim n α)","subjects":["5","11"],"theorem":"Green51.green_51.lower"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $A + A$ need not contain a coset of dimension $n - \\sqrt{n}$ [Gr13]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«51»","statement":"∃ α > 0, α ≤ 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Green51.guaranteedMaxCosetDim n α) < ↑n - √↑n","subjects":["5","11"],"theorem":"Green51.green_51.upper"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $A + A$ need not contain an arithmetic progression of length $\\sim \\exp(c (\\log N)^{2/3})$ [Ruz91]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«51»","statement":"∀ (α : ℝ),\n  0 < α →\n    α < 1 / 2 →\n      ∃ c > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Green51.guaranteedMaxAPLength N α) ≤ Real.exp (c * Real.log ↑N ^ (2 / 3))","subjects":["5","11"],"theorem":"Green51.green_51.upper_ap"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $A + A$ must contain an arithmetic progression of length $\\sim \\exp(c (\\log N)^{1/2})$ [Gr02]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«51»","statement":"∀ (α : ℝ),\n  0 < α →\n    α ≤ 1 →\n      ∃ c > 0, ∀ᶠ (N : ℕ) in Filter.atTop, Real.exp (c * Real.log ↑N ^ (1 / 2)) ≤ ↑(Green51.guaranteedMaxAPLength N α)","subjects":["5","11"],"theorem":"Green51.green_51.lower_ap"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Suppose that $A \\subset \\mathbb{F}_2^n$ is a set of density $\\alpha$. What is the largest size of coset\nguaranteed to be contained in $2A$?\n\nWe phrase this by asking for the exact function $F(\\alpha, n)$ giving the maximum dimension\nof a guaranteed coset.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«51»","statement":"sorry = Green51.guaranteedMaxCosetDim","subjects":["5","11"],"theorem":"Green51.green_51"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $A \\subset \\mathbb{F}_2^n$ has density $\\alpha > 1/2 - C/\\sqrt{n}$.\nDoes $A + A$ contain a subspace of co-dimension $O_C(1)$? [Sa11, Question 5.1]\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«51»","statement":"True ↔\n  ∀ (k : ℝ),\n    0 < k → ∃ c, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ α > 1 / 2 - k / √↑n, α ≤ 1 → n ≤ Green51.guaranteedMaxCosetDim n α + c","subjects":["5","11"],"theorem":"Green51.green_51.one_half"},{"answerKinds":[],"category":"research open","docstring":"Does Ulam's sequence have positive density?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«7»","statement":"True ↔ ∀ (a : ℕ → ℕ), Erdos342.IsUlamSequence a → (Set.range a).upperDensity > 0","subjects":["11","42"],"theorem":"Green7.green_7.variants.positive_density"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Suppose that $A \\subset \\mathbb{F}_2^n$ is a set with $|A + A| \\leq K|A|$. Is it true that $A$\nis covered by $K^{O(1)}$ translates of a subspace of size $\\leq |A|$?\n\nSolved by [GGM25].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«49»","statement":"True ↔\n  ∃ C > 0,\n    ∀ (n : ℕ) (A : Finset (𝔽₂ n)),\n      A.Nonempty → ∀ K ≥ 1, ↑(A + A).card ≤ K * ↑A.card → ∃ W T, Nat.card ↥W ≤ A.card ∧ ↑T.card ≤ K ^ C ∧ ↑A ⊆ ↑T + ↑W","subjects":["5","11"],"theorem":"Green49.green_49"},{"answerKinds":[],"category":"test","docstring":"In the degenerate case $k = 0$ the equation reads $0 = 0$, so the empty tuple is a monochromatic\nsolution for every colouring and `minColours` takes its junk value `0`. The bound asserted by\n`green_21` is therefore vacuously satisfied at $k = 0$, and the content of the problem is\nunaffected.\n","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«21»","statement":"∀ (a : Fin 0 → ℤ), Green21.minColours a = 0","subjects":["5","11"],"theorem":"Green21.minColours_eq_zero"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $a_1, \\dots, a_k$ are integers which do not satisfy Rado's condition: thus if\n$\\sum_{i \\in I} a_i = 0$ then $I = \\emptyset$. It then follows from Rado's theorem that the\nequation $a_1x_1 + \\cdots + a_kx_k = 0$ is not partition regular. Write $c(a_1, \\dots, a_k)$ for\nthe least number of colours required in order to colour $\\mathbb{N}$ so that there is no\nmonochromatic solution to $a_1x_1 + \\cdots + a_kx_k = 0$. Is $c(a_1, \\dots, a_k)$ bounded in\nterms of $k$ only?\n\nThis problem, which is known as Rado's boundedness conjecture, dates back to 1933 [Ra33]. It is\nopen for all $k \\geq 4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«21»","statement":"True ↔ ∃ B, ∀ (k : ℕ) (a : Fin k → ℤ), ¬Green21.RadoCondition a → Green21.minColours a ≤ B k","subjects":["5","11"],"theorem":"Green21.green_21"},{"answerKinds":[],"category":"research open","docstring":"A question [FoKl06, Conjecture 5] of Fox and Kleitman, which they call a 'modular analogue' of\nRado's Boundedness Conjecture. Let $p$ be a prime, and suppose that $a_1, \\dots, a_k$ are\nintegers with $\\sum_{i \\in I} a_i \\equiv 0 \\pmod p$ only when $I = \\emptyset$. Does there exist\nan $f(k)$-colouring of $(\\mathbb{Z}/p\\mathbb{Z})^*$ with no monochromatic solution to\n$a_1x_1 + \\cdots + a_kx_k = 0$? This seems to be open even when $k = 3$; Green [Gr24] suspects\nthe answer may be negative.\n\nThe point of the question is that the number of colours $f(k)$ must not depend on $p$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«21»","statement":"True ↔\n  ∃ f,\n    ∀ (k p : ℕ),\n      Nat.Prime p →\n        ∀ (a : Fin k → ℤ),\n          (∀ (I : Finset (Fin k)), ↑p ∣ ∑ i ∈ I, a i → I = ∅) →\n            ∃ col, ∀ (x : Fin k → (ZMod p)ˣ), (∀ (i j : Fin k), col (x i) = col (x j)) → ∑ i, ↑(a i) * ↑(x i) ≠ 0","subjects":["5","11"],"theorem":"Green21.green_21.variants.fox_kleitman_modular"},{"answerKinds":[],"category":"test","docstring":"A case where the infimum defining `minColours` is a genuine minimum rather than the junk value of\n`sInf ∅`. The coefficients $(1, 1, 1)$ fail `RadoCondition`, and since the $x_i$ are positive the\nequation $x_1 + x_2 + x_3 = 0$ has no solutions at all, so a single colour suffices; no colouring\nof $\\mathbb{N}$ into `Fin 0` exists, so $0$ is not attainable.\n","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«21»","statement":"Green21.minColours ![1, 1, 1] = 1","subjects":["5","11"],"theorem":"Green21.minColours_ones"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Green [Gr24] is not sure that the constant $24$ of [FoKl06] is sharp, and remarks that it might\nbe interesting to determine the sharp constant.\n\nThe largest value of $c(a_1, a_2, a_3)$ is attained, since by `green_21.variants.fox_kleitman`\nthe values form a non-empty set of naturals bounded above by $24$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«21»","statement":"IsGreatest {c | ∃ a, ¬Green21.RadoCondition a ∧ Green21.minColours a = c} sorry","subjects":["5","11"],"theorem":"Green21.green_21.variants.fox_kleitman_sharp"},{"answerKinds":[],"category":"research open","docstring":"Milićević [ElJo23, Conjecture 11.1] conjectures the following 2-adic variant. For any\n$k \\in \\mathbb{N}$, there exists $K = K(k)$ such that the following is true. Let $r$ be a\npositive integer, and let $a_1, \\dots, a_k \\in \\mathbb{Z}/2^r\\mathbb{Z}$. Let $d$ be the largest\ninteger such that $\\sum_{i \\in I} a_i \\equiv 0 \\pmod{2^d}$ for some non-empty subset\n$I \\subset [k]$. Then there is a $K$-colouring of $\\mathbb{Z}/2^r\\mathbb{Z}$ such that all\nmonochromatic solutions $x = (x_1, \\dots, x_k)$ to the equation\n$a_1x_1 + \\cdots + a_kx_k = 0$ satisfy $x_i \\equiv 0 \\pmod{2^{r-d}}$ for all $i = 1, \\dots, k$.\n\nMilićević remarks that, if true, this would imply the Rado boundedness conjecture by a\ncompactness argument.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«21»","statement":"∀ (k : ℕ),\n  ∃ K,\n    ∀ (r : ℕ),\n      0 < r →\n        ∀ (a : Fin k → ZMod (2 ^ r)),\n          ∃ col,\n            ∀ (x : Fin k → ZMod (2 ^ r)),\n              (∀ (i j : Fin k), col (x i) = col (x j)) →\n                ∑ i, a i * x i = 0 → ∀ (i : Fin k), 2 ^ (r - Green21.maxDepth a) ∣ (x i).val","subjects":["5","11"],"theorem":"Green21.green_21.variants.milicevic"},{"answerKinds":[],"category":"test","docstring":"A tuple failing `RadoCondition` has no zero coefficient, since a singleton is a non-empty subset.\n","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«21»","statement":"∀ {k : ℕ} {a : Fin k → ℤ}, ¬Green21.RadoCondition a → ∀ (i : Fin k), a i ≠ 0","subjects":["5","11"],"theorem":"Green21.ne_zero_of_not_radoCondition"},{"answerKinds":[],"category":"test","docstring":"Not satisfying `RadoCondition` is Green's phrasing \"if $\\sum_{i \\in I} a_i = 0$ then\n$I = \\emptyset$\".\n","hasSorryFreeProof":true,"module":"FormalConjectures.GreensOpenProblems.«21»","statement":"∀ {k : ℕ} (a : Fin k → ℤ), ¬Green21.RadoCondition a ↔ ∀ (I : Finset (Fin k)), ∑ i ∈ I, a i = 0 → I = ∅","subjects":["5","11"],"theorem":"Green21.not_radoCondition_iff"},{"answerKinds":[],"category":"research solved","docstring":"The answer was shown to be affirmative for $k = 3$ by Fox and Kleitman [FoKl06], who showed that\n$c(a_1, a_2, a_3) \\leq 24$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«21»","statement":"∀ (a : Fin 3 → ℤ), ¬Green21.RadoCondition a → Green21.minColours a ≤ 24","subjects":["5","11"],"theorem":"Green21.green_21.variants.fox_kleitman"},{"answerKinds":[],"category":"research solved","docstring":"Best known upper bound: $y \\ll x^2$ [Iw78]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«46»","statement":"Green46.maxY =O[Filter.atTop] Green46.bestUpper","subjects":["11"],"theorem":"Green46.green_46.variants.upper"},{"answerKinds":[],"category":"research open","docstring":"It seems very likely that we must have $y \\ll x^{1+o(1)}$ [Gr24]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«46»","statement":"∃ o, (o =o[Filter.atTop] fun x => 1) ∧ Green46.maxY =O[Filter.atTop] fun x => ↑x ^ (1 + o x)","subjects":["11"],"theorem":"Green46.green_46.improve_upper_conjectured"},{"answerKinds":[],"category":"research solved","docstring":"Best known lower bound: $y \\gg x \\frac{\\log x \\log \\log \\log x}{\\log \\log x}$ [Ra38]. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«46»","statement":"Green46.bestLower =O[Filter.atTop] Green46.maxY","subjects":["11"],"theorem":"Green46.green_46.variants.lower"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"We conjecture that the best-known upper bound can be improved. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«46»","statement":"have ans := sorry;\nans =o[Filter.atTop] Green46.bestUpper ∧ Green46.maxY =O[Filter.atTop] ans","subjects":["11"],"theorem":"Green46.green_46.improve_upper"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"We conjecture that the best-known lower bound can be improved. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«46»","statement":"have ans := sorry;\nGreen46.bestLower =o[Filter.atTop] ans ∧ ans =O[Filter.atTop] Green46.maxY","subjects":["11"],"theorem":"Green46.green_46.improve_lower"},{"answerKinds":[],"category":"research solved","docstring":"The answer is YES for 4-term progressions [BJP14].\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«15»","statement":"∃ K f, LipschitzWith K f ∧ {x | ∃ n, (↑n, f n) = x}.IsAPOfLengthFree 4","subjects":["5","11"],"theorem":"Green15.green_15_ap4"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a Lipschitz function $f : \\mathbb{N} \\to \\mathbb{Z}$ whose graph\n$\\Gamma = \\{(n, f(n)) : n \\in \\mathbb{N}\\} \\subseteq \\mathbb{Z}^2$ is free of 3-term progressions?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«15»","statement":"True ↔ ∃ K f, LipschitzWith K f ∧ {x | ∃ n, (↑n, f n) = x}.IsAPOfLengthFree 3","subjects":["5","11"],"theorem":"Green15.green_15"},{"answerKinds":[],"category":"research open","docstring":"Suppose that $A$ is an open subset of $[0, 1]^2$ with measure $\\alpha$. Are there four points in\n$A$ determining an axis-parallel rectangle with area $\\gt c \\alpha^2$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«85»","statement":"True ↔\n  ∃ c > 0,\n    ∀ (A : Set (ℝ × ℝ)),\n      IsOpen A →\n        A ⊆ Set.Icc 0 1 ×ˢ Set.Icc 0 1 →\n          A.Nonempty →\n            have α := (MeasureTheory.volume A).toReal;\n            ∃ x₁ x₂ y₁ y₂, {(x₁, y₁), (x₂, y₁), (x₂, y₂), (x₁, y₂)} ⊆ A ∧ c * α ^ 2 ≤ |x₁ - x₂| * |y₁ - y₂|","subjects":["28","52"],"theorem":"Green85.green_85"},{"answerKinds":[],"category":"research solved","docstring":"From [Gr24] \"It is quite easy to show using Cauchy-Schwarz that there must be such a rectangle with\narea $\\gg \\alpha^2 (\\log 1/\\alpha)^{-1}$.\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«85»","statement":"∃ c > 0,\n  ∀ᶠ (α : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n    ∀ (A : Set (ℝ × ℝ)),\n      IsOpen A →\n        A ⊆ Set.Icc 0 1 ×ˢ Set.Icc 0 1 →\n          A.Nonempty →\n            α = (MeasureTheory.volume A).toReal →\n              ∃ x₁ x₂ y₁ y₂,\n                {(x₁, y₁), (x₂, y₁), (x₂, y₂), (x₁, y₂)} ⊆ A ∧ c * α ^ 2 * (Real.log (1 / α))⁻¹ ≤ |x₁ - x₂| * |y₁ - y₂|","subjects":["28","52"],"theorem":"Green85.green_85_loose"},{"answerKinds":[],"category":"research open","docstring":"[Ma21] showed that $3.13 \\leq C$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«19»","statement":"Green19.C ≥ 3.13","subjects":["5","11"],"theorem":"Green19.green_19.lower"},{"answerKinds":[],"category":"research solved","docstring":"What is $C$, the infimum of all exponents $c$ for which the following is true, uniformly for\n$0 < \\alpha < 1$? Suppose that $A \\subset \\mathbb{F}_2^n \\times \\mathbb{F}_2^n$ is a set of density\n$\\alpha$. Write $N := 2^n$. Then there is some $d \\neq 0$ such that $A$ contains $\\gg \\alpha^c N^2$\ncorners $(x,y), (x,y+d), (x+d,y)$.\n\nThis question has been resolved by [FSS20], showing that $C = 4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«19»","statement":"Green19.C = 4","subjects":["5","11"],"theorem":"Green19.green_19"},{"answerKinds":[],"category":"research open","docstring":"[Ma21] showed that $C \\leq 4$. ","hasSorryFreeProof":false,"module":"FormalConjectures.GreensOpenProblems.«19»","statement":"Green19.C ≤ 4","subjects":["5","11"],"theorem":"Green19.green_19.upper"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53000»","statement":"OeisA53000.a 3 = 2","subjects":["11"],"theorem":"OeisA53000.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53000»","statement":"OeisA53000.a 0 = 2","subjects":["11"],"theorem":"OeisA53000.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) \\le 1 + \\phi(n)$ for $n > 0$. This improves on Oppermann's conjecture, which says\n$a(n) < n$.\n- Thomas Ordowski, Dec 17 2014\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«53000»","statement":"∀ (n : ℕ), 0 < n → OeisA53000.a n ≤ 1 + n.totient","subjects":["11"],"theorem":"OeisA53000.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53000»","statement":"OeisA53000.a 4 = 1","subjects":["11"],"theorem":"OeisA53000.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53000»","statement":"OeisA53000.a 2 = 1","subjects":["11"],"theorem":"OeisA53000.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53000»","statement":"OeisA53000.a 1 = 1","subjects":["11"],"theorem":"OeisA53000.a_1"},{"answerKinds":[],"category":"research open","docstring":"Are all terms integers? ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«78590»","statement":"∀ (n : ℕ), 3 ≤ n → OeisA78590.a (n - 2) ∣ 2 ^ OeisA78590.a (n - 1) + 1","subjects":["11"],"theorem":"OeisA78590.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78590»","statement":"OeisA78590.a 5 = 171","subjects":["11"],"theorem":"OeisA78590.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78590»","statement":"OeisA78590.a 2 = 1","subjects":["11"],"theorem":"OeisA78590.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78590»","statement":"OeisA78590.a 4 = 9","subjects":["11"],"theorem":"OeisA78590.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78590»","statement":"OeisA78590.a 1 = 1","subjects":["11"],"theorem":"OeisA78590.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78590»","statement":"OeisA78590.a 3 = 3","subjects":["11"],"theorem":"OeisA78590.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«307865»","statement":"OeisA307865.a 0 = 0","subjects":["11"],"theorem":"OeisA307865.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«307865»","statement":"OeisA307865.a 4 = 0","subjects":["11"],"theorem":"OeisA307865.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«307865»","statement":"OeisA307865.a 2 = 2","subjects":["11"],"theorem":"OeisA307865.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«307865»","statement":"OeisA307865.a 1 = 1","subjects":["11"],"theorem":"OeisA307865.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«307865»","statement":"OeisA307865.a 3 = 3","subjects":["11"],"theorem":"OeisA307865.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: if $2n+1$ is an absolute Euler pseudoprime, then $a(n) = 0$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/307865.wip.lean#L190"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«307865»","statement":"∀ {n : ℕ}, OeisA307865.IsAbsoluteEulerPseudoprime (2 * n + 1) → OeisA307865.a n = 0","subjects":["11"],"theorem":"OeisA307865.a_eq_zero_of_pseudoprime"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100478»","statement":"OeisA100478.a 0 = 1","subjects":["11"],"theorem":"OeisA100478.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100478»","statement":"OeisA100478.a 1 = 1","subjects":["11"],"theorem":"OeisA100478.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100478»","statement":"OeisA100478.a 3 = 1","subjects":["11"],"theorem":"OeisA100478.a_3"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Starting with other values of $a(1)$, $a(2)$, $a(3)$, $a(4)$, $a(5)$ what behaviors are possible?\nDoes the sequence always stick at a single integer after some point, or can it go into a loop,\nor is there a third pattern?\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a100478-eventual-periodicity/blob/642eed0ffee26415528ab8c48c5181826be04860/lean/OeisA100478FC.lean#L158-L168"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«100478»","statement":"∀ (v : Fin 5 → ℕ),\n  (∀ (i : Fin 5), v i > 0) → True = ∃ N, ∃ P > 0, ∀ n ≥ N, OeisA100478.aGeneral v (n + P) = OeisA100478.aGeneral v n","subjects":["11"],"theorem":"OeisA100478.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100478»","statement":"OeisA100478.a 2 = 1","subjects":["11"],"theorem":"OeisA100478.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100478»","statement":"OeisA100478.a 4 = 1","subjects":["11"],"theorem":"OeisA100478.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38098»","statement":"OeisA38098.a 2 = 4","subjects":["11"],"theorem":"OeisA38098.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38098»","statement":"OeisA38098.a 3 = 9","subjects":["11"],"theorem":"OeisA38098.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38098»","statement":"OeisA38098.a 1 = 0","subjects":["11"],"theorem":"OeisA38098.a_1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture (i): for any integer $k > 2$, the sequence $\\pi(n^k)/n^k$ ($n = 2, 3, \\ldots$) is strictly\ndecreasing, where $\\pi(x)$ denotes the number of primes not exceeding $x$.\n- Zhi-Wei Sun, Oct 17 2015\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38098»","statement":"∀ (k : ℕ), 2 < k → ∀ (n : ℕ), 2 ≤ n → ↑((n + 1) ^ k).primeCounting / (↑n + 1) ^ k < ↑(n ^ k).primeCounting / ↑n ^ k","subjects":["11"],"theorem":"OeisA38098.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture (ii): all the numbers $\\pi(n^2)/n^2$ ($n = 1, 2, 3, \\ldots$) are pairwise distinct.\nMoreover, we have $\\pi(n^2)/n^2 > \\pi((n+1)^2)/(n+1)^2$ for all $n > 15646$.\n- Zhi-Wei Sun, Oct 17 2015\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38098»","statement":"(∀ (m n : ℕ), 1 ≤ m → 1 ≤ n → ↑(m ^ 2).primeCounting / ↑m ^ 2 = ↑(n ^ 2).primeCounting / ↑n ^ 2 → m = n) ∧\n  ∀ (n : ℕ), 15646 < n → ↑((n + 1) ^ 2).primeCounting / (↑n + 1) ^ 2 < ↑(n ^ 2).primeCounting / ↑n ^ 2","subjects":["11"],"theorem":"OeisA38098.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38098»","statement":"OeisA38098.a 4 = 18","subjects":["11"],"theorem":"OeisA38098.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38098»","statement":"OeisA38098.a 5 = 30","subjects":["11"],"theorem":"OeisA38098.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«24356»","statement":"OeisA24356.a 1 = 2","subjects":["11"],"theorem":"OeisA24356.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«24356»","statement":"OeisA24356.a 2 = 1","subjects":["11"],"theorem":"OeisA24356.a_2"},{"answerKinds":[],"category":"research open","docstring":"\"I conjecture that $a(4)$ is the only zero. - _Jon Perry_, Mar 22 2004\"","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«24356»","statement":"∀ (n : ℕ), OeisA24356.a n = 0 → n = 4","subjects":["11","15"],"theorem":"OeisA24356.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«24356»","statement":"OeisA24356.a 0 = 1","subjects":["11"],"theorem":"OeisA24356.a_0"},{"answerKinds":[],"category":"research open","docstring":"**Zhi-Wei Sun's Conjecture (A281976)**: Any integer $n \\geq 0$ can be written as $x^2 + y^2 + z^2 + w^2$\nwith $x, y, z, w$ nonnegative integers and $z \\leq w$, such that both $x$ and $x + 24y$ are squares.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«281976»","statement":"∀ (n : ℕ), OeisA281976.A n","subjects":["11"],"theorem":"OeisA281976.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«281976»","statement":"OeisA281976.A 2","subjects":["11"],"theorem":"OeisA281976.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«281976»","statement":"OeisA281976.A 8","subjects":["11"],"theorem":"OeisA281976.a_8"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«281976»","statement":"OeisA281976.A 3","subjects":["11"],"theorem":"OeisA281976.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«281976»","statement":"OeisA281976.A 0","subjects":["11"],"theorem":"OeisA281976.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«281976»","statement":"OeisA281976.A 23","subjects":["11"],"theorem":"OeisA281976.a_23"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«281976»","statement":"OeisA281976.A 1","subjects":["11"],"theorem":"OeisA281976.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«281976»","statement":"OeisA281976.A 12","subjects":["11"],"theorem":"OeisA281976.a_12"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«281976»","statement":"OeisA281976.A 4","subjects":["11"],"theorem":"OeisA281976.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«281976»","statement":"OeisA281976.A 24","subjects":["11"],"theorem":"OeisA281976.a_24"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«116150»","statement":"OeisA116150.a 4 = 130","subjects":["11"],"theorem":"OeisA116150.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«116150»","statement":"OeisA116150.a 1 = 1","subjects":["11"],"theorem":"OeisA116150.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«116150»","statement":"OeisA116150.a 3 = 33","subjects":["11"],"theorem":"OeisA116150.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«116150»","statement":"OeisA116150.a 2 = 14","subjects":["11"],"theorem":"OeisA116150.a_2"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"First primes are $a(11) = 264353$ and $a(17) = 193622861$.\nAdditional primes: $a(71)$, $a(91)$, $a(431)$.\nWhat is the next prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«116150»","statement":"sorry = OeisA116150.a (sInf {n | 431 < n ∧ Nat.Prime (OeisA116150.a n)})","subjects":["11"],"theorem":"OeisA116150.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«116150»","statement":"OeisA116150.a 5 = 341","subjects":["11"],"theorem":"OeisA116150.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117545»","statement":"OeisA117545.a 1 = 2","subjects":["11"],"theorem":"OeisA117545.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117545»","statement":"OeisA117545.a 3 = 1","subjects":["11"],"theorem":"OeisA117545.a_3"},{"answerKinds":[],"category":"research open","docstring":"Is $a(n)$ defined for all $n \\ge 1$?\nThat is, for every $n \\ge 1$, does there exist $k > 0$ such that $|\\Phi_k(n)|$ is prime?","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«117545»","statement":"∀ (n : ℕ), 0 < n → ∃ k > 0, Nat.Prime (Polynomial.eval (↑n) (Polynomial.cyclotomic k ℤ)).natAbs","subjects":["11"],"theorem":"OeisA117545.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117545»","statement":"OeisA117545.a 2 = 2","subjects":["11"],"theorem":"OeisA117545.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117545»","statement":"OeisA117545.a 4 = 1","subjects":["11"],"theorem":"OeisA117545.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«86766»","statement":"OeisA86766.a 3 = 1","subjects":["11"],"theorem":"OeisA86766.a_3"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the smallest odd prime $p$ such that $(10^{p^2}-1)/(10^p-1)$ is a prime number\n(and $a(10^{p-1})$ could be nonzero)?\n- _Farideh Firoozbakht_, Jan 07 2015\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«86766»","statement":"sorry =\n  if h : ∃ p, Nat.Prime p ∧ 2 < p ∧ Nat.Prime ((10 ^ p ^ 2 - 1) / (10 ^ p - 1)) then\n    some (sInf {p | Nat.Prime p ∧ 2 < p ∧ Nat.Prime ((10 ^ p ^ 2 - 1) / (10 ^ p - 1))})\n  else none","subjects":["11"],"theorem":"OeisA86766.conjecture3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«86766»","statement":"OeisA86766.a 1 = 1","subjects":["11"],"theorem":"OeisA86766.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«86766»","statement":"OeisA86766.a 2 = 3","subjects":["11"],"theorem":"OeisA86766.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«86766»","statement":"OeisA86766.a 4 = 1","subjects":["11"],"theorem":"OeisA86766.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«86766»","statement":"OeisA86766.a 0 = 0","subjects":["11"],"theorem":"OeisA86766.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: If $n$ is not of the form $10^m$ then $a(n)$ is nonzero.\n- _Farideh Firoozbakht_, Jan 07 2015\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«86766»","statement":"∀ (n : ℕ), 0 < n → (∀ (m : ℕ), n ≠ 10 ^ m) → OeisA86766.a n ≠ 0","subjects":["11"],"theorem":"OeisA86766.conjecture2"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the smallest integer $m > 1$ such that $a(10^m)$ is nonzero?\n- _Farideh Firoozbakht_, Jan 07 2015\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«86766»","statement":"sorry = if h : ∃ m, 1 < m ∧ OeisA86766.a (10 ^ m) ≠ 0 then some (sInf {m | 1 < m ∧ OeisA86766.a (10 ^ m) ≠ 0}) else none","subjects":["11"],"theorem":"OeisA86766.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179537»","statement":"OeisA179537.a 3 = -575","subjects":["11"],"theorem":"OeisA179537.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179537»","statement":"OeisA179537.a 2 = -63","subjects":["11"],"theorem":"OeisA179537.a_2"},{"answerKinds":[],"category":"research open","docstring":"$\\sum_{k=0}^{n-1} (42k + 37) (-1)^k a(k) \\equiv 0 \\pmod n$ for all $n \\ge 1$.\n- _Zhi-Wei Sun_, Jul 17 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«179537»","statement":"∀ (n : ℕ), 1 ≤ n → ↑n ∣ ∑ k ∈ Finset.range n, (42 * ↑k + 37) * (-1) ^ k * OeisA179537.a k","subjects":["11"],"theorem":"OeisA179537.conjecture3"},{"answerKinds":[],"category":"research open","docstring":"$\\sum_{k=0}^{p-1} (42k + 37) (-1)^k a(k) \\equiv p(21(p/7) + 16) \\pmod{p^2}$ for any prime $p \\ne 7$.\n- _Zhi-Wei Sun_, Jul 17 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«179537»","statement":"∀ (p : ℕ) [Fact (Nat.Prime p)],\n  p ≠ 7 →\n    ∑ k ∈ Finset.range p, (42 * ↑k + 37) * (-1) ^ k * OeisA179537.a k ≡ ↑p * (21 * legendreSym 7 ↑p + 16) [ZMOD ↑p ^ 2]","subjects":["11"],"theorem":"OeisA179537.conjecture4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179537»","statement":"OeisA179537.a 4 = 6913","subjects":["11"],"theorem":"OeisA179537.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179537»","statement":"OeisA179537.a 1 = 1","subjects":["11"],"theorem":"OeisA179537.a_1"},{"answerKinds":[],"category":"research open","docstring":"If $p$ is a prime with $(p/7) = 1$ and $p = x^2 + 7y^2$ with $x, y$ integers, then\n$\\sum_{k=0}^{p-1} (-1)^k a(k) \\equiv 4x^2 - 2p \\pmod{p^2}$.\n- _Zhi-Wei Sun_, Jul 17 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«179537»","statement":"∀ (p : ℕ) [Fact (Nat.Prime p)],\n  p ≠ 7 →\n    legendreSym 7 ↑p = 1 →\n      ∀ (x y : ℤ),\n        ↑p = x ^ 2 + 7 * y ^ 2 → ∑ k ∈ Finset.range p, (-1) ^ k * OeisA179537.a k ≡ 4 * x ^ 2 - 2 * ↑p [ZMOD ↑p ^ 2]","subjects":["11"],"theorem":"OeisA179537.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179537»","statement":"OeisA179537.a 0 = 1","subjects":["11"],"theorem":"OeisA179537.a_0"},{"answerKinds":[],"category":"research open","docstring":"If $p$ is a prime with $(p/7) = -1$, then\n$\\sum_{k=0}^{p-1} (-1)^k a(k) \\equiv 0 \\pmod{p^2}$.\n- _Zhi-Wei Sun_, Jul 17 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«179537»","statement":"∀ (p : ℕ) [Fact (Nat.Prime p)],\n  p ≠ 7 → legendreSym 7 ↑p = -1 → ∑ k ∈ Finset.range p, (-1) ^ k * OeisA179537.a k ≡ 0 [ZMOD ↑p ^ 2]","subjects":["11"],"theorem":"OeisA179537.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1223»","statement":"OeisA1223.a 0 = 0","subjects":["11"],"theorem":"OeisA1223.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1223»","statement":"OeisA1223.a 1 = 1","subjects":["11"],"theorem":"OeisA1223.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1223»","statement":"OeisA1223.a 3 = 2","subjects":["11"],"theorem":"OeisA1223.a_3"},{"answerKinds":[],"category":"research open","docstring":"Any subsequence a(n .. n+m) with n > 2 (as to exclude the\nuntypical primes 2 and 3) should occur infinitely many times at other starting points k.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«1223»","statement":"∀ (n m : ℕ), n ≥ 3 → {k | OeisA1223.gapSubsequence k (m + 1) = OeisA1223.gapSubsequence n (m + 1)}.Infinite","subjects":["11"],"theorem":"OeisA1223.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1223»","statement":"OeisA1223.a 2 = 2","subjects":["11"],"theorem":"OeisA1223.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«79727»","statement":"OeisA79727.a 3 = 8225","subjects":["11"],"theorem":"OeisA79727.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«79727»","statement":"OeisA79727.a 0 = 1","subjects":["11"],"theorem":"OeisA79727.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 1 (Peter Bala, 2024):\nIf prime $p$ is in A003625 then $a(p^2) \\equiv 8 + p^2 \\pmod{p^3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«79727»","statement":"∀ (p : ℕ), OeisA3625.A p → OeisA79727.a (p ^ 2) ≡ 8 + p ^ 2 [MOD p ^ 3]","subjects":["11"],"theorem":"OeisA79727.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 4 (Peter Bala, 2024):\nIf $n$ is a product of distinct primes from A003625 then $a((n-1)/2)$ is divisible by $n^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«79727»","statement":"∀ (S : Finset ℕ), (∀ p ∈ S, OeisA3625.A p) → (∏ p ∈ S, p) ^ 2 ∣ OeisA79727.a ((∏ p ∈ S, p - 1) / 2)","subjects":["11"],"theorem":"OeisA79727.conjecture4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«79727»","statement":"OeisA79727.a 4 = 351225","subjects":["11"],"theorem":"OeisA79727.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 3 (Peter Bala, 2024):\nIf prime $p$ is in A003625 then $a((p^2-1)/2) \\equiv p^2 \\pmod{p^4}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«79727»","statement":"∀ (p : ℕ), OeisA3625.A p → OeisA79727.a ((p ^ 2 - 1) / 2) ≡ p ^ 2 [MOD p ^ 4]","subjects":["11"],"theorem":"OeisA79727.conjecture3"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 2 (Peter Bala, 2024):\nIf prime $p$ is in A003625 then $a(p(p-1)) \\equiv p^2 \\pmod{p^3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«79727»","statement":"∀ (p : ℕ), OeisA3625.A p → OeisA79727.a (p * (p - 1)) ≡ p ^ 2 [MOD p ^ 3]","subjects":["11"],"theorem":"OeisA79727.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«79727»","statement":"OeisA79727.a 2 = 225","subjects":["11"],"theorem":"OeisA79727.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«79727»","statement":"OeisA79727.a 1 = 9","subjects":["11"],"theorem":"OeisA79727.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 16. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157237»","statement":"OeisA157237.a 16 = 1","subjects":["11"],"theorem":"OeisA157237.a_16"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 19. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157237»","statement":"OeisA157237.a 19 = 2","subjects":["11"],"theorem":"OeisA157237.a_19"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157237»","statement":"OeisA157237.a 1 = 0","subjects":["11"],"theorem":"OeisA157237.a_1"},{"answerKinds":[],"category":"research open","docstring":"On Feb. 24, 2009, Zhi-Wei Sun conjectured that $a(n) = 0$ if and only if $n < 16$ or\n$n \\in \\{18, 21, 24, 51, 84, 1011, 59586\\}$; in other words, except for\n$35, 41, 47, 101, 167, 2021, 119171$, any odd integer greater than $30$ can be written as the\nsum of a prime congruent to $1 \\bmod 6$, a positive power of $2$ and eleven times a positive\npower of $2$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«157237»","statement":"∀ (n : ℕ), 0 < n → (OeisA157237.a n = 0 ↔ n ≤ 15 ∨ n = 18 ∨ n = 21 ∨ n = 24 ∨ n = 51 ∨ n = 84 ∨ n = 1011 ∨ n = 59586)","subjects":["11"],"theorem":"OeisA157237.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 17. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157237»","statement":"OeisA157237.a 17 = 1","subjects":["11"],"theorem":"OeisA157237.a_17"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 18. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157237»","statement":"OeisA157237.a 18 = 0","subjects":["11"],"theorem":"OeisA157237.a_18"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157237»","statement":"OeisA157237.a 2 = 0","subjects":["11"],"theorem":"OeisA157237.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«289411»","statement":"OeisA289411.a 2 = 0","subjects":["11"],"theorem":"OeisA289411.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«289411»","statement":"OeisA289411.a 4 = 0","subjects":["11"],"theorem":"OeisA289411.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«289411»","statement":"OeisA289411.a 1 = 1","subjects":["11"],"theorem":"OeisA289411.a_1"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: For $k \\ge 1$, let $m_k = 10^k / 2 - 1$. Then for $i = 0, \\ldots, m_k$, we have $a(m_k - i) = a(m_k + i)$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/289411.wip.lean#L167"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«289411»","statement":"∀ (k : ℕ),\n  0 < k →\n    have m_k := 10 ^ k / 2 - 1;\n    ∀ i ≤ m_k, OeisA289411.a (m_k - i) = OeisA289411.a (m_k + i)","subjects":["11"],"theorem":"OeisA289411.a_symm"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«289411»","statement":"OeisA289411.a 0 = 0","subjects":["11"],"theorem":"OeisA289411.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«289411»","statement":"OeisA289411.a 3 = 1","subjects":["11"],"theorem":"OeisA289411.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture (3): For each positive integer $n$, prime $p$, and $0 \\le k < p$,\n$\\mathrm{ord}_p(a(np)) = \\mathrm{ord}_p(a(np + k))$.\n\nAnswer: true, see linked proof.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a129365-conjectures/blob/9c0201540c337733d6b8afb2aff209f5489c122a/lean/OeisA129365FC.lean#L234-L395"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«129365»","statement":"∀ (n p k : ℕ),\n  0 < n → Nat.Prime p → k < p → padicValRat p (OeisA129365.a (n * p)) = padicValRat p (OeisA129365.a (n * p + k))","subjects":["11"],"theorem":"OeisA129365.conjecture3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«129365»","statement":"OeisA129365.a 3 = 1","subjects":["11"],"theorem":"OeisA129365.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture (1): $a(n)$ is always an integer (the denominator divides the numerator).\n\nAnswer: true, see linked proof.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a129365-conjectures/blob/9c0201540c337733d6b8afb2aff209f5489c122a/lean/OeisA129365FC.lean#L234-L395"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«129365»","statement":"∀ (n : ℕ), 0 < n → ∏ k ∈ Finset.Icc 1 n, (n / k).factorial ^ k ∣ ∏ j ∈ Finset.Icc 1 n, ∏ k ∈ Finset.Icc 1 n, j.gcd k","subjects":["11"],"theorem":"OeisA129365.conjecture1"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture (4): Let $b(n) = \\mathrm{A004125}(n) = \\sum_{k=1}^n (n \\bmod k)$. Then\n$\\mathrm{ord}_p(a(np)) = \\sum_{i \\ge 0} b(\\lfloor n/p^i \\rfloor)$.\n\nAnswer: true, see linked proof.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a129365-conjectures/blob/9c0201540c337733d6b8afb2aff209f5489c122a/lean/OeisA129365FC.lean#L234-L395"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«129365»","statement":"∀ (n p : ℕ), 0 < n → Nat.Prime p → padicValRat p (OeisA129365.a (n * p)) = ∑' (i : ℕ), ↑(OeisA129365.b (n / p ^ i))","subjects":["11"],"theorem":"OeisA129365.conjecture4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture (2): If $p$ is a prime, then $p \\mid a(n)$ if and only if $p \\le n/3$.\n\nAnswer: true, see linked proof.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a129365-conjectures/blob/9c0201540c337733d6b8afb2aff209f5489c122a/lean/OeisA129365FC.lean#L234-L395"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«129365»","statement":"∀ (n p : ℕ), 0 < n → Nat.Prime p → ((∃ m, OeisA129365.a n = ↑m ∧ p ∣ m) ↔ p ≤ n / 3)","subjects":["11"],"theorem":"OeisA129365.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«129365»","statement":"OeisA129365.a 0 = 1","subjects":["11"],"theorem":"OeisA129365.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«129365»","statement":"OeisA129365.a 4 = 1","subjects":["11"],"theorem":"OeisA129365.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«129365»","statement":"OeisA129365.a 2 = 1","subjects":["11"],"theorem":"OeisA129365.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«129365»","statement":"OeisA129365.a 1 = 1","subjects":["11"],"theorem":"OeisA129365.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3162»","statement":"OeisA3162.a 3 = 3","subjects":["11"],"theorem":"OeisA3162.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3162»","statement":"OeisA3162.a 2 = 1","subjects":["11"],"theorem":"OeisA3162.a_2"},{"answerKinds":[],"category":"research open","docstring":"Let $b(n) = a(2n-1)$. Then the supercongruence $b(n p^k) \\equiv b(n p^{k-1}) \\pmod{p^{3k}}$\nholds for positive integers $n$ and $k$ and all primes $p \\ge 5$.\n- Zhi-Wei Sun, Nov 16 2019\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«3162»","statement":"∀ (n k p : ℕ),\n  0 < n →\n    0 < k →\n      Nat.Prime p → 5 ≤ p → (OeisA3162.b (n * p ^ k)).num ≡ (OeisA3162.b (n * p ^ (k - 1))).num [ZMOD ↑p ^ (3 * k)]","subjects":["11"],"theorem":"OeisA3162.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3162»","statement":"OeisA3162.a 4 = 6","subjects":["11"],"theorem":"OeisA3162.a_4"},{"answerKinds":[],"category":"textbook","docstring":"$a(n)$ is an integer for all $n \\ge 0$.\n- Solution to Problem E2384 by H. W. Gould, Amer. Math. Monthly, 81 (1974), 170-171\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«3162»","statement":"∀ (n : ℕ), (OeisA3162.a n).den = 1","subjects":["11"],"theorem":"OeisA3162.a_is_integer"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3162»","statement":"OeisA3162.a 1 = 1","subjects":["11"],"theorem":"OeisA3162.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3162»","statement":"OeisA3162.a 0 = 1","subjects":["11"],"theorem":"OeisA3162.a_0"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«46969»","statement":"bernoulli' 10 = 5 / 66","subjects":["11"],"theorem":"OeisA46969.bernoulli'_ten"},{"answerKinds":[],"category":"research open","docstring":"Conjecture II: if $\\frac{a(n)}{12}$ is prime, then $\\frac{a(n-1)}{12} - (n-1)$,\n$\\frac{a(n)}{12} - n$ and $\\frac{a(n+2)}{12} - (n+2)$ are multiples of 6.\n- Lorenzo Sauras Altuzarra, Oct 13 2020\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«46969»","statement":"∀ (n : ℕ),\n  2 ≤ n →\n    12 ∣ OeisA46969.a n →\n      Nat.Prime (OeisA46969.a n / 12) →\n        12 ∣ OeisA46969.a (n - 1) →\n          12 ∣ OeisA46969.a (n + 2) →\n            6 ∣ ↑(OeisA46969.a (n - 1)) / 12 - (↑n - 1) ∧\n              6 ∣ ↑(OeisA46969.a n) / 12 - ↑n ∧ 6 ∣ ↑(OeisA46969.a (n + 2)) / 12 - (↑n + 2)","subjects":["11"],"theorem":"OeisA46969.conjecture2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«46969»","statement":"OeisA46969.a 0 = 0","subjects":["11"],"theorem":"OeisA46969.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«46969»","statement":"OeisA46969.a 4 = 1680","subjects":["11"],"theorem":"OeisA46969.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«46969»","statement":"OeisA46969.a 2 = 360","subjects":["11"],"theorem":"OeisA46969.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«46969»","statement":"OeisA46969.a 1 = 12","subjects":["11"],"theorem":"OeisA46969.a_1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«46969»","statement":"bernoulli' 8 = -1 / 30","subjects":["11"],"theorem":"OeisA46969.bernoulli'_eight"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«46969»","statement":"OeisA46969.a 5 = 1188","subjects":["11"],"theorem":"OeisA46969.a_5"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«46969»","statement":"bernoulli' 6 = 1 / 42","subjects":["11"],"theorem":"OeisA46969.bernoulli'_six"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«46969»","statement":"OeisA46969.a 3 = 1260","subjects":["11"],"theorem":"OeisA46969.a_3"},{"answerKinds":[],"category":"research open","docstring":"Conjecture I: if $n > 2$, then $\\frac{a(\\text{A005382}(n))}{12}$ is prime,\nwhere A005382 is the sequence of primes $p$ such that $2p-1$ is also prime.\n- Lorenzo Sauras Altuzarra, Oct 13 2020\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«46969»","statement":"∀ (n : ℕ), 2 < n → Nat.Prime (OeisA46969.a (OeisA46969.a005382 n) / 12)","subjects":["11"],"theorem":"OeisA46969.conjecture1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53175»","statement":"OeisA53175.a 0 = 1","subjects":["11"],"theorem":"OeisA53175.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53175»","statement":"OeisA53175.a 1 = 8","subjects":["11"],"theorem":"OeisA53175.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53175»","statement":"OeisA53175.a 3 = 896","subjects":["11"],"theorem":"OeisA53175.a_3"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: let $P(n)$ be the $(n+1) \\times (n+1)$ Hankel-type determinant with $(i,j)$-entry\nequal to $a(i+j)$ for all $i,j = 0, \\ldots, n$. Then $P(n)/2^{n(n+3)}$ is a positive odd integer.\n- Zhi-Wei Sun, Aug 14 2013\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«53175»","statement":"∀ (n : ℕ),\n  have detP := (OeisA53175.hankelMatrix n).det;\n  have pow2 := 2 ^ (n * (n + 3));\n  pow2 ∣ detP ∧ 0 < detP / pow2 ∧ detP / pow2 % 2 = 1","subjects":["11","15"],"theorem":"OeisA53175.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53175»","statement":"OeisA53175.a 5 = 137728","subjects":["11"],"theorem":"OeisA53175.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53175»","statement":"OeisA53175.a 2 = 80","subjects":["11"],"theorem":"OeisA53175.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53175»","statement":"OeisA53175.a 4 = 10816","subjects":["11"],"theorem":"OeisA53175.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«271591»","statement":"OeisA271591.a 5 = 0","subjects":["11"],"theorem":"OeisA271591.a_5"},{"answerKinds":[],"category":"research solved","docstring":"It is conjectured that after the first two 0's, the number of consecutive 0's is only 4 or 5, and the number of consecutive 1's is only 3 or 4 (tested up to $n = 10^4$).\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/271591.wip.lean#L497"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«271591»","statement":"(∀ (n L : ℕ), OeisA271591.IsMaximalRun 0 n L → L = 4 ∨ L = 5) ∧\n  ∀ (n L : ℕ), OeisA271591.IsMaximalRun 1 n L → L = 3 ∨ L = 4","subjects":["11"],"theorem":"OeisA271591.maximal_run_lengths"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«271591»","statement":"OeisA271591.a 4 = 0","subjects":["11"],"theorem":"OeisA271591.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«271591»","statement":"OeisA271591.a 6 = 1","subjects":["11"],"theorem":"OeisA271591.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«271591»","statement":"OeisA271591.a 7 = 1","subjects":["11"],"theorem":"OeisA271591.a_7"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«271591»","statement":"OeisA271591.a 8 = 1","subjects":["11"],"theorem":"OeisA271591.a_8"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64169»","statement":"OeisA64169.a 4 = 13","subjects":["11"],"theorem":"OeisA64169.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64169»","statement":"OeisA64169.a 2 = 1","subjects":["11"],"theorem":"OeisA64169.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64169»","statement":"OeisA64169.a 5 = 77","subjects":["11"],"theorem":"OeisA64169.a_5"},{"answerKinds":[],"category":"research open","docstring":"\"Conjecture: for $n > 2$, $n$ divides $a(n-2)$ if and only if $n$ is a prime.\nChecked up to 20000. - _Amiram Eldar_ and _Thomas Ordowski_, Jul 27 2019\"","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«64169»","statement":"∀ (n : ℕ), 2 < n → (↑n ∣ OeisA64169.a (n - 2) ↔ Nat.Prime n)","subjects":["11"],"theorem":"OeisA64169.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64169»","statement":"OeisA64169.a 3 = 5","subjects":["11"],"theorem":"OeisA64169.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64169»","statement":"OeisA64169.a 1 = 0","subjects":["11"],"theorem":"OeisA64169.a_1"},{"answerKinds":[],"category":"test","docstring":"$1$ is a practical number. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5153»","statement":"OeisA5153.A 1","subjects":["11"],"theorem":"OeisA5153.a_1"},{"answerKinds":[],"category":"test","docstring":"$8$ is a practical number. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5153»","statement":"OeisA5153.A 8","subjects":["11"],"theorem":"OeisA5153.a_8"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical\nnumber.\n- Hal M. Switkay, Jan 28 2023\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«5153»","statement":"∀ (n : ℕ), 3 ≤ n → Odd n → ∃ p q, Nat.Prime p ∧ OeisA5153.A q ∧ n = p + q","subjects":["11"],"theorem":"OeisA5153.conjecture"},{"answerKinds":[],"category":"test","docstring":"$2$ is a practical number. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5153»","statement":"OeisA5153.A 2","subjects":["11"],"theorem":"OeisA5153.a_2"},{"answerKinds":[],"category":"test","docstring":"$4$ is a practical number. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5153»","statement":"OeisA5153.A 4","subjects":["11"],"theorem":"OeisA5153.a_4"},{"answerKinds":[],"category":"test","docstring":"$6$ is a practical number. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5153»","statement":"OeisA5153.A 6","subjects":["11"],"theorem":"OeisA5153.a_6"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: for $n > 3$, $\\gcd(n, a(n-1)) = \\text{A089026}(n)$.\n- Amiram Eldar and Thomas Ordowski, Jul 28 2019\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«7406»","statement":"∀ (n : ℕ), 3 < n → n.gcd (OeisA7406.a (n - 1)) = OeisA89026.a n","subjects":["11"],"theorem":"OeisA7406.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7406»","statement":"OeisA7406.a 2 = 5","subjects":["11"],"theorem":"OeisA7406.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7406»","statement":"OeisA7406.a 3 = 49","subjects":["11"],"theorem":"OeisA7406.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7406»","statement":"OeisA7406.a 0 = 0","subjects":["11"],"theorem":"OeisA7406.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7406»","statement":"OeisA7406.a 1 = 1","subjects":["11"],"theorem":"OeisA7406.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7406»","statement":"OeisA7406.a 4 = 205","subjects":["11"],"theorem":"OeisA7406.a_4"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«6697»","statement":"∀ (b : Bool), List.count false (OeisA6697.morphism b) = if b = true then 0 else 2","subjects":["68"],"subsets":["FC100SolvedSet1"],"theorem":"OeisA6697.count_false_morphism"},{"answerKinds":[],"category":"research solved","docstring":"**Conjecture (A6697)**: The generating function for the number of subwords of length $n$\nin the infinite word generated by $a \\mapsto aab, b \\mapsto b$ is\n$$\\sum_{n \\geq 0} a_n x^n = 1 + \\frac{1}{1-x} + \\frac{1}{(1-x)^2}\\left(\\frac{1}{1-x} -\n  \\sum_{k \\geq 0} x^{2^{k+1} + k}\\right).$$\n\nEquivalently, a(n) equals the n-th coefficient of this generating function.\n\nHowever, the reference quoted in the OEIS sequence\n\n> J.-P. Allouche and J. Shallit, \"On the subword complexity of the fixed point of a → aab, b → b,\n> and generalizations,\" arXiv:1605.02361 [math.CO], 2016.\n\nprovides an explicit formula\n$$ a_n = \\sum_{i=0}^{n} \\min(2^i,n-i+1). $$\n\nIf one takes this as a definition of a(n) instead,\nit becomes straightforward to prove the conjecture.\nSee https://github.com/AxiomMath/gdm-formal-conjectures/blob/main/docs/OeisA6697.md\nfor a formal proof of the generating function using this definition.\n\nHence, a formalization of [arXiv:1605.02361](https://arxiv.org/abs/1605.02361)\nwould complete a formal proof as below.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/AxiomMath/gdm-formal-conjectures/blob/main/OeisA6697/solution.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«6697»","statement":"∀ (n : ℕ),\n  ↑(OeisA6697.a n) =\n    (PowerSeries.coeff n)\n      ((1 - PowerSeries.X)⁻¹ +\n        PowerSeries.X * (1 - PowerSeries.X)⁻¹ ^ 2 *\n          ((1 - PowerSeries.X)⁻¹ - ∑' (k : ℕ), PowerSeries.X ^ (2 ^ (k + 1) + k)))","subjects":["68"],"theorem":"OeisA6697.conjecture"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«6697»","statement":"∀ (b : Bool), List.count true (OeisA6697.morphism b) = 1","subjects":["68"],"theorem":"OeisA6697.count_true_morphism"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«6697»","statement":"∀ (n : ℕ), List.count true (OeisA6697.finiteWord n) = 2 ^ n - 1","subjects":["68"],"theorem":"OeisA6697.count_true_finiteWord"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«6697»","statement":"∀ (n : ℕ), (OeisA6697.finiteWord n).length = 2 ^ (n + 1) - 1","subjects":["68"],"theorem":"OeisA6697.length_finiteWord"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«6697»","statement":"∀ (n : ℕ), List.count false (OeisA6697.finiteWord n) = 2 ^ n","subjects":["68"],"theorem":"OeisA6697.count_false_finiteWord"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«243106»","statement":"OeisA243106.a 4 = 8910","subjects":["11"],"theorem":"OeisA243106.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«243106»","statement":"OeisA243106.a 5 = -91090","subjects":["11"],"theorem":"OeisA243106.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«243106»","statement":"OeisA243106.a 2 = -90","subjects":["11"],"theorem":"OeisA243106.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«243106»","statement":"OeisA243106.a 3 = -1090","subjects":["11"],"theorem":"OeisA243106.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«243106»","statement":"OeisA243106.a 1 = 10","subjects":["11"],"theorem":"OeisA243106.a_1"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: When expressed in base $b \\ge 5$, the absolute value of any partial sum $\\sum \\pm b^k$ only contains digits belonging to $\\{0, 1, b-2, b-1\\}$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/243106.wip.lean#L140"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«243106»","statement":"∀ (b n : ℕ),\n  b ≥ 5 →\n    ∀ (σ : ℕ → ℤ),\n      (∀ k ∈ Finset.Icc 1 n, σ k = 1 ∨ σ k = -1) →\n        have x := ∑ k ∈ Finset.Icc 1 n, σ k * ↑b ^ k;\n        ∀ d ∈ b.digits x.natAbs, d = 0 ∨ d = 1 ∨ d = b - 2 ∨ d = b - 1","subjects":["11"],"theorem":"OeisA243106.digits_sum_restricted"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«308734»","statement":"OeisA308734.A 3","subjects":["11"],"theorem":"OeisA308734.a_3"},{"answerKinds":[],"category":"research open","docstring":"**Zhi-Wei Sun's Four-Square Conjecture (A308734)**: Any integer $n > 1$ can be written as\n$(2^a \\cdot 3^b)^2 + (2^c \\cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«308734»","statement":"∀ (n : ℕ), 1 < n → OeisA308734.A n","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"OeisA308734.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«308734»","statement":"OeisA308734.A 4","subjects":["11"],"theorem":"OeisA308734.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«308734»","statement":"OeisA308734.A 6","subjects":["11"],"theorem":"OeisA308734.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«308734»","statement":"OeisA308734.A 5","subjects":["11"],"theorem":"OeisA308734.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«308734»","statement":"OeisA308734.A 2","subjects":["11"],"theorem":"OeisA308734.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119563»","statement":"OeisA119563.a 3 = 263","subjects":["11"],"theorem":"OeisA119563.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119563»","statement":"OeisA119563.a 0 = 2","subjects":["11"],"theorem":"OeisA119563.a_0"},{"answerKinds":[],"category":"research open","docstring":"The first 5 entries are primes. Are there infinitely many primes in this sequence?","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«119563»","statement":"{n | Nat.Prime (OeisA119563.a n)}.Infinite","subjects":["11"],"theorem":"OeisA119563.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119563»","statement":"OeisA119563.a 4 = 65551","subjects":["11"],"theorem":"OeisA119563.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119563»","statement":"OeisA119563.a 2 = 19","subjects":["11"],"theorem":"OeisA119563.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119563»","statement":"OeisA119563.a 1 = 5","subjects":["11"],"theorem":"OeisA119563.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«41»","statement":"OeisA41.a 1 = 1","subjects":["11"],"theorem":"OeisA41.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«41»","statement":"OeisA41.a 2 = 2","subjects":["11"],"theorem":"OeisA41.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«41»","statement":"OeisA41.a 4 = 5","subjects":["11"],"theorem":"OeisA41.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«41»","statement":"OeisA41.a 0 = 1","subjects":["11"],"theorem":"OeisA41.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«41»","statement":"OeisA41.a 3 = 3","subjects":["11"],"theorem":"OeisA41.a_3"},{"answerKinds":[],"category":"research open","docstring":"There are no partition numbers $a(k)$ of the form $x^m$, with $x,m$ integers $>1$.\nSee comment by Zhi-Wei Sun (Dec 02 2013).\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«41»","statement":"True ↔ ∀ (k : ℕ), ¬(OeisA41.a k).IsPerfectPower","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"OeisA41.noPowerPartitionNumber"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«41»","statement":"OeisA41.a 5 = 7","subjects":["11"],"theorem":"OeisA41.a_5"},{"answerKinds":[],"category":"research open","docstring":"Conjecture (Peter Bala, 2022):\nThe supercongruences $a(n \\cdot p^k) \\equiv a(n \\cdot p^{k-1}) \\pmod{p^{3k}}$ hold\nfor the integer-indexed extension $a(n)$ for all $n \\in \\mathbb{Z} \\setminus \\{0\\}$, primes $p \\ge 5$,\nand $k \\ge 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«141057»","statement":"∀ (p k : ℕ) (n : ℤ),\n  Nat.Prime p →\n    5 ≤ p → 1 ≤ k → n ≠ 0 → OeisA141057.aInt (n * ↑p ^ k) ≡ OeisA141057.aInt (n * ↑p ^ (k - 1)) [ZMOD ↑p ^ (3 * k)]","subjects":["11"],"theorem":"OeisA141057.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«141057»","statement":"OeisA141057.a 4 = 6219","subjects":["11"],"theorem":"OeisA141057.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `aInt` at -2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«141057»","statement":"OeisA141057.aInt (-2) = 255","subjects":["11"],"theorem":"OeisA141057.aInt_neg_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: the supercongruences $a(n \\cdot p^k) \\equiv a(n \\cdot p^{k-1}) \\pmod{p^{3k}}$ hold\nfor primes $p \\ge 5$ and positive integers $n$ and $k$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«141057»","statement":"∀ (p k n : ℕ),\n  Nat.Prime p →\n    5 ≤ p → 1 ≤ k → 1 ≤ n → ↑(OeisA141057.a (n * p ^ k)) ≡ ↑(OeisA141057.a (n * p ^ (k - 1))) [ZMOD ↑p ^ (3 * k)]","subjects":["11"],"theorem":"OeisA141057.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«141057»","statement":"OeisA141057.a 0 = 1","subjects":["11"],"theorem":"OeisA141057.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«141057»","statement":"OeisA141057.a 3 = 381","subjects":["11"],"theorem":"OeisA141057.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `aInt` at -1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«141057»","statement":"OeisA141057.aInt (-1) = -1","subjects":["11"],"theorem":"OeisA141057.aInt_neg_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«141057»","statement":"OeisA141057.a 1 = 3","subjects":["11"],"theorem":"OeisA141057.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«141057»","statement":"OeisA141057.a 2 = 27","subjects":["11"],"theorem":"OeisA141057.a_2"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108081»","statement":"OeisA108081.a 0 = 1","subjects":["5"],"theorem":"OeisA108081.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108081»","statement":"OeisA108081.a 4 = 92","subjects":["5"],"theorem":"OeisA108081.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108081»","statement":"OeisA108081.a 2 = 7","subjects":["5"],"theorem":"OeisA108081.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108081»","statement":"OeisA108081.a 1 = 2","subjects":["5"],"theorem":"OeisA108081.a_1"},{"answerKinds":[],"category":"research open","docstring":"\"The number of words of length $n$ for $n \\le 12$ is given by $a(n+1)$. Is this always true?\"\n\nFormalized as $|X_n| = a(n-1)$ for $n \\ge 1$, because the sequence values $a(0)=1, a(1)=2, a(2)=7$\nmatch the examples given for word lengths $n=1, 2, 3$ respectively.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108081»","statement":"∀ n ≥ 1, (OeisA108081.xN n).ncard = OeisA108081.a (n - 1)","subjects":["5"],"theorem":"OeisA108081.count_words_in_x_is_a_shifted"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108081»","statement":"OeisA108081.a 3 = 25","subjects":["5"],"theorem":"OeisA108081.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105020»","statement":"OeisA105020.a 2 = 4","subjects":["11"],"theorem":"OeisA105020.a_2"},{"answerKinds":[],"category":"research open","docstring":"A \"Goldbach Conjecture\" for this sequence: when there are $n$ terms between consecutive odd\nintegers $2n+1$ and $2n+3$ for $n > 0$, at least one will be the product of 2 primes\n(not necessarily distinct). Example: $n=3$ for consecutive odd integers $a(7) = 7$ and\n$a(11) = 9$ and of the 3 sequence entries $a(8) = 12$, $a(9) = 15$ and $a(10) = 16$ between\nthem, one is the product of 2 primes $a(9) = 15=3*5$. - _Michael Hiebl_, Jul 15 2007\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«105020»","statement":"∀ (n i j : ℕ),\n  1 ≤ n →\n    OeisA105020.a i = 2 * n + 1 →\n      OeisA105020.a j = 2 * n + 3 → j = i + n + 1 → ∃ k, i < k ∧ k < j ∧ (OeisA105020.a k).IsSemiprime","subjects":["11"],"theorem":"OeisA105020.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105020»","statement":"OeisA105020.a 1 = 3","subjects":["11"],"theorem":"OeisA105020.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105020»","statement":"OeisA105020.a 3 = 5","subjects":["11"],"theorem":"OeisA105020.a_3"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105020»","statement":"OeisA105020.a 0 = 1","subjects":["11"],"theorem":"OeisA105020.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105020»","statement":"OeisA105020.a 4 = 8","subjects":["11"],"theorem":"OeisA105020.a_4"},{"answerKinds":[],"category":"research open","docstring":"For $n$ large enough, does $a(n) > \\sqrt{n}$ always hold?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«71532»","statement":"∃ N, ∀ (n : ℕ), N ≤ n → ↑(OeisA71532.a n) > √↑n","subjects":["11"],"theorem":"OeisA71532.conjecture2"},{"answerKinds":[],"category":"research open","docstring":"Is $a(n) > 0$ for all $n > 2$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«71532»","statement":"∀ (n : ℕ), 2 < n → 0 < OeisA71532.a n","subjects":["11"],"theorem":"OeisA71532.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71532»","statement":"OeisA71532.a 1 = 1","subjects":["11"],"theorem":"OeisA71532.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71532»","statement":"OeisA71532.a 4 = 2","subjects":["11"],"theorem":"OeisA71532.a_4"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Conjecture: the constant $C$ in $a(n) \\sim C \\log(n)^2$ is approximately $1.4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«71532»","statement":"have C := sorry;\n|C - 1.4| < 0.1 ∧ Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(OeisA71532.a n)) fun n => C * Real.log ↑n ^ 2","subjects":["11"],"theorem":"OeisA71532.conjecture3_value"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71532»","statement":"OeisA71532.a 5 = 3","subjects":["11"],"theorem":"OeisA71532.a_5"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: asymptotically, $a(n) \\sim C \\log(n)^2$ for some constant $C > 0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«71532»","statement":"∃ C, 0 < C ∧ Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(OeisA71532.a n)) fun n => C * Real.log ↑n ^ 2","subjects":["11"],"theorem":"OeisA71532.conjecture3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71532»","statement":"OeisA71532.a 2 = 0","subjects":["11"],"theorem":"OeisA71532.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71532»","statement":"OeisA71532.a 3 = 1","subjects":["11"],"theorem":"OeisA71532.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109227»","statement":"OeisA109227.a 3 = 1101","subjects":["11"],"theorem":"OeisA109227.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109227»","statement":"OeisA109227.a 1 = 1","subjects":["11"],"theorem":"OeisA109227.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109227»","statement":"OeisA109227.a 2 = 11","subjects":["11"],"theorem":"OeisA109227.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109227»","statement":"OeisA109227.a 4 = 110101","subjects":["11"],"theorem":"OeisA109227.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(2)$ and $a(121)$ are primes. Are there any more?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«109227»","statement":"True ↔ ∃ n > 0, n ≠ 2 ∧ n ≠ 121 ∧ Nat.Prime (OeisA109227.a n)","subjects":["11"],"theorem":"OeisA109227.conjecture"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109227»","statement":"OeisA109227.a 0 = 0","subjects":["11"],"theorem":"OeisA109227.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113019»","statement":"OeisA113019.a 2 = 1","subjects":["11"],"theorem":"OeisA113019.a_2"},{"answerKinds":["Prop"],"category":"research solved","docstring":"$n=1$ and $32$ are fixed points. Are there any others?\n\nYes: 9^9 = 387420489 is also a fixed point. - [Kenta Kitamura](https://oeis.org/wiki/User:Kenta_Kitamura), Aug 14 2026\n","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113019»","statement":"False ↔ ∀ (n : ℕ), OeisA113019.a n = n → n = 1 ∨ n = 32","subjects":["11"],"theorem":"OeisA113019.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113019»","statement":"OeisA113019.a 1 = 1","subjects":["11"],"theorem":"OeisA113019.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113019»","statement":"OeisA113019.a 3 = 1","subjects":["11"],"theorem":"OeisA113019.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113019»","statement":"OeisA113019.a 0 = 1","subjects":["11"],"theorem":"OeisA113019.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113019»","statement":"OeisA113019.a 4 = 1","subjects":["11"],"theorem":"OeisA113019.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«176477»","statement":"OeisA176477.a 2 = 181","subjects":["11"],"theorem":"OeisA176477.a_2"},{"answerKinds":[],"category":"research open","docstring":"$a(n)$ is odd if and only if $n = 2, 2^2, 2^3, \\dots$.\n- _Zhi-Wei Sun_, Apr 06 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«176477»","statement":"∀ (n : ℕ), 1 ≤ n → ((OeisA176477.a n).den = 1 ∧ Odd (OeisA176477.a n).num ↔ ∃ m, 1 ≤ m ∧ n = 2 ^ m)","subjects":["11"],"theorem":"OeisA176477.conjecture2"},{"answerKinds":[],"category":"research open","docstring":"Each term $a(n)$ is a positive integer.\n- _Zhi-Wei Sun_, Apr 06 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«176477»","statement":"∀ (n : ℕ), 1 ≤ n → (OeisA176477.a n).den = 1 ∧ 0 < OeisA176477.a n","subjects":["11"],"theorem":"OeisA176477.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«176477»","statement":"OeisA176477.a 1 = 2","subjects":["11"],"theorem":"OeisA176477.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109671»","statement":"OeisA109671.a 3 = 2","subjects":["11"],"theorem":"OeisA109671.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109671»","statement":"OeisA109671.a 1 = 1","subjects":["11"],"theorem":"OeisA109671.a_1"},{"answerKinds":[],"category":"research open","docstring":"Does the sequence contain every positive integer (cf. A169741)?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«109671»","statement":"True ↔ ∀ (m : ℕ), 0 < m → ∃ n, 0 < n ∧ OeisA109671.a n = m","subjects":["11"],"theorem":"OeisA109671.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109671»","statement":"OeisA109671.a 5 = 1","subjects":["11"],"theorem":"OeisA109671.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109671»","statement":"OeisA109671.a 2 = 1","subjects":["11"],"theorem":"OeisA109671.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109671»","statement":"OeisA109671.a 4 = 1","subjects":["11"],"theorem":"OeisA109671.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114137»","statement":"OeisA114137.a 3 = 1","subjects":["11"],"theorem":"OeisA114137.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114137»","statement":"OeisA114137.a 1 = 7","subjects":["11"],"theorem":"OeisA114137.a_1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114137»","statement":"∀ (n val : ℕ),\n  val.IsSemiprime ∧ Odd val →\n    2 ^ n < val → (∀ (x : ℕ), 2 ^ n < x → x < val → ¬(x.IsSemiprime ∧ Odd x)) → OeisA114137.a n = val - 2 ^ n","subjects":["11"],"theorem":"OeisA114137.a_eq_of"},{"answerKinds":[],"category":"research open","docstring":"In this powers of 2 sequence, does 1 occur infinitely often?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«114137»","statement":"True ↔ {n | OeisA114137.a n = 1}.Infinite","subjects":["11"],"theorem":"OeisA114137.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"Does every odd number occur?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«114137»","statement":"True ↔ ∀ (k : ℕ), Odd k → ∃ n, OeisA114137.a n = k","subjects":["11"],"theorem":"OeisA114137.conjecture2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114137»","statement":"OeisA114137.a 5 = 1","subjects":["11"],"theorem":"OeisA114137.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114137»","statement":"OeisA114137.a 2 = 5","subjects":["11"],"theorem":"OeisA114137.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114137»","statement":"OeisA114137.a 4 = 5","subjects":["11"],"theorem":"OeisA114137.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179524»","statement":"OeisA179524.a 2 = -15","subjects":["11"],"theorem":"OeisA179524.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179524»","statement":"OeisA179524.a 3 = -143","subjects":["11"],"theorem":"OeisA179524.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179524»","statement":"OeisA179524.a 1 = 1","subjects":["11"],"theorem":"OeisA179524.a_1"},{"answerKinds":[],"category":"research open","docstring":"$\\sum_{k=0}^{n-1}(20k+17)a(k) \\equiv 0 \\pmod n$ for all $n=1,2,3,\\dots$.\n- _Zhi-Wei Sun_, Jul 01 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«179524»","statement":"∀ (n : ℕ), 1 ≤ n → ↑n ∣ ∑ k ∈ Finset.range n, (20 * ↑k + 17) * OeisA179524.a k","subjects":["11"],"theorem":"OeisA179524.conjecture4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179524»","statement":"OeisA179524.a 0 = 1","subjects":["11"],"theorem":"OeisA179524.a_0"},{"answerKinds":[],"category":"research open","docstring":"If $p$ is a prime with $p \\equiv 1, 9 \\pmod{20}$ and $p = x^2 + 5y^2$ with $x, y$ integers,\nthen $\\sum_{k=0}^{p-1} a(k) \\equiv 4x^2 - 2p \\pmod{p^2}$.\n- _Zhi-Wei Sun_, Jul 01 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«179524»","statement":"∀ (p : ℕ),\n  Nat.Prime p →\n    ↑p ≡ 1 [ZMOD 20] ∨ ↑p ≡ 9 [ZMOD 20] →\n      ∀ (x y : ℤ), ↑p = x ^ 2 + 5 * y ^ 2 → ∑ k ∈ Finset.range p, OeisA179524.a k ≡ 4 * x ^ 2 - 2 * ↑p [ZMOD ↑p ^ 2]","subjects":["11"],"theorem":"OeisA179524.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"If $p$ is a prime with $p \\equiv 3, 7 \\pmod{20}$ and $2p = x^2 + 5y^2$ with $x, y$ integers,\nthen $\\sum_{k=0}^{p-1} a(k) \\equiv 2x^2 - 2p \\pmod{p^2}$.\n- _Zhi-Wei Sun_, Jul 01 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«179524»","statement":"∀ (p : ℕ),\n  Nat.Prime p →\n    ↑p ≡ 3 [ZMOD 20] ∨ ↑p ≡ 7 [ZMOD 20] →\n      ∀ (x y : ℤ), 2 * ↑p = x ^ 2 + 5 * y ^ 2 → ∑ k ∈ Finset.range p, OeisA179524.a k ≡ 2 * x ^ 2 - 2 * ↑p [ZMOD ↑p ^ 2]","subjects":["11"],"theorem":"OeisA179524.conjecture2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«179524»","statement":"OeisA179524.a 4 = 1","subjects":["11"],"theorem":"OeisA179524.a_4"},{"answerKinds":[],"category":"research open","docstring":"$\\sum_{k=0}^{p-1}(20k+17)a(k) \\equiv p(10(-1/p)+7) \\pmod{p^2}$ for any odd prime $p$.\n- _Zhi-Wei Sun_, Jul 01 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«179524»","statement":"∀ (p : ℕ) [hp : Fact (Nat.Prime p)],\n  p ≠ 2 → ∑ k ∈ Finset.range p, (20 * ↑k + 17) * OeisA179524.a k ≡ ↑p * (10 * legendreSym p (-1) + 7) [ZMOD ↑p ^ 2]","subjects":["11"],"theorem":"OeisA179524.conjecture5"},{"answerKinds":[],"category":"research open","docstring":"If $p$ is a prime with $p \\equiv 11, 13, 17, 19 \\pmod{20}$,\nthen $\\sum_{k=0}^{p-1} a(k) \\equiv 0 \\pmod{p^2}$.\n- _Zhi-Wei Sun_, Jul 01 2010\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«179524»","statement":"∀ (p : ℕ),\n  Nat.Prime p →\n    ↑p ≡ 11 [ZMOD 20] ∨ ↑p ≡ 13 [ZMOD 20] ∨ ↑p ≡ 17 [ZMOD 20] ∨ ↑p ≡ 19 [ZMOD 20] →\n      ∑ k ∈ Finset.range p, OeisA179524.a k ≡ 0 [ZMOD ↑p ^ 2]","subjects":["11"],"theorem":"OeisA179524.conjecture3"},{"answerKinds":[],"category":"research open","docstring":"Is $1155$ the last odd number in this sequence?\n($1155$ is the $59$th term starting from $1$, corresponding to $a(58) = 1155$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108864»","statement":"True ↔ ∀ n > 58, Even (OeisA108864.a n)","subjects":["11"],"theorem":"OeisA108864.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108864»","statement":"OeisA108864.a 2 = 3","subjects":["11"],"theorem":"OeisA108864.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108864»","statement":"OeisA108864.a 4 = 5","subjects":["11"],"theorem":"OeisA108864.a_4"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108864»","statement":"OeisA108864.a 0 = 1","subjects":["11"],"theorem":"OeisA108864.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108864»","statement":"OeisA108864.a 3 = 4","subjects":["11"],"theorem":"OeisA108864.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108864»","statement":"OeisA108864.a 1 = 2","subjects":["11"],"theorem":"OeisA108864.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«194806»","statement":"OeisA194806.a 2 = 2","subjects":["11"],"theorem":"OeisA194806.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«194806»","statement":"OeisA194806.a 5 = 4","subjects":["11"],"theorem":"OeisA194806.a_5"},{"answerKinds":[],"category":"research solved","docstring":"Is $a(n) / \\pi(n)$ bounded as $n \\to \\infty$? - _Robert Israel_, Jan 09 2017\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/194806.wip.lean#L472"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«194806»","statement":"∃ C, ∀ (n : ℕ), 2 ≤ n → ↑(OeisA194806.a n) / ↑n.primeCounting ≤ C","subjects":["11"],"theorem":"OeisA194806.a_div_prime_counting_bounded"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«194806»","statement":"OeisA194806.a 1 = 1","subjects":["11"],"theorem":"OeisA194806.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«194806»","statement":"OeisA194806.a 3 = 3","subjects":["11"],"theorem":"OeisA194806.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«194806»","statement":"OeisA194806.a 4 = 3","subjects":["11"],"theorem":"OeisA194806.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110475»","statement":"OeisA110475.a 2 = 0","subjects":["11"],"theorem":"OeisA110475.a_2"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that $1,2,3,4,5,6,7,9,11$ are the only positive integers\nwhich cannot be represented as the sum of two elements of indices $n$ such that $a(n) = 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«110475»","statement":"∀ m > 0, m ∉ OeisA110475.exceptionalSet ↔ ∃ x y, OeisA110475.a x = 1 ∧ OeisA110475.a y = 1 ∧ m = x + y","subjects":["11"],"theorem":"OeisA110475.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110475»","statement":"OeisA110475.a 5 = 0","subjects":["11"],"theorem":"OeisA110475.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110475»","statement":"OeisA110475.a 4 = 1","subjects":["11"],"theorem":"OeisA110475.a_4"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110475»","statement":"OeisA110475.a 1 = 0","subjects":["11"],"theorem":"OeisA110475.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110475»","statement":"OeisA110475.a 3 = 0","subjects":["11"],"theorem":"OeisA110475.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108»","statement":"OeisA108.a 3 = 5","subjects":["11"],"theorem":"OeisA108.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108»","statement":"OeisA108.a 0 = 1","subjects":["11"],"theorem":"OeisA108.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108»","statement":"OeisA108.a 4 = 14","subjects":["11"],"theorem":"OeisA108.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: All the rational numbers $\\sum_{i=j}^k \\frac{1}{a(i)}$ with\n$0 < \\min\\{2,k\\} \\le j \\le k$ have pairwise distinct fractional parts.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/108.wip.lean#L255"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108»","statement":"∀ ⦃j₁ k₁ j₂ k₂ : ℕ⦄,\n  OeisA108.IndexCond j₁ k₁ →\n    OeisA108.IndexCond j₂ k₂ →\n      (j₁, k₁) ≠ (j₂, k₂) →\n        OeisA108.fracPart (OeisA108.catalanReciprocalSum j₁ k₁) ≠\n          OeisA108.fracPart (OeisA108.catalanReciprocalSum j₂ k₂)","subjects":["11"],"theorem":"OeisA108.catalanReciprocalSum_fracPart_inj"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108»","statement":"OeisA108.a 2 = 2","subjects":["11"],"theorem":"OeisA108.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108»","statement":"OeisA108.a 1 = 1","subjects":["11"],"theorem":"OeisA108.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«111114»","statement":"OeisA111114.a 3 = 2","subjects":["11"],"theorem":"OeisA111114.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«111114»","statement":"OeisA111114.a 2 = 3","subjects":["11"],"theorem":"OeisA111114.a_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: As $n \\rightarrow \\infty$, there are infinitely many n's such that\n$a(n)$ is greater than $a(n+1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«111114»","statement":"∃ᶠ (n : ℕ) in Filter.atTop, OeisA111114.a n > OeisA111114.a (n + 1)","subjects":["11"],"theorem":"OeisA111114.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«111114»","statement":"OeisA111114.a 5 = 3","subjects":["11"],"theorem":"OeisA111114.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«111114»","statement":"OeisA111114.a 4 = 3","subjects":["11"],"theorem":"OeisA111114.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«120424»","statement":"OeisA120424.a 0 = 1","subjects":["11"],"theorem":"OeisA120424.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«120424»","statement":"OeisA120424.a 3 = 5","subjects":["11"],"theorem":"OeisA120424.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«120424»","statement":"OeisA120424.a 1 = 3","subjects":["11"],"theorem":"OeisA120424.a_1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture (1): The natural density of even terms in the sequence is $1/2$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«120424»","statement":"Filter.Tendsto (fun n => ↑{k ∈ Finset.range n | OeisA120424.a k % 2 = 0}.card / ↑n) Filter.atTop (nhds (1 / 2))","subjects":["11"],"theorem":"OeisA120424.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture (2): There are infinitely many consecutive pairs that differ by 1.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«120424»","statement":"{n | OeisA120424.a (n + 1) = OeisA120424.a n + 1 ∨ OeisA120424.a n = OeisA120424.a (n + 1) + 1}.Infinite","subjects":["11"],"theorem":"OeisA120424.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«120424»","statement":"OeisA120424.a 2 = 4","subjects":["11"],"theorem":"OeisA120424.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«120424»","statement":"OeisA120424.a 4 = 7","subjects":["11"],"theorem":"OeisA120424.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105033»","statement":"OeisA105033.a 1 = 1","subjects":["11"],"theorem":"OeisA105033.a_1"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105033»","statement":"OeisA105033.a 0 = 0","subjects":["11"],"theorem":"OeisA105033.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105033»","statement":"OeisA105033.a 3 = 3","subjects":["11"],"theorem":"OeisA105033.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105033»","statement":"OeisA105033.a 2 = 0","subjects":["11"],"theorem":"OeisA105033.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105033»","statement":"OeisA105033.a 4 = 2","subjects":["11"],"theorem":"OeisA105033.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1157»","statement":"OeisA1157.a 4 = 21","subjects":["11"],"theorem":"OeisA1157.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: For each k = 2,3,..., all the rational numbers\n$\\frac{\\sigma_k(n)}{n^k} = \\sum_{d|n} \\frac{1}{d^k}$ (n = 1,2,3,...) have pairwise distinct\nfractional parts. - Zhi-Wei Sun, Oct 15 2015\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«1157»","statement":"∀ (k : ℕ),\n  2 ≤ k →\n    ∀ (n₁ n₂ : ℕ),\n      0 < n₁ →\n        0 < n₂ →\n          n₁ ≠ n₂ →\n            Int.fract (↑((ArithmeticFunction.sigma k) n₁) / ↑n₁ ^ k) ≠\n              Int.fract (↑((ArithmeticFunction.sigma k) n₂) / ↑n₂ ^ k)","subjects":["11"],"theorem":"OeisA1157.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1157»","statement":"OeisA1157.a 1 = 1","subjects":["11"],"theorem":"OeisA1157.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1157»","statement":"OeisA1157.a 3 = 10","subjects":["11"],"theorem":"OeisA1157.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1157»","statement":"OeisA1157.a 2 = 5","subjects":["11"],"theorem":"OeisA1157.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1157»","statement":"OeisA1157.a 5 = 26","subjects":["11"],"theorem":"OeisA1157.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76495»","statement":"OeisA76495.a 3 = 4","subjects":["11"],"theorem":"OeisA76495.a_3"},{"answerKinds":[],"category":"research open","docstring":"At present, the 0 entry for $n = 5$ is only a conjecture.\nThat is, it is conjectured that there is no positive integer $x$ such that\n$\\sigma_1(x) \\bmod x = 5$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«76495»","statement":"OeisA76495.a 5 = 0","subjects":["11"],"theorem":"OeisA76495.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76495»","statement":"OeisA76495.a 4 = 9","subjects":["11"],"theorem":"OeisA76495.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 6. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76495»","statement":"OeisA76495.a 6 = 25","subjects":["11"],"theorem":"OeisA76495.a_6"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76495»","statement":"OeisA76495.a 2 = 20","subjects":["11"],"theorem":"OeisA76495.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76495»","statement":"OeisA76495.a 1 = 2","subjects":["11"],"theorem":"OeisA76495.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«306477»","statement":"OeisA306477.A 2","subjects":["11"],"theorem":"OeisA306477.a_2"},{"answerKinds":[],"category":"research open","docstring":"**Zhi-Wei Sun's 2-4-6-8 Conjecture (A306477)**: Any integer $n > 0$ can be written as\n$\\binom{w+2}{2} + \\binom{x+3}{4} + \\binom{y+5}{6} + \\binom{z+7}{8}$ for nonnegative integers $w, x, y, z$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«306477»","statement":"∀ (n : ℕ), 0 < n → OeisA306477.A n","subjects":["11"],"theorem":"OeisA306477.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«306477»","statement":"OeisA306477.A 5","subjects":["11"],"theorem":"OeisA306477.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«306477»","statement":"OeisA306477.A 4","subjects":["11"],"theorem":"OeisA306477.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«306477»","statement":"OeisA306477.A 6","subjects":["11"],"theorem":"OeisA306477.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«306477»","statement":"OeisA306477.A 1","subjects":["11"],"theorem":"OeisA306477.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«306477»","statement":"OeisA306477.A 3","subjects":["11"],"theorem":"OeisA306477.a_3"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108129»","statement":"∀ {m n : ℕ}, IsLeast {m | m ≠ 0 ∧ Nat.Prime ((2 * n - 1) * 2 ^ m - 1)} m → OeisA108129.a n = ↑m","subjects":["11"],"theorem":"OeisA108129.a_of_isLeast"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108129»","statement":"OeisA108129.a 3 = 2","subjects":["11"],"theorem":"OeisA108129.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108129»","statement":"OeisA108129.a 1 = 2","subjects":["11"],"theorem":"OeisA108129.a_1"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that the integer $k = 509203$ is the smallest Riesel number,\nthat is, the first $n$ such that $a(n) = -1$ is $254602$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108129»","statement":"OeisA108129.a 254602 = -1 ∧ ∀ (n : ℕ), 1 ≤ n ∧ n < 254602 → OeisA108129.a n ≠ -1","subjects":["11"],"theorem":"OeisA108129.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108129»","statement":"OeisA108129.a 2 = 1","subjects":["11"],"theorem":"OeisA108129.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108129»","statement":"OeisA108129.a 4 = 1","subjects":["11"],"theorem":"OeisA108129.a_4"},{"answerKinds":[],"category":"research solved","docstring":"**Zhi-Wei Sun's Conjecture (A303639)**: any integer $n > 1$ can be written as\n$a^2 + b^2 + \\binom{2c+1}{c} + \\binom{2d+1}{d}$ with $a, b, c, d$ nonnegative integers.\nSun checked this for $n$ up to $6 \\cdot 10^8$.\n\nThis is false: $n = 800322180$ admits no such representation, so $a(800322180) = 0$. Since\n$\\binom{33}{16} > 800322180$, only $c, d \\le 15$ are possible, and each of the resulting $136$\nremainders $800322180 - \\binom{2c+1}{c} - \\binom{2d+1}{d}$ is divisible by some prime\n$p \\equiv 3 \\pmod 4$ to an odd power, hence is not a sum of two squares by Fermat's two-square\ntheorem.\n\nThe counterexample is recorded as an approved comment on the OEIS entry; the Lean proof linked\nbelow formalises this argument, and was produced by Claude Opus 5 prompted by Sunsu Jeong\n([DCLXAI](https://github.com/DCLXAI)).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/DCLXAI/a303639-counterexample/blob/8e21c57c622b05a1815bcf9927458ebb3973ef2e/lean/A303639/Counterexample.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«303639»","statement":"¬∀ (n : ℕ), 1 < n → OeisA303639.A n","subjects":["11"],"theorem":"OeisA303639.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303639»","statement":"OeisA303639.A 5","subjects":["11"],"theorem":"OeisA303639.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303639»","statement":"OeisA303639.A 2","subjects":["11"],"theorem":"OeisA303639.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303639»","statement":"OeisA303639.A 4","subjects":["11"],"theorem":"OeisA303639.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303639»","statement":"OeisA303639.A 6","subjects":["11"],"theorem":"OeisA303639.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303639»","statement":"¬OeisA303639.A 1","subjects":["11"],"theorem":"OeisA303639.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303639»","statement":"OeisA303639.A 3","subjects":["11"],"theorem":"OeisA303639.a_3"},{"answerKinds":[],"category":"research solved","docstring":"The counterexample witnessing that A303639 vanishes: $a(800322180) = 0$. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/DCLXAI/a303639-counterexample/blob/8e21c57c622b05a1815bcf9927458ebb3973ef2e/lean/A303639/Counterexample.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«303639»","statement":"¬OeisA303639.A 800322180","subjects":["11"],"theorem":"OeisA303639.conjecture.counterexample"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«89026»","statement":"OeisA89026.a 3 = 3","subjects":["11"],"theorem":"OeisA89026.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«89026»","statement":"OeisA89026.a 2 = 2","subjects":["11"],"theorem":"OeisA89026.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«89026»","statement":"OeisA89026.a 5 = 5","subjects":["11"],"theorem":"OeisA89026.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«89026»","statement":"OeisA89026.a 4 = 1","subjects":["11"],"theorem":"OeisA89026.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«89026»","statement":"OeisA89026.a 1 = 1","subjects":["11"],"theorem":"OeisA89026.a_1"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«89026»","statement":"OeisA89026.a 0 = 1","subjects":["11"],"theorem":"OeisA89026.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113271»","statement":"OeisA113271.a 4 = 593","subjects":["11"],"theorem":"OeisA113271.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113271»","statement":"OeisA113271.a 2 = 9","subjects":["11"],"theorem":"OeisA113271.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113271»","statement":"OeisA113271.a 1 = 3","subjects":["11"],"theorem":"OeisA113271.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113271»","statement":"OeisA113271.a 3 = 41","subjects":["11"],"theorem":"OeisA113271.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113271»","statement":"OeisA113271.a 0 = 1","subjects":["11"],"theorem":"OeisA113271.a_0"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"The smallest primes in this (always odd) sequence are $a(1) = 3$, $a(3) = 41$ and $a(5) = 543$.\nWhat is the next prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113271»","statement":"sorry = OeisA113271.a (sInf {n | 5 < n ∧ Nat.Prime (OeisA113271.a n)})","subjects":["11"],"theorem":"OeisA113271.conjecture1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224515»","statement":"OeisA224515.a 1 = 4","subjects":["11"],"theorem":"OeisA224515.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224515»","statement":"OeisA224515.a 2 = 3","subjects":["11"],"theorem":"OeisA224515.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224515»","statement":"OeisA224515.a 4 = 23","subjects":["11"],"theorem":"OeisA224515.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224515»","statement":"OeisA224515.a 0 = 0","subjects":["11"],"theorem":"OeisA224515.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224515»","statement":"OeisA224515.a 3 = 24","subjects":["11"],"theorem":"OeisA224515.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(n) \\ge 0$, i.e., for every $n$ there exists $k$ such that $\\sqrt{k^2 \\oplus (k+1)^2} = 2n+1$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/224515.wip.lean#L268"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«224515»","statement":"∀ (n : ℕ), ∃ k, (k ^ 2).xor ((k + 1) ^ 2) = (2 * n + 1) ^ 2","subjects":["11"],"theorem":"OeisA224515.exists_xor_sq_eq"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224515»","statement":"∀ (n val : ℕ),\n  (val ^ 2).xor ((val + 1) ^ 2) = (2 * n + 1) ^ 2 →\n    (∀ k < val, (k ^ 2).xor ((k + 1) ^ 2) ≠ (2 * n + 1) ^ 2) → OeisA224515.a n = val","subjects":["11"],"theorem":"OeisA224515.A224515_eq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167604»","statement":"OeisA167604.a 4 = 11","subjects":["11"],"theorem":"OeisA167604.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167604»","statement":"OeisA167604.a 2 = 3","subjects":["11"],"theorem":"OeisA167604.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167604»","statement":"OeisA167604.a 1 = 2","subjects":["11"],"theorem":"OeisA167604.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167604»","statement":"OeisA167604.a 3 = 5","subjects":["11"],"theorem":"OeisA167604.a_3"},{"answerKinds":[],"category":"research open","docstring":"Does Chua's sequence contain every prime? ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«167604»","statement":"True ↔ ∀ (p : ℕ), Nat.Prime p → ∃ n ≥ 1, OeisA167604.a n = p","subjects":["11"],"theorem":"OeisA167604.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«22030»","statement":"OeisA22030.a 1 = 16","subjects":["11"],"theorem":"OeisA22030.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«22030»","statement":"OeisA22030.a 0 = 4","subjects":["11"],"theorem":"OeisA22030.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«22030»","statement":"OeisA22030.a 4 = 984","subjects":["11"],"theorem":"OeisA22030.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«22030»","statement":"OeisA22030.a 2 = 63","subjects":["11"],"theorem":"OeisA22030.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«22030»","statement":"OeisA22030.a 5 = 3889","subjects":["11"],"theorem":"OeisA22030.a_5"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) = 4 a(n-1) - a(n-3) + a(n-4)$.\n- Colin Barker, Feb 16 2012\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«22030»","statement":"∀ (n : ℕ), 4 ≤ n → OeisA22030.a n = 4 * OeisA22030.a (n - 1) - OeisA22030.a (n - 3) + OeisA22030.a (n - 4)","subjects":["11"],"theorem":"OeisA22030.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«22030»","statement":"OeisA22030.a 3 = 249","subjects":["11"],"theorem":"OeisA22030.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«340737»","statement":"OeisA340737.a 5 = 193","subjects":["11"],"theorem":"OeisA340737.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«340737»","statement":"OeisA340737.a 2 = 5","subjects":["11"],"theorem":"OeisA340737.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«340737»","statement":"OeisA340737.a 3 = 19","subjects":["11"],"theorem":"OeisA340737.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«340737»","statement":"OeisA340737.a 1 = 3","subjects":["11"],"theorem":"OeisA340737.a_1"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $\\lim_{n \\to \\infty} a(n) / b(n) = e$, where $b(n)$ is the companion denominator sequence.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/340737.wip.lean#L438"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«340737»","statement":"Filter.Tendsto (fun n => ↑(OeisA340737.a n) / ↑(OeisA340737.b n)) Filter.atTop (nhds (Real.exp 1))","subjects":["11"],"theorem":"OeisA340737.tendsto_exp_one"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«340737»","statement":"OeisA340737.a 4 = 49","subjects":["11"],"theorem":"OeisA340737.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«101779»","statement":"OeisA101779.a 1 = 2","subjects":["11"],"theorem":"OeisA101779.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«101779»","statement":"OeisA101779.a 2 = 2","subjects":["11"],"theorem":"OeisA101779.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«101779»","statement":"OeisA101779.a 4 = 5","subjects":["11"],"theorem":"OeisA101779.a_4"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured k always exists.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«101779»","statement":"∀ (n : ℕ), 1 ≤ n → ∃ k, OeisA101779.Ak n k","subjects":["11"],"theorem":"OeisA101779.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«101779»","statement":"OeisA101779.a 3 = 3","subjects":["11"],"theorem":"OeisA101779.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(n) = [x^n y^n z^n] (1+x+y+z)^{2n} (1+x+y-z)^n (1+x-y+z)^n$. - _Peter Bala_, Apr 10 2022\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/2897.wip.lean#L408"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«2897»","statement":"∀ (n : ℕ), ↑(OeisA2897.a n) = MvPolynomial.coeff (OeisA2897.xyzPowN n) (OeisA2897.pPoly n)","subjects":["11"],"theorem":"OeisA2897.a_eq_coeff"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2897»","statement":"OeisA2897.a 1 = 8","subjects":["11"],"theorem":"OeisA2897.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2897»","statement":"OeisA2897.a 3 = 8000","subjects":["11"],"theorem":"OeisA2897.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2897»","statement":"OeisA2897.a 2 = 216","subjects":["11"],"theorem":"OeisA2897.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2897»","statement":"OeisA2897.a 4 = 343000","subjects":["11"],"theorem":"OeisA2897.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2897»","statement":"OeisA2897.a 0 = 1","subjects":["11"],"theorem":"OeisA2897.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64313»","statement":"OeisA64313.a 1 = 0","subjects":["51"],"theorem":"OeisA64313.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64313»","statement":"OeisA64313.a 4 = 1","subjects":["51"],"theorem":"OeisA64313.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64313»","statement":"OeisA64313.a 6 = 2","subjects":["51"],"theorem":"OeisA64313.a_6"},{"answerKinds":[],"category":"research open","docstring":"\"Usually (perhaps always?) $\\lfloor n^2 / (4\\pi) - \\pi / 12 \\rfloor$ for a polygon of circumference $n$.\nNote that the area of a circle with circumference $C$ is $C^2 / (4\\pi)$.\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«64313»","statement":"∀ (n : ℕ), 3 ≤ n → OeisA64313.a n = ⌊↑n ^ 2 / (4 * Real.pi) - Real.pi / 12⌋.toNat","subjects":["51"],"theorem":"OeisA64313.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64313»","statement":"OeisA64313.a 2 = 0","subjects":["51"],"theorem":"OeisA64313.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«64313»","statement":"OeisA64313.a 3 = 0","subjects":["51"],"theorem":"OeisA64313.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69922»","statement":"OeisA69922.a 3 = 2","subjects":["11"],"theorem":"OeisA69922.a_3"},{"answerKinds":[],"category":"research open","docstring":"Question: for any $n > 0$, is there at least one prime $p$ such that $n^n \\le p \\le n^n + n^2$?\nIn this case, that would be stronger than the Schinzel conjecture: \"for $m > 1$ there's at least\none prime $p$ such that $m \\le p \\le m + \\log(m)^2$\" since $n^2 < \\log(n^n)^2 = n^2 \\log(n)^2$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«69922»","statement":"∀ (n : ℕ), 0 < n → 1 ≤ OeisA69922.a n","subjects":["11"],"theorem":"OeisA69922.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69922»","statement":"OeisA69922.a 4 = 4","subjects":["11"],"theorem":"OeisA69922.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69922»","statement":"OeisA69922.a 5 = 1","subjects":["11"],"theorem":"OeisA69922.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69922»","statement":"OeisA69922.a 2 = 2","subjects":["11"],"theorem":"OeisA69922.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69922»","statement":"OeisA69922.a 1 = 1","subjects":["11"],"theorem":"OeisA69922.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185150»","statement":"OeisA185150.a 3 = 2","subjects":["11"],"theorem":"OeisA185150.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185150»","statement":"OeisA185150.a 1 = 1","subjects":["11"],"theorem":"OeisA185150.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185150»","statement":"OeisA185150.a 2 = 1","subjects":["11"],"theorem":"OeisA185150.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185150»","statement":"OeisA185150.a 4 = 3","subjects":["11"],"theorem":"OeisA185150.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) > 0$ for all $n > 0$.\n- _Zhi-Wei Sun_, Dec 29 2012\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«185150»","statement":"∀ (n : ℕ), 0 < n → 0 < OeisA185150.a n","subjects":["11"],"theorem":"OeisA185150.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185150»","statement":"OeisA185150.a 0 = 0","subjects":["11"],"theorem":"OeisA185150.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87455»","statement":"OeisA87455.a 2 = -1","subjects":["11"],"theorem":"OeisA87455.a_2"},{"answerKinds":[],"category":"research open","docstring":"It is an open question whether or not this sequence satisfies Benford's law\n[Berger-Hill, 2017; Arno Berger, email, Jan 06 2017]. - N. J. A. Sloane, Feb 08 2017","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«87455»","statement":"True ↔ OeisA87455.SatisfiesBenford OeisA87455.a","subjects":["11","60"],"theorem":"OeisA87455.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87455»","statement":"OeisA87455.a 1 = 1","subjects":["11"],"theorem":"OeisA87455.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87455»","statement":"OeisA87455.a 3 = -5","subjects":["11"],"theorem":"OeisA87455.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87455»","statement":"OeisA87455.a 0 = 1","subjects":["11"],"theorem":"OeisA87455.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87455»","statement":"OeisA87455.a 4 = -7","subjects":["11"],"theorem":"OeisA87455.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51903»","statement":"OeisA51903.a 2 = 1","subjects":["11"],"theorem":"OeisA51903.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51903»","statement":"OeisA51903.a 3 = 1","subjects":["11"],"theorem":"OeisA51903.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51903»","statement":"OeisA51903.a 1 = 0","subjects":["11"],"theorem":"OeisA51903.a_1"},{"answerKinds":[],"category":"research open","docstring":"Are there composite numbers $n > 4$ such that $n \\equiv a(n) \\pmod{\\phi(n)}$?\n- Thomas Ordowski, Dec 02 2019\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«51903»","statement":"True ↔ ∃ n, 4 < n ∧ ¬Nat.Prime n ∧ n.totient ∣ n - OeisA51903.a n","subjects":["11"],"theorem":"OeisA51903.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"Are there odd numbers $n$ such that $a(n) > 1$ and $n \\equiv a(n) \\pmod{\\lambda(n)}$?\n(Equivalently, odd numbers $n$ such that $a(n) > 1$ and $b^n \\equiv b^{a(n)} \\pmod n$ for all $b$.)\n- Thomas Ordowski, Dec 02 2019\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«51903»","statement":"True ↔ ∃ n, Odd n ∧ 1 < OeisA51903.a n ∧ ∀ (b : ℕ), b ^ n ≡ b ^ OeisA51903.a n [MOD n]","subjects":["11"],"theorem":"OeisA51903.conjecture2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51903»","statement":"OeisA51903.a 4 = 2","subjects":["11"],"theorem":"OeisA51903.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51903»","statement":"OeisA51903.a 5 = 1","subjects":["11"],"theorem":"OeisA51903.a_5"},{"answerKinds":[],"category":"research open","docstring":"Are there odd numbers $n$ such that $a(n) > 1$ and $n \\equiv a(n) \\pmod{\\operatorname{ord}_n(2)}$?\n(Equivalently, odd numbers $n$ such that $a(n) > 1$ and $2^n \\equiv 2^{a(n)} \\pmod n$.)\n- Thomas Ordowski, Dec 02 2019\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«51903»","statement":"True ↔ ∃ n, Odd n ∧ 1 < OeisA51903.a n ∧ 2 ^ n ≡ 2 ^ OeisA51903.a n [MOD n]","subjects":["11"],"theorem":"OeisA51903.conjecture3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114362»","statement":"OeisA114362.a 2 = 6","subjects":["11"],"theorem":"OeisA114362.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114362»","statement":"OeisA114362.a 1 = 2","subjects":["11"],"theorem":"OeisA114362.a_1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: if an integer $n > 1$ is odd, then $\\zeta(2n)/\\zeta(n)^2$ is irrational.\nCf. W. Kohnen (link) and my conjecture in A348829. - Thomas Ordowski, Jan 05 2022\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«114362»","statement":"∀ (n : ℕ), 1 < n → Odd n → Irrational (riemannZeta (2 * ↑n) / riemannZeta ↑n ^ 2).re","subjects":["11"],"theorem":"OeisA114362.conjecture1"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture:\n$\\frac{1 - t(n)}{1 + t(n)} = \\frac{1}{2^n} + \\frac{1}{3^n} + \\frac{1}{5^n} + \\frac{1}{7^n} +\n  O(\\frac{1}{11^n})$,\nwhere $t(n) = \\zeta(2n)/\\zeta(n)^2$. Cf. A348829. - Thomas Ordowski, Nov 13 2022\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/chy4pro/formal-conjectures/blob/872759d0b464254d868f107fe7f9cf762900f57d/FormalConjectures/OEIS/114362.lean#L646"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«114362»","statement":"(fun n =>\n    (1 - OeisA114362.t n) / (1 + OeisA114362.t n) - (1 / 2 ^ n + 1 / 3 ^ n + 1 / 5 ^ n + 1 / 7 ^ n)) =O[Filter.atTop]\n  fun n => 1 / 11 ^ n","subjects":["11"],"theorem":"OeisA114362.conjecture2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114362»","statement":"OeisA114362.a 0 = 2","subjects":["11"],"theorem":"OeisA114362.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114362»","statement":"OeisA114362.a 3 = 691","subjects":["11"],"theorem":"OeisA114362.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80101»","statement":"OeisA80101.a 2 = 1","subjects":["11"],"theorem":"OeisA80101.a_2"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that $a(n) \\le 2$ for all $n$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«80101»","statement":"∀ (n : ℕ), OeisA80101.a n ≤ 2","subjects":["11"],"theorem":"OeisA80101.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80101»","statement":"OeisA80101.a 1 = 0","subjects":["11"],"theorem":"OeisA80101.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80101»","statement":"OeisA80101.a 3 = 0","subjects":["11"],"theorem":"OeisA80101.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80101»","statement":"OeisA80101.a 4 = 2","subjects":["11"],"theorem":"OeisA80101.a_4"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«107247»","statement":"OeisA107247.a 0 = 0","subjects":["11"],"theorem":"OeisA107247.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«107247»","statement":"OeisA107247.a 3 = 0","subjects":["11"],"theorem":"OeisA107247.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«107247»","statement":"OeisA107247.a 1 = 0","subjects":["11"],"theorem":"OeisA107247.a_1"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Primes in this sequence include: $a(8) = 2$, which is next?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«107247»","statement":"sorry = OeisA107247.a (sInf {n | 8 < n ∧ Nat.Prime (OeisA107247.a n)})","subjects":["11"],"theorem":"OeisA107247.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«107247»","statement":"OeisA107247.a 2 = 0","subjects":["11"],"theorem":"OeisA107247.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«107247»","statement":"OeisA107247.a 4 = 0","subjects":["11"],"theorem":"OeisA107247.a_4"},{"answerKinds":[],"category":"textbook","docstring":"Primes in this sequence include: $a(8) = 2$.\nSemiprimes in this sequence include: $a(9) = 6 = 2 * 3$, $a(10) = 22 = 2 * 11$,\n$a(11) = 86 = 2 * 43$, $a(13) = 1366 = 2 * 683$, $a(14) = 5462 = 2 * 2731$,\n$a(16) = 87382 = 2 * 43691$, $a(17) = 348503 = 37 * 9419$,\n$a(27) = 358201316657 = 71 * 5045088967$.\n(Note: The OEIS comment uses 1-based indexing, so their indices are shifted by +1 compared\nto this formalization).\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«107247»","statement":"Nat.Prime (OeisA107247.a 8) ∧\n  (OeisA107247.a 9).IsSemiprime ∧\n    (OeisA107247.a 10).IsSemiprime ∧\n      (OeisA107247.a 11).IsSemiprime ∧\n        (OeisA107247.a 13).IsSemiprime ∧\n          (OeisA107247.a 14).IsSemiprime ∧\n            (OeisA107247.a 16).IsSemiprime ∧ (OeisA107247.a 17).IsSemiprime ∧ (OeisA107247.a 27).IsSemiprime","subjects":["11"],"theorem":"OeisA107247.known_prime_and_semiprimes"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: For prime $p$ such that $p-2$ is not a prime, $a(p-1) = p$.\n- _Bill McEachen_, Sep 26 2025\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«135508»","statement":"∀ (p : ℕ), Nat.Prime p → ¬Nat.Prime (p - 2) → OeisA135508.a (p - 1) = p","subjects":["11"],"theorem":"OeisA135508.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«135508»","statement":"OeisA135508.a 2 = 3","subjects":["11"],"theorem":"OeisA135508.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«135508»","statement":"OeisA135508.a 4 = 1","subjects":["11"],"theorem":"OeisA135508.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«135508»","statement":"OeisA135508.a 0 = 0","subjects":["11"],"theorem":"OeisA135508.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«135508»","statement":"OeisA135508.a 3 = 1","subjects":["11"],"theorem":"OeisA135508.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«135508»","statement":"OeisA135508.a 1 = 2","subjects":["11"],"theorem":"OeisA135508.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«248802»","statement":"OeisA248802.a 2 = 67","subjects":["11"],"theorem":"OeisA248802.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«248802»","statement":"OeisA248802.a 1 = 19","subjects":["11"],"theorem":"OeisA248802.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«248802»","statement":"OeisA248802.a 3 = 13","subjects":["11"],"theorem":"OeisA248802.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«248802»","statement":"OeisA248802.a 0 = 11","subjects":["11"],"theorem":"OeisA248802.a_0"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture 1: $a(10n+2) = 67$ for $n \\ge 0$. - _Chai Wah Wu_, Oct 21 2019\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/248802.wip.lean#L208"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«248802»","statement":"∀ (n : ℕ), OeisA248802.a (10 * n + 2) = 67","subjects":["11"],"theorem":"OeisA248802.a_ten_mul_add_two_eq"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture 4: $a(58n+26) = 1399$ for $n \\ge 0$ and when it is not covered by Conjectures 1-3. - _Chai Wah Wu_, Oct 21 2019\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/248802.wip.lean#L982"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«248802»","statement":"∀ (n : ℕ),\n  ¬OeisA248802.CoveredByC1 (58 * n + 26) ∧\n      ¬OeisA248802.CoveredByC2 (58 * n + 26) ∧ ¬OeisA248802.CoveredByC3 (58 * n + 26) →\n    OeisA248802.a (58 * n + 26) = 1399","subjects":["11"],"theorem":"OeisA248802.a_fifty_eight_mul_add_twenty_six_eq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«248802»","statement":"OeisA248802.a 4 = 262147","subjects":["11"],"theorem":"OeisA248802.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112521»","statement":"OeisA112521.a 2 = 0","subjects":["11"],"theorem":"OeisA112521.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112521»","statement":"OeisA112521.a 3 = 6","subjects":["11"],"theorem":"OeisA112521.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112521»","statement":"OeisA112521.a 1 = 1","subjects":["11"],"theorem":"OeisA112521.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112521»","statement":"OeisA112521.a 0 = 0","subjects":["11"],"theorem":"OeisA112521.a_0"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: Starting with $n=1$, $a(n)$ is the main diagonal of the array $T(n, k)$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a112521-formal-conjectures/blob/d4ee80e997c20209a1f18b7b9aa3521150d5474d/lean/OeisA112521Proof.lean#L876-L885"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«112521»","statement":"∀ n ≥ 1, ↑(OeisA112521.a n) = OeisA112521.T n n","subjects":["11"],"theorem":"OeisA112521.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112521»","statement":"OeisA112521.a 4 = 4","subjects":["11"],"theorem":"OeisA112521.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112521»","statement":"OeisA112521.a 5 = 60","subjects":["11"],"theorem":"OeisA112521.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«48153»","statement":"OeisA48153.a 4 = 2","subjects":["11"],"theorem":"OeisA48153.a_4"},{"answerKinds":[],"category":"research open","docstring":"\"Conjecture: $a(n) <= \\frac{n^2-1}{2}$. - _Aspen A.M. Meissner_, Mar 06 2025\"","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«48153»","statement":"∀ (n : ℕ), 1 ≤ n → OeisA48153.a n ≤ (n ^ 2 - 1) / 2","subjects":["11"],"theorem":"OeisA48153.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«48153»","statement":"OeisA48153.a 3 = 2","subjects":["11"],"theorem":"OeisA48153.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«48153»","statement":"OeisA48153.a 5 = 10","subjects":["11"],"theorem":"OeisA48153.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«48153»","statement":"OeisA48153.a 1 = 0","subjects":["11"],"theorem":"OeisA48153.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«48153»","statement":"OeisA48153.a 2 = 1","subjects":["11"],"theorem":"OeisA48153.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114831»","statement":"OeisA114831.a 2 = 2","subjects":["11"],"theorem":"OeisA114831.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114831»","statement":"OeisA114831.a 3 = 3","subjects":["11"],"theorem":"OeisA114831.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114831»","statement":"OeisA114831.a 1 = 1","subjects":["11"],"theorem":"OeisA114831.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114831»","statement":"OeisA114831.a 4 = 5","subjects":["11"],"theorem":"OeisA114831.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture based on OEIS A114831: What is this sequence, asymptotically?\nIf the limit exists, the ratio of consecutive terms must tend to $\\sqrt{3}$:\n$$ \\lim_{n \\to \\infty} \\frac{a(n+1)}{a(n)} = \\sqrt{3}. $$\nThat's because $a(n)$ is positive, monotonically increasing ($a(n) > a(n-1)$)\nand $a(n+2) \\geq a(n+1) + a(n)$.\nSo $a(n)$ grows exponentially, at least as fast as the Fibonnaci numbers.\nAssuming $\\frac{a(n+1)}{a(n)}$ tend to a limit L, solving for L in the definition of $a(n)$\ngives $L=\\sqrt{3}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a114831-asymptotic/blob/54cdeeed2ef5838e3aa61a3a228e6867802d20df/lean/OeisA114831FC.lean#L252-L285"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«114831»","statement":"Filter.Tendsto (fun n => ↑(OeisA114831.a (n + 1)) / ↑(OeisA114831.a n)) Filter.atTop (nhds √3)","subjects":["11"],"theorem":"OeisA114831.conjecture3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108306»","statement":"OeisA108306.a 3 = 81","subjects":["11"],"theorem":"OeisA108306.a_3"},{"answerKinds":[],"category":"textbook","docstring":"The sequence is the INVERT transform of (1, 5, 10, 20, 40, 80, 160, ...) and can be obtained\nby extracting the upper left terms of matrix powers of [(1,5); (1,2)].\nThese results are a case (a=5, b=2) of the general conjecture below.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108306»","statement":"∀ (n : ℕ), OeisA108306.a n = (OeisA108306.m ^ (n + 1)) 0 0","subjects":["11"],"theorem":"OeisA108306.a_is_invert_transform_case"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108306»","statement":"OeisA108306.a 1 = 6","subjects":["11"],"theorem":"OeisA108306.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108306»","statement":"OeisA108306.a 2 = 21","subjects":["11"],"theorem":"OeisA108306.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108306»","statement":"OeisA108306.a 4 = 306","subjects":["11"],"theorem":"OeisA108306.a_4"},{"answerKinds":[],"category":"research solved","docstring":"The conjecture: The INVERT transform of a sequence starting\n$(1, a, ab, ab^2, ab^3, \\ldots)$ is equivalent to extracting the upper left terms\nof powers of the 2x2 matrix [(1,a); (1,b)].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-108306/blob/2e01e81b20a56880993e174ee4af0f0d7af37bdd/lean/OeisA108306FC.lean#L101-L106"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108306»","statement":"∀ (a_val b_val n : ℕ), OeisA108306.invertSeqD a_val b_val n = (OeisA108306.genMatrix a_val b_val ^ n) 0 0","subjects":["11"],"theorem":"OeisA108306.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108306»","statement":"OeisA108306.a 0 = 1","subjects":["11"],"theorem":"OeisA108306.a_0"},{"answerKinds":[],"category":"research open","docstring":"\"Conjecture: $a(n) < n^2$ for $n > 1$. - _Thomas Ordowski_, Dec 19 2016\"","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«34694»","statement":"∀ (n : ℕ), 1 < n → OeisA34694.a n < n ^ 2","subjects":["11"],"theorem":"OeisA34694.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34694»","statement":"OeisA34694.a 5 = 11","subjects":["11"],"theorem":"OeisA34694.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34694»","statement":"OeisA34694.a 2 = 3","subjects":["11"],"theorem":"OeisA34694.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34694»","statement":"OeisA34694.a 4 = 5","subjects":["11"],"theorem":"OeisA34694.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34694»","statement":"OeisA34694.a 1 = 2","subjects":["11"],"theorem":"OeisA34694.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34694»","statement":"OeisA34694.a 3 = 7","subjects":["11"],"theorem":"OeisA34694.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Non-primitive terms have the form $m \\cdot s$ where $m$ is primitive and $s$ is\nsquarefree with $\\gcd(m, s) = 1$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«63880»","statement":"∀ {n : ℕ}, OeisA63880.A n → ∃ m s, OeisA63880.IsPrimitiveTerm m ∧ Squarefree s ∧ m.Coprime s ∧ n = m * s","subjects":["11"],"theorem":"OeisA63880.exists_primitive_of_a"},{"answerKinds":[],"category":"test","docstring":"$108$ is in the sequence A063880. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«63880»","statement":"OeisA63880.A 108","subjects":["11"],"theorem":"OeisA63880.a_108"},{"answerKinds":[],"category":"textbook","docstring":"All primitive terms are powerful numbers. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«63880»","statement":"∀ {n : ℕ}, OeisA63880.IsPrimitiveTerm n → n.Powerful","subjects":["11"],"theorem":"OeisA63880.powerful_of_isPrimitiveTerm"},{"answerKinds":[],"category":"research open","docstring":"All members of the sequence satisfy $n \\equiv 108 \\pmod{216}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«63880»","statement":"∀ {n : ℕ}, OeisA63880.A n → n % 216 = 108","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"OeisA63880.mod_216_of_a"},{"answerKinds":[],"category":"test","docstring":"$540$ is in the sequence A063880. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«63880»","statement":"OeisA63880.A 540","subjects":["11"],"theorem":"OeisA63880.a_540"},{"answerKinds":[],"category":"textbook","docstring":"If $m$ is a primitive term and $s$ is squarefree with $\\gcd(m, s) = 1$, then $m \\cdot s$\nis in the sequence. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«63880»","statement":"∀ (m s : ℕ), OeisA63880.IsPrimitiveTerm m → Squarefree s → m.Coprime s → OeisA63880.A (m * s)","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"OeisA63880.a_of_primitive_mul_squarefree"},{"answerKinds":[],"category":"research open","docstring":"$108$ is the only primitive term. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«63880»","statement":"∀ {n : ℕ}, OeisA63880.IsPrimitiveTerm n → n = 108","subjects":["11"],"theorem":"OeisA63880.unique_primitive_108"},{"answerKinds":[],"category":"test","docstring":"$108$ is a primitive term. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«63880»","statement":"OeisA63880.IsPrimitiveTerm 108","subjects":["11"],"theorem":"OeisA63880.isPrimitiveTerm_108"},{"answerKinds":[],"category":"test","docstring":"$756$ is in the sequence A063880. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«63880»","statement":"OeisA63880.A 756","subjects":["11"],"theorem":"OeisA63880.a_756"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182126»","statement":"OeisA182126.a 3 = 2","subjects":["11"],"theorem":"OeisA182126.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182126»","statement":"OeisA182126.a 2 = 1","subjects":["11"],"theorem":"OeisA182126.a_2"},{"answerKinds":[],"category":"research open","docstring":"Are 2, 7, 11, 13, 29 the only primes in this sequence?\n- _Hugo Pfoertner_, Sep 22 2025\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«182126»","statement":"∀ (n : ℕ), 0 < n → (Nat.Prime (OeisA182126.a n) ↔ OeisA182126.a n ∈ [2, 7, 11, 13, 29])","subjects":["11"],"theorem":"OeisA182126.conjecture3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182126»","statement":"OeisA182126.a 4 = 12","subjects":["11"],"theorem":"OeisA182126.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182126»","statement":"OeisA182126.a 1 = 1","subjects":["11"],"theorem":"OeisA182126.a_1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: For $x > 10^9$, the most frequent value in $a(n)$, $n=1\\dots x$, has form $120k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«182126»","statement":"∀ (x : ℕ), 10 ^ 9 < x → ∀ (v₀ : ℕ), OeisA182126.IsMostFrequent x v₀ → 120 ∣ v₀","subjects":["11"],"theorem":"OeisA182126.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"Let $b = \\mathrm{prime}(n+2) - \\mathrm{prime}(n)$ and $c = \\mathrm{prime}(n+2) - \\mathrm{prime}(n+1)$.\nConjecture: for $n > 61$, $a(n) = b \\cdot c$.\n- _Charles R Greathouse IV_, May 11 2012\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«182126»","statement":"∀ (n : ℕ),\n  61 < n →\n    have b := OeisA182126.prime (n + 2) - OeisA182126.prime n;\n    have c := OeisA182126.prime (n + 2) - OeisA182126.prime (n + 1);\n    OeisA182126.a n = b * c","subjects":["11"],"theorem":"OeisA182126.conjecture2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«325046»","statement":"OeisA325046.a 0 = 1","subjects":["11"],"theorem":"OeisA325046.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«325046»","statement":"OeisA325046.a 1 = 2","subjects":["11"],"theorem":"OeisA325046.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«325046»","statement":"OeisA325046.a 3 = 4","subjects":["11"],"theorem":"OeisA325046.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: Odd terms occur only at positions $n(n+1)$ for $n \\ge 0$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/325046.wip.lean#L166"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«325046»","statement":"∀ (N : ℕ), OeisA325046.a N % 2 = 1 → ∃ k, N = k * (k + 1)","subjects":["11"],"theorem":"OeisA325046.odd_a_implies_pronic"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«325046»","statement":"OeisA325046.a 2 = 3","subjects":["11"],"theorem":"OeisA325046.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«325046»","statement":"OeisA325046.a 4 = 6","subjects":["11"],"theorem":"OeisA325046.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjectures: a(2^n-1)=a(3*2^n-1)=1. This formalizes the equality part a(2^n-1) = a(3*2^n-1). ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a112970-formal-conjectures/blob/71ae72f443bd9bee7d958f0a19d9f9ec5ab82af5/lean/OeisA112970FC.lean#L56-L64"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«112970»","statement":"∀ (n : ℕ), OeisA112970.a (2 ^ n - 1) = OeisA112970.a (3 * 2 ^ n - 1)","subjects":["11"],"theorem":"OeisA112970.conjecture2"},{"answerKinds":[],"category":"research solved","docstring":"Conjectures: a(2^n-1)=a(3*2^n-1)=1. This formalizes the value part a(2^n-1)=1. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a112970-formal-conjectures/blob/71ae72f443bd9bee7d958f0a19d9f9ec5ab82af5/lean/OeisA112970FC.lean#L48-L54"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«112970»","statement":"∀ (n : ℕ), OeisA112970.a (2 ^ n - 1) = 1","subjects":["11"],"theorem":"OeisA112970.conjecture3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112970»","statement":"OeisA112970.a 4 = 2","subjects":["11"],"theorem":"OeisA112970.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjectures: a(2^n)=a(2^(n+1)+1)=A033638(n).\nThis formalizes the equality a(2^n) = a(2^(n+1)+1).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a112970-formal-conjectures/blob/71ae72f443bd9bee7d958f0a19d9f9ec5ab82af5/lean/OeisA112970FC.lean#L43-L46"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«112970»","statement":"∀ (n : ℕ), OeisA112970.a (2 ^ n) = OeisA112970.a (2 ^ (n + 1) + 1)","subjects":["11"],"theorem":"OeisA112970.conjecture1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112970»","statement":"OeisA112970.a 0 = 1","subjects":["11"],"theorem":"OeisA112970.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112970»","statement":"OeisA112970.a 3 = 1","subjects":["11"],"theorem":"OeisA112970.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112970»","statement":"OeisA112970.a 1 = 1","subjects":["11"],"theorem":"OeisA112970.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«112970»","statement":"OeisA112970.a 2 = 1","subjects":["11"],"theorem":"OeisA112970.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«228828»","statement":"OeisA228828.a 2 = 7","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"OeisA228828.a_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: the sequence A228828 is infinite.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«228828»","statement":"{x | ∃ n, OeisA228828.a n = x}.Infinite","subjects":["11"],"theorem":"OeisA228828.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«228828»","statement":"OeisA228828.a 1 = 3","subjects":["11"],"theorem":"OeisA228828.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«228828»","statement":"OeisA228828.a 0 = 2","subjects":["11"],"theorem":"OeisA228828.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103662»","statement":"OeisA103662.a 1 = 2","subjects":["11"],"theorem":"OeisA103662.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103662»","statement":"OeisA103662.a 3 = 8","subjects":["11"],"theorem":"OeisA103662.a_3"},{"answerKinds":[],"category":"research open","docstring":"$a(40)$, if it exists, is not known.\n\nThis claim is rooted in the finiteness conjecture. The most direct mathematical expression\nof the open problem concerning $a(40)$ is the negation of the existence of a valid base.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«103662»","statement":"¬∃ b, OeisA103662.IsValidZerolessPower 40 b","subjects":["11"],"theorem":"OeisA103662.conjecture.variants.a_40"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103662»","statement":"OeisA103662.a 2 = 4","subjects":["11"],"theorem":"OeisA103662.a_2"},{"answerKinds":[],"category":"research open","docstring":"For statistical reasons it is conjectured that the sequence is finite.\nThis is formalized as the assertion that for large enough $n$, no valid zeroless power exists,\nwhich in our definition results in $a(n) = 0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«103662»","statement":"∃ N, ∀ n > N, OeisA103662.a n = 0","subjects":["11"],"theorem":"OeisA103662.conjecture"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103662»","statement":"OeisA103662.a 0 = 1","subjects":["11"],"theorem":"OeisA103662.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«92243»","statement":"OeisA92243.a 3 = 1","subjects":["11"],"theorem":"OeisA92243.a_3"},{"answerKinds":[],"category":"research open","docstring":"Is the score $a(n)$ bounded from above?","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«92243»","statement":"True ↔ ∃ B, ∀ (n : ℕ), OeisA92243.a n ≤ B","subjects":["11"],"theorem":"OeisA92243.conjecture3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«92243»","statement":"OeisA92243.a 1 = 0","subjects":["11"],"theorem":"OeisA92243.a_1"},{"answerKinds":[],"category":"research open","docstring":"Is the score $a(n) < 0$ for infinitely many values of $n$?","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«92243»","statement":"True ↔ {n | OeisA92243.a n < 0}.Infinite","subjects":["11"],"theorem":"OeisA92243.conjecture5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«92243»","statement":"OeisA92243.a 2 = 1","subjects":["11"],"theorem":"OeisA92243.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«92243»","statement":"OeisA92243.a 0 = 0","subjects":["11"],"theorem":"OeisA92243.a_0"},{"answerKinds":[],"category":"research open","docstring":"Is the score $a(n)$ bounded from below?","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«92243»","statement":"True ↔ ∃ B, ∀ (n : ℕ), B ≤ OeisA92243.a n","subjects":["11"],"theorem":"OeisA92243.conjecture2"},{"answerKinds":[],"category":"research open","docstring":"Is the score $a(n) > 0$ for infinitely many values of $n$?","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«92243»","statement":"True ↔ {n | OeisA92243.a n > 0}.Infinite","subjects":["11"],"theorem":"OeisA92243.conjecture4"},{"answerKinds":[],"category":"research open","docstring":"Is the score $a(n) > 0$ for some $n > 250000$?","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«92243»","statement":"True ↔ ∃ n > 250000, OeisA92243.a n > 0","subjects":["11"],"theorem":"OeisA92243.conjecture1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113010»","statement":"OeisA113010.a 4 = 1","subjects":["11"],"theorem":"OeisA113010.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113010»","statement":"OeisA113010.a 2 = 1","subjects":["11"],"theorem":"OeisA113010.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113010»","statement":"OeisA113010.a 3 = 1","subjects":["11"],"theorem":"OeisA113010.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113010»","statement":"OeisA113010.a 1 = 1","subjects":["11"],"theorem":"OeisA113010.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113010»","statement":"OeisA113010.a 0 = 1","subjects":["11"],"theorem":"OeisA113010.a_0"},{"answerKinds":[],"category":"research open","docstring":"$n=1$ and $32$ are two fixed points. Are there any others?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113010»","statement":"True ↔ ∀ (n : ℕ), OeisA113010.a n = n ∧ n > 0 → n = 1 ∨ n = 32","subjects":["11"],"theorem":"OeisA113010.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«49473»","statement":"OeisA49473.a 0 = 0","subjects":["11"],"theorem":"OeisA49473.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«49473»","statement":"OeisA49473.a 1 = 1","subjects":["11"],"theorem":"OeisA49473.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«49473»","statement":"OeisA49473.a 4 = 3","subjects":["11"],"theorem":"OeisA49473.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«49473»","statement":"OeisA49473.a 5 = 4","subjects":["11"],"theorem":"OeisA49473.a_5"},{"answerKinds":[],"category":"research open","docstring":"Let $s(n) = \\zeta(3) - \\sum_{k=1}^n \\frac{1}{k^3}$.\nConjecture: for $n \\ge 1$, $s(a(n)) < \\frac{1}{n^2} < s(a(n)-1)$, and the difference sequence of\nA049473 consists solely of $0$'s and $1$'s, in positions given by the nonhomogeneous Beatty\nsequences A001954 and A001953, respectively.\n- Clark Kimberling, Oct 05 2014\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«49473»","statement":"(∀ (n : ℕ), 1 ≤ n → OeisA49473.s (OeisA49473.a n) < 1 / ↑n ^ 2 ∧ 1 / ↑n ^ 2 < OeisA49473.s (OeisA49473.a n - 1)) ∧\n  ∀ (n : ℕ),\n    1 ≤ n →\n      have diff := OeisA49473.a n - OeisA49473.a (n - 1);\n      (diff = 0 ↔ n - 1 ∈ OeisA49473.A001954) ∧ (diff = 1 ↔ n - 1 ∈ OeisA49473.A001953)","subjects":["11"],"theorem":"OeisA49473.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«49473»","statement":"OeisA49473.a 2 = 1","subjects":["11"],"theorem":"OeisA49473.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«49473»","statement":"OeisA49473.a 3 = 2","subjects":["11"],"theorem":"OeisA49473.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«267581»","statement":"OeisA267581.a 4 = 26","subjects":["11"],"theorem":"OeisA267581.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«267581»","statement":"OeisA267581.a 0 = 1","subjects":["11"],"theorem":"OeisA267581.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«267581»","statement":"OeisA267581.a 3 = 13","subjects":["11"],"theorem":"OeisA267581.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«267581»","statement":"OeisA267581.a 1 = 3","subjects":["11"],"theorem":"OeisA267581.a_1"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(n) = 2 a(n-1) + 1 - \\lfloor (1/2)^{2^{n+1} \\bmod n} \\rfloor$ for $n \\ge 2$. - _Andres Cicuttin_, Mar 29 2016\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/267581.wip.lean#L190"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«267581»","statement":"∀ (n : ℕ), 2 ≤ n → OeisA267581.a n = 2 * OeisA267581.a (n - 1) + 1 - OeisA267581.oeisFloorTerm n","subjects":["11"],"theorem":"OeisA267581.a_recurrence"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«267581»","statement":"OeisA267581.a 2 = 6","subjects":["11"],"theorem":"OeisA267581.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«28859»","statement":"OeisA28859.a 0 = 1","subjects":["11"],"theorem":"OeisA28859.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«28859»","statement":"OeisA28859.a 3 = 22","subjects":["11"],"theorem":"OeisA28859.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«28859»","statement":"OeisA28859.a 1 = 3","subjects":["11"],"theorem":"OeisA28859.a_1"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: The sequence a(n) is also the number of compositions of $n$ into positive integers\nsuch that adjacent parts and the largest part differ by at most 1. - _Gus Wiseman_, May 19 2020\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/28859.wip.lean#L402"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«28859»","statement":"∀ (n : ℕ),\n  let L := n + 1;\n  let Sequence := Fin L → ℕ;\n  have S :=\n    {σ |\n      L > 0 ∧\n        (∀ (i : Fin L), σ i > 0) ∧\n          have max_val := Finset.univ.sup σ;\n          (∀ (k : ℕ), 1 ≤ k ∧ k ≤ max_val → ∃ i, σ i = k) ∧ ∀ (i j : Fin L), i < j → ↑j ≠ ↑i + 1 → σ i ≥ σ j};\n  ∃ F, ↑F = S ∧ F.card = OeisA28859.a n","subjects":["11"],"theorem":"OeisA28859.exists_finset_sequence"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«28859»","statement":"OeisA28859.a 2 = 8","subjects":["11"],"theorem":"OeisA28859.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«28859»","statement":"OeisA28859.a 4 = 60","subjects":["11"],"theorem":"OeisA28859.a_4"},{"answerKinds":[],"category":"research open","docstring":"According to the \"k-tuple\" conjecture, $a(n)$ is the initial term of the\nlexicographically earliest increasing arithmetic progression of $n$ primes;\nthe corresponding common differences are given by A061558.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«7918»","statement":"∀ (n : ℕ), 0 < n → OeisA7918.a n = sInf {p0 | ∃ d, OeisA7918.isApOfNPrimes n p0 d}","subjects":["11"],"theorem":"OeisA7918.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"If $n > 1$, then $a(n) < n^{n^{1/n}}$.\n- Thomas Ordowski, Feb 23 2023\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«7918»","statement":"∀ (n : ℕ), 1 < n → ↑(OeisA7918.a n) < ↑n ^ ↑n ^ (1 / ↑n)","subjects":["11"],"theorem":"OeisA7918.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7918»","statement":"OeisA7918.a 2 = 2","subjects":["11"],"theorem":"OeisA7918.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7918»","statement":"OeisA7918.a 3 = 3","subjects":["11"],"theorem":"OeisA7918.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7918»","statement":"OeisA7918.a 1 = 2","subjects":["11"],"theorem":"OeisA7918.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7918»","statement":"OeisA7918.a 0 = 2","subjects":["11"],"theorem":"OeisA7918.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«70823»","statement":"OeisA70823.a 5 = 243","subjects":["11"],"theorem":"OeisA70823.a_5"},{"answerKinds":["Prop"],"category":"research solved","docstring":"$a(n) \\equiv 0 \\pmod 3$ if $n > 2$. Is $a(n)$ always of the form $2^j \\cdot 3^k \\cdot s$\nwhere $s$ is a squarefree number?\n\nAnswer: False, $a(20)$ is divisible by $13^2$ but not by $13^3$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a070823-counterexample/blob/51399770e734616c6463be034e41f7469991d752/lean/OeisA70823CounterexampleFC.lean#L72-L81"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«70823»","statement":"False ↔ ∀ (n : ℕ), 2 < n → OeisA70823.a n ≡ 0 [MOD 3] ∧ ∃ j k s, OeisA70823.a n = 2 ^ j * 3 ^ k * s ∧ Squarefree s","subjects":["11"],"theorem":"OeisA70823.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«70823»","statement":"OeisA70823.a 2 = 1","subjects":["11"],"theorem":"OeisA70823.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«70823»","statement":"OeisA70823.a 3 = 9","subjects":["11"],"theorem":"OeisA70823.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«70823»","statement":"OeisA70823.a 1 = 0","subjects":["11"],"theorem":"OeisA70823.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«70823»","statement":"OeisA70823.a 4 = 72","subjects":["11"],"theorem":"OeisA70823.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113250»","statement":"OeisA113250.a 3 = 64","subjects":["11"],"theorem":"OeisA113250.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113250»","statement":"OeisA113250.a 2 = 32","subjects":["11"],"theorem":"OeisA113250.a_2"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(m, 2n+1)$ is a perfect square for all $m$ (see A113249).\nSpecialized to $m = 4$, which is A113250.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a113249-family-square-terms-lean/blob/9b999db08344184285e8050c2722c845fd5f5309/lean/OeisA113249FamilyFC.lean#L106-L114"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113250»","statement":"∀ (n : ℕ), IsSquare (OeisA113250.a (2 * n + 1))","subjects":["11"],"theorem":"OeisA113250.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113250»","statement":"OeisA113250.a 4 = -256","subjects":["11"],"theorem":"OeisA113250.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113250»","statement":"OeisA113250.a 1 = 4","subjects":["11"],"theorem":"OeisA113250.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113250»","statement":"OeisA113250.a 0 = -1","subjects":["11"],"theorem":"OeisA113250.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«72200»","statement":"OeisA72200.a 4 = 26","subjects":["11"],"theorem":"OeisA72200.a_4"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that $a(24) = 0$ since no factorial less than $10000$ contained just 24 sixes.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«72200»","statement":"OeisA72200.a 24 = 0","subjects":["11"],"theorem":"OeisA72200.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«72200»","statement":"OeisA72200.a 1 = 3","subjects":["11"],"theorem":"OeisA72200.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«72200»","statement":"OeisA72200.a 3 = 23","subjects":["11"],"theorem":"OeisA72200.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«72200»","statement":"OeisA72200.a 2 = 15","subjects":["11"],"theorem":"OeisA72200.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«72200»","statement":"OeisA72200.a 5 = 32","subjects":["11"],"theorem":"OeisA72200.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«17666»","statement":"OeisA17666.a 2 = 2","subjects":["11"],"theorem":"OeisA17666.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«17666»","statement":"OeisA17666.a 3 = 3","subjects":["11"],"theorem":"OeisA17666.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«17666»","statement":"OeisA17666.a 1 = 1","subjects":["11"],"theorem":"OeisA17666.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«17666»","statement":"OeisA17666.a 0 = 1","subjects":["11"],"theorem":"OeisA17666.a_0"},{"answerKinds":[],"category":"research open","docstring":"If $a(n)$ is in A005153, then $n$ is in A005153.\n- Jaycob Coleman, Sep 27 2014\n\nWe require $0 < n$ because $a(0) = 1$ is in A005153 (practical numbers), but $0$ is not.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«17666»","statement":"∀ (n : ℕ), 0 < n → OeisA5153.A (OeisA17666.a n) → OeisA5153.A n","subjects":["11"],"theorem":"OeisA17666.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«17666»","statement":"OeisA17666.a 4 = 4","subjects":["11"],"theorem":"OeisA17666.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«232174»","statement":"OeisA232174.A 4","subjects":["11"],"theorem":"OeisA232174.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«232174»","statement":"OeisA232174.A 6","subjects":["11"],"theorem":"OeisA232174.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«232174»","statement":"OeisA232174.A 5","subjects":["11"],"theorem":"OeisA232174.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«232174»","statement":"OeisA232174.A 2","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"OeisA232174.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«232174»","statement":"OeisA232174.A 3","subjects":["11"],"theorem":"OeisA232174.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«232174»","statement":"OeisA232174.A 8","subjects":["11"],"theorem":"OeisA232174.a_8"},{"answerKinds":[],"category":"research open","docstring":"**Zhi-Wei Sun's Conjecture (A232174)**: Any integer $n > 1$ can be written as $x + y$ with\n$x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«232174»","statement":"∀ (n : ℕ), 1 < n → OeisA232174.A n","subjects":["11"],"theorem":"OeisA232174.conjecture"},{"answerKinds":[],"category":"textbook","docstring":"$a(10^k) = 10^k$ for all $k$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«4290»","statement":"∀ (k : ℕ), OeisA4290.a (10 ^ k) = 10 ^ k","subjects":["11"],"theorem":"OeisA4290.a_ten_pow"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«4290»","statement":"OeisA4290.a 1 = 1","subjects":["11"],"theorem":"OeisA4290.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«4290»","statement":"OeisA4290.a 2 = 10","subjects":["11"],"theorem":"OeisA4290.a_2"},{"answerKinds":[],"category":"research open","docstring":"It is known that $a(10^k - 1) = (10^{9k} - 1) / 9$ for all $k$.\nIs $a(n) < a(10^k - 1)$ for all $n < 10^k - 1$?\n- David Radcliffe, Aug 01 2025\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«4290»","statement":"∀ (k n : ℕ), n < 10 ^ k - 1 → OeisA4290.a n < OeisA4290.a (10 ^ k - 1)","subjects":["11"],"theorem":"OeisA4290.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«4290»","statement":"OeisA4290.a 0 = 0","subjects":["11"],"theorem":"OeisA4290.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87207»","statement":"OeisA87207.a 0 = 0","subjects":["11"],"theorem":"OeisA87207.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87207»","statement":"OeisA87207.a 1 = 0","subjects":["11"],"theorem":"OeisA87207.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87207»","statement":"OeisA87207.a 3 = 2","subjects":["11"],"theorem":"OeisA87207.a_3"},{"answerKinds":[],"category":"research open","docstring":"Starting at any $n$ and iterating the map $n \\mapsto a(n)$, we will always reach $0$.\n- _Antti Karttunen_, Jun 18,20 2017\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«87207»","statement":"∀ (n : ℕ), ∃ k, OeisA87207.a^[k] n = 0","subjects":["11"],"theorem":"OeisA87207.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87207»","statement":"OeisA87207.a 2 = 1","subjects":["11"],"theorem":"OeisA87207.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87207»","statement":"OeisA87207.a 4 = 1","subjects":["11"],"theorem":"OeisA87207.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100475»","statement":"OeisA100475.a 3 = 5","subjects":["11"],"theorem":"OeisA100475.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100475»","statement":"OeisA100475.a 1 = 2","subjects":["11"],"theorem":"OeisA100475.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100475»","statement":"OeisA100475.a 2 = 3","subjects":["11"],"theorem":"OeisA100475.a_2"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100475»","statement":"∀ (n : ℕ),\n  OeisA100475.a (n + 1) =\n    if OeisA100475.a n = 0 then 0 else OeisA100475.reverseDigits (Nat.nth Nat.Prime (OeisA100475.a n - 1))","subjects":["11"],"theorem":"OeisA100475.a_succ"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100475»","statement":"OeisA100475.a 4 = 11","subjects":["11"],"theorem":"OeisA100475.a_4"},{"answerKinds":["Prop"],"category":"research open","docstring":"Starting at other than $a(n) = 1$, does this sequence ever go into a loop?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«100475»","statement":"∀ (x : ℕ), x ≠ 1 → sorry = OeisA100475.IsUltimatelyPeriodic (OeisA100475.aStartAt x)","subjects":["11"],"theorem":"OeisA100475.conjecture"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100475»","statement":"OeisA100475.a 0 = 1","subjects":["11"],"theorem":"OeisA100475.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1359»","statement":"OeisA1359.a 1 = 3","subjects":["11"],"theorem":"OeisA1359.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1359»","statement":"OeisA1359.a 0 = 0","subjects":["11"],"theorem":"OeisA1359.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1359»","statement":"OeisA1359.a 4 = 17","subjects":["11"],"theorem":"OeisA1359.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1359»","statement":"OeisA1359.a 2 = 5","subjects":["11"],"theorem":"OeisA1359.a_2"},{"answerKinds":[],"category":"research open","docstring":"Primes $p_k$ such that $p_k! \\equiv 1 \\pmod{p_{k+1}}$ with the exception of $p_{991} = 7841$ and\nother unknown primes $p_k$ for which $(p_k+1)(p_k+2)\\cdots(p_{k+1}-2) \\equiv 1 \\pmod{p_{k+1}}$\nwhere $p_{k+1} - p_k > 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«1359»","statement":"∀ k > 1,\n  have Pk := Nat.nth Nat.Prime (k - 1);\n  have Pk_succ := Nat.nth Nat.Prime k;\n  have Congruence := Pk.factorial ≡ 1 [MOD Pk_succ];\n  have IsLesserTwinPrime := Nat.Prime (Pk + 2);\n  have Wk_prod := ∏ i ∈ Finset.Icc (Pk + 1) (Pk_succ - 2), i;\n  Congruence ↔ IsLesserTwinPrime ∨ k = 991 ∨ Pk_succ - Pk > 2 ∧ Wk_prod ≡ 1 [MOD Pk_succ]","subjects":["11"],"theorem":"OeisA1359.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1359»","statement":"OeisA1359.a 3 = 11","subjects":["11"],"theorem":"OeisA1359.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«239957»","statement":"OeisA239957.A 2","subjects":["11"],"theorem":"OeisA239957.a_2"},{"answerKinds":[],"category":"research open","docstring":"**Zhi-Wei Sun's Conjecture (A239957)**: Every prime $p$ has a primitive root $0 < g < p$ of the\nform $k^2 + 1$, where $k$ is an integer.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«239957»","statement":"∀ (p : ℕ), Nat.Prime p → OeisA239957.A p","subjects":["11"],"theorem":"OeisA239957.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102371»","statement":"OeisA102371.a 4 = 12","subjects":["11"],"theorem":"OeisA102371.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102371»","statement":"OeisA102371.a 2 = 2","subjects":["11"],"theorem":"OeisA102371.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102371»","statement":"OeisA102371.a 5 = 29","subjects":["11"],"theorem":"OeisA102371.a_5"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Do we have $a(n) = 2^n - 1 - \\operatorname{A105033}(n-1)$ for $n \\ge 1$?\n\nNote: The OEIS comment suggests $n-1$ which means this applies at least for $n \\ge 1$,\nas we assume $\\operatorname{A105033}(\\mathbb{N})$ is defined on $\\mathbb{N}$.\nWe include the case $n=1$ which relies on $A105033(0)$, which is 0.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a102371/blob/d80fba9/lean/OeisA102371FC.lean#L490-L497"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«102371»","statement":"True ↔ ∀ (n : ℕ), 0 < n → OeisA102371.a n = 2 ^ n - 1 - OeisA105033.a (n - 1)","subjects":["11"],"theorem":"OeisA102371.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102371»","statement":"OeisA102371.a 3 = 7","subjects":["11"],"theorem":"OeisA102371.a_3"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102371»","statement":"OeisA102371.a 1 = 1","subjects":["11"],"theorem":"OeisA102371.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«258667»","statement":"OeisA258667.a 4 = 0","subjects":["11"],"theorem":"OeisA258667.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(n) \\sim e^{-2} \\cdot n! / (n-2)$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/258667.wip.lean#L427"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«258667»","statement":"Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(OeisA258667.a n)) OeisA258667.asymptoticTerm","subjects":["11"],"theorem":"OeisA258667.a_is_equivalent_asymptoticTerm"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«258667»","statement":"OeisA258667.a 1 = 0","subjects":["11"],"theorem":"OeisA258667.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«258667»","statement":"OeisA258667.a 3 = 0","subjects":["11"],"theorem":"OeisA258667.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«258667»","statement":"OeisA258667.a 2 = 0","subjects":["11"],"theorem":"OeisA258667.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«258667»","statement":"OeisA258667.a 5 = 0","subjects":["11"],"theorem":"OeisA258667.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78729»","statement":"OeisA78729.a 2 = 1","subjects":["11"],"theorem":"OeisA78729.a_2"},{"answerKinds":[],"category":"research open","docstring":"$(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2$, which is never prime. Hence $a(4) = 0$.\nConjecture: $a(n) = 0$ if and only if $n = 4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«78729»","statement":"∀ (n : ℕ), 0 < n → (OeisA78729.a n = 0 ↔ n = 4)","subjects":["11"],"theorem":"OeisA78729.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78729»","statement":"OeisA78729.a 5 = 2","subjects":["11"],"theorem":"OeisA78729.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 6. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78729»","statement":"OeisA78729.a 6 = 2","subjects":["11"],"theorem":"OeisA78729.a_6"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78729»","statement":"OeisA78729.a 1 = 1","subjects":["11"],"theorem":"OeisA78729.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78729»","statement":"OeisA78729.a 3 = 2","subjects":["11"],"theorem":"OeisA78729.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51293»","statement":"OeisA51293.a 5 = 15","subjects":["11"],"theorem":"OeisA51293.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51293»","statement":"OeisA51293.a 2 = 2","subjects":["11"],"theorem":"OeisA51293.a_2"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(n) = 2^{n+1}/n \\cdot (1 + 1/n + 3/n^2 + 13/n^3 + 75/n^4 + 541/n^5 + o(1/n^5))$. - _Benoit Cloitre_, Oct 20 2002\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/51293.wip.lean#L503"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«51293»","statement":"Filter.Tendsto\n  (fun n =>\n    (OeisA51293.aReal n - 2 ^ (n + 1) / ↑n * (1 + 1 / ↑n + 3 / ↑n ^ 2 + 13 / ↑n ^ 3 + 75 / ↑n ^ 4 + 541 / ↑n ^ 5)) /\n      (2 ^ (n + 1) / ↑n ^ 6))\n  Filter.atTop (nhds 0)","subjects":["11"],"theorem":"OeisA51293.tendsto_aReal_asymptotic"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51293»","statement":"OeisA51293.a 3 = 5","subjects":["11"],"theorem":"OeisA51293.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51293»","statement":"OeisA51293.a 1 = 1","subjects":["11"],"theorem":"OeisA51293.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«51293»","statement":"OeisA51293.a 4 = 8","subjects":["11"],"theorem":"OeisA51293.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n)/A006880(n) \\rightarrow 1.77...$\nwhere A006880(n) is the number of primes $\\le 10^n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«115366»","statement":"∃ L, Filter.Tendsto (fun n => ↑(OeisA115366.a n) / ↑(10 ^ n).primeCounting') Filter.atTop (nhds L) ∧ 1.77 ≤ L ∧ L ≤ 1.78","subjects":["11"],"theorem":"OeisA115366.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«115366»","statement":"OeisA115366.a 2 = 50","subjects":["11"],"theorem":"OeisA115366.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«115366»","statement":"OeisA115366.a 0 = 1","subjects":["11"],"theorem":"OeisA115366.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«115366»","statement":"OeisA115366.a 1 = 9","subjects":["11"],"theorem":"OeisA115366.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«115366»","statement":"OeisA115366.a 3 = 313","subjects":["11"],"theorem":"OeisA115366.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«130911»","statement":"OeisA130911.a 3 = -1","subjects":["11"],"theorem":"OeisA130911.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«130911»","statement":"OeisA130911.a 2 = 0","subjects":["11"],"theorem":"OeisA130911.a_2"},{"answerKinds":[],"category":"research open","docstring":"Shevelev conjectures that $a(n) \\ge 0$ for $n > 3$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«130911»","statement":"∀ (n : ℕ), 3 < n → OeisA130911.a n ≥ 0","subjects":["11"],"theorem":"OeisA130911.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«130911»","statement":"OeisA130911.a 4 = 0","subjects":["11"],"theorem":"OeisA130911.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«130911»","statement":"OeisA130911.a 1 = 1","subjects":["11"],"theorem":"OeisA130911.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«130911»","statement":"OeisA130911.a 0 = 0","subjects":["11"],"theorem":"OeisA130911.a_0"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103151»","statement":"OeisA103151.a 1 = 0","subjects":["11"],"theorem":"OeisA103151.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103151»","statement":"OeisA103151.a 3 = 0","subjects":["11"],"theorem":"OeisA103151.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103151»","statement":"OeisA103151.a 2 = 0","subjects":["11"],"theorem":"OeisA103151.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103151»","statement":"OeisA103151.a 5 = 1","subjects":["11"],"theorem":"OeisA103151.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103151»","statement":"OeisA103151.a 4 = 1","subjects":["11"],"theorem":"OeisA103151.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: all items for $n \\ge 4$ are greater than or equal to $1$. This is a stronger\nconjecture than the Goldbach conjecture.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«103151»","statement":"∀ n ≥ 4, OeisA103151.a n ≥ 1","subjects":["11"],"theorem":"OeisA103151.conjecture"},{"answerKinds":[],"category":"textbook","docstring":"$a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108301»","statement":"Nat.Prime (OeisA108301.a 0) ∧\n  Nat.Prime (OeisA108301.a 1) ∧\n    Nat.Prime (OeisA108301.a 5) ∧\n      Nat.Prime (OeisA108301.a 6) ∧ Nat.Prime (OeisA108301.a 7) ∧ Nat.Prime (OeisA108301.a 11)","subjects":["11"],"theorem":"OeisA108301.primes_in_a"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108301»","statement":"OeisA108301.a 4 = 26","subjects":["11"],"theorem":"OeisA108301.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108301»","statement":"OeisA108301.a 1 = 5","subjects":["11"],"theorem":"OeisA108301.a_1"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108301»","statement":"OeisA108301.a 0 = 3","subjects":["11"],"theorem":"OeisA108301.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108301»","statement":"OeisA108301.a 3 = 14","subjects":["11"],"theorem":"OeisA108301.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108301»","statement":"OeisA108301.a 2 = 8","subjects":["11"],"theorem":"OeisA108301.a_2"},{"answerKinds":[],"category":"research open","docstring":"$a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes. Are there any more? ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108301»","statement":"True ↔ ∃ n > 11, Nat.Prime (OeisA108301.a n)","subjects":["11"],"theorem":"OeisA108301.conjecture"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102847»","statement":"OeisA102847.a 0 = 1","subjects":["11"],"theorem":"OeisA102847.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102847»","statement":"OeisA102847.a 1 = 3","subjects":["11"],"theorem":"OeisA102847.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102847»","statement":"OeisA102847.a 4 = 15131","subjects":["11"],"theorem":"OeisA102847.a_4"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Prime for $a(1) = 3$, $a(2) = 11$, $a(4) = 15131$; semiprime for $a(3) = 123 = 3 * 41$,\n$a(5) = 228947163 = 3 * 76315721$.\n$a(6)$, added by Jonathan Vos Post, has 4 prime factors.\n$a(7) = 41 * 811^2 * 106693969 * 317171188688357726699 * 8272236925540996054440172449761$.\nWhen is the next prime in the sequence?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«102847»","statement":"sorry = sInf {n | 4 < n ∧ Nat.Prime (OeisA102847.a n)}","subjects":["11"],"theorem":"OeisA102847.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102847»","statement":"OeisA102847.a 2 = 11","subjects":["11"],"theorem":"OeisA102847.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102847»","statement":"OeisA102847.a 3 = 123","subjects":["11"],"theorem":"OeisA102847.a_3"},{"answerKinds":[],"category":"research open","docstring":"**Zhi-Wei Sun's Conjecture (A287616)**: Any nonnegative integer can be written as the sum of\na triangular number $x(x+1)/2$, a generalized pentagonal number $y(3y+1)/2$, and a generalized\nheptagonal number $z(5z+1)/2$, where $x, y, z$ are nonnegative integers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«287616»","statement":"∀ (n : ℕ), OeisA287616.A n","subjects":["11"],"theorem":"OeisA287616.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«287616»","statement":"OeisA287616.A 2","subjects":["11"],"theorem":"OeisA287616.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«287616»","statement":"OeisA287616.A 3","subjects":["11"],"theorem":"OeisA287616.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«287616»","statement":"OeisA287616.A 0","subjects":["11"],"theorem":"OeisA287616.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«287616»","statement":"OeisA287616.A 1","subjects":["11"],"theorem":"OeisA287616.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«287616»","statement":"OeisA287616.A 4","subjects":["11"],"theorem":"OeisA287616.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«382590»","statement":"OeisA382590.a 2 = 3","subjects":["11"],"theorem":"OeisA382590.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«382590»","statement":"OeisA382590.a 3 = 5","subjects":["11"],"theorem":"OeisA382590.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«382590»","statement":"OeisA382590.a 1 = 2","subjects":["11"],"theorem":"OeisA382590.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«382590»","statement":"OeisA382590.a 0 = 1","subjects":["11"],"theorem":"OeisA382590.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«382590»","statement":"OeisA382590.a 4 = 8","subjects":["11"],"theorem":"OeisA382590.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: For any $k > 1$, if you take the $k$-th prime factor of each term, you get an eventually periodic sequence. - _Pontus von Brömssen_, Mar 30 2025\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/382590.wip.lean#L281"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«382590»","statement":"∀ k ≥ 2,\n  ∃ N₀,\n    ∃ p > 0,\n      ∀ n ≥ N₀, OeisA382590.kthPrimeFactor k (OeisA382590.a (n + p)) = OeisA382590.kthPrimeFactor k (OeisA382590.a n)","subjects":["11"],"theorem":"OeisA382590.kthPrimeFactor_periodic"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"Nat.nth Nat.Prime 8 = 23","subjects":["11"],"theorem":"OeisA117027.nth_prime_eight"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"OeisA117027.a 3 = -48","subjects":["11"],"theorem":"OeisA117027.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"OeisA117027.a 0 = 0","subjects":["11"],"theorem":"OeisA117027.a_0"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"Nat.nth Nat.Prime 10 = 31","subjects":["11"],"theorem":"OeisA117027.nth_prime_ten"},{"answerKinds":[],"category":"research open","docstring":"This suggests the ratio is approaching a limit close to 0.87.\n\nFormalized as: The sequence of ratios $P(N)/Neg(N)$ converges to a limit L,\nand L is in the interval (0.8, 0.9).\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«117027»","statement":"∃ L, Filter.Tendsto OeisA117027.ratioSeq Filter.atTop (nhds L) ∧ 0.8 < L ∧ L < 0.9","subjects":["11"],"theorem":"OeisA117027.conjecture"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"Nat.nth Nat.Prime 5 = 13","subjects":["11"],"theorem":"OeisA117027.nth_prime_five"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"Nat.nth Nat.Prime 9 = 29","subjects":["11"],"theorem":"OeisA117027.nth_prime_nine"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"Nat.nth Nat.Prime 11 = 37","subjects":["11"],"theorem":"OeisA117027.nth_prime_eleven"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"OeisA117027.a 2 = -12","subjects":["11"],"theorem":"OeisA117027.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"OeisA117027.a 1 = -1","subjects":["11"],"theorem":"OeisA117027.a_1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"Nat.nth Nat.Prime 6 = 17","subjects":["11"],"theorem":"OeisA117027.nth_prime_six"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117027»","statement":"Nat.nth Nat.Prime 7 = 19","subjects":["11"],"theorem":"OeisA117027.nth_prime_seven"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«323557»","statement":"OeisA323557.a 1 = 0","subjects":["11"],"theorem":"OeisA323557.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«323557»","statement":"OeisA323557.a 3 = -2","subjects":["11"],"theorem":"OeisA323557.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: Odd terms occur only at positions $n(n+1)$ for $n \\ge 0$ (verified for initial 32600 terms).\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/323557.wip.lean#L193"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«323557»","statement":"∀ (m : ℕ), Odd (OeisA323557.a m) → ∃ n, m = n * (n + 1)","subjects":["11"],"theorem":"OeisA323557.odd_a_implies_pronic"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«323557»","statement":"OeisA323557.a 0 = 1","subjects":["11"],"theorem":"OeisA323557.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«323557»","statement":"OeisA323557.a 4 = 2","subjects":["11"],"theorem":"OeisA323557.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«323557»","statement":"OeisA323557.a 2 = 3","subjects":["11"],"theorem":"OeisA323557.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«228143»","statement":"OeisA228143.a 4 = 674708032182398976","subjects":["11"],"theorem":"OeisA228143.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«228143»","statement":"OeisA228143.a 2 = 161856","subjects":["11"],"theorem":"OeisA228143.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«228143»","statement":"OeisA228143.a 3 = 39002646528","subjects":["11"],"theorem":"OeisA228143.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«228143»","statement":"OeisA228143.a 1 = 48","subjects":["11"],"theorem":"OeisA228143.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«228143»","statement":"OeisA228143.a 0 = 1","subjects":["11"],"theorem":"OeisA228143.a_0"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: if $A(x) = 1 + 48x + 161856 x^2 + \\cdots$ denotes the o.g.f. then $A(x/3)^{1/8}$ has integer coefficients. - _Peter Bala_, Apr 22 2018\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/228143.wip.lean#L698"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«228143»","statement":"∃ C, (PowerSeries.map (Int.castRingHom ℚ)) (C ^ 8) = OeisA228143.ogfAScaled","subjects":["11"],"theorem":"OeisA228143.exists_power_series_eighth_pow_eq"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117531»","statement":"OeisA117531.a 4 = 3","subjects":["11"],"theorem":"OeisA117531.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117531»","statement":"OeisA117531.a 2 = 2","subjects":["11"],"theorem":"OeisA117531.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117531»","statement":"OeisA117531.a 1 = 1","subjects":["11"],"theorem":"OeisA117531.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117531»","statement":"OeisA117531.a 3 = 3","subjects":["11"],"theorem":"OeisA117531.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«117531»","statement":"OeisA117531.a 0 = 0","subjects":["11"],"theorem":"OeisA117531.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) < n$ for $n > 13$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«117531»","statement":"∀ n > 13, OeisA117531.a n < n","subjects":["11"],"theorem":"OeisA117531.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«180017»","statement":"OeisA180017.a 2 = 1","subjects":["11"],"theorem":"OeisA180017.a_2"},{"answerKinds":[],"category":"research open","docstring":"\"This sequence is positive on average, since 1/log(3) > 1/log(4). Do all integers appear\ninfinitely often?\" - Charles R Greathouse IV, Feb 07 2013","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«180017»","statement":"∀ (z : ℤ), {n | OeisA180017.a n = z}.Infinite","subjects":["11"],"theorem":"OeisA180017.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«180017»","statement":"OeisA180017.a 1 = 0","subjects":["11"],"theorem":"OeisA180017.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«180017»","statement":"OeisA180017.a 3 = -1","subjects":["11"],"theorem":"OeisA180017.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«180017»","statement":"OeisA180017.a 0 = 0","subjects":["11"],"theorem":"OeisA180017.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«180017»","statement":"OeisA180017.a 4 = 1","subjects":["11"],"theorem":"OeisA180017.a_4"},{"answerKinds":[],"category":"research solved","docstring":"The forward direction: if $k = a(n)$ for $n \\ge 2$, then $k^4 - 1$ divides $2^k - 1$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1146»","statement":"∀ (n : ℕ), 2 ≤ n → OeisA1146.a n ^ 4 - 1 ∣ 2 ^ OeisA1146.a n - 1","subjects":["11"],"theorem":"OeisA1146.divisibility_fact"},{"answerKinds":[],"category":"research open","docstring":"I conjecture that { $a(n)$ ; $n>1$ } are the numbers such that $n^4-1$ divides $2^n-1$,\nintersection of A247219 and A247165. - M. F. Hasler, Jul 25 2015\nThis formalizes the reverse direction.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«1146»","statement":"∀ (k : ℕ), k ^ 4 - 1 ∣ 2 ^ k - 1 → k > 1 → ∃ n, 2 ≤ n ∧ k = OeisA1146.a n","subjects":["11"],"theorem":"OeisA1146.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1146»","statement":"OeisA1146.a 0 = 2","subjects":["11"],"theorem":"OeisA1146.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1146»","statement":"OeisA1146.a 3 = 256","subjects":["11"],"theorem":"OeisA1146.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1146»","statement":"OeisA1146.a 1 = 4","subjects":["11"],"theorem":"OeisA1146.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1146»","statement":"OeisA1146.a 2 = 16","subjects":["11"],"theorem":"OeisA1146.a_2"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1146»","statement":"∀ (n : ℕ), 2 ≤ n → n + 2 ≤ 2 ^ n","subjects":["11"],"theorem":"OeisA1146.n_add_two_le_two_pow"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«166944»","statement":"OeisA166944.a 1 = 2","subjects":["11"],"theorem":"OeisA166944.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«166944»","statement":"OeisA166944.a 3 = 5","subjects":["11"],"theorem":"OeisA166944.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«166944»","statement":"OeisA166944.a 4 = 6","subjects":["11"],"theorem":"OeisA166944.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«166944»","statement":"OeisA166944.a 2 = 4","subjects":["11"],"theorem":"OeisA166944.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«166944»","statement":"OeisA166944.a 5 = 9","subjects":["11"],"theorem":"OeisA166944.a_5"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: Every record of differences $a(n)-a(n-1)$ more than 5 is the greater of twin primes\n(A006512).","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«166944»","statement":"∀ (R : ℕ), 5 < R → OeisA166944.IsDifferenceRecord R → OeisA166944.IsGreaterTwinPrime R","subjects":["11"],"theorem":"OeisA166944.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87571»","statement":"OeisA87571.a 3 = 3","subjects":["11"],"theorem":"OeisA87571.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87571»","statement":"OeisA87571.a 1 = 0","subjects":["11"],"theorem":"OeisA87571.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87571»","statement":"OeisA87571.a 2 = 2","subjects":["11"],"theorem":"OeisA87571.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87571»","statement":"OeisA87571.a 4 = 43","subjects":["11"],"theorem":"OeisA87571.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: There are infinitely many composite numbers $n$ such that $a(n)$ is nonzero.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«87571»","statement":"∀ (M : ℕ), ∃ n > M, 1 < n ∧ ¬Nat.Prime n ∧ OeisA87571.a n ≠ 0","subjects":["11"],"theorem":"OeisA87571.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87571»","statement":"OeisA87571.a 0 = 0","subjects":["11"],"theorem":"OeisA87571.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.b 4 = 646","subjects":["11"],"theorem":"OeisA109074.b_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.a 0 = 1","subjects":["11"],"theorem":"OeisA109074.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.b 0 = 1","subjects":["11"],"theorem":"OeisA109074.b_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.b 3 = 26","subjects":["11"],"theorem":"OeisA109074.b_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.a 1 = 1","subjects":["11"],"theorem":"OeisA109074.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.b 1 = 1","subjects":["11"],"theorem":"OeisA109074.b_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.a 4 = 323","subjects":["11"],"theorem":"OeisA109074.a_4"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that\n$\\binom{6n-2}{2n} / \\left(2 \\binom{4n-1}{2n}\\right) = A005156(n+1)/A005156(n)$,\nwhere the OEIS comment reads A005156 as 1-based; with the 0-indexed `b` this is\n`frac (n + 1) = b (n + 1) / b n`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«109074»","statement":"∀ (n : ℕ), OeisA109074.frac (n + 1) = ↑(OeisA109074.b (n + 1)) / ↑(OeisA109074.b n)","subjects":["11"],"theorem":"OeisA109074.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.a 2 = 3","subjects":["11"],"theorem":"OeisA109074.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.a 3 = 26","subjects":["11"],"theorem":"OeisA109074.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109074»","statement":"OeisA109074.b 2 = 3","subjects":["11"],"theorem":"OeisA109074.b_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: the sequence contains 8 zeros.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«182510»","statement":"{n | OeisA182510.a n = 0}.ncard = 8","subjects":["11"],"theorem":"OeisA182510.conjecture1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182510»","statement":"OeisA182510.a 1 = 1","subjects":["11"],"theorem":"OeisA182510.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182510»","statement":"OeisA182510.a 3 = -1","subjects":["11"],"theorem":"OeisA182510.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182510»","statement":"OeisA182510.a 0 = 0","subjects":["11"],"theorem":"OeisA182510.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182510»","statement":"OeisA182510.a 4 = -8","subjects":["11"],"theorem":"OeisA182510.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182510»","statement":"OeisA182510.a 6 = 0","subjects":["11"],"theorem":"OeisA182510.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182510»","statement":"OeisA182510.a 2 = 3","subjects":["11"],"theorem":"OeisA182510.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«182510»","statement":"OeisA182510.a 5 = -2","subjects":["11"],"theorem":"OeisA182510.a_5"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: more positive terms than negative.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«182510»","statement":"∃ d_pos d_neg, {n | 0 < OeisA182510.a n}.HasDensity d_pos ∧ {n | OeisA182510.a n < 0}.HasDensity d_neg ∧ d_neg < d_pos","subjects":["11"],"theorem":"OeisA182510.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«96535»","statement":"OeisA96535.a 2 = 0","subjects":["11"],"theorem":"OeisA96535.a_2"},{"answerKinds":[],"category":"research open","docstring":"All numbers appear infinitely often, i.e., for every number $k \\ge 0$ and every frequency $f > 0$\nthere is an index $i$ such that $a(i) = k$ is the $f$-th occurrence of $k$ in the sequence.\n- _Klaus Brockhaus_, Aug 29 2006\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«96535»","statement":"∀ (k N : ℕ), ∃ i > N, OeisA96535.a i = k","subjects":["11"],"theorem":"OeisA96535.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«96535»","statement":"OeisA96535.a 1 = 1","subjects":["11"],"theorem":"OeisA96535.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«96535»","statement":"OeisA96535.a 3 = 1","subjects":["11"],"theorem":"OeisA96535.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«96535»","statement":"OeisA96535.a 0 = 1","subjects":["11"],"theorem":"OeisA96535.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«96535»","statement":"OeisA96535.a 4 = 1","subjects":["11"],"theorem":"OeisA96535.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«372761»","statement":"OeisA372761.a 5 = 7","subjects":["11"],"theorem":"OeisA372761.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«372761»","statement":"OeisA372761.a 7 = 31","subjects":["11"],"theorem":"OeisA372761.a_7"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«372761»","statement":"OeisA372761.a 3 = 11","subjects":["11"],"theorem":"OeisA372761.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: Except for 3 and 5, all odd primes appear in the sequence once. - _Thomas Scheuerle_, May 11 2024\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/372761.wip.lean#L733"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«372761»","statement":"∀ (p : ℕ), Nat.Prime p ∧ p % 2 = 1 ∧ p ≠ 3 ∧ p ≠ 5 → ∃! n, n ≥ 3 ∧ OeisA372761.a n = p","subjects":["11"],"theorem":"OeisA372761.exists_unique_a_eq_prime"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«372761»","statement":"OeisA372761.a 4 = 4","subjects":["11"],"theorem":"OeisA372761.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«372761»","statement":"OeisA372761.a 6 = 13","subjects":["11"],"theorem":"OeisA372761.a_6"},{"answerKinds":[],"category":"research open","docstring":"Every integer at least two reaches a home prime. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«37274»","statement":"∀ (n : ℕ), 2 ≤ n → OeisA37274.ReachesPrime n","subjects":["11"],"theorem":"OeisA37274.home_prime_conjecture"},{"answerKinds":[],"category":"test","docstring":"The first step in the trajectory from $25$ is $25\\mapsto55$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«37274»","statement":"OeisA37274.primeFactorSplice 25 = 55","subjects":["11"],"theorem":"OeisA37274.primeFactorSplice_25"},{"answerKinds":[],"category":"test","docstring":"The third step in the trajectory from $25$ is $511\\mapsto773$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«37274»","statement":"OeisA37274.primeFactorSplice 511 = 773","subjects":["11"],"theorem":"OeisA37274.primeFactorSplice_511"},{"answerKinds":[],"category":"test","docstring":"A prime is a fixed point of prime-factor splicing. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«37274»","statement":"∀ {p : ℕ}, Nat.Prime p → OeisA37274.primeFactorSplice p = p","subjects":["11"],"theorem":"OeisA37274.primeFactorSplice_prime"},{"answerKinds":[],"category":"test","docstring":"The second step in the trajectory from $25$ is $55\\mapsto511$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«37274»","statement":"OeisA37274.primeFactorSplice 55 = 511","subjects":["11"],"theorem":"OeisA37274.primeFactorSplice_55"},{"answerKinds":[],"category":"test","docstring":"The trajectory from $25$ reaches the prime $773$ after three steps. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«37274»","statement":"OeisA37274.ReachesPrime 25","subjects":["11"],"theorem":"OeisA37274.reachesPrime_25"},{"answerKinds":[],"category":"test","docstring":"$65$ is in the sequence A56777. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«56777»","statement":"OeisA56777.A 65","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"OeisA56777.a_65"},{"answerKinds":[],"category":"textbook","docstring":"Numbers coming from prime quadruples satisfy $n \\equiv 9 \\pmod{100}$,\nexcept the first value \"65\". ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«56777»","statement":"∀ {n : ℕ}, 65 < n → OeisA56777.ComesFromPrimeQuadruple n → n % 100 = 9","subjects":["11"],"theorem":"OeisA56777.mod_100_of_comesFromPrimeQuadruple"},{"answerKinds":[],"category":"textbook","docstring":"Numbers coming from prime quadruples satisfy $n \\equiv 65 \\pmod{72}$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«56777»","statement":"∀ {n : ℕ}, OeisA56777.ComesFromPrimeQuadruple n → n % 72 = 65","subjects":["11"],"theorem":"OeisA56777.mod_72_of_comesFromPrimeQuadruple"},{"answerKinds":[],"category":"research open","docstring":"All members of the sequence A56777 come from prime quadruples. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«56777»","statement":"∀ {n : ℕ}, OeisA56777.A n → OeisA56777.ComesFromPrimeQuadruple n","subjects":["11"],"theorem":"OeisA56777.comesFromPrimeQuadruple_of_a"},{"answerKinds":[],"category":"test","docstring":"$11009$ is in the sequence A56777. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«56777»","statement":"OeisA56777.A 11009","subjects":["11"],"theorem":"OeisA56777.a_11009"},{"answerKinds":[],"category":"test","docstring":"$209$ is in the sequence A56777. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«56777»","statement":"OeisA56777.A 209","subjects":["11"],"theorem":"OeisA56777.a_209"},{"answerKinds":[],"category":"textbook","docstring":"Numbers coming from prime quadruples are in the sequence A56777. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«56777»","statement":"∀ {n : ℕ}, OeisA56777.ComesFromPrimeQuadruple n → OeisA56777.A n","subjects":["11"],"theorem":"OeisA56777.a_of_comesFromPrimeQuadruple"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80170»","statement":"¬OeisA80170.A 5","subjects":["11"],"theorem":"OeisA80170.a_5"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: The gcd condition is equivalent to the prime power condition.\nThis has been conjectured by Ralf Stephan.\n\nBoth the natural-language proof and its Lean 4 formalization were carried out\nby the KLMM MechMath Agent Team; see the `formal_proof` attribute.\n\n*References:*\n- [Ralf Stephan, *Prove or Disprove. 100 Conjectures from the OEIS*, 2004, Conjecture 17 (arXiv:math/0409509)](https://arxiv.org/abs/math/0409509)\n- [Dakai Guo et al., *A Greatest Common Divisor Criterion of Certain Binomial Coefficients*, 2026 (arXiv:2606.22997)](https://arxiv.org/abs/2606.22997)\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/guodk/formal-conjectures/blob/0720658844d76a50d48e4baa152eef14d4462907/FormalConjectures/OEIS/80170.lean#L1823"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«80170»","statement":"∀ (k : ℕ), 2 ≤ k → (OeisA80170.A k ↔ OeisA80170.B (k + 1))","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"OeisA80170.gcdCondition_iff_primePowerCondition"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80170»","statement":"¬OeisA80170.A 4","subjects":["11"],"theorem":"OeisA80170.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80170»","statement":"¬OeisA80170.A 6","subjects":["11"],"theorem":"OeisA80170.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80170»","statement":"¬OeisA80170.A 3","subjects":["11"],"theorem":"OeisA80170.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80170»","statement":"¬OeisA80170.A 2","subjects":["11"],"theorem":"OeisA80170.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109905»","statement":"OeisA109905.a 4 = 5","subjects":["11"],"theorem":"OeisA109905.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109905»","statement":"OeisA109905.a 5 = 7","subjects":["11"],"theorem":"OeisA109905.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109905»","statement":"OeisA109905.a 2 = 2","subjects":["11"],"theorem":"OeisA109905.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109905»","statement":"OeisA109905.a 3 = 3","subjects":["11"],"theorem":"OeisA109905.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109905»","statement":"OeisA109905.a 1 = 0","subjects":["11"],"theorem":"OeisA109905.a_1"},{"answerKinds":[],"category":"research open","docstring":"$a(n) = 0$ for $n = 1$, $6$, $30$ and $54$. Are there any others?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«109905»","statement":"True ↔ {n | n > 0 ∧ OeisA109905.a n = 0} = {1, 6, 30, 54}","subjects":["11"],"theorem":"OeisA109905.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«211417»","statement":"OeisA211417.a 3 = 43880754270176401422739454033276880","subjects":["11"],"theorem":"OeisA211417.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: \"More generally, for r >= 1, we conjecture that there exists a constant D(r) such that\nD(r)*a(n)/Product_{i = 1..r, i coprime to 30} (30*n - i) is integral for all n.\"\n- _Peter Bala_, Aug 28 2025\n\nThis generalizes `thirty_mul_sub_one_dvd_a` (the $r = 1$ case where $D(1) = 1$).\n\n**Proof sketch** (kernel-checked development at the `formal_proof` permalink below, which\nproves the non-vacuous form with the explicit positive witness $D = (r!)^{r^2}$):\nLegendre-valuation analysis of the Landau step function\n$\\Delta(x) = \\lfloor 30x \\rfloor + \\lfloor x \\rfloor - \\lfloor 15x \\rfloor -\n\\lfloor 10x \\rfloor - \\lfloor 6x \\rfloor$, whose values lie in $\\{0, 1\\}$. Three\ningredients: a unit-class rigidity lemma ($\\Delta_c = 1$ for every unit class $c$ modulo\n$30$), a witness-uniqueness bound for prime powers $p^k > r$ (at most one factor of the\ndivisor product is divisible by $p^k$), and a uniform low-layer budget for $p^k \\le r$\nabsorbed by the witness constant.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/chy4pro/formal-conjectures/blob/fbc6706451b0e80787580d91ee6c252c121e0165/FormalConjectures/OEIS/211417.lean#L651"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«211417»","statement":"∀ (r : ℕ), 1 ≤ r → ∃ D, 0 < D ∧ ∀ (n : ℕ), OeisA211417.divisorProduct n r ∣ D * ↑(OeisA211417.a n)","subjects":["11"],"theorem":"OeisA211417.general_divisibility"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«211417»","statement":"OeisA211417.a 1 = 77636318760","subjects":["11"],"theorem":"OeisA211417.a_1"},{"answerKinds":[],"category":"research solved","docstring":"It appears that $a(n)/(30n - 1)$ is integral for all $n$ (checked up to $n = 1000$). - _Peter Bala_, Aug 28 2025\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/211417.wip.lean#L243"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«211417»","statement":"∀ (n : ℕ), 30 * ↑n - 1 ∣ ↑(OeisA211417.a n)","subjects":["11"],"theorem":"OeisA211417.thirty_mul_sub_one_dvd_a"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«211417»","statement":"OeisA211417.a 2 = 53837289804317953893960","subjects":["11"],"theorem":"OeisA211417.a_2"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: \"7*a(n)/(2*n + 1) ... [is an] integer for all n (checked up to n = 1000).\"\n- _Peter Bala_, Aug 28 2025\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a211417/blob/cad1fb228b5b80573cab3eeb92c8be57fd73c506/lean/OeisA211417FC.lean#L866-L868"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«211417»","statement":"∀ (n : ℕ), 2 * ↑n + 1 ∣ 7 * ↑(OeisA211417.a n)","subjects":["11"],"theorem":"OeisA211417.seven_mul_a_dvd_two_mul_add_one"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«211417»","statement":"OeisA211417.a 4 = 38113558705192522309151157825210540422513019720","subjects":["11"],"theorem":"OeisA211417.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«211417»","statement":"OeisA211417.a 0 = 1","subjects":["11"],"theorem":"OeisA211417.a_0"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: \"42*a(n)/((2*n + 1)*(3*n + 1)*(5*n + 1)) [is an] integer for all n\n(checked up to n = 1000).\" - _Peter Bala_, Aug 28 2025\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a211417/blob/cad1fb228b5b80573cab3eeb92c8be57fd73c506/lean/OeisA211417FC.lean#L1514-L1524"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«211417»","statement":"∀ (n : ℕ), (2 * ↑n + 1) * (3 * ↑n + 1) * (5 * ↑n + 1) ∣ 42 * ↑(OeisA211417.a n)","subjects":["11"],"theorem":"OeisA211417.forty_two_mul_a_dvd_product"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: \"a(n)/(5*n + 1) ... [is an] integer for all n (checked up to n = 1000).\"\n- _Peter Bala_, Aug 28 2025\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a211417/blob/cad1fb228b5b80573cab3eeb92c8be57fd73c506/lean/OeisA211417FC.lean#L1446-L1448"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«211417»","statement":"∀ (n : ℕ), 5 * ↑n + 1 ∣ ↑(OeisA211417.a n)","subjects":["11"],"theorem":"OeisA211417.a_dvd_five_mul_add_one"},{"answerKinds":[],"category":"research open","docstring":"Supercongruence: \"a(p^k) == a(p^(k-1)) ( mod p^(3*k) ) for any prime p >= 5 and any positive\ninteger k.\" - _Peter Bala_, Jan 24 2020\n\nMore generally, \"the congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) may hold for any\nprime p >= 5 and any positive integers n and k.\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«211417»","statement":"∀ (p k : ℕ), Nat.Prime p → 5 ≤ p → 0 < k → ↑p ^ (3 * k) ∣ ↑(OeisA211417.a (p ^ k)) - ↑(OeisA211417.a (p ^ (k - 1)))","subjects":["11"],"theorem":"OeisA211417.supercongruence"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: \"a(n)/(3*n + 1) ... [is an] integer for all n (checked up to n = 1000).\"\n- _Peter Bala_, Aug 28 2025\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a211417/blob/cad1fb228b5b80573cab3eeb92c8be57fd73c506/lean/OeisA211417FC.lean#L1173-L1175"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«211417»","statement":"∀ (n : ℕ), 3 * ↑n + 1 ∣ ↑(OeisA211417.a n)","subjects":["11"],"theorem":"OeisA211417.a_dvd_three_mul_add_one"},{"answerKinds":[],"category":"research open","docstring":"Stronger conjecture: Let $\\pi(n)$ be the prime counting function (A000720).\nThen $\\pi(n) \\ge a(n) \\ge \\pi(n)/5$ for $n > 1$, with the following equalities:\n$\\pi(2) = a(2)$, $\\pi(10) = a(10)$ and $a(12) = \\pi(12)/5$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«69004»","statement":"(∀ (n : ℕ), 1 < n → n.primeCounting ≥ OeisA69004.a n) ∧\n  (∀ (n : ℕ), 1 < n → 5 * OeisA69004.a n ≥ n.primeCounting) ∧\n    Nat.primeCounting 2 = OeisA69004.a 2 ∧\n      Nat.primeCounting 10 = OeisA69004.a 10 ∧ 5 * OeisA69004.a 12 = Nat.primeCounting 12","subjects":["11"],"theorem":"OeisA69004.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69004»","statement":"OeisA69004.a 5 = 2","subjects":["11"],"theorem":"OeisA69004.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69004»","statement":"OeisA69004.a 4 = 1","subjects":["11"],"theorem":"OeisA69004.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) > 0$ for all $n > 1$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«69004»","statement":"∀ (n : ℕ), 1 < n → 0 < OeisA69004.a n","subjects":["11"],"theorem":"OeisA69004.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69004»","statement":"OeisA69004.a 3 = 1","subjects":["11"],"theorem":"OeisA69004.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69004»","statement":"OeisA69004.a 1 = 0","subjects":["11"],"theorem":"OeisA69004.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69004»","statement":"OeisA69004.a 2 = 1","subjects":["11"],"theorem":"OeisA69004.a_2"},{"answerKinds":[],"category":"research open","docstring":"All the terms in this sequence have exactly two prime factors.\nThis conjecture is true for the first 133 terms.\n- [Dmitry Kamenetsky](https://oeis.org/wiki/User:Dmitry_Kamenetsky), Jan 06 2019\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38771»","statement":"∀ (n : ℕ), (OeisA38771.a n).IsSemiprime","subjects":["11"],"theorem":"OeisA38771.conjecture2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38771»","statement":"OeisA38771.a 2 = 25","subjects":["11"],"theorem":"OeisA38771.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38771»","statement":"OeisA38771.a 1 = 9","subjects":["11"],"theorem":"OeisA38771.a_1"},{"answerKinds":[],"category":"textbook","docstring":"$a(n) \\ne 0$ for all $n$ (i.e., a suitable composite $c$ always exists).\nThe following more general statement follows from Dirichlet's theorem\non primes in arithmetic progressions:\n  there doesn't exist a > 0 natural number such that p - a is prime for every prime p > a.\n\nChoose q prime such that q is coprime with a, and p > a + q prime such that q | p - a\n(such a p exists from Dirichlet's theorem). Then p - a is composite, a contradiction.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38771»","statement":"∀ (n : ℕ), OeisA38771.a n ≠ 0","subjects":["11"],"theorem":"OeisA38771.a_n_exists"},{"answerKinds":[],"category":"research open","docstring":"Conjecture:\n$\\liminf_{n \\to \\infty} \\frac{a(n)}{p_{n+1}^2} = 1 <$\n$\\limsup_{n \\to \\infty} \\frac{a(n)}{p_{n+1}^2} = 2$.\n- Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38771»","statement":"have p_next_sq := fun n => ↑(Nat.nth Nat.Prime n) ^ 2;\nhave seq := fun n => ↑(OeisA38771.a n) / p_next_sq n;\nFilter.liminf seq Filter.atTop = 1 ∧ Filter.limsup seq Filter.atTop = 2","subjects":["11"],"theorem":"OeisA38771.conjecture1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38771»","statement":"OeisA38771.a 3 = 49","subjects":["11"],"theorem":"OeisA38771.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38771»","statement":"OeisA38771.a 0 = 4","subjects":["11"],"theorem":"OeisA38771.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«81091»","statement":"OeisA81091.A 7","subjects":["11"],"theorem":"OeisA81091.a_7"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«81091»","statement":"OeisA81091.A 13","subjects":["11"],"theorem":"OeisA81091.a_13"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«81091»","statement":"OeisA81091.A 19","subjects":["11"],"theorem":"OeisA81091.a_19"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«81091»","statement":"OeisA81091.A 11","subjects":["11"],"theorem":"OeisA81091.a_11"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«81091»","statement":"OeisA81091.A 37","subjects":["11"],"theorem":"OeisA81091.a_37"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture (A81091)**: There are infinite primes of the form $2^n + 2^i + 1$,\nwith $0 < i < n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«81091»","statement":"True ↔ {p | OeisA81091.A p}.Infinite","subjects":["11"],"theorem":"OeisA81091.conjectureA81091"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113252»","statement":"OeisA113252.a 2 = 92","subjects":["11"],"theorem":"OeisA113252.a_2"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(m, 2n+1)$ is a perfect square for all $m, n$ (see A113249).\nSpecialized for $m=6$, which is A113252.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a113249-family-square-terms-lean/blob/9b999db08344184285e8050c2722c845fd5f5309/lean/OeisA113249FamilyFC.lean#L116-L124"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113252»","statement":"∀ (n : ℕ), IsSquare (OeisA113252.a (2 * n + 1))","subjects":["11"],"theorem":"OeisA113252.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113252»","statement":"OeisA113252.a 4 = -3856","subjects":["11"],"theorem":"OeisA113252.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113252»","statement":"OeisA113252.a 1 = 4","subjects":["11"],"theorem":"OeisA113252.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113252»","statement":"OeisA113252.a 0 = -1","subjects":["11"],"theorem":"OeisA113252.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113252»","statement":"OeisA113252.a 3 = 784","subjects":["11"],"theorem":"OeisA113252.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113254»","statement":"OeisA113254.a 4 = -15616","subjects":["11"],"theorem":"OeisA113254.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(8, 2n+1)$ is a perfect square for all $n$ (see A113249).\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/113254.wip.lean#L130"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113254»","statement":"∀ (n : ℕ), IsSquare (OeisA113254.a (2 * n + 1))","subjects":["11"],"theorem":"OeisA113254.a_odd_is_square"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113254»","statement":"OeisA113254.a 2 = 176","subjects":["11"],"theorem":"OeisA113254.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113254»","statement":"OeisA113254.a 1 = 4","subjects":["11"],"theorem":"OeisA113254.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113254»","statement":"OeisA113254.a 3 = 3136","subjects":["11"],"theorem":"OeisA113254.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113254»","statement":"OeisA113254.a 0 = -1","subjects":["11"],"theorem":"OeisA113254.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«145355»","statement":"OeisA145355.a 3 = 1","subjects":["11"],"theorem":"OeisA145355.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«145355»","statement":"OeisA145355.a 2 = 1","subjects":["11"],"theorem":"OeisA145355.a_2"},{"answerKinds":[],"category":"research open","docstring":"This sequence suggests that the distance between a factorial and the closest power is\ntightly bounded.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«145355»","statement":"∃ C, ∀ (n : ℕ), 2 ≤ n → OeisA145355.a n ≤ C","subjects":["11"],"theorem":"OeisA145355.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«145355»","statement":"OeisA145355.a 5 = 11","subjects":["11"],"theorem":"OeisA145355.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«145355»","statement":"OeisA145355.a 4 = 5","subjects":["11"],"theorem":"OeisA145355.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363102»","statement":"OeisA363102.a 4 = 7","subjects":["11"],"theorem":"OeisA363102.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363102»","statement":"OeisA363102.a 6 = 17","subjects":["11"],"theorem":"OeisA363102.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363102»","statement":"OeisA363102.a 7 = 47","subjects":["11"],"theorem":"OeisA363102.a_7"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: The sequence contains only 1's and primes.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/363102.wip.lean#L283"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«363102»","statement":"∀ (n : ℕ), 3 ≤ n → OeisA363102.a n = 1 ∨ Nat.Prime (OeisA363102.a n)","subjects":["11"],"theorem":"OeisA363102.a_eq_one_or_prime"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363102»","statement":"OeisA363102.a 3 = 7","subjects":["11"],"theorem":"OeisA363102.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363102»","statement":"OeisA363102.a 5 = 23","subjects":["11"],"theorem":"OeisA363102.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2454»","statement":"OeisA2454.a 3 = 2304","subjects":["11"],"theorem":"OeisA2454.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2454»","statement":"OeisA2454.a 0 = 1","subjects":["11"],"theorem":"OeisA2454.a_0"},{"answerKinds":[],"category":"research open","docstring":"Let $\\zeta$ be a primitive $(2n+1)$-th root of unity. Then the permanent of the\n$2n \\times 2n$ matrix $[m(j,k)]_{j,k=1..2n}$ is $a(n)/(2n+1) = ((2n)!!)^2/(2n+1)$,\nwhere $m(j,k)$ is $1$ or $(1+\\zeta^{j-k})/(1-\\zeta^{j-k})$ according as $j = k$ or not.\n- Zhi-Wei Sun, Dec 21 2021","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«2454»","statement":"∀ (n : ℕ),\n  have N := 2 * n;\n  have K := N + 1;\n  ∀ (ζ : ℂ),\n    IsPrimitiveRoot ζ K →\n      (Matrix.permanent fun j k =>\n          if j = k then 1\n          else\n            have pow := ↑↑j - ↑↑k;\n            (1 + ζ ^ pow) / (1 - ζ ^ pow)) =\n        ↑(OeisA2454.a n) / ↑K","subjects":["11","15"],"theorem":"OeisA2454.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2454»","statement":"OeisA2454.a 4 = 147456","subjects":["11"],"theorem":"OeisA2454.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2454»","statement":"OeisA2454.a 2 = 64","subjects":["11"],"theorem":"OeisA2454.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2454»","statement":"OeisA2454.a 1 = 4","subjects":["11"],"theorem":"OeisA2454.a_1"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"The smallest prime in this sequence is $a(2) = 5$. What is the next prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113257»","statement":"sorry = OeisA113257.a (sInf {n | 2 < n ∧ Nat.Prime (OeisA113257.a n)})","subjects":["11"],"theorem":"OeisA113257.conjecture1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113257»","statement":"OeisA113257.a 1 = 1","subjects":["11"],"theorem":"OeisA113257.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113257»","statement":"OeisA113257.a 4 = 268722","subjects":["11"],"theorem":"OeisA113257.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113257»","statement":"OeisA113257.a 5 = 4682453347","subjects":["11"],"theorem":"OeisA113257.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113257»","statement":"OeisA113257.a 2 = 5","subjects":["11"],"theorem":"OeisA113257.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113257»","statement":"OeisA113257.a 3 = 266","subjects":["11"],"theorem":"OeisA113257.a_3"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the first square value after 1?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113257»","statement":"sorry = OeisA113257.a (sInf {n | 1 < n ∧ IsSquare (OeisA113257.a n)})","subjects":["11"],"theorem":"OeisA113257.conjecture2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«52709»","statement":"OeisA52709.a 4 = 9","subjects":["5","11"],"theorem":"OeisA52709.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: for $n > 0$, $a(n)$ is also the number of sequences of length $n - 1$ covering an\ninitial interval of positive integers and avoiding three terms\n$(\\dots, x, \\dots, y, \\dots, z, \\dots)$ such that $x \\le y \\le z$.\n- Gus Wiseman, Jun 17 2021\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«52709»","statement":"∀ (n : ℕ),\n  0 < n →\n    ∀ [inst : Fintype ↑(OeisA52709.sequencesCountedByA052709 n)],\n      OeisA52709.a n = Fintype.card ↑(OeisA52709.sequencesCountedByA052709 n)","subjects":["5","11"],"theorem":"OeisA52709.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«52709»","statement":"OeisA52709.a 0 = 0","subjects":["5","11"],"theorem":"OeisA52709.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«52709»","statement":"OeisA52709.a 3 = 3","subjects":["5","11"],"theorem":"OeisA52709.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«52709»","statement":"OeisA52709.a 5 = 31","subjects":["5","11"],"theorem":"OeisA52709.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«52709»","statement":"OeisA52709.a 1 = 1","subjects":["5","11"],"theorem":"OeisA52709.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«52709»","statement":"OeisA52709.a 2 = 1","subjects":["5","11"],"theorem":"OeisA52709.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"OeisA110854.a 2 = 0","subjects":["11"],"theorem":"OeisA110854.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"OeisA110854.a 3 = 0","subjects":["11"],"theorem":"OeisA110854.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"OeisA110854.a 1 = 1","subjects":["11"],"theorem":"OeisA110854.a_1"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"OeisA110854.a 0 = 0","subjects":["11"],"theorem":"OeisA110854.a_0"},{"answerKinds":[],"category":"research open","docstring":"Do the absolute values cover A004275?\nA004275 is $1$ together with the nonnegative even numbers.\nThe conjecture asks whether every member of A004275 occurs as $|a(n)|$ for some\nterm of the sequence.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«110854»","statement":"∀ (d : ℕ), d = 1 ∨ Even d → ∃ n > 0, d = (OeisA110854.a n).natAbs","subjects":["11"],"theorem":"OeisA110854.conjecture"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"Nat.nth Nat.Prime 8 = 23","subjects":["11"],"theorem":"OeisA110854.nth_prime_eight"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"Nat.nth Nat.Prime 6 = 17","subjects":["11"],"theorem":"OeisA110854.nth_prime_six"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"OeisA110854.a 4 = 4","subjects":["11"],"theorem":"OeisA110854.a_4"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"Nat.nth Nat.Prime 5 = 13","subjects":["11"],"theorem":"OeisA110854.nth_prime_five"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"Nat.nth Nat.Prime 9 = 29","subjects":["11"],"theorem":"OeisA110854.nth_prime_nine"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110854»","statement":"Nat.nth Nat.Prime 7 = 19","subjects":["11"],"theorem":"OeisA110854.nth_prime_seven"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3161»","statement":"OeisA3161.a 4 = 36","subjects":["11"],"theorem":"OeisA3161.a_4"},{"answerKinds":[],"category":"research open","docstring":"Let $b(n) = a(2n-1)$. Then the supercongruence $b(n p^k) \\equiv b(n p^{k-1}) \\pmod{p^{3k}}$\nholds for positive integers $n$ and $k$ and all primes $p \\ge 5$.\n- Zhi-Wei Sun, Nov 16 2019","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«3161»","statement":"∀ (n k p : ℕ),\n  0 < n →\n    0 < k → Nat.Prime p → 5 ≤ p → ↑(OeisA3161.b (n * p ^ k)) ≡ ↑(OeisA3161.b (n * p ^ (k - 1))) [ZMOD ↑p ^ (3 * k)]","subjects":["11"],"theorem":"OeisA3161.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3161»","statement":"OeisA3161.a 0 = 1","subjects":["11"],"theorem":"OeisA3161.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3161»","statement":"OeisA3161.a 3 = 9","subjects":["11"],"theorem":"OeisA3161.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3161»","statement":"OeisA3161.a 1 = 1","subjects":["11"],"theorem":"OeisA3161.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3161»","statement":"OeisA3161.a 2 = 2","subjects":["11"],"theorem":"OeisA3161.a_2"},{"answerKinds":[],"category":"research open","docstring":"\"$a(31) = a(177147) = 311$. Is there any solution to $a(n) = n$?\n- _Franklin T. Adams-Watters_, Dec 18 2006\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«67599»","statement":"True ↔ ∃ n, 2 ≤ n ∧ OeisA67599.a n = n","subjects":["11"],"theorem":"OeisA67599.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67599»","statement":"OeisA67599.a 5 = 51","subjects":["11"],"theorem":"OeisA67599.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67599»","statement":"OeisA67599.a 2 = 21","subjects":["11"],"theorem":"OeisA67599.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67599»","statement":"OeisA67599.a 4 = 22","subjects":["11"],"theorem":"OeisA67599.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67599»","statement":"OeisA67599.a 6 = 2131","subjects":["11"],"theorem":"OeisA67599.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67599»","statement":"OeisA67599.a 3 = 31","subjects":["11"],"theorem":"OeisA67599.a_3"},{"answerKinds":[],"category":"research open","docstring":"Simon Colton conjectures that the number of refactorable numbers less than $x$ is at least\n$\\frac{x}{2\\log x}$. This is an asymptotic claim, so we state it for sufficiently large $x$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«111291»","statement":"∀ᶠ (x : ℝ) in Filter.atTop, ↑(OeisA111291.countRefactorable x) ≥ x / (2 * Real.log x)","subjects":["11"],"theorem":"OeisA111291.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«111291»","statement":"OeisA111291.a 2 = 16","subjects":["11"],"theorem":"OeisA111291.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«111291»","statement":"OeisA111291.a 0 = 1","subjects":["11"],"theorem":"OeisA111291.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«111291»","statement":"OeisA111291.a 1 = 4","subjects":["11"],"theorem":"OeisA111291.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«111291»","statement":"OeisA111291.a 3 = 92","subjects":["11"],"theorem":"OeisA111291.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«175386»","statement":"OeisA175386.a 3 = 6","subjects":["11"],"theorem":"OeisA175386.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«175386»","statement":"OeisA175386.a 2 = 2","subjects":["11"],"theorem":"OeisA175386.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«175386»","statement":"OeisA175386.a 5 = 5","subjects":["11"],"theorem":"OeisA175386.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«175386»","statement":"OeisA175386.a 4 = 4","subjects":["11"],"theorem":"OeisA175386.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«175386»","statement":"OeisA175386.a 1 = 1","subjects":["11"],"theorem":"OeisA175386.a_1"},{"answerKinds":[],"category":"research solved","docstring":"We conjecture that $\\sum_{i=1}^{n} \\frac{1}{i} \\binom{2n-i-1}{i-1}$ is not an integer for $n > 1$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/175386.wip.lean#L304"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«175386»","statement":"∀ (n : ℕ), 1 < n → OeisA175386.a n ≠ 1","subjects":["11"],"theorem":"OeisA175386.a_ne_one"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80326»","statement":"OeisA80326.a 4 = 6","subjects":["11"],"theorem":"OeisA80326.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) = \\text{primorial}(n)$ for infinitely many $n$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«80326»","statement":"{n | OeisA80326.a n = primorial n}.Infinite","subjects":["11"],"theorem":"OeisA80326.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80326»","statement":"OeisA80326.a 3 = 6","subjects":["11"],"theorem":"OeisA80326.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80326»","statement":"OeisA80326.a 5 = 30","subjects":["11"],"theorem":"OeisA80326.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80326»","statement":"OeisA80326.a 1 = 1","subjects":["11"],"theorem":"OeisA80326.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«80326»","statement":"OeisA80326.a 2 = 2","subjects":["11"],"theorem":"OeisA80326.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«256012»","statement":"OeisA256012.a 1 = 0","subjects":["11"],"theorem":"OeisA256012.a_1"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(n) > 0$ for $n > 23$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/256012.wip.lean#L91"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«256012»","statement":"∀ n > 23, OeisA256012.a n > 0","subjects":["11"],"theorem":"OeisA256012.a_pos_of_gt"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«256012»","statement":"OeisA256012.a 0 = 1","subjects":["11"],"theorem":"OeisA256012.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«256012»","statement":"OeisA256012.a 3 = 0","subjects":["11"],"theorem":"OeisA256012.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«256012»","statement":"OeisA256012.a 2 = 0","subjects":["11"],"theorem":"OeisA256012.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«256012»","statement":"OeisA256012.a 4 = 1","subjects":["11"],"theorem":"OeisA256012.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100800»","statement":"OeisA100800.a 2 = 4","subjects":["11"],"theorem":"OeisA100800.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100800»","statement":"OeisA100800.a 5 = 10","subjects":["11"],"theorem":"OeisA100800.a_5"},{"answerKinds":[],"category":"research open","docstring":"A100800 Conjecture: No term is zero. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«100800»","statement":"∀ (n : ℕ), n ≠ 0 → OeisA100800.a n ≠ 0","subjects":["11"],"theorem":"OeisA100800.conjecture"},{"answerKinds":[],"category":"API","docstring":"If `n` already divides `f n`, the search stops immediately and `a n = f n`.  ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100800»","statement":"∀ {n : ℕ}, n ∣ OeisA100800.f n → OeisA100800.a n = OeisA100800.f n","subjects":["11"],"theorem":"OeisA100800.a_of_dvd"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100800»","statement":"OeisA100800.a 1 = 2","subjects":["11"],"theorem":"OeisA100800.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100800»","statement":"OeisA100800.a 3 = 6","subjects":["11"],"theorem":"OeisA100800.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100800»","statement":"OeisA100800.a 4 = 8","subjects":["11"],"theorem":"OeisA100800.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109845»","statement":"OeisA109845.a 4 = 929","subjects":["11"],"theorem":"OeisA109845.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109845»","statement":"OeisA109845.a 1 = 3","subjects":["11"],"theorem":"OeisA109845.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109845»","statement":"OeisA109845.a 0 = 2","subjects":["11"],"theorem":"OeisA109845.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109845»","statement":"OeisA109845.a 3 = 31","subjects":["11"],"theorem":"OeisA109845.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109845»","statement":"OeisA109845.a 2 = 5","subjects":["11"],"theorem":"OeisA109845.a_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: There are infinitely many primes in this sequence.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«109845»","statement":"{n | Nat.Prime (OeisA109845.a n)}.Infinite","subjects":["11"],"theorem":"OeisA109845.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93456»","statement":"OeisA93456.a 3 = 24","subjects":["11"],"theorem":"OeisA93456.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93456»","statement":"OeisA93456.a 0 = 1","subjects":["11"],"theorem":"OeisA93456.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: There are finitely many numbers such that $a(n)$ is not $\\equiv 0 \\pmod{a(n-1)}$.\n(Also mentioned in A093455.)","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«93456»","statement":"{n | 1 < n ∧ ¬OeisA93456.a (n - 1) ∣ OeisA93456.a n}.Finite","subjects":["11"],"theorem":"OeisA93456.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93456»","statement":"OeisA93456.a 4 = 720","subjects":["11"],"theorem":"OeisA93456.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93456»","statement":"OeisA93456.a 2 = 1","subjects":["11"],"theorem":"OeisA93456.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93456»","statement":"OeisA93456.a 1 = 1","subjects":["11"],"theorem":"OeisA93456.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«181546»","statement":"OeisA181546.a 2 = 2","subjects":["11"],"theorem":"OeisA181546.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«181546»","statement":"OeisA181546.a 3 = 17","subjects":["11"],"theorem":"OeisA181546.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«181546»","statement":"OeisA181546.a 1 = 1","subjects":["11"],"theorem":"OeisA181546.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«181546»","statement":"OeisA181546.a 0 = 1","subjects":["11"],"theorem":"OeisA181546.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: Given $F(n,L) = \\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\binom{n-k}{k}^L$, then\n$\\lim_{n\\to\\infty} F(n+1,L)/F(n,L) = (\\mathrm{Fibonacci}(L)\\sqrt{5} + \\mathrm{Lucas}(L))/2$ for\n$L \\ge 0$ where $\\mathrm{Fibonacci}(n) = \\mathrm{A000045}(n)$ and\n$\\mathrm{Lucas}(n) = \\mathrm{A000032}(n)$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«181546»","statement":"∀ (L : ℕ),\n  Filter.Tendsto (fun n => ↑(OeisA181546.F (n + 1) L) / ↑(OeisA181546.F n L)) Filter.atTop\n    (nhds (OeisA181546.limitValue L))","subjects":["11"],"theorem":"OeisA181546.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«181546»","statement":"OeisA181546.a 4 = 83","subjects":["11"],"theorem":"OeisA181546.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185895»","statement":"OeisA185895.a 0 = 1","subjects":["11"],"theorem":"OeisA185895.a_0"},{"answerKinds":[],"category":"research open","docstring":"The coefficients $c(n)$ of $A(x)^2 = (\\sum_{n \\ge 0} a(n) x^n)^2$ differ in sign from $c(n-1)$\nif and only if $n$ is a triangular number.\n- _Peter Bala_, Mar 17 2022\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«185895»","statement":"∀ (n : ℕ), 0 < n → (OeisA185895.c n * OeisA185895.c (n - 1) < 0 ↔ OeisA185895.IsTriangular n)","subjects":["11"],"theorem":"OeisA185895.conjecture2"},{"answerKinds":[],"category":"research open","docstring":"$a(n)$ differs in sign from $a(n-1)$ if and only if $n$ is a triangular number\n(checked up to $n = 1225 = (50 \\cdot 51)/2$).\n- _Peter Bala_, Mar 17 2022\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«185895»","statement":"∀ (n : ℕ), 0 < n → (OeisA185895.a n * OeisA185895.a (n - 1) < 0 ↔ OeisA185895.IsTriangular n)","subjects":["11"],"theorem":"OeisA185895.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185895»","statement":"OeisA185895.a 1 = -1","subjects":["11"],"theorem":"OeisA185895.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185895»","statement":"OeisA185895.a 4 = 3","subjects":["11"],"theorem":"OeisA185895.a_4"},{"answerKinds":[],"category":"research open","docstring":"The Gauss congruences $a(n \\cdot p^k) \\equiv a(n \\cdot p^{k-1}) \\pmod{p^k}$ hold\nfor all primes $p$ and positive integers $n$ and $k$.\n- _Peter Bala_, Mar 17 2022\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«185895»","statement":"∀ (p : ℕ),\n  Nat.Prime p → ∀ (n k : ℕ), 0 < n → 0 < k → OeisA185895.a (n * p ^ k) ≡ OeisA185895.a (n * p ^ (k - 1)) [ZMOD ↑p ^ k]","subjects":["11"],"theorem":"OeisA185895.conjecture3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185895»","statement":"OeisA185895.a 2 = -1","subjects":["11"],"theorem":"OeisA185895.a_2"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185895»","statement":"∀ (a b : ℚ) (n m : ℕ),\n  Polynomial.C a * Polynomial.X ^ n * (Polynomial.C b * Polynomial.X ^ m) =\n    Polynomial.C (a * b) * Polynomial.X ^ (n + m)","subjects":["11"],"theorem":"OeisA185895.C_mul_X_pow_mul"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«185895»","statement":"OeisA185895.a 3 = 2","subjects":["11"],"theorem":"OeisA185895.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture (for $n > 2$): if $n \\mid a(n-1) + 2^{n-2}$, then $n$ is a prime with primitive root 2 (A001122). - _Amiram Eldar_ and _Thomas Ordowski_, Jan 19 2020\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/91669.wip.lean#L250"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«91669»","statement":"∀ n > 2, n ∣ OeisA91669.a (n - 1) + 2 ^ (n - 2) → Nat.Prime n ∧ IsPrimitiveRoot 2 n.totient","subjects":["11"],"theorem":"OeisA91669.prime_and_primitive_root_of_dvd"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91669»","statement":"OeisA91669.a 3 = 2","subjects":["11"],"theorem":"OeisA91669.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91669»","statement":"OeisA91669.a 1 = 1","subjects":["11"],"theorem":"OeisA91669.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91669»","statement":"OeisA91669.a 5 = 42","subjects":["11"],"theorem":"OeisA91669.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91669»","statement":"OeisA91669.a 2 = 1","subjects":["11"],"theorem":"OeisA91669.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91669»","statement":"OeisA91669.a 4 = 7","subjects":["11"],"theorem":"OeisA91669.a_4"},{"answerKinds":[],"category":"research open","docstring":"(25,27) is the smallest pair of prime powers (q,q+2) such that both q and q+2 are not primes,\nconjecture: there are more (but not < 10^6).\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113609»","statement":"True ↔\n  ∃ q ≥ 1000000,\n    OeisA113609.IsOeisPrimePower q ∧ OeisA113609.IsOeisPrimePower (q + 2) ∧ ¬Nat.Prime q ∧ ¬Nat.Prime (q + 2)","subjects":["11"],"theorem":"OeisA113609.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113609»","statement":"OeisA113609.a 5 = 4","subjects":["11"],"theorem":"OeisA113609.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113609»","statement":"OeisA113609.a 2 = 2","subjects":["11"],"theorem":"OeisA113609.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113609»","statement":"OeisA113609.a 4 = 3","subjects":["11"],"theorem":"OeisA113609.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113609»","statement":"OeisA113609.a 1 = 1","subjects":["11"],"theorem":"OeisA113609.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113609»","statement":"OeisA113609.a 3 = 3","subjects":["11"],"theorem":"OeisA113609.a_3"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that $a(n)>0$ for all $n>122$.\nProving this would also prove Legendre's conjecture that there is a prime\nbetween $n^2$ and $(n+1)^2$. - _T. D. Noe_, Feb 28 2007","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«91591»","statement":"∀ n > 122, OeisA91591.a n > 0","subjects":["11"],"theorem":"OeisA91591.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91591»","statement":"OeisA91591.a 2 = 1","subjects":["11"],"theorem":"OeisA91591.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91591»","statement":"OeisA91591.a 3 = 1","subjects":["11"],"theorem":"OeisA91591.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91591»","statement":"OeisA91591.a 0 = 0","subjects":["11"],"theorem":"OeisA91591.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91591»","statement":"OeisA91591.a 1 = 0","subjects":["11"],"theorem":"OeisA91591.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«91591»","statement":"OeisA91591.a 4 = 1","subjects":["11"],"theorem":"OeisA91591.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«83753»","statement":"OeisA83753.a 1 = 1","subjects":["11"],"theorem":"OeisA83753.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«83753»","statement":"OeisA83753.a 3 = 4","subjects":["11"],"theorem":"OeisA83753.a_3"},{"answerKinds":[],"category":"research open","docstring":"There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural\nnumber.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«83753»","statement":"∀ (m k : ℕ), 1 < m → OeisA83753.IsDecimalPalindrome m → 5 ≤ k → ¬∃ x, m = x ^ k","subjects":["11"],"theorem":"OeisA83753.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«83753»","statement":"OeisA83753.a 2 = 2","subjects":["11"],"theorem":"OeisA83753.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«83753»","statement":"OeisA83753.a 4 = 6","subjects":["11"],"theorem":"OeisA83753.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114216»","statement":"OeisA114216.a 4 = 5","subjects":["11"],"theorem":"OeisA114216.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114216»","statement":"OeisA114216.a 1 = 1","subjects":["11"],"theorem":"OeisA114216.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114216»","statement":"OeisA114216.a 0 = 0","subjects":["11"],"theorem":"OeisA114216.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114216»","statement":"OeisA114216.a 3 = 3","subjects":["11"],"theorem":"OeisA114216.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114216»","statement":"OeisA114216.a 2 = 1","subjects":["11"],"theorem":"OeisA114216.a_2"},{"answerKinds":[],"category":"research open","docstring":"Is $a(33900)$ the last term equal to $1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«114216»","statement":"True ↔ ∀ n > 33900, OeisA114216.a n ≠ 1","subjects":["11"],"theorem":"OeisA114216.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«114216»","statement":"OeisA114216.a 5 = 1","subjects":["11"],"theorem":"OeisA114216.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93818»","statement":"OeisA93818.a 4 = 1","subjects":["11"],"theorem":"OeisA93818.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93818»","statement":"OeisA93818.a 2 = 1","subjects":["11"],"theorem":"OeisA93818.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93818»","statement":"OeisA93818.a 3 = 1","subjects":["11"],"theorem":"OeisA93818.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93818»","statement":"OeisA93818.a 1 = 1","subjects":["11"],"theorem":"OeisA93818.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«93818»","statement":"OeisA93818.a 0 = 1","subjects":["11"],"theorem":"OeisA93818.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: Every odd prime occurs as a term in the sequence.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«93818»","statement":"∀ (p : ℕ), Nat.Prime p → p ≠ 2 → ∃ n > 0, OeisA93818.a n = p","subjects":["11"],"theorem":"OeisA93818.conjecture"},{"answerKinds":[],"category":"research solved","docstring":"**Conjecture from Creighton Dement (A100434)**:\nLet the auxiliary sequences c, d, e, f, g, b be defined as specified.\nThen for all $n \\ge 0$, $g(n) + a(n) = b(n)$.\n\n**Proof outline**:\nstrong two-step induction on the paired recurrences. The even/odd cases are coupled\nthrough closed identities between consecutive terms of the relevant pair of sequences\n(here $g, a$), and the induction step is discharged by `rfl`-level unfolding of the definitions\nplus linear arithmetic. (Numerically verified for $n < 600$ before formalization.)\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/chy4pro/formal-conjectures/blob/32f88077a444b83741f1db6734390eebd3678ecf/FormalConjectures/OEIS/100434.lean#L335"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«100434»","statement":"∀ (n : ℕ), OeisA100434.g n + OeisA100434.a n = OeisA100434.b n","subjects":["11"],"theorem":"OeisA100434.conjecture3"},{"answerKinds":[],"category":"textbook","docstring":"For all $n \\ge 0$, we have $a(2n) = - c(2n+1)$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100434»","statement":"∀ (n : ℕ), OeisA100434.a (2 * n) = -OeisA100434.c (2 * n + 1)","subjects":["11"],"theorem":"OeisA100434.a_even"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100434»","statement":"OeisA100434.a 3 = -24","subjects":["11"],"theorem":"OeisA100434.a_3"},{"answerKinds":[],"category":"research solved","docstring":"**Conjecture from Creighton Dement (A100434)**:\nLet the auxiliary sequences c, d, e, f, g, b be defined as specified.\nThen for all $n \\ge 0$, $c(n) + d(n) = b(n)$.\n\n**Proof outline**:\nstrong two-step induction on the paired recurrences. The even/odd cases are coupled\nthrough closed identities between consecutive terms of the relevant pair of sequences\n(here $c, d$), and the induction step is discharged by `rfl`-level unfolding of the definitions\nplus linear arithmetic. (Numerically verified for $n < 600$ before formalization.)\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/chy4pro/formal-conjectures/blob/32f88077a444b83741f1db6734390eebd3678ecf/FormalConjectures/OEIS/100434.lean#L287"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«100434»","statement":"∀ (n : ℕ), OeisA100434.c n + OeisA100434.d n = OeisA100434.b n","subjects":["11"],"theorem":"OeisA100434.conjecture1"},{"answerKinds":[],"category":"research solved","docstring":"**Conjecture from Creighton Dement (A100434)**:\nLet the auxiliary sequences c, d, e, f, g, b be defined as specified.\nThen for all $n \\ge 0$, $e(n) + f(n) = b(n)$.\n\n**Proof outline**:\nstrong two-step induction on the paired recurrences. The even/odd cases are coupled\nthrough closed identities between consecutive terms of the relevant pair of sequences\n(here $e, f$), and the induction step is discharged by `rfl`-level unfolding of the definitions\nplus linear arithmetic. (Numerically verified for $n < 600$ before formalization.)\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/chy4pro/formal-conjectures/blob/32f88077a444b83741f1db6734390eebd3678ecf/FormalConjectures/OEIS/100434.lean#L306"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«100434»","statement":"∀ (n : ℕ), OeisA100434.e n + OeisA100434.f n = OeisA100434.b n","subjects":["11"],"theorem":"OeisA100434.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100434»","statement":"OeisA100434.a 0 = 3","subjects":["11"],"theorem":"OeisA100434.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100434»","statement":"OeisA100434.a 4 = 99","subjects":["11"],"theorem":"OeisA100434.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100434»","statement":"OeisA100434.a 2 = -17","subjects":["11"],"theorem":"OeisA100434.a_2"},{"answerKinds":[],"category":"textbook","docstring":"For all $n \\ge 0$, we have $a(2n+1) = d(2n+1)$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100434»","statement":"∀ (n : ℕ), OeisA100434.a (2 * n + 1) = OeisA100434.d (2 * n + 1)","subjects":["11"],"theorem":"OeisA100434.a_odd"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100434»","statement":"OeisA100434.a 1 = 4","subjects":["11"],"theorem":"OeisA100434.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157225»","statement":"OeisA157225.a 2 = 0","subjects":["11"],"theorem":"OeisA157225.a_2"},{"answerKinds":[],"category":"research solved","docstring":"On Feb. 24, 2009, Zhi-Wei Sun conjectured that $a(n) = 0$ if and only if $n < 11$ or\n$n \\in \\{13, 16, 992\\}$; in other words, except for $25, 31, 1983$, any odd integer greater\nthan $20$ can be written as the sum of a prime congruent to $5 \\bmod 6$, a positive power of $2$\nand seven times a positive power of $2$.\n\nAnswer: false, for n = 716993899 we have a(n) = 0.\nSee T. Adamczewski, OEIS Open: How many conjectures can language models turn into theorems?,\n[arxiv/2608.11941](https://arxiv.org/pdf/2608.11941).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/epoch-research/LeanOpenProblems-results/blob/main/runs/oeis-full-50usd-ant-j0j0g4uzligm1k41/oeis_157225_conjecture_0/Submission/Spec.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«157225»","statement":"∀ (n : ℕ), 0 < n → (OeisA157225.a n = 0 ↔ n < 11 ∨ n = 13 ∨ n = 16 ∨ n = 992)","subjects":["11"],"theorem":"OeisA157225.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 11. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157225»","statement":"OeisA157225.a 11 = 1","subjects":["11"],"theorem":"OeisA157225.a_11"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157225»","statement":"OeisA157225.a 1 = 0","subjects":["11"],"theorem":"OeisA157225.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 12. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157225»","statement":"OeisA157225.a 12 = 1","subjects":["11"],"theorem":"OeisA157225.a_12"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 14. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157225»","statement":"OeisA157225.a 14 = 2","subjects":["11"],"theorem":"OeisA157225.a_14"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 13. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«157225»","statement":"OeisA157225.a 13 = 0","subjects":["11"],"theorem":"OeisA157225.a_13"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2326»","statement":"OeisA2326.a 3 = 3","subjects":["11"],"theorem":"OeisA2326.a_3"},{"answerKinds":[],"category":"research open","docstring":"If $p$ is an odd prime then $a((p^3-1)/2) = p \\cdot a((p^2-1)/2)$.\nBecause otherwise $a((p^3-1)/2) < p \\cdot a((p^2-1)/2)$ iff $a((p^3-1)/2) = a((p-1)/2)$\nfor a prime $p$. Equivalently $p^3$ divides $2^{p-1}-1$, but no such prime $p$ is known.\n- Thomas Ordowski, Feb 10 2014\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«2326»","statement":"∀ (p : ℕ), Nat.Prime p → p ≠ 2 → OeisA2326.a ((p ^ 3 - 1) / 2) = p * OeisA2326.a ((p ^ 2 - 1) / 2)","subjects":["11"],"theorem":"OeisA2326.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"A generalization of the previous conjecture: For each $k \\ge 2$, if $p$ is an odd prime\nthen $a((p^{k+1}-1)/2) = p \\cdot a((p^k-1)/2)$.\nComputer testing of this generalized conjecture shows that there is no counterexample for $k$\nand $p$ both up to 1000.\n- [Ahmad J. Masad](https://oeis.org/wiki/User:Ahmad_J._Masad), Oct 17 2020\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«2326»","statement":"∀ (k : ℕ),\n  2 ≤ k → ∀ (p : ℕ), Nat.Prime p → p ≠ 2 → OeisA2326.a ((p ^ (k + 1) - 1) / 2) = p * OeisA2326.a ((p ^ k - 1) / 2)","subjects":["11"],"theorem":"OeisA2326.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2326»","statement":"OeisA2326.a 0 = 1","subjects":["11"],"theorem":"OeisA2326.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2326»","statement":"OeisA2326.a 4 = 6","subjects":["11"],"theorem":"OeisA2326.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2326»","statement":"OeisA2326.a 2 = 4","subjects":["11"],"theorem":"OeisA2326.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2326»","statement":"OeisA2326.a 1 = 2","subjects":["11"],"theorem":"OeisA2326.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105565»","statement":"OeisA105565.a 2 = 1","subjects":["11"],"theorem":"OeisA105565.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105565»","statement":"OeisA105565.a 5 = 1","subjects":["11"],"theorem":"OeisA105565.a_5"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $\\beta-2 < S(n)-\\alpha n < \\beta-1$.\nThe constants $\\alpha$ and $\\beta$ are as defined in the formula section.\n\nSolved by OpenAI Codex, prompted by Adam Haig. A complete Lean 4 proof is\nlinked by the `formal_proof` attribute below.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/HaigAd/formal-conjectures/blob/327b99914ce787ab41c67ba645626982e15b0124/FormalConjectures/OEIS/105565.lean#L595"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«105565»","statement":"∀ (n : ℕ),\n  1 ≤ n →\n    OeisA105565.betaConst - 2 < OeisA105565.s n - OeisA105565.alphaConst * ↑n ∧\n      OeisA105565.s n - OeisA105565.alphaConst * ↑n < OeisA105565.betaConst - 1","subjects":["11"],"theorem":"OeisA105565.conjecture"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105565»","statement":"OeisA105565.a 1 = 0","subjects":["11"],"theorem":"OeisA105565.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105565»","statement":"OeisA105565.a 3 = 1","subjects":["11"],"theorem":"OeisA105565.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105565»","statement":"OeisA105565.a 4 = 0","subjects":["11"],"theorem":"OeisA105565.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«40»","statement":"OeisA40.a 0 = 0","subjects":["11"],"theorem":"OeisA40.a_0"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture from Thomas Ordowski (2023)**:\n$\\log \\log a(n+1) - \\log \\log a(n) < 1/n$ for $n > 0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«40»","statement":"∀ (n : ℕ), 0 < n → Real.log (Real.log ↑(OeisA40.a (n + 1))) - Real.log (Real.log ↑(OeisA40.a n)) < 1 / ↑n","subjects":["11"],"theorem":"OeisA40.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«40»","statement":"OeisA40.a 4 = 7","subjects":["11"],"theorem":"OeisA40.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«40»","statement":"OeisA40.a 2 = 3","subjects":["11"],"theorem":"OeisA40.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«40»","statement":"OeisA40.a 1 = 2","subjects":["11"],"theorem":"OeisA40.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«40»","statement":"OeisA40.a 3 = 5","subjects":["11"],"theorem":"OeisA40.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«77408»","statement":"OeisA77408.a 5 = 3856","subjects":["11"],"theorem":"OeisA77408.a_5"},{"answerKinds":[],"category":"research open","docstring":"$103$ is conjectured to be the smallest number such that the Reverse and Add! algorithm in base $3$\ndoes not lead to a palindrome. Its trajectory is conjectured to never reach a palindrome.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«77408»","statement":"∀ (n : ℕ), ¬OeisA77408.IsBasePalindrome 3 (OeisA77408.a n)","subjects":["11"],"theorem":"OeisA77408.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«77408»","statement":"OeisA77408.a 2 = 436","subjects":["11"],"theorem":"OeisA77408.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«77408»","statement":"OeisA77408.a 3 = 776","subjects":["11"],"theorem":"OeisA77408.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«77408»","statement":"OeisA77408.a 0 = 103","subjects":["11"],"theorem":"OeisA77408.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«77408»","statement":"OeisA77408.a 1 = 230","subjects":["11"],"theorem":"OeisA77408.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«77408»","statement":"OeisA77408.a 4 = 2424","subjects":["11"],"theorem":"OeisA77408.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«104320»","statement":"OeisA104320.a 3 = 0","subjects":["11"],"theorem":"OeisA104320.a_3"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«104320»","statement":"OeisA104320.a 0 = 0","subjects":["11"],"theorem":"OeisA104320.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture from N. J. A. Sloane: $a(n) > 0$ for $n > 15$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«104320»","statement":"∀ (n : ℕ), 15 < n → OeisA104320.a n > 0","subjects":["11"],"theorem":"OeisA104320.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«104320»","statement":"OeisA104320.a 4 = 0","subjects":["11"],"theorem":"OeisA104320.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«104320»","statement":"OeisA104320.a 2 = 0","subjects":["11"],"theorem":"OeisA104320.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«104320»","statement":"OeisA104320.a 1 = 0","subjects":["11"],"theorem":"OeisA104320.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113255»","statement":"OeisA113255.a 3 = 5329","subjects":["11"],"theorem":"OeisA113255.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113255»","statement":"OeisA113255.a 0 = -1","subjects":["11"],"theorem":"OeisA113255.a_0"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(m, 2n+1)$ is a perfect square for all $m, n$ (see A113249).\nSpecialized for $m=9$, which is A113255.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a113249-family-square-terms-lean/blob/9b999db08344184285e8050c2722c845fd5f5309/lean/OeisA113249FamilyFC.lean#L126-L134"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113255»","statement":"∀ (n : ℕ), IsSquare (OeisA113255.a (2 * n + 1))","subjects":["11"],"theorem":"OeisA113255.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113255»","statement":"OeisA113255.a 4 = -26581","subjects":["11"],"theorem":"OeisA113255.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113255»","statement":"OeisA113255.a 2 = 227","subjects":["11"],"theorem":"OeisA113255.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113255»","statement":"OeisA113255.a 1 = 4","subjects":["11"],"theorem":"OeisA113255.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103311»","statement":"OeisA103311.a 0 = 0","subjects":["11"],"theorem":"OeisA103311.a_0"},{"answerKinds":[],"category":"research solved","docstring":"Fibonacci transform satisfying $|a(n)| = F(n+1)$.\n\nConjecture: all elements in absolute value are Fibonacci numbers.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/103311.wip.lean#L218"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«103311»","statement":"∀ (n : ℕ), ∃ m, (OeisA103311.a n).natAbs = Nat.fib m","subjects":["11"],"theorem":"OeisA103311.a_abs_eq_fib"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103311»","statement":"OeisA103311.a 1 = 1","subjects":["11"],"theorem":"OeisA103311.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103311»","statement":"OeisA103311.a 4 = -2","subjects":["11"],"theorem":"OeisA103311.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103311»","statement":"OeisA103311.a 2 = 1","subjects":["11"],"theorem":"OeisA103311.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103311»","statement":"OeisA103311.a 3 = 0","subjects":["11"],"theorem":"OeisA103311.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«70518»","statement":"OeisA70518.a 3 = 13","subjects":["11"],"theorem":"OeisA70518.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«70518»","statement":"OeisA70518.a 2 = 3","subjects":["11"],"theorem":"OeisA70518.a_2"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"$a(28341)$ is divisible by $283411^2$. What is the next $n$ such that $a(n)$ is not squarefree?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«70518»","statement":"sorry =\n  if h : ∃ n, 28341 < n ∧ ¬Squarefree (OeisA70518.a n) then some (sInf {n | 28341 < n ∧ ¬Squarefree (OeisA70518.a n)})\n  else none","subjects":["11"],"theorem":"OeisA70518.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«70518»","statement":"OeisA70518.a 5 = 781","subjects":["11"],"theorem":"OeisA70518.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«70518»","statement":"OeisA70518.a 1 = 0","subjects":["11"],"theorem":"OeisA70518.a_1"},{"answerKinds":[],"category":"test","docstring":"$4$ is in the sequence A067720. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67720»","statement":"OeisA67720.A 4","subjects":["11"],"theorem":"OeisA67720.a_4"},{"answerKinds":[],"category":"test","docstring":"$6$ is in the sequence A067720. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67720»","statement":"OeisA67720.A 6","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"OeisA67720.a_6"},{"answerKinds":[],"category":"test","docstring":"$10$ is in the sequence A067720. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67720»","statement":"OeisA67720.A 10","subjects":["11"],"theorem":"OeisA67720.a_10"},{"answerKinds":[],"category":"research open","docstring":"For members of the sequence other than $8$, we have $k + 1$ is prime. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«67720»","statement":"∀ {k : ℕ}, OeisA67720.A k → k ≠ 8 → Nat.Prime (k + 1)","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"OeisA67720.prime_add_one_of_a"},{"answerKinds":[],"category":"test","docstring":"$2$ is in the sequence A067720. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67720»","statement":"OeisA67720.A 2","subjects":["11"],"theorem":"OeisA67720.a_2"},{"answerKinds":[],"category":"test","docstring":"$1$ is in the sequence A067720. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67720»","statement":"OeisA67720.A 1","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"OeisA67720.a_1"},{"answerKinds":[],"category":"textbook","docstring":"If $k + 1$ and $k^2 + 1$ are both prime, then $k$ is in the sequence. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67720»","statement":"∀ {k : ℕ}, Nat.Prime (k + 1) → Nat.Prime (k ^ 2 + 1) → OeisA67720.A k","subjects":["11"],"theorem":"OeisA67720.a_of_primes"},{"answerKinds":[],"category":"test","docstring":"$8$ is in the sequence A067720. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67720»","statement":"OeisA67720.A 8","subjects":["11"],"theorem":"OeisA67720.a_8"},{"answerKinds":[],"category":"test","docstring":"$3$ is in the sequence A003625. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3625»","statement":"OeisA3625.A 3","subjects":["11"],"theorem":"OeisA3625.a_3"},{"answerKinds":[],"category":"test","docstring":"$13$ is in the sequence A003625. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3625»","statement":"OeisA3625.A 13","subjects":["11"],"theorem":"OeisA3625.a_13"},{"answerKinds":[],"category":"test","docstring":"$17$ is in the sequence A003625. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3625»","statement":"OeisA3625.A 17","subjects":["11"],"theorem":"OeisA3625.a_17"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: Represents primes $p$ where the polynomial $x^2 + x + 2$ is irreducible over $\\text{GF}(p)$.\n- _Federico Provvedi_, Jul 21 2018\n\nAnswer: true, the equivalence is classical (complete the square: $4(x^2+x+2) = (2x+1)^2 + 7$,\nthen use quadratic reciprocity).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a003625-irreducibility/blob/d6c9f90827805142d81eee4e3d9099c8b48cbcc8/lean/OeisA3625FC.lean#L103-L104"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«3625»","statement":"∀ (p : ℕ), Nat.Prime p → (OeisA3625.A p ↔ Irreducible (Polynomial.X ^ 2 + Polynomial.X + 2))","subjects":["11"],"theorem":"OeisA3625.conjecture"},{"answerKinds":[],"category":"test","docstring":"$5$ is in the sequence A003625. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3625»","statement":"OeisA3625.A 5","subjects":["11"],"theorem":"OeisA3625.a_5"},{"answerKinds":[],"category":"test","docstring":"$19$ is in the sequence A003625. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«3625»","statement":"OeisA3625.A 19","subjects":["11"],"theorem":"OeisA3625.a_19"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"Nat.nth Nat.Prime 10 = 31","subjects":["11"],"theorem":"OeisA100474.nth_prime_ten"},{"answerKinds":[],"category":"research open","docstring":"After $a(2) = 5$, is there another prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«100474»","statement":"True ↔ ∃ n > 2, Nat.Prime (OeisA100474.a n)","subjects":["11"],"theorem":"OeisA100474.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"OeisA100474.a 5 = 58546472","subjects":["11"],"theorem":"OeisA100474.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"OeisA100474.a 2 = 5","subjects":["11"],"theorem":"OeisA100474.a_2"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"Nat.nth Nat.Prime 11 = 37","subjects":["11"],"theorem":"OeisA100474.nth_prime_eleven"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"OeisA100474.a 4 = 96197","subjects":["11"],"theorem":"OeisA100474.a_4"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"Nat.nth Nat.Prime 7 = 19","subjects":["11"],"theorem":"OeisA100474.nth_prime_seven"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"Nat.nth Nat.Prime 6 = 17","subjects":["11"],"theorem":"OeisA100474.nth_prime_six"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"Nat.nth Nat.Prime 13 = 43","subjects":["11"],"theorem":"OeisA100474.nth_prime_thirteen"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"Nat.nth Nat.Prime 12 = 41","subjects":["11"],"theorem":"OeisA100474.nth_prime_twelve"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"Nat.nth Nat.Prime 5 = 13","subjects":["11"],"theorem":"OeisA100474.nth_prime_five"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"Nat.nth Nat.Prime 9 = 29","subjects":["11"],"theorem":"OeisA100474.nth_prime_nine"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"OeisA100474.a 1 = 1","subjects":["11"],"theorem":"OeisA100474.a_1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"Nat.nth Nat.Prime 8 = 23","subjects":["11"],"theorem":"OeisA100474.nth_prime_eight"},{"answerKinds":["non-Prop"],"category":"research solved","docstring":"What is the next semiprime in the sequence after $a(11)$?\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a100474-next-semiprime/blob/d7781751918e63fd1268b15525bf9554c92fdd4d/lean/OeisA100474NextSemiprimeFC.lean#L1285-L1288"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«100474»","statement":"3852669607062814427999374038085094563026983841699038416757537720951140990693348082633155462564082456461927363575765861495986901576629 =\n  OeisA100474.a (sInf {n | 11 < n ∧ (OeisA100474.a n).IsSemiprime})","subjects":["11"],"theorem":"OeisA100474.next_semiprime"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«100474»","statement":"OeisA100474.a 3 = 380","subjects":["11"],"theorem":"OeisA100474.a_3"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.S 6 = 30","subjects":["11"],"theorem":"OeisA167918.S_6"},{"answerKinds":[],"category":"research open","docstring":"Open problem: Whether the ratio $f(n, k)$ is bounded, where $k = a(n)$.\n\nWe assume $a(n) \\ne 0$ (i.e., that a suitable $k > n$ always exists), as `sInf` evaluates to $0$\non an empty set.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«167918»","statement":"(∀ n > 0, OeisA167918.a n ≠ 0) → ∃ C, ∀ n > 0, OeisA167918.S (OeisA167918.a n) / OeisA167918.S n ≤ C","subjects":["11"],"theorem":"OeisA167918.conjecture2"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.P 8 = 19","subjects":["11"],"theorem":"OeisA167918.P_8"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.P 5 = 11","subjects":["11"],"theorem":"OeisA167918.P_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.a 4 = 7","subjects":["11"],"theorem":"OeisA167918.a_4"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.P 2 = 3","subjects":["11"],"theorem":"OeisA167918.P_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.a 2 = 5","subjects":["11"],"theorem":"OeisA167918.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.a 1 = 6","subjects":["11"],"theorem":"OeisA167918.a_1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.P 4 = 7","subjects":["11"],"theorem":"OeisA167918.P_4"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.S 4 = 18","subjects":["11"],"theorem":"OeisA167918.S_4"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.S 2 = 8","subjects":["11"],"theorem":"OeisA167918.S_2"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.P 3 = 5","subjects":["11"],"theorem":"OeisA167918.P_3"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.S 1 = 5","subjects":["11"],"theorem":"OeisA167918.S_1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.S 3 = 12","subjects":["11"],"theorem":"OeisA167918.S_3"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $f(n, k) = 2$ for infinitely many cases, where $k = a(n)$.\n\nWe assume $a(n) \\ne 0$ (i.e., that a suitable $k > n$ always exists), as `sInf` evaluates to $0$\non an empty set.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«167918»","statement":"∀ (M : ℕ), (∀ n > 0, OeisA167918.a n ≠ 0) → ∃ n ≥ M, n > 0 ∧ OeisA167918.S (OeisA167918.a n) = 2 * OeisA167918.S n","subjects":["11"],"theorem":"OeisA167918.conjecture1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.P 6 = 13","subjects":["11"],"theorem":"OeisA167918.P_6"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.S 7 = 36","subjects":["11"],"theorem":"OeisA167918.S_7"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.P 1 = 2","subjects":["11"],"theorem":"OeisA167918.P_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.a 3 = 5","subjects":["11"],"theorem":"OeisA167918.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.a 0 = 0","subjects":["11"],"theorem":"OeisA167918.a_0"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.S 5 = 24","subjects":["11"],"theorem":"OeisA167918.S_5"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«167918»","statement":"OeisA167918.P 7 = 17","subjects":["11"],"theorem":"OeisA167918.P_7"},{"answerKinds":[],"category":"research open","docstring":"In April 2009, _Zhi-Wei Sun_ conjectured that $a(n) > 0$ for every $n = 0, 1, 2, 3, \\dots$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«160324»","statement":"∀ (n : ℕ), 0 < OeisA160324.a n","subjects":["11"],"theorem":"OeisA160324.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"For each integer $m > 2$, any natural number $n$ can be written in the form\n$p_{m+1}(x_1) + \\cdots + p_{2m}(x_m)$ with $x_1, \\dots, x_m$ nonnegative integers, where\n$p_k(x) = (k-2)x(x-1)/2 + x$ ($x=0,1,2,\\dots$) are $k$-gonal numbers.\n- _Zhi-Wei Sun_, Aug 15 2009\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«160324»","statement":"∀ m > 2, ∀ (n : ℕ), ∃ x, n = ∑ i, OeisA160324.polygonalNumber (m + ↑i + 1) (x i)","subjects":["11"],"theorem":"OeisA160324.conjecture2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture (Zhi-Wei Sun, Aug 21 2009):\nFor each integer $m > 2$, all sufficiently large integers $n$ can be expressed in the form\n$p_{m+1}(x_1) + p_{m+2}(x_2) + p_{m+3}(x_3)$ with $x_1, x_2, x_3 \\in \\mathbb{N}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«160324»","statement":"∀ (m : ℕ),\n  2 < m →\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∃ x1 x2 x3,\n        n =\n          OeisA160324.polygonalNumber (m + 1) x1 + OeisA160324.polygonalNumber (m + 2) x2 +\n            OeisA160324.polygonalNumber (m + 3) x3","subjects":["11"],"theorem":"OeisA160324.conjecture5"},{"answerKinds":[],"category":"research open","docstring":"The sequence contains every positive integer.\n- _Zhi-Wei Sun_, Sep 04 2009\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«160324»","statement":"∀ (k : ℕ), 0 < k → ∃ n, OeisA160324.a n = k","subjects":["11"],"theorem":"OeisA160324.conjecture3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«160324»","statement":"OeisA160324.a 2 = 3","subjects":["11"],"theorem":"OeisA160324.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«160324»","statement":"OeisA160324.a 3 = 1","subjects":["11"],"theorem":"OeisA160324.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«160324»","statement":"OeisA160324.a 1 = 3","subjects":["11"],"theorem":"OeisA160324.a_1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture (Zhi-Wei Sun, Aug 21 2009):\nFor any integer $m > 2$, each natural number $n$ can be expressed as\n$p_{m+1}(x_1) + p_{m+2}(x_2) + p_{m+3}(x_3) + r$ with $x_1, x_2, x_3 \\in \\mathbb{N}$ and\n$r \\in \\{0, \\dots, m-3\\}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«160324»","statement":"∀ (m : ℕ),\n  2 < m →\n    ∀ (n : ℕ),\n      ∃ x1 x2 x3,\n        ∃ r ≤ m - 3,\n          n =\n            OeisA160324.polygonalNumber (m + 1) x1 + OeisA160324.polygonalNumber (m + 2) x2 +\n                OeisA160324.polygonalNumber (m + 3) x3 +\n              r","subjects":["11"],"theorem":"OeisA160324.conjecture4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«160324»","statement":"OeisA160324.a 0 = 1","subjects":["11"],"theorem":"OeisA160324.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1818»","statement":"OeisA1818.a 4 = 11025","subjects":["11"],"theorem":"OeisA1818.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1818»","statement":"OeisA1818.a 1 = 1","subjects":["11"],"theorem":"OeisA1818.a_1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 1: For any primitive $2n$-th root $\\zeta$ of unity, the permanent of the $2n \\times 2n$\nmatrix $[m(j,k)]_{j,k=1..2n}$ coincides with $a(n) = ((2n-1)!!)^2$, where $m(j,k)$ is\n$(1+\\zeta^{j-k})/(1-\\zeta^{j-k})$ if $j \\neq k$, and $1$ otherwise.\n- Zhi-Wei Sun, Dec 21 2021\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«1818»","statement":"∀ (n : ℕ),\n  1 ≤ n →\n    ∀ (ζ : ℂ),\n      IsPrimitiveRoot ζ (2 * n) →\n        (Matrix.permanent fun i j => if i = j then 1 else (1 + ζ ^ (↑↑i - ↑↑j)) / (1 - ζ ^ (↑↑i - ↑↑j))) =\n          ↑(OeisA1818.a n)","subjects":["11","15"],"theorem":"OeisA1818.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 2: Let $p$ be an odd prime. Then the permanent of the $(p-1) \\times (p-1)$ matrix\n$[f(j,k)]_{j,k=1..p-1}$ is congruent to $a((p-1)/2) = ((p-2)!!)^2 \\pmod{p^2}$,\nwhere $f(j,k)$ is $(j+k)/(j-k)$ if $j \\neq k$, and $f(j,k) = 1$ otherwise.\n- Zhi-Wei Sun, Dec 22 2021\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«1818»","statement":"∀ {p : ℕ},\n  Nat.Prime p →\n    p ≠ 2 →\n      have N := p - 1;\n      let R := ZMod (p ^ 2);\n      let Idx := Fin N;\n      have M := fun i j => OeisA1818.fEntry (↑i + 1) (↑j + 1);\n      M.permanent = ↑(OeisA1818.a ((p - 1) / 2))","subjects":["11","15"],"theorem":"OeisA1818.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1818»","statement":"OeisA1818.a 0 = 1","subjects":["11"],"theorem":"OeisA1818.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1818»","statement":"OeisA1818.a 3 = 225","subjects":["11"],"theorem":"OeisA1818.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«1818»","statement":"OeisA1818.a 2 = 9","subjects":["11"],"theorem":"OeisA1818.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 1 = 1","subjects":["11"],"theorem":"OeisA260194.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 3 = 2","subjects":["11"],"theorem":"OeisA260194.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 0 = 1","subjects":["11"],"theorem":"OeisA260194.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 9 = 12","subjects":["11"],"theorem":"OeisA260194.a_9"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 4 = 3","subjects":["11"],"theorem":"OeisA260194.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 6 = 6","subjects":["11"],"theorem":"OeisA260194.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 2 = 1","subjects":["11"],"theorem":"OeisA260194.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 5 = 4","subjects":["11"],"theorem":"OeisA260194.a_5"},{"answerKinds":[],"category":"research open","docstring":"Does every positive integer occur as a difference in this sequence?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«260194»","statement":"True ↔ ∀ d > 0, ∃ n, OeisA260194.a (n + 1) = OeisA260194.a n + d","subjects":["11"],"theorem":"OeisA260194.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 7 = 9","subjects":["11"],"theorem":"OeisA260194.a_7"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«260194»","statement":"OeisA260194.a 8 = 10","subjects":["11"],"theorem":"OeisA260194.a_8"},{"answerKinds":[],"category":"research open","docstring":"The only positive integer $n$ such that $a(n)$ is a perfect square is $n=38$.\n- Carlos Eduardo Olivieri, Mar 09 2015\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«7468»","statement":"∀ (n : ℕ), 0 < n → IsSquare (OeisA7468.a n) → n = 38","subjects":["11"],"theorem":"OeisA7468.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7468»","statement":"OeisA7468.a 2 = 8","subjects":["11"],"theorem":"OeisA7468.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7468»","statement":"OeisA7468.a 0 = 0","subjects":["11"],"theorem":"OeisA7468.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7468»","statement":"OeisA7468.a 1 = 2","subjects":["11"],"theorem":"OeisA7468.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7468»","statement":"OeisA7468.a 3 = 31","subjects":["11"],"theorem":"OeisA7468.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119591»","statement":"OeisA119591.a 4 = 1","subjects":["11"],"theorem":"OeisA119591.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 6. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119591»","statement":"OeisA119591.a 6 = 1","subjects":["11"],"theorem":"OeisA119591.a_6"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119591»","statement":"OeisA119591.a 5 = 4","subjects":["11"],"theorem":"OeisA119591.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119591»","statement":"OeisA119591.a 2 = 1","subjects":["11"],"theorem":"OeisA119591.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119591»","statement":"OeisA119591.a 3 = 1","subjects":["11"],"theorem":"OeisA119591.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119591»","statement":"OeisA119591.a 1 = 0","subjects":["11"],"theorem":"OeisA119591.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«119591»","statement":"OeisA119591.a 0 = 0","subjects":["11"],"theorem":"OeisA119591.a_0"},{"answerKinds":[],"category":"research open","docstring":"Is $a(n)$ defined for all $n \\ge 2$?\nThat is, does there exist $k > 0$ such that $2 \\cdot n^k - 1$ is prime?","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«119591»","statement":"∀ (n : ℕ), 2 ≤ n → ∃ k > 0, Nat.Prime (2 * n ^ k - 1)","subjects":["11"],"theorem":"OeisA119591.conjecture"},{"answerKinds":[],"category":"test","docstring":"`3` is prime, so the interior sum is empty and `f 3 = a046644 3 / 2 = 1`. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«317940»","statement":"OeisA317940.a 3 = 1","subjects":["11"],"theorem":"OeisA317940.a_3"},{"answerKinds":[],"category":"research solved","docstring":"\"No negative terms among the first 2^20 terms. Is the sequence nonnegative?\"\n\nEquivalently, the rational Dirichlet square root `f` is nonnegative at every\npositive index. Informally, the proof constructs a strictly positive\nmultiplicative square root prime-power by prime-power using formal power\nseries, and then identifies it with the recursively defined sequence `f`.\n\nThe proof was obtained by exploiting the multiplicativity of A046644 and\nreducing the problem to its values on prime powers. The resulting convolution\nrecurrence for the prime-power coefficients was recognized as a\nformal-power-series identity. A positive coefficient sequence was constructed\nthrough a differential equation for the generating series, extended\nmultiplicatively to all positive integers, and finally identified with the\nrecursively defined Dirichlet square root by uniqueness.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://domthedeveloper.github.io/crl/math/a317940/proof/A317940_verified.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«317940»","statement":"∀ n > 0, OeisA317940.f n ≥ 0","subjects":["11"],"theorem":"OeisA317940.f_nonnegative"},{"answerKinds":[],"category":"test","docstring":"`5` is prime, so the interior sum is empty and `f 5 = a046644 5 / 2 = 1`. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«317940»","statement":"OeisA317940.a 5 = 1","subjects":["11"],"theorem":"OeisA317940.a_5"},{"answerKinds":[],"category":"test","docstring":"`f 1 = 1` is the base case of the recurrence, so `a 1 = 1`. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«317940»","statement":"OeisA317940.a 1 = 1","subjects":["11"],"theorem":"OeisA317940.a_1"},{"answerKinds":[],"category":"test","docstring":"`2` is prime, so the interior sum is empty and `f 2 = a046644 2 / 2 = 1`. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«317940»","statement":"OeisA317940.a 2 = 1","subjects":["11"],"theorem":"OeisA317940.a_2"},{"answerKinds":[],"category":"test","docstring":"`4` is the first index with a nonempty interior sum, giving `f 4 = 7 / 2`. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«317940»","statement":"OeisA317940.a 4 = 7","subjects":["11"],"theorem":"OeisA317940.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105801»","statement":"OeisA105801.a 2 = 2","subjects":["11"],"theorem":"OeisA105801.a_2"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: for every $k > 0$ there is an index $m$ such that all the $a(n)$ with $n > m$\nhave the same residue $\\bmod 3^k$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a105801-lean/blob/68642a80db062bee7061437b60d31c3e7d626595/lean/OeisA105801FC.lean#L323-L351"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«105801»","statement":"∀ (k : ℕ), 0 < k → ∃ m, ∀ (n : ℕ), m < n → OeisA105801.a n ≡ OeisA105801.a (m + 1) [MOD 3 ^ k]","subjects":["11"],"theorem":"OeisA105801.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105801»","statement":"OeisA105801.a 5 = 8","subjects":["11"],"theorem":"OeisA105801.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105801»","statement":"OeisA105801.a 4 = 6","subjects":["11"],"theorem":"OeisA105801.a_4"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105801»","statement":"OeisA105801.a 1 = 1","subjects":["11"],"theorem":"OeisA105801.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105801»","statement":"OeisA105801.a 3 = 10","subjects":["11"],"theorem":"OeisA105801.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38107»","statement":"OeisA38107.a 1 = 0","subjects":["11"],"theorem":"OeisA38107.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38107»","statement":"OeisA38107.a 0 = 0","subjects":["11"],"theorem":"OeisA38107.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38107»","statement":"OeisA38107.a 4 = 6","subjects":["11"],"theorem":"OeisA38107.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38107»","statement":"OeisA38107.a 2 = 2","subjects":["11"],"theorem":"OeisA38107.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38107»","statement":"OeisA38107.a 5 = 9","subjects":["11"],"theorem":"OeisA38107.a_5"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: all the numbers $\\sum_{i=j}^k \\frac{1}{a(i)}$ with $1 < j \\le k$ have pairwise distinct\nfractional parts.\n- Zhi-Wei Sun, Sep 24 2015\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38107»","statement":"∀ (j k j' k' : ℕ),\n  1 < j →\n    j ≤ k →\n      1 < j' →\n        j' ≤ k' →\n          Int.fract (∑ i ∈ Finset.Icc j k, 1 / ↑(OeisA38107.a i)) =\n              Int.fract (∑ i ∈ Finset.Icc j' k', 1 / ↑(OeisA38107.a i)) →\n            j = j' ∧ k = k'","subjects":["11"],"theorem":"OeisA38107.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38107»","statement":"OeisA38107.a 3 = 4","subjects":["11"],"theorem":"OeisA38107.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«227582»","statement":"OeisA227582.a 3 = 14","subjects":["11"],"theorem":"OeisA227582.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«227582»","statement":"OeisA227582.a 5 = 35","subjects":["11"],"theorem":"OeisA227582.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«227582»","statement":"OeisA227582.a 1 = 2","subjects":["11"],"theorem":"OeisA227582.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«227582»","statement":"OeisA227582.a 2 = 7","subjects":["11"],"theorem":"OeisA227582.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«227582»","statement":"OeisA227582.a 4 = 23","subjects":["11"],"theorem":"OeisA227582.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture (from A227581): $a(n) = \\lfloor 1/(2 H(n) - H(n^2 + n - 1) - \\gamma) \\rfloor$,\nwhere $H$ denotes harmonic numbers and $\\gamma$ denotes the Euler-Mascheroni constant.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/227582.wip.lean#L282"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«227582»","statement":"∀ (n : ℕ),\n  0 < n → OeisA227582.a n = ⌊1 / (2 * ↑(harmonic n) - ↑(harmonic (n * n + n - 1)) - Real.eulerMascheroniConstant)⌋.toNat","subjects":["11"],"theorem":"OeisA227582.a_eq_floor_harmonic_expr"},{"answerKinds":[],"category":"API","docstring":"The linear recurrence:\n$$b(n + 7) = 2 b(n + 6) - b(n + 5) + b(n + 2) - 2 b(n + 1) + b(n)$$\nholds for `baseSeq`. This follows directly from `LinearRecurrence.is_sol_mkSol`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«227582»","statement":"∀ (n : ℕ),\n  OeisA227582.baseSeq (n + 7) =\n    2 * OeisA227582.baseSeq (n + 6) - OeisA227582.baseSeq (n + 5) + OeisA227582.baseSeq (n + 2) -\n        2 * OeisA227582.baseSeq (n + 1) +\n      OeisA227582.baseSeq n","subjects":["11"],"theorem":"OeisA227582.baseSeq_recurrence"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69923»","statement":"OeisA69923.a 3 = 2","subjects":["11"],"theorem":"OeisA69923.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69923»","statement":"OeisA69923.a 1 = 2","subjects":["11"],"theorem":"OeisA69923.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69923»","statement":"OeisA69923.a 2 = 2","subjects":["11"],"theorem":"OeisA69923.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«69923»","statement":"OeisA69923.a 4 = 3","subjects":["11"],"theorem":"OeisA69923.a_4"},{"answerKinds":[],"category":"research open","docstring":"For any $n > 0$, is there always at least one prime $p$ such that\n$2^n \\le p \\le 2^n + \\mathrm{prime}(n)$?\n(checked up to $n = 250$). In this case, that would be stronger than the Schinzel conjecture:\n\"for $m > 1$ there's at least one prime $p$ such that $m \\le p \\le m + \\log(m)^2$\" since,\nfor $n > 2$, $\\mathrm{prime}(n) < \\log(2^n)^2 = n^2 \\log(2)$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«69923»","statement":"∀ (n : ℕ), 0 < n → 1 ≤ OeisA69923.a n","subjects":["11"],"theorem":"OeisA69923.conjecture"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103885»","statement":"OeisA103885.a 0 = 1","subjects":["11"],"theorem":"OeisA103885.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103885»","statement":"OeisA103885.a 1 = 2","subjects":["11"],"theorem":"OeisA103885.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103885»","statement":"OeisA103885.a 4 = 1408","subjects":["11"],"theorem":"OeisA103885.a_4"},{"answerKinds":[],"category":"research open","docstring":"The recurrence given below can be rewritten in the form\n$$(2n+1)(2n+2)P(2,n)a(n+1) - (2n-1)(2n-2)P(2,-n)a(n-1) = Q(2,n^2)a(n),$$\nwhere the polynomial $Q(2,n) = 4(55n^2 - 34n + 3)$ and the polynomial $P(2,n) = 5n^2 - 5n + 1$\nsatisfies the symmetry condition $P(2,n) = P(2,1-n)$ and has real zeros.\nMore generally, for fixed $m = 1,2,3, \\ldots$, we conjecture that the sequence $b(n) := a(mn)$\nsatisfies a recurrence of the form\n$$( \\prod_{k = 1}^{2m} (2mn + k) )P(2m,n)b(n+1) + (-1)^m( \\prod_{k = 1}^{2*m} (2mn - k) )\n  P(2m,-n)b(n-1) = Q(2m,n^2)b(n),$$\nwhere the polynomials $P(2m,n)$ and $Q(2m,n)$ have degree $2m$. Conjecturally, the polynomial\n$P(2m,n) = P(2m,1-n)$ and has real zeros in the interval [0, 1].\nThe $4m$ zeros of the polynomial $Q(2m,n^2)$ seem to belong to the interval $[-1, 1]$ and\n$4m - 2$ of these zeros appear to be approximated by the rational numbers\n$\\pm k/(3m)$, where $1 \\le k \\le 3m - 2$, $k$ not a multiple of $3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«103885»","statement":"∀ (m : ℕ),\n  1 ≤ m →\n    ∃ P Q,\n      P.degree = ↑(2 * m) ∧\n        Q.degree = ↑(2 * m) ∧\n          (∀ (n : ℕ),\n              1 ≤ n →\n                OeisA103885.prodFactorPlus m n * Polynomial.eval (↑n) P * OeisA103885.aSubsequenceReal m (n + 1) +\n                    (-1) ^ m * OeisA103885.prodFactorMinus m n * Polynomial.eval (-↑n) P *\n                      OeisA103885.aSubsequenceReal m (n - 1) =\n                  Polynomial.eval (↑n ^ 2) Q * OeisA103885.aSubsequenceReal m n) ∧\n            (∀ (x : ℝ), Polynomial.eval x P = Polynomial.eval (1 - x) P) ∧\n              (∀ (z : ℂ), Polynomial.eval z (Polynomial.map (algebraMap ℝ ℂ) P) = 0 → z.im = 0 ∧ z.re ∈ Set.Icc 0 1) ∧\n                ∀ (z : ℂ),\n                  Polynomial.eval (z ^ 2) (Polynomial.map (algebraMap ℝ ℂ) Q) = 0 → z.im = 0 ∧ z.re ∈ Set.Icc (-1) 1","subjects":["11"],"theorem":"OeisA103885.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103885»","statement":"OeisA103885.a 2 = 16","subjects":["11"],"theorem":"OeisA103885.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103885»","statement":"OeisA103885.a 3 = 146","subjects":["11"],"theorem":"OeisA103885.a_3"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"∀ (f : ℕ → ℝ), (Finset.Icc 1 3).sum f = f 1 + f 2 + f 3","subjects":["11"],"theorem":"OeisA102722.a_sum_3"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"∀ (f : ℕ → ℝ), (Finset.Icc 1 4).sum f = f 1 + f 2 + f 3 + f 4","subjects":["11"],"theorem":"OeisA102722.a_sum_4"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"OeisA102722.a 1 = 0","subjects":["11"],"theorem":"OeisA102722.a_1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"∀ (f : ℕ → ℝ), (Finset.Icc 1 1).sum f = f 1","subjects":["11"],"theorem":"OeisA102722.a_sum_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"OeisA102722.a 4 = 0","subjects":["11"],"theorem":"OeisA102722.a_4"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"∀ (f : ℕ → ℝ), (Finset.Icc 1 5).sum f = f 1 + f 2 + f 3 + f 4 + f 5","subjects":["11"],"theorem":"OeisA102722.a_sum_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"OeisA102722.a 5 = 1","subjects":["11"],"theorem":"OeisA102722.a_5"},{"answerKinds":[],"category":"research solved","docstring":"A102722 Conjecture: $a(n) \\sim (1-\\gamma)n$. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-102722-asymptotic/blob/0e3bf1bc6dfd04627f926b07fb8d25f4395a5072/lean/OEIS102722FC.lean#L18-L24"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«102722»","statement":"Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(OeisA102722.a n)) fun n => (1 - Real.eulerMascheroniConstant) * ↑n","subjects":["11"],"theorem":"OeisA102722.conjecture"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"∀ (f : ℕ → ℝ), (Finset.Icc 1 2).sum f = f 1 + f 2","subjects":["11"],"theorem":"OeisA102722.a_sum_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"OeisA102722.a 2 = 0","subjects":["11"],"theorem":"OeisA102722.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«102722»","statement":"OeisA102722.a 3 = 0","subjects":["11"],"theorem":"OeisA102722.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108569»","statement":"OeisA108569.a 4 = 32","subjects":["11"],"theorem":"OeisA108569.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108569»","statement":"OeisA108569.a 1 = 4","subjects":["11"],"theorem":"OeisA108569.a_1"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108569»","statement":"OeisA108569.a 0 = 1","subjects":["11"],"theorem":"OeisA108569.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108569»","statement":"OeisA108569.a 3 = 16","subjects":["11"],"theorem":"OeisA108569.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108569»","statement":"OeisA108569.a 2 = 8","subjects":["11"],"theorem":"OeisA108569.a_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: Except for the first term all terms are even. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108569»","statement":"∀ (n : ℕ), 0 < n → Even (OeisA108569.a n)","subjects":["11"],"theorem":"OeisA108569.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«309132»","statement":"OeisA309132.a 4 = 16","subjects":["11"],"theorem":"OeisA309132.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: composite numbers $n$ such that $a(n)$ is squarefree are only the Carmichael numbers (A002997). - _Thomas Ordowski_, Jul 15 2019\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/309132.wip.lean#L353"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«309132»","statement":"∀ (n : ℕ), OeisA309132.IsComposite n ∧ Squarefree (OeisA309132.a n) ↔ OeisA309132.IsCarmichaelNumber n","subjects":["11"],"theorem":"OeisA309132.carmichael_iff_squarefree_a"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«309132»","statement":"OeisA309132.a 1 = 1","subjects":["11"],"theorem":"OeisA309132.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«309132»","statement":"OeisA309132.a 3 = 1","subjects":["11"],"theorem":"OeisA309132.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«309132»","statement":"OeisA309132.a 2 = 1","subjects":["11"],"theorem":"OeisA309132.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«309132»","statement":"OeisA309132.a 5 = 1","subjects":["11"],"theorem":"OeisA309132.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113258»","statement":"OeisA113258.a 1 = 1","subjects":["11"],"theorem":"OeisA113258.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113258»","statement":"OeisA113258.a 4 = 125","subjects":["11"],"theorem":"OeisA113258.a_4"},{"answerKinds":[],"category":"research open","docstring":"Is there a nontrivial power after $a(4) = 5^3$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113258»","statement":"True ↔ ∃ n > 4, ∃ b > 1, ∃ e > 1, OeisA113258.a n = b ^ e","subjects":["11"],"theorem":"OeisA113258.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113258»","statement":"OeisA113258.a 2 = 3","subjects":["11"],"theorem":"OeisA113258.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113258»","statement":"OeisA113258.a 3 = 11","subjects":["11"],"theorem":"OeisA113258.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109909»","statement":"OeisA109909.a 1 = 0","subjects":["11"],"theorem":"OeisA109909.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109909»","statement":"OeisA109909.a 3 = 0","subjects":["11"],"theorem":"OeisA109909.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109909»","statement":"OeisA109909.a 2 = 0","subjects":["11"],"theorem":"OeisA109909.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109909»","statement":"OeisA109909.a 5 = 2","subjects":["11"],"theorem":"OeisA109909.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109909»","statement":"OeisA109909.a 4 = 2","subjects":["11"],"theorem":"OeisA109909.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) > 0$ for $n > 3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«109909»","statement":"∀ n > 3, OeisA109909.a n > 0","subjects":["11"],"theorem":"OeisA109909.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67857»","statement":"OeisA67857.a 1 = 1","subjects":["11"],"theorem":"OeisA67857.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67857»","statement":"OeisA67857.a 4 = 14","subjects":["11"],"theorem":"OeisA67857.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67857»","statement":"OeisA67857.a 2 = 1","subjects":["11"],"theorem":"OeisA67857.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67857»","statement":"OeisA67857.a 5 = 154","subjects":["11"],"theorem":"OeisA67857.a_5"},{"answerKinds":[],"category":"research open","docstring":"The terms are not all positive. The first negative one is\n$a(30) = -22690644647302814715858124800000$.\nConjecture: $a(n) < 0$ if and only if A001221(n) is an odd number $\\ge 3$.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«67857»","statement":"∀ (n : ℕ),\n  0 < n →\n    (OeisA67857.a n < 0 ↔ Odd (ArithmeticFunction.cardDistinctFactors n) ∧ 3 ≤ ArithmeticFunction.cardDistinctFactors n)","subjects":["11"],"theorem":"OeisA67857.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«67857»","statement":"OeisA67857.a 3 = 5","subjects":["11"],"theorem":"OeisA67857.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224»","statement":"OeisA224.a 4 = 2","subjects":["11"],"theorem":"OeisA224.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224»","statement":"OeisA224.a 2 = 2","subjects":["11"],"theorem":"OeisA224.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224»","statement":"OeisA224.a 3 = 2","subjects":["11"],"theorem":"OeisA224.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224»","statement":"OeisA224.a 1 = 1","subjects":["11"],"theorem":"OeisA224.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«224»","statement":"OeisA224.a 0 = 1","subjects":["11"],"theorem":"OeisA224.a_0"},{"answerKinds":[],"category":"research open","docstring":"$n^2 \\equiv 1 \\pmod{a(n)(a(n)-1)}$ if and only if $n$ is an odd prime.\n- Thomas Ordowski, Jun 08 2017\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«224»","statement":"∀ (n : ℕ), 1 < n → (Nat.Prime n ∧ n ≠ 2 ↔ n ^ 2 ≡ 1 [MOD OeisA224.a n * (OeisA224.a n - 1)])","subjects":["11"],"theorem":"OeisA224.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108211»","statement":"OeisA108211.a 4 = 257","subjects":["11"],"theorem":"OeisA108211.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108211»","statement":"OeisA108211.a 5 = 401","subjects":["11"],"theorem":"OeisA108211.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108211»","statement":"OeisA108211.a 2 = 65","subjects":["11"],"theorem":"OeisA108211.a_2"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108211»","statement":"OeisA108211.a 1 = 17","subjects":["11"],"theorem":"OeisA108211.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108211»","statement":"OeisA108211.a 3 = 145","subjects":["11"],"theorem":"OeisA108211.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture:\n$$a(n) = \\left\\lfloor \\frac{1}{\\frac{1}{4n} - \\log(2) +\n  \\frac{1}{n+1} + \\frac{1}{n+2} + \\dots + \\frac{1}{2n}} \\right\\rfloor.$$\n\n**Proof sketch** (certificate style; the kernel-checked development lives at the\n`formal_proof` permalink below). Write $T(n) = \\log 2 - (H(2n) - H(n))$ for the harmonic\ntail defect. The proof sandwiches $T(n)$ between two explicit telescoping bounds — $h(n)$\nfrom below and $h(n) + 60/(4n+1)^7$ from above, where $h$ telescopes a degree-7 rational\ncertificate. The two resulting inequalities reduce to polynomial coefficient-nonnegativity\nfacts discharged by elementary tactics, after which the reciprocal lands in\n$[16n^2 + 1,\\, 16n^2 + 2)$ and the floor evaluates exactly.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/chy4pro/formal-conjectures/blob/f24f80aeaa3d5073bf4a54ed9daa102a5e0f1fad/FormalConjectures/OEIS/108211.lean#L540"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108211»","statement":"∀ n > 0, ↑(OeisA108211.a n) = ↑⌊1 / ((4 * ↑n)⁻¹ - Real.log 2 + ∑ k ∈ Finset.Icc (n + 1) (2 * n), (↑k)⁻¹)⌋","subjects":["11"],"theorem":"OeisA108211.conjecture"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture 1 / Theorem: \"a(n) is the number of odd divisors of n except the 'e' odd\ndivisors described in A005279.\" - _Omar E. Pol_, Dec 21 2024.\n\"The conjecture 1 is true. For a proof see A379288.\" - _Hartmut F. W. Hoft_, Jan 21 2025.\nEquivalently, $a(n) = \\text{A001227}(n) - \\text{A239657}(n)$. - _Omar E. Pol_, Mar 23 2014\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«237271»","statement":"∀ (n : ℕ), 0 < n → OeisA237271.a n = OeisA237271.A001227 n - OeisA237271.A239657 n","subjects":["11"],"theorem":"OeisA237271.conjecture_1"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture 5: \"a(A000384(n)) is odd.\" - _Omar E. Pol_, Oct 21 2025\n\nThat is, the number of parts in the symmetric representation of $\\sigma(n(2n-1))$ is always odd,\nwhere $n(2n-1)$ is the $n$-th hexagonal number.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a237271-square-hexagonal-parity/blob/430c09114d4ad3a3a2654b9816c8bfdc3cdf38de/lean/OeisA237271ParityFC.lean#L26-L32"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«237271»","statement":"∀ (n : ℕ), 0 < n → Odd (OeisA237271.a (n * (2 * n - 1)))","subjects":["11"],"theorem":"OeisA237271.conjecture_5"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture 4: \"a(A000290(n)) is odd.\" - _Omar E. Pol_, Oct 21 2025\n\nThat is, the number of parts in the symmetric representation of $\\sigma(n^2)$ is always odd.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a237271-square-hexagonal-parity/blob/430c09114d4ad3a3a2654b9816c8bfdc3cdf38de/lean/OeisA237271ParityFC.lean#L19-L24"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«237271»","statement":"∀ (n : ℕ), 0 < n → Odd (OeisA237271.a (n ^ 2))","subjects":["11"],"theorem":"OeisA237271.conjecture_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«237271»","statement":"OeisA237271.a 3 = 2","subjects":["11"],"theorem":"OeisA237271.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Theorem: \"a(p^k) = k + 1, where p is an odd prime and k >= 0.\"\n- _Hartmut F. W. Hoft_, Dec 26 2016\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«237271»","statement":"∀ (p k : ℕ), Nat.Prime p → Odd p → OeisA237271.a (p ^ k) = k + 1","subjects":["11"],"theorem":"OeisA237271.a_odd_prime_pow"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«237271»","statement":"OeisA237271.a 4 = 1","subjects":["11"],"theorem":"OeisA237271.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture 3: \"a(n) is the number of divisors p of n such that p is greater than\ntwice the adjacent previous divisor of n. The divisors p give the n-th row of A379288.\"\n- _Omar E. Pol_, Aug 02 2025\n\nNote: this is equivalent to `a_eq_num2DenseSublists` (Conjecture 2), since the divisors that\nstart a new 2-dense sublist are exactly those greater than twice their predecessor (plus the\nsmallest divisor).\n","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«237271»","statement":"∀ (n : ℕ),\n  OeisA237271.a n =\n    1 +\n      List.countP (fun pair => decide (pair.2 > 2 * pair.1))\n        ((OeisA237271.sortedDivisorsList n).zip (OeisA237271.sortedDivisorsList n).tail)","subjects":["11"],"theorem":"OeisA237271.conjecture_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«237271»","statement":"OeisA237271.a 5 = 2","subjects":["11"],"theorem":"OeisA237271.a_5"},{"answerKinds":[],"category":"research open","docstring":"Observation: \"a(A002997(n)) >= 3, at least for 1 <= n <= 10000.\"\n- _Omar E. Pol_, Oct 21 2025\n\nThat is, $a(k) \\ge 3$ for every Carmichael number $k$.\nA002997 is the sequence of Carmichael numbers: the composite numbers $k$ such that\n$b^{k-1} \\equiv 1 \\pmod k$ for every $b$ coprime to $k$. This is `IsCarmichael`,\nwhich also forces $k$ to be composite.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«237271»","statement":"∀ (k : ℕ), IsCarmichael k → 3 ≤ OeisA237271.a k","subjects":["11"],"theorem":"OeisA237271.observation_carmichael"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«237271»","statement":"OeisA237271.a 2 = 1","subjects":["11"],"theorem":"OeisA237271.a_2"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture 2: \"a(n) is the number of 2-dense sublists of divisors of n.\nWe call '2-dense sublists of divisors of n' to the maximal sublists of divisors of n whose terms\nincrease by a factor of at most 2.\" - _Omar E. Pol_, Jul 31 2025\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/237271.wip.lean#L102"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«237271»","statement":"∀ (n : ℕ), OeisA237271.a n = OeisA237271.num2DenseSublists n","subjects":["11"],"theorem":"OeisA237271.conjecture_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«237271»","statement":"OeisA237271.a 1 = 1","subjects":["11"],"theorem":"OeisA237271.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«945»","statement":"OeisA945.a 0 = 1","subjects":["11"],"theorem":"OeisA945.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«945»","statement":"OeisA945.a 1 = 2","subjects":["11"],"theorem":"OeisA945.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«945»","statement":"OeisA945.a 3 = 7","subjects":["11"],"theorem":"OeisA945.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«945»","statement":"OeisA945.a 7 = 5","subjects":["11"],"theorem":"OeisA945.a_7"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«945»","statement":"OeisA945.a 5 = 13","subjects":["11"],"theorem":"OeisA945.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«945»","statement":"OeisA945.a 2 = 3","subjects":["11"],"theorem":"OeisA945.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«945»","statement":"OeisA945.a 4 = 43","subjects":["11"],"theorem":"OeisA945.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«945»","statement":"OeisA945.a 6 = 53","subjects":["11"],"theorem":"OeisA945.a_6"},{"answerKinds":[],"category":"research open","docstring":"\"Does the sequence ... contain every prime? ... [It] was considered by Guy and Nowakowski\nand later by Shanks, [Wagstaff93] computed the sequence through the 43rd term. The\ncomputational problem inherent in continuing the sequence further is the enormous size of the\nnumbers that must be factored. Already the number $a(1) \\cdots a(43) + 1$ has 180 digits.\"\n- [CrandallPomerance01]\n\nSee also [Mullin63].\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«945»","statement":"True ↔ ∀ (p : ℕ), Nat.Prime p → ∃ n ≥ 1, OeisA945.a n = p","subjects":["11"],"theorem":"OeisA945.every_prime_occurs"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«278070»","statement":"OeisA278070.a 0 = 1","subjects":["11"],"theorem":"OeisA278070.a_0"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $a(n+k) \\equiv a(n) \\pmod{k}$ for all $n$ and $k$. - _Peter Bala_, Mar 12 2023\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/278070.wip.lean#L68"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«278070»","statement":"∀ (n k : ℕ), OeisA278070.a (n + k) ≡ OeisA278070.a n [MOD k]","subjects":["11"],"theorem":"OeisA278070.a_modeq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«278070»","statement":"OeisA278070.a 4 = 1457","subjects":["11"],"theorem":"OeisA278070.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«278070»","statement":"OeisA278070.a 2 = 11","subjects":["11"],"theorem":"OeisA278070.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«278070»","statement":"OeisA278070.a 1 = 2","subjects":["11"],"theorem":"OeisA278070.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«278070»","statement":"OeisA278070.a 3 = 106","subjects":["11"],"theorem":"OeisA278070.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53067»","statement":"OeisA53067.a 4 = 78910","subjects":["11"],"theorem":"OeisA53067.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53067»","statement":"OeisA53067.a 2 = 23","subjects":["11"],"theorem":"OeisA53067.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53067»","statement":"OeisA53067.a 1 = 1","subjects":["11"],"theorem":"OeisA53067.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53067»","statement":"OeisA53067.a 5 = 1112131415","subjects":["11"],"theorem":"OeisA53067.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«53067»","statement":"OeisA53067.a 3 = 456","subjects":["11"],"theorem":"OeisA53067.a_3"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"\"The second term is a prime. When is the next prime, if there is another?\n- _N. J. A. Sloane_, Dec 16 2016\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«53067»","statement":"sorry =\n  if h : ∃ n, 2 < n ∧ Nat.Prime (OeisA53067.a n) then some (sInf {n | 2 < n ∧ Nat.Prime (OeisA53067.a n)}) else none","subjects":["11"],"theorem":"OeisA53067.conjecture"},{"answerKinds":[],"category":"research open","docstring":"For each $n = 1, 2, 3, \\dots$ the polynomial\n$a_n(x) = \\sum_{k=0}^n \\binom{n}{k}^2 \\binom{n+k}{k} x^k$\nis irreducible over the field of rational numbers.\n- Zhi-Wei Sun, Mar 21 2013\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«5258»","statement":"∀ (n : ℕ), 1 ≤ n → Irreducible (OeisA5258.aperyPoly n)","subjects":["11","12"],"theorem":"OeisA5258.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5258»","statement":"OeisA5258.a 2 = 19","subjects":["11"],"theorem":"OeisA5258.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5258»","statement":"OeisA5258.a 3 = 147","subjects":["11"],"theorem":"OeisA5258.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5258»","statement":"OeisA5258.a 0 = 1","subjects":["11"],"theorem":"OeisA5258.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5258»","statement":"OeisA5258.a 1 = 3","subjects":["11"],"theorem":"OeisA5258.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«5258»","statement":"OeisA5258.a 4 = 1251","subjects":["11"],"theorem":"OeisA5258.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: let $a_p(n)$ be the length of the period of the sequence $k^p \\bmod n$ where $p$ is a prime, then $a_p(n) = n/p$ if $n \\equiv 0 \\pmod{p^2}$, else $a_p(n) = n$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/282779.wip.lean#L104"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«282779»","statement":"∀ (p n : ℕ), Nat.Prime p → n > 0 → OeisA282779.periodOfPowerMod p n = if p ^ 2 ∣ n then n / p else n","subjects":["11"],"theorem":"OeisA282779.periodOfPowerMod_eq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«282779»","statement":"OeisA282779.a 4 = 4","subjects":["11"],"theorem":"OeisA282779.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«282779»","statement":"OeisA282779.a 2 = 2","subjects":["11"],"theorem":"OeisA282779.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«282779»","statement":"OeisA282779.a 1 = 1","subjects":["11"],"theorem":"OeisA282779.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«282779»","statement":"OeisA282779.a 5 = 5","subjects":["11"],"theorem":"OeisA282779.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«282779»","statement":"OeisA282779.a 3 = 3","subjects":["11"],"theorem":"OeisA282779.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78680»","statement":"OeisA78680.a 5 = 1","subjects":["11"],"theorem":"OeisA78680.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78680»","statement":"OeisA78680.a 4 = 2","subjects":["11"],"theorem":"OeisA78680.a_4"},{"answerKinds":[],"category":"research open","docstring":"There is a conjecture that the first zero is $n = 65536 = 2^{16}$ (which is equivalent to\nthe statement that $2^{2^k} + 1$ is composite for $k > 4$). - _T. D. Noe_, Feb 25 2011\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«78680»","statement":"OeisA78680.a (2 ^ 16) = 0 ∧ ∀ (n : ℕ), 1 ≤ n ∧ n < 2 ^ 16 → OeisA78680.a n ≠ 0","subjects":["11"],"theorem":"OeisA78680.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78680»","statement":"OeisA78680.a 3 = 1","subjects":["11"],"theorem":"OeisA78680.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78680»","statement":"OeisA78680.a 1 = 1","subjects":["11"],"theorem":"OeisA78680.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«78680»","statement":"OeisA78680.a 2 = 1","subjects":["11"],"theorem":"OeisA78680.a_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: unless $n! + 1$ is prime (i.e., $n \\in \\text{A002981}$), $a(n) = p q$ where $p$ is the\nleast prime $> \\sqrt{n!}$ such that $(p - 1) \\mid n!$ and $q = \\frac{n!}{p - 1} + 1$ is prime.\n- M. F. Hasler, Oct 04 2009\n\nWe assume $a(n) \\ne 0$ and $(p(n)).\\text{Prime}$ to ensure the `sInf` searches are non-empty\nand do not collapse to $0 = 0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«55487»","statement":"∀ (n : ℕ),\n  1 ≤ n →\n    ¬OeisA55487.isFactorialPrime n →\n      OeisA55487.a n ≠ 0 → Nat.Prime (OeisA55487.p n) → OeisA55487.a n = OeisA55487.p n * OeisA55487.q n","subjects":["11"],"theorem":"OeisA55487.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«55487»","statement":"OeisA55487.a 4 = 35","subjects":["11"],"theorem":"OeisA55487.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«55487»","statement":"OeisA55487.a 2 = 3","subjects":["11"],"theorem":"OeisA55487.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«55487»","statement":"OeisA55487.a 1 = 1","subjects":["11"],"theorem":"OeisA55487.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«55487»","statement":"OeisA55487.a 3 = 7","subjects":["11"],"theorem":"OeisA55487.a_3"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 2: For any $k \\ge 3$, there are infinitely many primes of the form $n^k + m^k + 1$\nfor $n, m \\ge 1$.\n- _Ulrich Krug_, 2009\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«159829»","statement":"∀ (k : ℕ), 3 ≤ k → {p | ∃ n m, 1 ≤ n ∧ 1 ≤ m ∧ Nat.Prime p ∧ p = n ^ k + m ^ k + 1}.Infinite","subjects":["11"],"theorem":"OeisA159829.conjecture2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«159829»","statement":"OeisA159829.a 2 = some 2","subjects":["11"],"theorem":"OeisA159829.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«159829»","statement":"OeisA159829.a 4 = some 2","subjects":["11"],"theorem":"OeisA159829.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«159829»","statement":"OeisA159829.a 1 = some 1","subjects":["11"],"theorem":"OeisA159829.a_1"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Conjecture 1: For any $k \\ge 3$, there are infinitely many primes of the form $n^k + m^k$\nfor $n, m \\ge 1$.\n- _Ulrich Krug_, 2009\n\nAnswer: No.\n- _Kenta Kitamura_, 2026\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/oeis-a159829-conjecture1-counterexample/blob/6632e626baa7f28ad14045aa7408a84178ec128d/lean/A159829Conjecture1FC.lean#L52-L61"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«159829»","statement":"False ↔ ∀ (k : ℕ), 3 ≤ k → {p | ∃ n m, 1 ≤ n ∧ 1 ≤ m ∧ Nat.Prime p ∧ p = n ^ k + m ^ k}.Infinite","subjects":["11"],"theorem":"OeisA159829.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«159829»","statement":"OeisA159829.a 3 = some 1","subjects":["11"],"theorem":"OeisA159829.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«300997»","statement":"OeisA300997.a 3 = 3","subjects":["11"],"theorem":"OeisA300997.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«300997»","statement":"OeisA300997.a 2 = 1","subjects":["11"],"theorem":"OeisA300997.a_2"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: It appears that the finite difference of this sequence only contains 1's and 2's.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/300997.wip.lean#L1153"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«300997»","statement":"∀ (n : ℕ), 1 ≤ n → OeisA300997.a (n + 1) = OeisA300997.a n + 1 ∨ OeisA300997.a (n + 1) = OeisA300997.a n + 2","subjects":["11"],"theorem":"OeisA300997.a_succ_eq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«300997»","statement":"OeisA300997.a 5 = 6","subjects":["11"],"theorem":"OeisA300997.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«300997»","statement":"OeisA300997.a 4 = 4","subjects":["11"],"theorem":"OeisA300997.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«300997»","statement":"OeisA300997.a 1 = 0","subjects":["11"],"theorem":"OeisA300997.a_1"},{"answerKinds":[],"category":"research open","docstring":"Wolfgang Haken (1977) conjectured that no term of this sequence is a perfect square,\nand estimated the probability that this conjecture is false to be smaller than $10^-9$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«11545»","statement":"∀ (n : ℕ), ¬IsSquare (OeisA11545.a n)","subjects":["11"],"theorem":"OeisA11545.conjecture1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«11545»","statement":"OeisA11545.a 1 = 31","subjects":["11"],"theorem":"OeisA11545.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«11545»","statement":"OeisA11545.a 3 = 3141","subjects":["11"],"theorem":"OeisA11545.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«11545»","statement":"OeisA11545.a 0 = 3","subjects":["11"],"theorem":"OeisA11545.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«11545»","statement":"OeisA11545.a 2 = 314","subjects":["11"],"theorem":"OeisA11545.a_2"},{"answerKinds":[],"category":"research open","docstring":"Number of collisions occurring in a system consisting of an infinitely massive,\nrigid wall at the origin, a ball with mass m stationary at position $x_1 > 0$,\nand a ball with mass $(10^2n)m$ at position $x_2 > x_1$ and rolling toward the origin,\nassuming perfectly elastic collisions and no friction.\n\nStrictly speaking, this property, which is equivalent to the statement that the interval\n$(m\\pi, \\pi/\\textrm{arctan}(1/m))$ does not contain an integer for all $m = 10^n$, is not\nknown to be true for sure. In other words, we do not know for certain that A332045 does not\ncontain a power of $10$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«11545»","statement":"∀ (n : ℕ), ¬∃ k, Real.pi * 10 ^ ↑n < ↑k ∧ ↑k < Real.pi / Real.arctan (1 / 10 ^ ↑n)","subjects":["11"],"theorem":"OeisA11545.conjecture2"},{"answerKinds":[],"category":"research open","docstring":"A038552 also gives the largest absolute value of negative fundamental discriminant\nfor each class number. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38552»","statement":"∀ {n k : ℕ}, OeisA38552.IsA038552 n k → OeisA38552.IsLargestNegFundDiscrForClassNumber k","subjects":["11"],"theorem":"OeisA38552.isA038552_eq_largestNegFundDisc"},{"answerKinds":[],"category":"research solved","docstring":"The Stark-Heegner theorem [Sta67] implies that the squarefree $k > 0$ such that\n$\\mathbb{Q}(\\sqrt{-k})$ has class number $1$ are exactly $\\{1, 2, 3, 7, 11, 19, 43, 67, 163\\}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38552»","statement":"{k | Squarefree k ∧ OeisA38552.HasClassNumber k 1} = {1, 2, 3, 7, 11, 19, 43, 67, 163}","subjects":["11"],"theorem":"OeisA38552.starkHeegner_classNumberOne"},{"answerKinds":[],"category":"research open","docstring":"All terms of A038552 are congruent to $19 \\pmod{24}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38552»","statement":"∀ {n k : ℕ}, OeisA38552.IsA038552 n k → k % 24 = 19","subjects":["11"],"theorem":"OeisA38552.mod_24_of_isA038552"},{"answerKinds":[],"category":"API","docstring":"$\\mathbb{Q}(\\sqrt{-163})$ has class number $1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38552»","statement":"OeisA38552.HasClassNumber 163 1","subjects":["11"],"theorem":"OeisA38552.hasClassNumber_163_1"},{"answerKinds":[],"category":"research open","docstring":"For even class number $n$, the $n$-th term of A038552 is odd. The source states this as:\nthe largest odd squarefree $k$ with $h(-k) = n$ is greater than the largest even one. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«38552»","statement":"∀ {n k : ℕ}, Even n → OeisA38552.IsA038552 n k → Odd k","subjects":["11"],"theorem":"OeisA38552.odd_of_isA038552"},{"answerKinds":[],"category":"test","docstring":"$163$ is the largest squarefree $k$ with class number $1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«38552»","statement":"OeisA38552.IsA038552 1 163","subjects":["11"],"theorem":"OeisA38552.isA038552_1_163"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110566»","statement":"OeisA110566.a 2 = 1","subjects":["11"],"theorem":"OeisA110566.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110566»","statement":"OeisA110566.a 5 = 1","subjects":["11"],"theorem":"OeisA110566.a_5"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that every odd number occurs in this sequence.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«110566»","statement":"∀ (m : ℕ), Odd m → ∃ n > 0, OeisA110566.a n = m","subjects":["11"],"theorem":"OeisA110566.conjecture"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110566»","statement":"OeisA110566.a 1 = 1","subjects":["11"],"theorem":"OeisA110566.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110566»","statement":"OeisA110566.a 3 = 1","subjects":["11"],"theorem":"OeisA110566.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110566»","statement":"OeisA110566.a 4 = 1","subjects":["11"],"theorem":"OeisA110566.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110566»","statement":"OeisA110566.a 6 = 3","subjects":["11"],"theorem":"OeisA110566.a_6"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71524»","statement":"OeisA71524.a 4 = 1","subjects":["11"],"theorem":"OeisA71524.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture (Generalization): For every $m \\in \\mathbb{N}$, the determinant `generalDet m n` is\nnonzero for all sufficiently large $n$.\n- _Zhi-Wei Sun_, Aug 26-27 2013\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«71524»","statement":"∀ (m : ℕ), ∃ N, ∀ (n : ℕ), N < n → OeisA71524.generalDet m n ≠ 0","subjects":["11","15"],"theorem":"OeisA71524.conjecture2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) = 0$ for no $n > 28$.\n- _Zhi-Wei Sun_, Aug 26 2013\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«71524»","statement":"∀ (n : ℕ), 28 < n → OeisA71524.a n ≠ 0","subjects":["11","15"],"theorem":"OeisA71524.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71524»","statement":"OeisA71524.a 3 = -1","subjects":["11"],"theorem":"OeisA71524.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71524»","statement":"OeisA71524.a 5 = 1","subjects":["11"],"theorem":"OeisA71524.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71524»","statement":"OeisA71524.a 1 = 1","subjects":["11"],"theorem":"OeisA71524.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«71524»","statement":"OeisA71524.a 2 = -1","subjects":["11"],"theorem":"OeisA71524.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2426»","statement":"OeisA2426.a 1 = 1","subjects":["11"],"theorem":"OeisA2426.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2426»","statement":"OeisA2426.a 3 = 7","subjects":["11"],"theorem":"OeisA2426.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2426»","statement":"OeisA2426.a 0 = 1","subjects":["11"],"theorem":"OeisA2426.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2426»","statement":"OeisA2426.a 4 = 19","subjects":["11"],"theorem":"OeisA2426.a_4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2426»","statement":"OeisA2426.a 2 = 3","subjects":["11"],"theorem":"OeisA2426.a_2"},{"answerKinds":[],"category":"research open","docstring":"An integer $n > 3$ is prime if and only if $a(n) \\equiv 1 \\pmod{n^2}$.\nWe have verified this for $n$ up to $8 \\cdot 10^5$, and proved that $a(p) \\equiv 1 \\pmod{p^2}$\nfor any prime $p > 3$ (cf. A277640).\n- Zhi-Wei Sun, Nov 30 2016","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«2426»","statement":"∀ (n : ℕ), 3 < n → (Nat.Prime n ↔ OeisA2426.a n ≡ 1 [MOD n ^ 2])","subjects":["11"],"theorem":"OeisA2426.conjecture"},{"answerKinds":[],"category":"test","docstring":"a(1) = 15. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87719»","statement":"OeisA87719.a 1 = 15","subjects":["11"],"theorem":"OeisA87719.a_1"},{"answerKinds":[],"category":"textbook","docstring":"We have the following formula: $a(n) = 3^n + 3 * 2^n + 6$ for $n \\geq 1$. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/pull/1894/commits/7a286754f623759d69a3dd18f482c53c1d70959b"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«87719»","statement":"∀ {n : ℕ}, n ≥ 1 → OeisA87719.a n = 3 ^ n + 3 * 2 ^ n + 6","subjects":["11"],"theorem":"OeisA87719.a_formula"},{"answerKinds":[],"category":"textbook","docstring":"There exists m such that countExceeding n m > countNotExceeding n m. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«87719»","statement":"∀ (n : ℕ), ∃ m, OeisA87719.countExceeding n m > OeisA87719.countNotExceeding n m","subjects":["11"],"theorem":"OeisA87719.a_exists"},{"answerKinds":[],"category":"test","docstring":"a(2) = 27. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87719»","statement":"OeisA87719.a 2 = 27","subjects":["11"],"theorem":"OeisA87719.a_2"},{"answerKinds":[],"category":"test","docstring":"a(3) = 57. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«87719»","statement":"OeisA87719.a 3 = 57","subjects":["11"],"theorem":"OeisA87719.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«357513»","statement":"OeisA357513.a 5 = 1335793103","subjects":["11"],"theorem":"OeisA357513.a_5"},{"answerKinds":[],"category":"research open","docstring":"We conjecture that $u(p-1) == 0 (mod p^4)$ for all primes $p$,\nwith a finite number of exceptions that depend on $m$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«357513»","statement":"∀ (m : ℕ), ∃ exceptions, ∀ (p : ℕ), Nat.Prime p → p ∉ exceptions → ↑(OeisA357513.u m (p - 1)) = 0","subjects":["11"],"theorem":"OeisA357513.general_supercongruence"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«357513»","statement":"(∀ (p : ℕ), Nat.Prime p → p ≥ 3 → p ≠ 7 → ↑(OeisA357513.a (p - 1)) ≡ 0 [ZMOD ↑p ^ 4]) →\n  ∃ exceptions, ∀ (p : ℕ), Nat.Prime p → p ∉ exceptions → ↑(OeisA357513.u 1 (p - 1)) = 0","subjects":["11"],"theorem":"OeisA357513.general_supercongruence_one_of_a357513_supercongruence"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«357513»","statement":"OeisA357513.a 4 = 956875","subjects":["11"],"theorem":"OeisA357513.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«357513»","statement":"OeisA357513.a 0 = 0","subjects":["11"],"theorem":"OeisA357513.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«357513»","statement":"OeisA357513.a 3 = 14651","subjects":["11"],"theorem":"OeisA357513.a_3"},{"answerKinds":[],"category":"research solved","docstring":"We have  $a(p-1) \\equiv 0 \\pmod{p^4}$ for all primes $p \\ge 3$ except $p=7$.\n\nproved by AlphaProof\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/commit/9c7f21e7d4445637538bc1817b058b9b3f31bd2b"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«357513»","statement":"∀ (p : ℕ), Nat.Prime p → p ≥ 3 → p ≠ 7 → ↑(OeisA357513.a (p - 1)) ≡ 0 [ZMOD ↑p ^ 4]","subjects":["11"],"theorem":"OeisA357513.a357513_supercongruence"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«357513»","statement":"OeisA357513.a 1 = 4","subjects":["11"],"theorem":"OeisA357513.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«357513»","statement":"OeisA357513.a 2 = 81","subjects":["11"],"theorem":"OeisA357513.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«306424»","statement":"OeisA306424.a 4 = 4","subjects":["11"],"theorem":"OeisA306424.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«306424»","statement":"OeisA306424.a 2 = 2","subjects":["11"],"theorem":"OeisA306424.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«306424»","statement":"OeisA306424.a 1 = 1","subjects":["11"],"theorem":"OeisA306424.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«306424»","statement":"OeisA306424.a 5 = 5","subjects":["11"],"theorem":"OeisA306424.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«306424»","statement":"OeisA306424.a 3 = 3","subjects":["11"],"theorem":"OeisA306424.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: The sequence is finite, with 43 being the last term.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/306424.wip.lean#L276"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«306424»","statement":"OeisA306424.Condition 43 ∧ ∀ (k : ℕ), 43 < k → ¬OeisA306424.Condition k","subjects":["11"],"theorem":"OeisA306424.forty_three_is_max"},{"answerKinds":[],"category":"research open","docstring":"Is $a(n) \\le 1$ for all $n$?","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«76141»","statement":"∀ (n : ℕ), OeisA76141.a n ≤ 1","subjects":["11"],"theorem":"OeisA76141.conjecture"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76141»","statement":"OeisA76141.a 0 = 1","subjects":["11"],"theorem":"OeisA76141.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76141»","statement":"OeisA76141.a 3 = 0","subjects":["11"],"theorem":"OeisA76141.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76141»","statement":"OeisA76141.a 1 = 1","subjects":["11"],"theorem":"OeisA76141.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76141»","statement":"OeisA76141.a 2 = 1","subjects":["11"],"theorem":"OeisA76141.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 5. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76141»","statement":"OeisA76141.a 5 = 0","subjects":["11"],"theorem":"OeisA76141.a_5"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«76141»","statement":"OeisA76141.a 4 = 1","subjects":["11"],"theorem":"OeisA76141.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363347»","statement":"OeisA363347.a 4 = 5","subjects":["11"],"theorem":"OeisA363347.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363347»","statement":"OeisA363347.a 6 = 11","subjects":["11"],"theorem":"OeisA363347.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363347»","statement":"OeisA363347.a 7 = 59","subjects":["11"],"theorem":"OeisA363347.a_7"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363347»","statement":"OeisA363347.a 3 = 11","subjects":["11"],"theorem":"OeisA363347.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: The sequence contains all prime numbers which end with a 1 or 9 (i.e., primes $p \\equiv 1$ or $9 \\pmod{10}$).\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/363347.wip.lean#L825"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«363347»","statement":"∀ (p : ℕ), Nat.Prime p ∧ (p ≡ 1 [MOD 10] ∨ p ≡ 9 [MOD 10]) → ∃ n, OeisA363347.a n = p","subjects":["11"],"theorem":"OeisA363347.exists_a_eq_prime"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«363347»","statement":"OeisA363347.a 5 = 31","subjects":["11"],"theorem":"OeisA363347.a_5"},{"answerKinds":[],"category":"research open","docstring":"Counter-conjecture to `a_isBigO`: $a(n) / (\\log n \\log \\log n)$ is unbounded. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«34693»","statement":"¬BddAbove (Set.range fun n => ↑(OeisA34693.a n) / (Real.log ↑n * Real.log (Real.log ↑n)))","subjects":["11"],"theorem":"OeisA34693.a_unbounded"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34693»","statement":"OeisA34693.a 1 = 1","subjects":["11"],"theorem":"OeisA34693.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34693»","statement":"OeisA34693.a 2 = 1","subjects":["11"],"theorem":"OeisA34693.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34693»","statement":"OeisA34693.a 7 = 4","subjects":["11"],"theorem":"OeisA34693.a_7"},{"answerKinds":[],"category":"research open","docstring":"A stronger conjecture: for every n there exists a number $k < 1 + n^{0.75}$ such that\n$nk + 1$ is a prime. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«34693»","statement":"∀ {n : ℕ}, 0 < n → ∃ k, ↑k < 1 + Real.nthRoot 4 ↑n ^ 3 ∧ Nat.Prime (n * k + 1)","subjects":["11"],"theorem":"OeisA34693.exists_k_stronger"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34693»","statement":"OeisA34693.a 0 = 0","subjects":["11"],"theorem":"OeisA34693.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34693»","statement":"OeisA34693.a 3 = 2","subjects":["11"],"theorem":"OeisA34693.a_3"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: for every $n > 1$ there exists a number $k < n$ such that $nk + 1$ is a prime. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«34693»","statement":"∀ {n : ℕ}, 1 < n → ∃ k < n, Nat.Prime (n * k + 1)","subjects":["11"],"theorem":"OeisA34693.exists_k"},{"answerKinds":[],"category":"research solved","docstring":"The expression $1 + n^{0.74}$ does not work as an upper bound. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«34693»","statement":"∃ n > 0, ∀ (k : ℕ), ↑k < 1 + Real.nthRoot 100 ↑n ^ 74 → ¬Nat.Prime (n * k + 1)","subjects":["11"],"theorem":"OeisA34693.exists_k_best_possible"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) = O(\\log(n)\\log(\\log(n)))$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«34693»","statement":"(fun n => ↑(OeisA34693.a n)) =O[Filter.atTop] fun n => Real.log ↑n * Real.log (Real.log ↑n)","subjects":["11"],"theorem":"OeisA34693.a_isBigO"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2407»","statement":"OeisA2407.A 127","subjects":["11"],"theorem":"OeisA2407.a_127"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2407»","statement":"OeisA2407.A 37","subjects":["11"],"theorem":"OeisA2407.a_37"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2407»","statement":"OeisA2407.A 61","subjects":["11"],"theorem":"OeisA2407.a_61"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2407»","statement":"OeisA2407.A 7","subjects":["11"],"theorem":"OeisA2407.a_7"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2407»","statement":"¬OeisA2407.A 91","subjects":["11"],"theorem":"OeisA2407.not_a_91"},{"answerKinds":[],"category":"research open","docstring":"This sequence is believed to be infinite.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«2407»","statement":"{p | OeisA2407.A p}.Infinite","subjects":["11"],"theorem":"OeisA2407.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«2407»","statement":"OeisA2407.A 19","subjects":["11"],"theorem":"OeisA2407.a_19"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«231201»","statement":"OeisA231201.A 53","subjects":["11"],"theorem":"OeisA231201.a_53"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«231201»","statement":"OeisA231201.A 2","subjects":["11"],"theorem":"OeisA231201.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«231201»","statement":"OeisA231201.A 4","subjects":["11"],"theorem":"OeisA231201.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«231201»","statement":"OeisA231201.A 8","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"OeisA231201.a_8"},{"answerKinds":[],"category":"research open","docstring":"The conjecture for sequence A231201: for any $n > 1$, there exist $x, y > 0$ such that $n = x + y$ and $2^x + y$ is prime. ","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«231201»","statement":"∀ (n : ℕ), 1 < n → OeisA231201.A n","subjects":["11"],"theorem":"OeisA231201.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«231201»","statement":"OeisA231201.A 3","subjects":["11"],"theorem":"OeisA231201.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«231201»","statement":"OeisA231201.A 5","subjects":["11"],"theorem":"OeisA231201.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109908»","statement":"OeisA109908.a 5 = 5","subjects":["11"],"theorem":"OeisA109908.a_5"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) > 0$ for $n > 3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«109908»","statement":"∀ n > 3, OeisA109908.a n > 0","subjects":["11"],"theorem":"OeisA109908.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109908»","statement":"OeisA109908.a 2 = 0","subjects":["11"],"theorem":"OeisA109908.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109908»","statement":"OeisA109908.a 3 = 0","subjects":["11"],"theorem":"OeisA109908.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109908»","statement":"OeisA109908.a 1 = 0","subjects":["11"],"theorem":"OeisA109908.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«109908»","statement":"OeisA109908.a 4 = 3","subjects":["11"],"theorem":"OeisA109908.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«358684»","statement":"OeisA358684.a 5 = 23","subjects":["11"],"theorem":"OeisA358684.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«358684»","statement":"OeisA358684.a 2 = 0","subjects":["11"],"theorem":"OeisA358684.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«358684»","statement":"OeisA358684.a 7 = 73","subjects":["11"],"theorem":"OeisA358684.a_7"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«358684»","statement":"OeisA358684.a 3 = 0","subjects":["11"],"theorem":"OeisA358684.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«358684»","statement":"OeisA358684.a 0 = 0","subjects":["11"],"theorem":"OeisA358684.a_0"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: the dyadic valuation of A93179(n) - 1 does not exceed 2^n - a(n).\n\nA93179(n) is minFac(fermatNumber n), the smallest prime factor of the n-th Fermat number.\nThe conjecture states that if $P_n$ is the smallest prime factor of the $n$-th Fermat number,\nthen $\\nu_2(P_n - 1) \\le 2^n - a(n)$.\nSubstituting the definition of $a(n)$, this is equivalent to $\\nu_2(P_n - 1) \\le \\lfloor \\log_2(P_n) \\rfloor$.\n\nThis is Conjecture 3.4 in [SA22].\n","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«358684»","statement":"∀ (n : ℕ), padicValNat 2 (n.fermatNumber.minFac - 1) ≤ 2 ^ n - OeisA358684.a n","subjects":["11"],"theorem":"OeisA358684.oeis_358684_conjecture_0"},{"answerKinds":[],"category":"API","docstring":"The minimization definition is equivalent to the closed form.\n","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«358684»","statement":"∀ (n : ℕ), OeisA358684.a n = OeisA358684.a' n","subjects":["11"],"theorem":"OeisA358684.a_equiv_a'"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«358684»","statement":"OeisA358684.a 1 = 0","subjects":["11"],"theorem":"OeisA358684.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«358684»","statement":"OeisA358684.a 4 = 0","subjects":["11"],"theorem":"OeisA358684.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«358684»","statement":"OeisA358684.a 6 = 46","subjects":["11"],"theorem":"OeisA358684.a_6"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«341254»","statement":"OeisA341254.a 1 = 4","subjects":["11"],"theorem":"OeisA341254.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«341254»","statement":"OeisA341254.a 3 = 12","subjects":["11"],"theorem":"OeisA341254.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«341254»","statement":"OeisA341254.a 2 = 8","subjects":["11"],"theorem":"OeisA341254.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«341254»","statement":"OeisA341254.a 5 = 21","subjects":["11"],"theorem":"OeisA341254.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«341254»","statement":"OeisA341254.a 4 = 16","subjects":["11"],"theorem":"OeisA341254.a_4"},{"answerKinds":[],"category":"research solved","docstring":"Conjecture: $1/4 < n r^2 - a(n) < 3$ for $n \\ge 1$, where $r = (2 + \\sqrt{5})/2$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/341254.wip.lean#L188"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«341254»","statement":"∀ (n : ℕ), 1 ≤ n → 1 / 4 < ↑n * OeisA341254.rSq - ↑(OeisA341254.a n) ∧ ↑n * OeisA341254.rSq - ↑(OeisA341254.a n) < 3","subjects":["11"],"theorem":"OeisA341254.a_bounds"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105210»","statement":"OeisA105210.a 1 = 393","subjects":["11"],"theorem":"OeisA105210.a_1"},{"answerKinds":[],"category":"research open","docstring":"Cormier and Selfridge found 5 starting values for which the sequences appear to not merge.\nThe sequences were checked up to 10^8.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«105210»","statement":"∀ (j k : ℕ),\n  j ∈ {1, 393, 412, 668, 932} →\n    k ∈ {1, 393, 412, 668, 932} → j ≠ k → OeisA105210.sequenceSet j ∩ OeisA105210.sequenceSet k = ∅","subjects":["11"],"theorem":"OeisA105210.conjecture_disjoint_starting_values"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105210»","statement":"OeisA105210.a 3 = 545","subjects":["11"],"theorem":"OeisA105210.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105210»","statement":"OeisA105210.a 2 = 528","subjects":["11"],"theorem":"OeisA105210.a_2"},{"answerKinds":[],"category":"research open","docstring":"This suggests that there may be infinitely many different (non-merging) sequences obtained\nby choosing different starting values.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«105210»","statement":"∃ K,\n  K.Infinite ∧\n    (∀ k ∈ K, 1 ≤ k) ∧ ∀ (j k : ℕ), j ∈ K → k ∈ K → j ≠ k → OeisA105210.sequenceSet j ∩ OeisA105210.sequenceSet k = ∅","subjects":["11"],"theorem":"OeisA105210.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105210»","statement":"OeisA105210.a 5 = 682","subjects":["11"],"theorem":"OeisA105210.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105210»","statement":"OeisA105210.a 4 = 660","subjects":["11"],"theorem":"OeisA105210.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7013»","statement":"OeisA7013.a 4 = 170141183460469231731687303715884105727","subjects":["11"],"theorem":"OeisA7013.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7013»","statement":"OeisA7013.a 2 = 7","subjects":["11"],"theorem":"OeisA7013.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7013»","statement":"OeisA7013.a 1 = 3","subjects":["11"],"theorem":"OeisA7013.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7013»","statement":"OeisA7013.a 3 = 127","subjects":["11"],"theorem":"OeisA7013.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«7013»","statement":"OeisA7013.a 0 = 2","subjects":["11"],"theorem":"OeisA7013.a_0"},{"answerKinds":[],"category":"research open","docstring":"Catalan-Mersenne conjecture: All terms of the Catalan-Mersenne sequence are prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«7013»","statement":"∀ (n : ℕ), Nat.Prime (OeisA7013.a n)","subjects":["11"],"theorem":"OeisA7013.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«368692»","statement":"OeisA368692.a 3 = 6700034035890000","subjects":["11"],"theorem":"OeisA368692.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«368692»","statement":"OeisA368692.a 2 = 54231252075","subjects":["11"],"theorem":"OeisA368692.a_2"},{"answerKinds":[],"category":"research solved","docstring":"It is conjectured here that $a(n)$ are integers.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/368692.wip.lean#L158"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«368692»","statement":"∀ (n : ℕ),\n  108 * ((4 * n + 2).factorial * (2 * n + 3).factorial * (6 * n + 5).factorial ^ 2) ∣\n    (12 * n + 6).factorial * (6 * n + 9).factorial","subjects":["11"],"theorem":"OeisA368692.a_is_int"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«368692»","statement":"OeisA368692.a 4 = 928978310614152999200","subjects":["11"],"theorem":"OeisA368692.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«368692»","statement":"OeisA368692.a 1 = 563108","subjects":["11"],"theorem":"OeisA368692.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«368692»","statement":"OeisA368692.a 0 = 14","subjects":["11"],"theorem":"OeisA368692.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«84046»","statement":"OeisA84046.a 2 = 2","subjects":["11"],"theorem":"OeisA84046.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«84046»","statement":"OeisA84046.a 3 = 5","subjects":["11"],"theorem":"OeisA84046.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«84046»","statement":"OeisA84046.a 1 = 2","subjects":["11"],"theorem":"OeisA84046.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«84046»","statement":"OeisA84046.a 0 = 0","subjects":["11"],"theorem":"OeisA84046.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: if a(k) = 0 then k is an even square.","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«84046»","statement":"∀ (k : ℕ), OeisA84046.a k = 0 → ∃ m, k = (2 * m) ^ 2","subjects":["11"],"theorem":"OeisA84046.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113213»","statement":"OeisA113213.a 4 = 3","subjects":["11"],"theorem":"OeisA113213.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113213»","statement":"OeisA113213.a 1 = 0","subjects":["11"],"theorem":"OeisA113213.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113213»","statement":"OeisA113213.a 3 = 3","subjects":["11"],"theorem":"OeisA113213.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113213»","statement":"OeisA113213.a 2 = 1","subjects":["11"],"theorem":"OeisA113213.a_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: $a(n) = O(n^3)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«113213»","statement":"(fun n => ↑(OeisA113213.a n)) =O[Filter.atTop] fun n => ↑n ^ 3","subjects":["11"],"theorem":"OeisA113213.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«113213»","statement":"OeisA113213.a 5 = 9","subjects":["11"],"theorem":"OeisA113213.a_5"},{"answerKinds":[],"category":"research open","docstring":"\"Conjecture: $1/\\det(M)$ is an integer only for n: 1 to 34, 36 and 38.\nAll denominators are powers of two (A000079). - _Robert G. Wilson v_, Aug 02 2015\"","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«60841»","statement":"(∀ (n : ℕ), 1 ≤ n → ((OeisA60841.lcmMatrix n).det⁻¹.den = 1 ↔ n ∈ OeisA60841.integerDetN)) ∧\n  ∀ (n : ℕ), 1 ≤ n → ∃ k, (OeisA60841.lcmMatrix n).det⁻¹.den = 2 ^ k","subjects":["11","15"],"theorem":"OeisA60841.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60841»","statement":"OeisA60841.a 3 = 18","subjects":["11","15"],"theorem":"OeisA60841.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60841»","statement":"OeisA60841.a 1 = 1","subjects":["11","15"],"theorem":"OeisA60841.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60841»","statement":"OeisA60841.a 2 = 4","subjects":["11","15"],"theorem":"OeisA60841.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60841»","statement":"OeisA60841.a 5 = 900","subjects":["11","15"],"theorem":"OeisA60841.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60841»","statement":"OeisA60841.a 4 = 144","subjects":["11","15"],"theorem":"OeisA60841.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60957»","statement":"OeisA60957.a 1 = 1","subjects":["5","11"],"theorem":"OeisA60957.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60957»","statement":"OeisA60957.a 0 = 1","subjects":["5","11"],"theorem":"OeisA60957.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60957»","statement":"OeisA60957.a 4 = 8","subjects":["5","11"],"theorem":"OeisA60957.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60957»","statement":"OeisA60957.a 2 = 2","subjects":["5","11"],"theorem":"OeisA60957.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60957»","statement":"OeisA60957.a 5 = 16","subjects":["5","11"],"theorem":"OeisA60957.a_5"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: let $p \\le n$ be prime. If $m$ and $p^a m$ are two such products, then so is $p^k m$\nfor all $0 < k < a$.\n- Yan Sheng Ang, Feb 13 2020\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«60957»","statement":"∀ (n p : ℕ),\n  Nat.Prime p →\n    p ≤ n →\n      ∀ (m a_exp : ℕ),\n        m ∈ OeisA60957.productsOfSubsets n →\n          p ^ a_exp * m ∈ OeisA60957.productsOfSubsets n →\n            ∀ (k : ℕ), 0 < k → k < a_exp → p ^ k * m ∈ OeisA60957.productsOfSubsets n","subjects":["5","11"],"theorem":"OeisA60957.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«60957»","statement":"OeisA60957.a 3 = 4","subjects":["5","11"],"theorem":"OeisA60957.a_3"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«62567»","statement":"∀ (k : ℕ), k ≠ 0 → k < 10 → OeisA62567.reverseNat k = k","subjects":["11"],"theorem":"OeisA62567.reverseNat_of_lt"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«62567»","statement":"OeisA62567.a 4 = 4","subjects":["11"],"theorem":"OeisA62567.a_4"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«62567»","statement":"∀ n > 0, n ∣ OeisA62567.reverseNat n → OeisA62567.a n = n","subjects":["11"],"theorem":"OeisA62567.a_eq_self"},{"answerKinds":[],"category":"research solved","docstring":"The conjecture that $a(3^n) = 10^{3^{n-2}} - 1$ for $n > 1$ was shown to be false for $4 < n < 21$\nby Farideh Firoozbakht, who conjectured that for all $n > 4$, $a(3^n) \\neq 10^{3^{n-2}} - 1$.\nThis latter conjecture is proved here.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/62567.wip.lean#L324"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«62567»","statement":"∀ (n : ℕ), 2 ≤ n → (OeisA62567.a (3 ^ n) = 10 ^ 3 ^ (n - 2) - 1 ↔ n = 2 ∨ n = 3 ∨ n = 4)","subjects":["11"],"theorem":"OeisA62567.a_three_pow_eq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«62567»","statement":"OeisA62567.a 1 = 1","subjects":["11"],"theorem":"OeisA62567.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«62567»","statement":"OeisA62567.a 3 = 3","subjects":["11"],"theorem":"OeisA62567.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«62567»","statement":"OeisA62567.a 2 = 2","subjects":["11"],"theorem":"OeisA62567.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«62567»","statement":"OeisA62567.a 5 = 5","subjects":["11"],"theorem":"OeisA62567.a_5"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108866»","statement":"OeisA108866.a 1 = 2","subjects":["11"],"theorem":"OeisA108866.a_1"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108866»","statement":"OeisA108866.a 0 = 0","subjects":["11"],"theorem":"OeisA108866.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108866»","statement":"OeisA108866.a 4 = 32","subjects":["11"],"theorem":"OeisA108866.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108866»","statement":"OeisA108866.a 2 = 4","subjects":["11"],"theorem":"OeisA108866.a_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: for $n > 3$,\n$\\textrm{numerator}(-2/n + \\sum_{k=1}^{n} \\frac{2^k}{k}) == 0 (\\textrm{mod} n^2)$\nif and only if n is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«108866»","statement":"∀ {n : ℕ}, n > 3 → ((OeisA108866.ratExpression n).num ≡ 0 [ZMOD ↑n ^ 2] ↔ Nat.Prime n)","subjects":["11"],"theorem":"OeisA108866.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«108866»","statement":"OeisA108866.a 3 = 20","subjects":["11"],"theorem":"OeisA108866.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103425»","statement":"OeisA103425.a 1 = 3","subjects":["11"],"theorem":"OeisA103425.a_1"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103425»","statement":"OeisA103425.a 0 = 1","subjects":["11"],"theorem":"OeisA103425.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103425»","statement":"OeisA103425.a 4 = 41","subjects":["11"],"theorem":"OeisA103425.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103425»","statement":"OeisA103425.a 2 = 5","subjects":["11"],"theorem":"OeisA103425.a_2"},{"answerKinds":[],"category":"research open","docstring":"The current sequence contains primes, including $3, 5, 41, 21523361$.\nIs there an $(a, b, c)$ weighted tribonacci sequence with $a, b, c$ relatively prime\nwhich is prime-free?\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«103425»","statement":"True ↔\n  ∃ a b c x, (a.gcd b).gcd c.natAbs = 1 ∧ OeisA103425.IsWeightedTribonacci a b c x ∧ ∀ (n : ℕ), ¬Nat.Prime (x n).natAbs","subjects":["11"],"theorem":"OeisA103425.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«103425»","statement":"OeisA103425.a 3 = 15","subjects":["11"],"theorem":"OeisA103425.a_3"},{"answerKinds":[],"category":"research open","docstring":"**Zhi-Wei Sun's 1680-Conjecture (A280831)**: Any nonnegative integer can be written as\n$x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«280831»","statement":"∀ (n : ℕ), OeisA280831.A n","subjects":["11"],"theorem":"OeisA280831.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«280831»","statement":"OeisA280831.A 0","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"OeisA280831.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«280831»","statement":"OeisA280831.A 3","subjects":["11"],"theorem":"OeisA280831.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«280831»","statement":"OeisA280831.A 1","subjects":["11"],"theorem":"OeisA280831.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«280831»","statement":"OeisA280831.A 2","subjects":["11"],"theorem":"OeisA280831.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«280831»","statement":"OeisA280831.A 7","subjects":["11"],"theorem":"OeisA280831.a_7"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«280831»","statement":"OeisA280831.A 4","subjects":["11"],"theorem":"OeisA280831.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«280831»","statement":"OeisA280831.A 95","subjects":["11"],"theorem":"OeisA280831.a_95"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105720»","statement":"OeisA105720.a 3 = 36","subjects":["11"],"theorem":"OeisA105720.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105720»","statement":"OeisA105720.a 1 = 5","subjects":["11"],"theorem":"OeisA105720.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105720»","statement":"OeisA105720.a 2 = 15","subjects":["11"],"theorem":"OeisA105720.a_2"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105720»","statement":"OeisA105720.a 4 = 67","subjects":["11"],"theorem":"OeisA105720.a_4"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105720»","statement":"Nat.nth Nat.Prime 7 = 19","subjects":["11"],"theorem":"OeisA105720.nth_prime_seven"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105720»","statement":"Nat.nth Nat.Prime 6 = 17","subjects":["11"],"theorem":"OeisA105720.nth_prime_six"},{"answerKinds":[],"category":"research open","docstring":"Terms are squares at only(?) three values of $n = 3, 6, 4072$:\ncorresponding terms are 6^2, 13^2, and 15735^2.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«105720»","statement":"∀ (n : ℕ), 0 < n → (IsSquare (OeisA105720.a n) ↔ n = 3 ∨ n = 6 ∨ n = 4072)","subjects":["11"],"theorem":"OeisA105720.conjecture"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105720»","statement":"Nat.nth Nat.Prime 5 = 13","subjects":["11"],"theorem":"OeisA105720.nth_prime_five"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105720»","statement":"OeisA105720.a 0 = 0","subjects":["11"],"theorem":"OeisA105720.a_0"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 4. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«153330»","statement":"OeisA153330.a 4 = some 3","subjects":["11"],"theorem":"OeisA153330.a_4"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 2: 1, 6 and 16 appear only once and 3 appears twice in the sequence,\ni.e., $a(1) = 1$, $a(2) = 6$, $a(4) = a(5) = 3$, and $a(8) = 16$.\n- _Ya-Ping Lu_, May 04 2024\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«153330»","statement":"OeisA153330.indices 1 = {1} ∧\n  OeisA153330.indices 6 = {2} ∧ OeisA153330.indices 16 = {8} ∧ OeisA153330.indices 3 = {4, 5}","subjects":["11"],"theorem":"OeisA153330.conjecture2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 1: More than half of the terms are 0.\n- _Ya-Ping Lu_, May 04 2024\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«153330»","statement":"1 / 2 < Filter.liminf (fun n => ↑{i ∈ Finset.Icc 1 n | OeisA153330.a i = some 0}.card / ↑n) Filter.atTop","subjects":["11"],"theorem":"OeisA153330.conjecture1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«153330»","statement":"OeisA153330.a 0 = none","subjects":["11"],"theorem":"OeisA153330.a_0"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 4 (Ya-Ping Lu, 2024):\nThe ratio of the number of terms with value $m$ to that of $-m$ approaches 1 as $n \\to \\infty$,\nfor any $m \\notin \\{1, 3, 6, 16\\}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«153330»","statement":"∀ (m : ℤ),\n  m ≠ 1 ∧ m ≠ 3 ∧ m ≠ 6 ∧ m ≠ 16 →\n    Filter.Tendsto\n      (fun n =>\n        ↑{i ∈ Finset.Icc 1 n | OeisA153330.a i = some m}.card /\n          ↑{i ∈ Finset.Icc 1 n | OeisA153330.a i = some (-m)}.card)\n      Filter.atTop (nhds 1)","subjects":["11"],"theorem":"OeisA153330.conjecture4"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«153330»","statement":"OeisA153330.a 1 = some 1","subjects":["11"],"theorem":"OeisA153330.a_1"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 3. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«153330»","statement":"OeisA153330.a 3 = some (-5)","subjects":["11"],"theorem":"OeisA153330.a_3"},{"answerKinds":[],"category":"test","docstring":"Value of the sequence `a` at 2. ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«153330»","statement":"OeisA153330.a 2 = some 6","subjects":["11"],"theorem":"OeisA153330.a_2"},{"answerKinds":[],"category":"research open","docstring":"Conjecture 3 (Ya-Ping Lu, 2024):\nExcept 1, 3 and 6, the absolute value of all terms can be written as $5x + 8y$ for $x, y \\in \\mathbb{N}$.\n(Note: in the OEIS comment, \"x and y are integers\" means $x$ and $y$ have the same sign,\ni.e., $|v| = 5x + 8y$ with $x, y \\ge 0$, since every integer is a $\\mathbb{Z}$-linear combination of 5 and 8).\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«153330»","statement":"∀ (n : ℕ) (v : ℤ), 0 < n → OeisA153330.a n = some v → v ≠ 1 ∧ v ≠ 3 ∧ v ≠ 6 → ∃ x y, v.natAbs = 5 * x + 8 * y","subjects":["11"],"theorem":"OeisA153330.conjecture3"},{"answerKinds":[],"category":"research open","docstring":"Conjecture: For any positive integer n, the polynomials Sum_{k=0}^n binomial(2k,k)^2*x^k\nand Sum_{k=0}^n binomial(2k,k)^2*x^k/(k+1) are irreducible over the field of rational numbers.\n- Zhi-Wei Sun, Mar 23 2013\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«115257»","statement":"∀ (n : ℕ), 1 ≤ n → Irreducible (OeisA115257.polyP n) ∧ Irreducible (OeisA115257.polyQ n)","subjects":["11"],"theorem":"OeisA115257.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«115257»","statement":"OeisA115257.a 2 = 41","subjects":["11"],"theorem":"OeisA115257.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«115257»","statement":"OeisA115257.a 0 = 1","subjects":["11"],"theorem":"OeisA115257.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«115257»","statement":"OeisA115257.a 1 = 5","subjects":["11"],"theorem":"OeisA115257.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«115257»","statement":"OeisA115257.a 3 = 441","subjects":["11"],"theorem":"OeisA115257.a_3"},{"answerKinds":[],"category":"research solved","docstring":"Moll's conjecture 5.5 extends to this sequence and takes the form:\n(i) the $2$-adic valuation $\\nu_2(a(n)) \\sim n/4$ as n -> oo.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/KitaKen1/oeis-a105751-two-adic/blob/a9692d90078c4dff1478acaaa70be2fba06a3ef9/lean/OeisA105751TwoAdicFC.lean#L980-L985"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«105751»","statement":"Filter.Tendsto (fun n => 4 * ↑(padicValInt 2 (OeisA105751.a n)) / ↑n) Filter.atTop (nhds 1)","subjects":["11"],"theorem":"OeisA105751.conjecture"},{"answerKinds":[],"category":"research open","docstring":"Moll's conjecture 5.5 extends to this sequence and takes the form:\n(ii) for the other primes of type $2$, the p-adic valuation\n$\\nu_p(a(n)) \\sim n/(p - 1)$ as $n \\rightarrow \\infty$.\n\n(Type 2 primes consists of primes p == 1 (mod 4))\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«105751»","statement":"∀ {p : ℕ},\n  Nat.Prime p →\n    p % 4 = 1 → Filter.Tendsto (fun n => (↑p - 1) * ↑(padicValInt p (OeisA105751.a n)) / ↑n) Filter.atTop (nhds 1)","subjects":["11"],"theorem":"OeisA105751.conjecture.variants.moll_p_mod_4_eq_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105751»","statement":"OeisA105751.a 2 = 3","subjects":["11"],"theorem":"OeisA105751.a_2"},{"answerKinds":[],"category":"textbook","docstring":"Numerical calculation suggests that a similar division holds in this case.\nType 1: primes p that do not divide any element of the sequence {$a(n)$}.\nIn this case, unlike in A105750, the set of type 1 primes is empty;\nthat is, every prime p divides some term of this sequence.\n\nNote: The triangular number n*(n+1)/2 divides $a(n)$ (see A164652).\nIn particular, if $p$ is an odd prime then $p$ divides $a(p)$. For $p=2$, $2$ divides $a(4)=-40$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«105751»","statement":"∀ (p : ℕ), Nat.Prime p → ∃ n, ↑p ∣ OeisA105751.a n","subjects":["11"],"theorem":"OeisA105751.prime_divides_some_term"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105751»","statement":"OeisA105751.a 3 = 0","subjects":["11"],"theorem":"OeisA105751.a_3"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105751»","statement":"OeisA105751.a 0 = 0","subjects":["11"],"theorem":"OeisA105751.a_0"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105751»","statement":"OeisA105751.a 1 = 1","subjects":["11"],"theorem":"OeisA105751.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«105751»","statement":"OeisA105751.a 4 = -40","subjects":["11"],"theorem":"OeisA105751.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303656»","statement":"OeisA303656.A 5","subjects":["11"],"theorem":"OeisA303656.a_5"},{"answerKinds":[],"category":"research open","docstring":"**Zhi-Wei Sun's Conjecture (A303656)**: Any integer $n > 1$ can be written as the sum of two\nsquares, a power of 3, and a power of 5.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«303656»","statement":"∀ (n : ℕ), 1 < n → OeisA303656.A n","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"OeisA303656.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303656»","statement":"OeisA303656.A 2","subjects":["11"],"theorem":"OeisA303656.a_2"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303656»","statement":"OeisA303656.A 3","subjects":["11"],"theorem":"OeisA303656.a_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303656»","statement":"OeisA303656.A 25","subjects":["11"],"theorem":"OeisA303656.a_25"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303656»","statement":"OeisA303656.A 4","subjects":["11"],"theorem":"OeisA303656.a_4"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«303656»","statement":"OeisA303656.A 6","subjects":["11"],"theorem":"OeisA303656.a_6"},{"answerKinds":[],"category":"research open","docstring":"Sierpinski's conjecture (1958) is precisely that $a(n) >= n$ for all $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.OEIS.«110835»","statement":"∀ n > 0, OeisA110835.a n ≥ n","subjects":["11"],"theorem":"OeisA110835.conjecture"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110835»","statement":"OeisA110835.a 2 = 4","subjects":["11"],"theorem":"OeisA110835.a_2"},{"answerKinds":[],"category":"test","docstring":"Term theorems verifying the first few values of the sequence against the official OEIS b-file ","hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110835»","statement":"OeisA110835.a 1 = 8","subjects":["11"],"theorem":"OeisA110835.a_1"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.OEIS.«110835»","statement":"OeisA110835.a 3 = 8","subjects":["11"],"theorem":"OeisA110835.a_3"},{"answerKinds":[],"category":"research open","docstring":"3) implies 2): if the coefficients of $f$ satisfy the $\\omega$-integrality condition for some superlinear $\\omega$,\nthen there exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\\mathbb{Z}[1/N]$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.LittProblems.«1»","statement":"∀ {n : ℕ} (f : PowerSeries ℚ) (g : MvRatFunc (Fin (n + 1)) ℚ),\n  LamLitt.IsSolutionOfAlgebraicODE n f g →\n    ∀ (ω : Nat.Primes → ℤ),\n      (LamLitt.omegaSuperlinear ω ∧ LamLitt.omegaIntegral ω fun x => (PowerSeries.coeff x) f) →\n        ∃ N, LamLitt.IsCoeffIntegralAdjointInvNat f N","subjects":["11","14"],"theorem":"LamLitt.lam_litt.variants.omega_integrality_implies_algebraicity"},{"answerKinds":[],"category":"research open","docstring":"2) implies 1): if the coefficients of $f$ are in $\\mathbb{Z}[1/N]$ for some $N$, then $f$ is algebraic over $\\mathbb{Q}[z]$.\nAlso the version of conjecture of Litt's problem 1 on his website.\n","hasSorryFreeProof":false,"module":"FormalConjectures.LittProblems.«1»","statement":"∀ {n : ℕ} (f : PowerSeries ℚ) (g : MvRatFunc (Fin (n + 1)) ℚ),\n  LamLitt.IsSolutionOfAlgebraicODE n f g →\n    ∀ (N : ℕ), LamLitt.IsCoeffIntegralAdjointInvNat f N → IsAlgebraic (Polynomial ℚ) f","subjects":["11","14"],"theorem":"LamLitt.lam_litt.variants.integrality_implies_algebraicity"},{"answerKinds":[],"category":"textbook","docstring":"Textbook implication: integrality (2) trivially implies ω(p)-integrality (3).\n","hasSorryFreeProof":true,"module":"FormalConjectures.LittProblems.«1»","statement":"∀ (f : PowerSeries ℚ) (N : ℕ),\n  LamLitt.IsCoeffIntegralAdjointInvNat f N →\n    ∃ ω, LamLitt.omegaSuperlinear ω ∧ LamLitt.omegaIntegral ω fun x => (PowerSeries.coeff x) f","subjects":["12"],"theorem":"LamLitt.lam_litt.variants.integrality_implies_omega_integrality"},{"answerKinds":[],"category":"research solved","docstring":"Eisenstein's theorem (1852): an algebraic power series over $\\mathbb{Q}[z]$ has bounded\ndenominators, i.e., there exists $N$ such that all coefficients lie in $\\mathbb{Z}[1/N]$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.LittProblems.«1»","statement":"∀ (f : PowerSeries ℚ), IsAlgebraic (Polynomial ℚ) f → ∃ N, LamLitt.IsCoeffIntegralAdjointInvNat f N","subjects":["12","13"],"theorem":"LamLitt.lam_litt.variants.eisenstein"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $x,y\\geq 1$ be integers such that, for all $n\\geq 1$, the set of primes dividing $x^{n}-1$ is\nequal to the set of primes dividing $y^n-1$. Must $x=y$?\n\nErdős asked this at a 1988 number theory conference in Banff.\n\nA positive answer was given by Corrales-Rodrigáñez and Schoof [CoSc97].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1214»","statement":"True ↔\n  ∀ (x y : ℕ), x ≥ 1 → y ≥ 1 → (∀ n ≥ 1, {p | Nat.Prime p ∧ p ∣ x ^ n - 1} = {p | Nat.Prime p ∧ p ∣ y ^ n - 1}) → x = y","subjects":["11"],"theorem":"Erdos1214.erdos_1214"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $A\\subseteq \\mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as\n$a_1 - a_2$ with $a_1, a_2\\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ has bounded\ngaps?\n\nThis is formalised here using the `answer(sorry)` mechanism. In order to solve this problem one\nhas to provide what the sufficient conditions are, and proof that they imply the desired condition.\nIf the condition is a solution to the problem is up to human judgement.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«332»","statement":"∀ (A : Set ℕ), sorry A → Erdos332.HasBoundedGaps (Erdos332.D_A A)","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos332.erdos_332"},{"answerKinds":[],"category":"research open","docstring":"Let $r\\geq 2$ and suppose that $A\\subseteq\\{1,\\ldots,N\\}$ is such that, for any\n$m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\\in A$.\nGive the best possible upper bound for $\\sum_{n\\in A}\\frac{1}{n}$.\n\nThe order is known — `∑ 1/n = Θ_r(log N / loglog N)` (see `erdos_538.matching_order`) —\nbut the sharp constant is not. This asks whether `maxMass r N` has a well-defined\nleading constant `c_r` in `c_r · log N / loglog N`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«538»","statement":"True ↔\n  ∀ (r : ℕ),\n    2 ≤ r →\n      ∃ c,\n        0 < c ∧\n          Filter.Tendsto (fun N => Erdos538.maxMass r N * Real.log (Real.log ↑N) / Real.log ↑N) Filter.atTop (nhds c)","subjects":["11"],"theorem":"Erdos538.erdos_538"},{"answerKinds":[],"category":"research solved","docstring":"The reciprocal sum has matching order `Θ_r(log N / loglog N)`: an explicit\nupper bound for every admissible `A`, together with a witnessing construction\nachieving the same order. This pins the order (up to the one iterated-logarithm\nfactor) but not the sharp constant asked for in `erdos_538`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-538/Research/FinalMatchingOrder.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«538»","statement":"∀ (r N : ℕ),\n  2 ≤ r →\n    2 ≤ N →\n      (∀ (A : Finset ℕ),\n          Erdos538.Admissible r N A →\n            Real.log (Real.log (↑N + 1)) * ↑(Erdos538.reciprocalMass A) ≤ 2 * ↑r * (1 + Real.log (↑N * ↑N))) ∧\n        ∃ A,\n          Erdos538.Admissible r N A ∧\n            Real.log (↑N + 1) ≤ 4 + 8192 * (↑(Nat.log 2 (Nat.log 2 N)) + 1) * ↑(Erdos538.reciprocalMass A)","subjects":["11"],"theorem":"Erdos538.erdos_538.matching_order"},{"answerKinds":[],"category":"research open","docstring":"Is every proportionately dissociated (infinite) set the union of a finite\nnumber of dissociated sets?","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«774»","statement":"True ↔\n  ∀ (A : Set ℕ), A.Infinite → A.IsProportionatelyDissociated → ∃ T, (∀ S ∈ T, S.IsDissociated) ∧ T.Finite ∧ A = ⋃₀ T","subjects":["5"],"theorem":"Erdos774.erdos_774"},{"answerKinds":[],"category":"research solved","docstring":"Every sequence with property Q has upper density at most `6 / π^2`.\n-","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/1e075c4f6e8a907b924647fa88238f978e941742/ErdosProblem1102PropertyQDensity.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1102»","statement":"∀ (A : ℕ → ℕ),\n  StrictMono A →\n    Erdos1102.HasPropertyQ (Set.range A) → Filter.limsup (fun j => ↑j / ↑(A j)) Filter.atTop ≤ 6 / Real.pi ^ 2","subjects":["11"],"theorem":"Erdos1102.erdos_1102.upper_density_Q"},{"answerKinds":[],"category":"research solved","docstring":"Conversely, for any function `f : ℕ → ℕ` that goes to infinity,\nthere exists a strictly increasing sequence `A = {a₁ < a₂ < …}`\nwith property P such that `(a_j / j) ≤ f(j)` for all `j`.\n-","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/1e075c4f6e8a907b924647fa88238f978e941742/ErdosProblem1102PropertyP.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1102»","statement":"∀ (f : ℕ → ℕ),\n  Filter.Tendsto f Filter.atTop Filter.atTop →\n    (∀ (n : ℕ), f n ≠ 0) → ∃ A, StrictMono A ∧ Erdos1102.HasPropertyP (Set.range A) ∧ ∀ (j : ℕ), ↑(A j) / ↑j ≤ ↑(f j)","subjects":["11"],"theorem":"Erdos1102.erdos_1102.exists_sequence_with_P"},{"answerKinds":[],"category":"research solved","docstring":"There exists an infinite sequence $A = {a₁ < a₂ < …} ⊂ \\mathsf{SF}$ where\n$\\mathsf{SF} := \\mathbb{N} \\setminus \\bigcup_{p} p^{2}\\mathbb{N}$, i.e. the set of\nsquarefree numbers. The set `A` has property `Q` and natural density `6 / π^2`.\nEquivalently, `(j / a_j) → 6/π^2` as `j → ∞`.\n-","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/1e075c4f6e8a907b924647fa88238f978e941742/ErdosProblem1102PropertyQDensity.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1102»","statement":"∃ A,\n  StrictMono A ∧\n    (∀ (j : ℕ), Squarefree (A j)) ∧\n      Erdos1102.HasPropertyQ (Set.range A) ∧ Filter.Tendsto (fun j => ↑j / ↑(A j)) Filter.atTop (nhds (6 / Real.pi ^ 2))","subjects":["11"],"theorem":"Erdos1102.erdos_1102.lower_density_Q_exists"},{"answerKinds":[],"category":"research solved","docstring":"If `A = {a₁ < a₂ < …}` has property P,\nthen `A` has natural density `0`.\nEquivalently, `(a_j / j) → ∞` as `j → ∞`.\n-","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/1e075c4f6e8a907b924647fa88238f978e941742/ErdosProblem1102PropertyP.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1102»","statement":"∀ (A : ℕ → ℕ),\n  StrictMono A → Erdos1102.HasPropertyP (Set.range A) → Filter.Tendsto (fun j => ↑(A j) / ↑j) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos1102.erdos_1102.density_zero_of_P"},{"answerKinds":[],"category":"research solved","docstring":"The statement for which Baumgartner actually writes a proof. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«198»","statement":"∀ (V : Type u_1) [inst : AddCommGroup V] [Module ℚ V],\n  ∃ X, (∀ (Y : Set V), Y.IsAPOfLength ⊤ → (X ∩ Y).Nonempty) ∧ ∀ (Y : Set V), Y.IsAPOfLength 3 → (X ∩ Y).ncard ≤ 2","subjects":["5"],"theorem":"Erdos198.baumgartner_headline"},{"answerKinds":[],"category":"research solved","docstring":"Let $V$ be a vector space over the rationals and let $k$ be a fixed\npositive integer. Then there is a set $X_k \\subseteq V$ such that $X_k$ meets\nevery infinite arithmetic progression in $V$ but $X_k$ intersects every\n$k$-element arithmetic progression in at most two points.\n\nAt the end of [Ba75] the author claims that by \"slightly modifying the method of [his proof]\", one\ncan prove this. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«198»","statement":"∀ (V : Type u_1) [inst : AddCommGroup V] [Module ℚ V] (k : ℕ),\n  ∃ X, (∀ (Y : Set V), Y.IsAPOfLength ⊤ → (X ∩ Y).Nonempty) ∧ ∀ (Y : Set V), Y.IsAPOfLength ↑k → (X ∩ Y).ncard ≤ 2","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos198.baumgartner_strong"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The answer is no; Erdős and Graham report this was proved by Baumgartner, presumably referring to\nthe paper [Ba75], which does not state this exactly, but the following simple construction is\nimplicit in [Ba75].\n\nLet $P_1,P_2,\\ldots$ be an enumeration of all countably many infinite arithmetic progressions. We\nchoose $a_1$ to be the minimal element of $P_1\\cap \\mathbb{N}$, and in general choose $a_n$ to be an\nelement of $P_n\\cap \\mathbb{N}$ such that $a_n>2a_{n-1}$. By construction $A=\\{a_1 < a_2 < \\cdots\\}$\ncontains at least one element from every infinite arithmetic progression, and is a lacunary set, so\nis certainly Sidon.\n\nAlphaProof has found the following explicit construction: $A = \\{ (n+1)!+n : n\\geq 0\\}$. This is a\nSidon set, and intersects every arithmetic progression, since for any $a,d\\in \\mathbb{N}$,\n$(a+d+1)!+(a+d)\\in A$, and $d$ divides $(a+d+1)!+d$.\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/33045ec97b08a40c7ff91f1e2a112f5e3b4725f8/FormalConjectures/ErdosProblems/198.lean#L168"},{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos198.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«198»","statement":"(∀ (A : Set ℕ), IsSidon A → ∃ Y, Y.IsAPOfLength ⊤ ∧ Y ⊆ Aᶜ) ↔ False","subjects":["5","11"],"theorem":"Erdos198.erdos_198"},{"answerKinds":[],"category":"research solved","docstring":"In fact one such sequence is $n! + n$.\n\nThis was found and proved by AlphaProof.\n\nIt also found $(n + 1)! + n$.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mzhorvath1/formal-conjectures/blob/21f6780f84b406de468389571eb01717b8072f09/FormalConjectures/ErdosProblems/198.lean#L84"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«198»","statement":"∃ A, A = {x | ∃ n, n.factorial + n = x} ∧ IsSidon A ∧ ∀ (Y : Set ℕ), Y.IsAPOfLength ⊤ → (A ∩ Y).Nonempty","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos198.erdos_198.variants.concrete"},{"answerKinds":[],"category":"research open","docstring":"If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges\nof $K_n$ contains a monochromatic copy of $K_k$, then\n$$\\frac{R(k)}{k2^{k/2}}\\to \\infty.$$\n\nIn [Er93] Erdős offers $100 for a proof of this and $1000 for a disproof, but says 'this last offer\nis to some extent phoney: I am sure that this is true (but I have been wrong before).'\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1029»","statement":"Filter.Tendsto (fun k => ↑(SimpleGraph.diagonalRamsey k) / (↑k * 2 ^ (↑k / 2))) Filter.atTop Filter.atTop","subjects":["5"],"theorem":"Erdos1029.erdos_1029"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many $n$ such that ${2n\\choose n}$ is coprime to $105$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«376»","statement":"True ↔ {n | n.centralBinom.Coprime 105}.Infinite","subjects":["11"],"theorem":"Erdos376.erdos_376"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown that, for any two odd primes $p$ and $q$,\nthere are infinite many $n$ such that ${2n\\choose n}$ is coprime to $pq$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«376»","statement":"∀ {p q : ℕ}, Nat.Prime p → Odd p → Nat.Prime q → Odd q → {n | n.centralBinom.Coprime (p * q)}.Infinite","subjects":["11"],"theorem":"Erdos376.erdos_376.variants.prime"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"For what values of $0 \\leq m < n$ is there a complete sequence\n$A = \\{a_1 \\leq a_2 \\leq \\cdots\\}$ of integers such that\n 1. $A$ remains complete after removing any $m$ elements, but\n 2. $A$ is not complete after removing any $n$ elements.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«348»","statement":"{x |\n    ∃ m n,\n      ∃ (_ : m < n),\n        ∃ a,\n          ∃ (_ : Monotone a) (_ :\n            ∀ (s : Finset ℕ), s.card = m → IsAddComplete (Set.range (Function.updateFinset a s 0))) (_ :\n            ∀ (t : Finset ℕ), t.card = n → ¬IsAddComplete (Set.range (Function.updateFinset a t 0))), (m, n) = x} =\n  sorry","subjects":["11"],"theorem":"Erdos348.erdos_348"},{"answerKinds":[],"category":"research open","docstring":"There is no consecutive triple of powerful numbers. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«364»","statement":"¬∃ n, n.Powerful ∧ (n + 1).Powerful ∧ (n + 2).Powerful","subjects":["11"],"theorem":"Erdos364.erdos_364"},{"answerKinds":[],"category":"textbook","docstring":"There is no quadruple of powerful numbers, since at least one of the four numbers must be\n$2 \\pmod{4}$, which cannot be powerful (since $2$ divides it, but $2^2$ does not).\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«364»","statement":"¬∃ n, n.Powerful ∧ (n + 1).Powerful ∧ (n + 2).Powerful ∧ (n + 3).Powerful","subjects":["11"],"theorem":"Erdos364.erdos_364.variants.weak"},{"answerKinds":[],"category":"research open","docstring":"Erdős [Er76d] conjectured a stronger statement: if $n_k$ is the $k$th powerful number,\nthen $n_{k+2} - n_k > n_k^c$ for some constant $c > 0$.\n\n[Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«364»","statement":"∃ c,\n  ∃ (_ : c > 0), ∀ (k : ℕ), ↑(Nat.nth Nat.Powerful (k + 2)) - ↑(Nat.nth Nat.Powerful k) > ↑(Nat.nth Nat.Powerful k) ^ c","subjects":["11"],"theorem":"Erdos364.erdos_364.variants.strong"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $A(x) \\le x^{o(1)}$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1073»","statement":"True ↔ ∃ o, o =o[Filter.atTop] 1 ∧ ∀ (x : ℕ), Erdos1073.A x ≤ ↑x ^ o x","subjects":["11"],"theorem":"Erdos1073.erdos_1073"},{"answerKinds":[],"category":"research solved","docstring":"**The dyadic fiber at $\\alpha = 2$.** For every $k$, the pair $(1/2^k, 2)$ is good: the\nsequence $\\lfloor 2^n / 2^k\\rfloor$ is additively complete because at index $n = m + k$ it equals\nthe exact power $2^m$, so its range contains all powers of two, which already form an additively\ncomplete set. Uses monotonicity `IsAddComplete.mono`. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/cepadugato/formal-conjectures/blob/erdos-349-integer-characterization-proof/FormalConjectures/ErdosProblems/349.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"∀ (k : ℕ), Erdos349.IsGoodPair (1 / 2 ^ k) 2","subjects":["11"],"theorem":"Erdos349.dyadic_two_isGoodPair"},{"answerKinds":[],"category":"research solved","docstring":"For $\\alpha > 2$ and any $t > 0$, the sequence $\\lfloor t\\alpha^n\\rfloor$ is not additively\ncomplete; equivalently $(t, \\alpha)$ is not a \"good pair\". A partial result on the open Erdős\nProblem 349: it complements `complete_for_alpha_in_Ioo_one_to_goldenRatio`.\n\nThe proof is recorded via the `formal_proof` mechanism rather than written inline, as it exceeds\nthe repository's proof-length guideline. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/cepadugato/formal-conjectures/blob/23c629bc2347864782ce88f957a64d6567b978a1/FormalConjectures/ErdosProblems/349.lean#L87"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"∀ (t α : ℝ), 0 < t → 2 < α → ¬Erdos349.IsGoodPair t α","subjects":["11"],"theorem":"Erdos349.alpha_gt_two_not_isGoodPair"},{"answerKinds":[],"category":"research solved","docstring":"**Binary expansion.** Every natural number $k$ is a sum of distinct powers of two: there is\na finite set $E$ of exponents with $k = \\sum_{i \\in E} 2^i$. Proved by strong induction:\nsubtract the largest power $2^m \\le k$, recurse on the remainder. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/cepadugato/formal-conjectures/blob/erdos-349-integer-characterization-proof/FormalConjectures/ErdosProblems/349.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"∀ (k : ℕ), ∃ E, k = ∑ i ∈ E, 2 ^ i","subjects":["11"],"theorem":"Erdos349.exists_finsetSum_two_pow"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"For what values of $t,\\alpha \\in (0,\\infty)$ is the sequence $\\lfloor t\\alpha^n\\rfloor$ complete\n(that is, all sufficiently large integers are the sum of distinct integers of the form $\\lfloor t\\alpha^n\\rfloor$)?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"{(t, α) | 0 < t ∧ 0 < α ∧ Erdos349.IsGoodPair t α} = sorry","subjects":["11"],"theorem":"Erdos349.erdos_349"},{"answerKinds":[],"category":"research solved","docstring":"**The pair $(1, 2)$ is good.** The powers of two $\\lfloor 1\\cdot 2^n\\rfloor = 2^n$ form an\nadditively complete set: every $k \\ge 1$ is a finite sum of distinct powers of two. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/cepadugato/formal-conjectures/blob/erdos-349-integer-characterization-proof/FormalConjectures/ErdosProblems/349.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"Erdos349.IsGoodPair 1 2","subjects":["11"],"theorem":"Erdos349.one_two_isGoodPair"},{"answerKinds":[],"category":"research open","docstring":"It seems likely that the sequence is complete for all\nfor all $t>0$ and all $1 < \\alpha < \\frac{1+\\sqrt{5}}{2}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"∀ (t α : ℝ), 0 < t → α ∈ Set.Ioo 1 ((1 + √5) / 2) → Erdos349.IsGoodPair t α","subjects":["11"],"theorem":"Erdos349.complete_for_alpha_in_Ioo_one_to_goldenRatio"},{"answerKinds":[],"category":"research open","docstring":"Is it true that the terms of the sequence $\\lfloor (3/2)^n\\rfloor$ are even infinitely often?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"True ↔ {n | Even ⌊(3 / 2) ^ n⌋}.Infinite","subjects":["11"],"theorem":"Erdos349.erdos_349.variants.floor_3_halves_even"},{"answerKinds":[],"category":"research solved","docstring":"For $0 < \\alpha \\le 1$ and any $t > 0$, $(t, \\alpha)$ is not a good pair: every term\n$\\lfloor t\\alpha^n\\rfloor$ lies in the finite interval $[0, \\lfloor t\\rfloor]$ (since\n$\\alpha^n \\le 1$), so every subset sum is bounded by the constant $\\sum_{i \\in [0,\\lfloor t\\rfloor]} i$,\nand no large integer can be a subset sum. A partial result on the open Erdős Problem 349,\ncomplementing the $2 < \\alpha$ and integer-coefficient cases. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/cepadugato/formal-conjectures/blob/erdos-349-integer-characterization-proof/FormalConjectures/ErdosProblems/349.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"∀ (t α : ℝ), 0 < t → 0 < α → α ≤ 1 → ¬Erdos349.IsGoodPair t α","subjects":["11"],"theorem":"Erdos349.alpha_le_one_not_isGoodPair"},{"answerKinds":[],"category":"research open","docstring":"Is it true that the terms of the sequence $\\lfloor (3/2)^n\\rfloor$ are odd infinitely\noften and even infinitely often?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"True ↔ {n | Odd ⌊(3 / 2) ^ n⌋}.Infinite","subjects":["11"],"theorem":"Erdos349.erdos_349.variants.floor_3_halves_odd"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős Problem 349, complete characterization on positive integer pairs.** For integers\n$t \\ge 1$, $\\alpha \\ge 1$, the pair $(t, \\alpha)$ is good (i.e. $\\lfloor t\\alpha^n\\rfloor$ is\nadditively complete) iff $(t, \\alpha) = (1, 2)$. Assembles the four partial results: $(1,2)$ is\ngood, $\\alpha \\le 1$ fails, $2 < \\alpha$ fails (`alpha_gt_two_not_isGoodPair`), and integer\n$t \\ge 2$ fails. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/cepadugato/formal-conjectures/blob/erdos-349-integer-characterization-proof/FormalConjectures/ErdosProblems/349.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"∀ (t α : ℤ), 1 ≤ t → 1 ≤ α → (Erdos349.IsGoodPair ↑t ↑α ↔ t = 1 ∧ α = 2)","subjects":["11"],"theorem":"Erdos349.integer_isGoodPair_iff"},{"answerKinds":[],"category":"research solved","docstring":"For any $k$ there exists some $t_k\\in (0,1)$ such that the set of $\\alpha$\nsuch that the sequence $\\lfloor t_k\\alpha^n\\rfloor$ is complete consists of at least $k$\ndisjoint line segments.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"∀ (k : ℕ),\n  ∃ t ∈ Set.Ioo 0 1,\n    ∃ ι,\n      ↑k ≤ Set.univ.encard ∧\n        ∃ I,\n          (∀ (i : ι), 2 ≤ (I i).encard ∧ (I i).Nonempty ∧ IsConnected (I i)) ∧\n            Pairwise (Function.onFun Disjoint I) ∧ ⋃ i, I i ⊆ {α | α > 0 ∧ Erdos349.IsGoodPair t α}","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos349.exists_t_for_k_disjoint_segments"},{"answerKinds":[],"category":"research solved","docstring":"**Integer leading coefficient $t \\ge 2$ blocks completeness.** For every integer base\n$\\alpha$, the pair $(t, \\alpha)$ with integer $t \\ge 2$ is not good: $\\lfloor t\\alpha^n\\rfloor =\nt\\alpha^n$ is a multiple of $t$, so every subset sum is too, but two consecutive large integers\ncannot both be multiples of $t$. Generalizes the parity obstruction ($t = 2$). A partial result\non Erdős Problem 349. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/cepadugato/formal-conjectures/blob/erdos-349-integer-characterization-proof/FormalConjectures/ErdosProblems/349.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«349»","statement":"∀ (t : ℤ), 2 ≤ t → ∀ (α : ℤ), ¬Erdos349.IsGoodPair ↑t ↑α","subjects":["11"],"theorem":"Erdos349.int_coeff_ge_two_not_isGoodPair"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$\nfor some prime $q$ and $k ≥ 0$, $l ≥ 0$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1065»","statement":"True ↔ {p | ∃ q k l, Nat.Prime p ∧ Nat.Prime q ∧ p = 2 ^ k * 3 ^ l * q + 1}.Infinite","subjects":["11"],"theorem":"Erdos1065.erdos_1065.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many primes $p$ such that $p = 2^k * q + 1$\nfor some prime $q$ and $k ≥ 0$?\n\nThis is mentioned as B46\nin [Unsolved Problems in Number Theory](https://doi.org/10.1007/978-0-387-26677-0)\nby *Richard K. Guy*\n ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1065»","statement":"True ↔ {p | ∃ q k, Nat.Prime p ∧ Nat.Prime q ∧ p = 2 ^ k * q + 1}.Infinite","subjects":["11"],"theorem":"Erdos1065.erdos_1065.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph\n(not necessarily induced) of $G$. Is there a graph on $n$ vertices with\n$$\\gg \\frac{2^{\\binom{n}{2}}}{n!}$$\nmany distinct unique subgraphs?\n\nBradač and Christoph [BrCh24] have proved the answer is no: if $f(n)$ is the maximum number of\nunique subgraphs in a graph on $n$ vertices then\n$$f(n) = o\\left(\\frac{2^{\\binom{n}{2}}}{n!}\\right).$$\n\nThe $\\gg$ below is read as: some constant $c>0$ works for arbitrarily large $n$. The negation\nof the proposition on the right is then exactly $f(n) = o(2^{\\binom{n}{2}}/n!)$, the form in\nwhich Bradač and Christoph [BrCh24] resolved the problem.\n\nThe linked file states the resolution in that negated form, as\n`Tendsto fSeq atTop (nhds 0)`. It counts the isomorphism classes occurring as unique subgraphs,\nwhereas `uniqueSubgraphCount` counts their representatives $G\\leq H$; uniqueness forces exactly\none representative per class, so the two counts agree.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos426.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«426»","statement":"False ↔ ∃ c, 0 < c ∧ ∃ᶠ (n : ℕ) in Filter.atTop, ∃ H, c * (2 ^ n.choose 2 / ↑n.factorial) ≤ ↑H.uniqueSubgraphCount","subjects":["5"],"theorem":"Erdos426.erdos_426"},{"answerKinds":[],"category":"test","docstring":"Sanity check: the empty graph `⊥` is a unique subgraph of itself. Its only subgraph is `⊥`\n(everything `≤ ⊥` equals `⊥`), which is isomorphic to `⊥` via the identity. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«426»","statement":"∀ {V : Type u_1}, ⊥.IsUniqueSubgraph ⊥","subjects":["5"],"theorem":"Erdos426.isUniqueSubgraph_bot_bot"},{"answerKinds":[],"category":"research open","docstring":"Let $S \\subseteq \\mathbb{R}$ be a set containing no solutions to $a + b = c$.\nMust there be a set $A \\subseteq \\mathbb{R} \\setminus S$ of cardinality continuum such that\n$A + A \\subseteq \\mathbb{R}\\setminus S$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«949»","statement":"True ↔ ∀ (S : Set ℝ), (∀ a ∈ S, ∀ b ∈ S, a + b ∉ S) → ∃ A ⊆ Sᶜ, Cardinal.mk ↑A = Cardinal.continuum ∧ A + A ⊆ Sᶜ","subjects":["5"],"theorem":"Erdos949.erdos_949"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $S\\sub \\mathbb{R}$ be a Sidon set. Must there be a set $A\\sub \\mathbb{R}∖S$ of cardinality\ncontinuum such that $A + A \\sub \\mathbb{R}∖S$? ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«949»","statement":"True ↔ ∀ (S : Set ℝ), IsSidon S → ∃ A ⊆ Sᶜ, Cardinal.mk ↑A = Cardinal.continuum ∧ A + A ⊆ Sᶜ","subjects":["5"],"theorem":"Erdos949.erdos_949.variants.sidon"},{"answerKinds":[],"category":"research open","docstring":"Erdős says that $f(n) = o(\\frac{n}{\\log n})$ has never been proved.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«126»","statement":"∀ (f : ℕ → ℕ), Erdos126.IsMaximalAddFactorsCard f → (fun n => ↑(f n)) =o[Filter.atTop] fun n => ↑n / Real.log ↑n","subjects":["11"],"theorem":"Erdos126.erdos_126.variants.isLittleO"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Turán proved [ErTu34] in their first joint paper that\n$$\n  \\log n \\ll f(n) \\ll \\frac{n}{\\log n}\n$$\n\n[ErTu34] Erdős, Paul and Turan, Paul, _On a Problem in the Elementary Theory of Numbers_. Amer. Math. Monthly (1934), 608-611.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«126»","statement":"∀ (f : ℕ → ℕ),\n  Erdos126.IsMaximalAddFactorsCard f →\n    ((fun n => Real.log ↑n) =O[Filter.atTop] fun n => ↑(f n)) ∧\n      (fun n => ↑(f n)) =O[Filter.atTop] fun n => ↑n / Real.log ↑n","subjects":["11"],"theorem":"Erdos126.erdos_126.variants.IsBigO"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n)$ be maximal such that if $A\\subseteq\\mathbb{N}$ has $|A| = n$ then\n$\\prod_{a\\neq b\\in A}(a + b)$ has at least $f(n)$ distinct prime factors.\nIs it true that $\\frac{f(n)}{\\log n} \\to\\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«126»","statement":"True ↔\n  ∀ (f : ℕ → ℕ),\n    Erdos126.IsMaximalAddFactorsCard f → Filter.Tendsto (fun n => ↑(f n) / Real.log ↑n) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos126.erdos_126"},{"answerKinds":[],"category":"research open","docstring":"Let $\\epsilon > 0$. Is there some set $A\\subset\\mathbb{N}$ of density $> 1 - \\epsilon$\nsuch that $a_1\\cdots a_r = b_1\\cdots b_s$ with $a_i, b_j\\in A$ can only hold when\n$r = s$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«786»","statement":"True ↔ ∀ ε > 0, ε ≤ 1 → ∃ A δ, 0 ∉ A ∧ 1 - ε < δ ∧ A.HasDensity δ ∧ Erdos786.Set.IsMulCardSet A","subjects":["11"],"theorem":"Erdos786.erdos_786.parts.i"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«786»","statement":"Erdos786.consecutivePrimesFrom 2 1 = {2, 3}","subjects":["11"],"theorem":"Erdos786.consecutivePrimesFrom_two_one"},{"answerKinds":[],"category":"research open","docstring":"Is there some set $A\\subset\\{1, ..., N\\}$ of size $\\geq (1 - o(1))N$ such that\n$a_1\\cdots a_r = b_1\\cdots b_s$ with $a_i, b_j\\in A$ can only hold when\n$r = s$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«786»","statement":"True ↔\n  ∃ A f,\n    ∃ (_ : f =o[Filter.atTop] 1),\n      ∀ (N : ℕ), A N ⊆ Set.Icc 1 (N + 1) ∧ (1 - f N) * ↑N ≤ ↑(A N).ncard ∧ Erdos786.Set.IsMulCardSet (A N)","subjects":["11"],"theorem":"Erdos786.erdos_786.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Let $\\epsilon > 0$ be given. Then, for a sufficiently large prime `p`, take the sequence of\nconsecutive primes $p_1 < \\cdots < p_k$ such that\n$$\n\\sum_{i=1}^k \\frac{1}{p_i} < 1 < \\sum_{i=1}^{k + 1} \\frac{1}{p_i},\n$$\nand let $A$ be the set of all naturals divisible by exactly one of $p_1, ..., p_k$ (with\nmultiplicity $1$). Then $A$ has density $\\frac{1}{e} - \\epsilon$ and has the property\nthat $a_1\\cdots a_r = b_1\\cdots b_s$ with $a_i, b_j\\in A$ can only hold when $r = s$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«786»","statement":"∀ (ε : ℝ),\n  0 < ε ∧ ε < 1 / Real.exp 1 →\n    ∀ᶠ (p : ℕ) in Filter.atTop,\n      Nat.Prime p →\n        ∃ k,\n          ∑ q ∈ Erdos786.consecutivePrimesFrom p k, 1 / ↑q < 1 ∧\n            1 < ∑ q ∈ Erdos786.consecutivePrimesFrom p (k + 1), 1 / ↑q ∧\n              {n | ∑ q ∈ Erdos786.consecutivePrimesFrom p k, n.factorization q = 1}.HasDensity (1 / Real.exp 1 - ε) ∧\n                Erdos786.Set.IsMulCardSet {n | ∑ q ∈ Erdos786.consecutivePrimesFrom p k, n.factorization q = 1}","subjects":["11"],"theorem":"Erdos786.erdos_786.parts.i.selfridge"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«786»","statement":"∀ {p : ℕ}, Nat.Prime p → Nat.nth (fun q => Nat.Prime q ∧ p ≤ q) 0 = p","subjects":["11"],"theorem":"Erdos786.nth_zero"},{"answerKinds":[],"category":"textbook","docstring":"An example of such a set with density $\\frac 1 4$ is given by the integers $\\equiv 2\\pmod{4}$\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«786»","statement":"∀ (A : Set ℕ), A = {n | n % 4 = 2} → A.HasDensity (1 / 4) ∧ Erdos786.Set.IsMulCardSet A","subjects":["11"],"theorem":"Erdos786.erdos_786.parts.i.example"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«786»","statement":"∀ {p : ℕ}, Nat.Prime p → Erdos786.consecutivePrimesFrom p 0 = {p}","subjects":["11"],"theorem":"Erdos786.consecutivePrimesFrom_zero"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er88c] notes that if the sequence grows rapidly to infinity (specifically, if\n$a_{n+1} \\geq C \\cdot a_n^2$ for some constant $C > 0$), then the series is irrational.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1051»","statement":"∀ (a : ℕ → ℤ), StrictMono a → (∃ C > 0, ∀ (n : ℕ), ↑(a (n + 1)) ≥ C * ↑(a n) ^ 2) → Irrational (Erdos1051.ErdosSeries a)","subjects":["11"],"theorem":"Erdos1051.erdos_1051.variants.rapid_growth"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that if $a_0 < a_1 < a_2 < \\cdots$ is a strictly increasing sequence\nof integers with $\\liminf a_n^{1/2^n} > 1$, then the series\n$\\sum_{n=0}^\\infty \\frac{1}{a_n \\cdot a_{n+1}}$ is irrational?\n\nThis was solved in the affirmative by Aletheia [Fe26]. This was extended by Barreto, Kang, Kim,\nKovač, and Zhang [BKKKZ26], who essentially give a complete answer: if $\\phi=\\frac{1+\\sqrt{5}}{2}$\nis the golden ratio and $1\\leq a_1 < a_2 < \\cdots$ is a monotonically increasing sequence of\nintegers such that $\\limsup a_n^{1/\\phi^{n}}=\\infty$ then $\\sum_{n=1}^\\infty \\frac{1}{a_na_{n+1}}$\nis irrational. Conversely, for any $1 < C < \\infty$ there exists a sequence of integers\n$1\\leq a_1<\\cdots$ such that $\\lim a_n^{1/\\phi^{n}}=C$ where this infinite sum is a rational number.\n\n(Further, more general, results are available in [BKKKZ26].)\n\nThis was formalized in Lean by Baretto.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://www.erdosproblems.com/forum/thread/1051"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1051»","statement":"True ↔ ∀ (a : ℕ → ℤ), StrictMono a → Erdos1051.GrowthCondition a → Irrational (Erdos1051.ErdosSeries a)","subjects":["11"],"theorem":"Erdos1051.erdos_1051"},{"answerKinds":[],"category":"research solved","docstring":"Show that, for any $n\\geq 5$, the binomial coefficient $\\binom{2n}{n}$ is not squarefree.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«175»","statement":"∀ (n : ℕ), 5 ≤ n → ¬Squarefree ((2 * n).choose n)","subjects":["11"],"theorem":"Erdos175.erdos_175"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that there exists $c > 0$ such that for infinitely many $n$ we have $M_n > n^c$?\n\nThe second question was answered by Beck [Be91], who proved that there exists some $c>0$ such that\n$\\max_{n\\leq N} M_n > N^c$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«119»","statement":"True ↔ ∀ (z : ℕ → ℂ), (∀ (i : ℕ), ‖z i‖ = 1) → ∃ c, ∃ (_ : c > 0), Infinite ↑{n | Erdos119.M z n > ↑n ^ c}","subjects":["30"],"theorem":"Erdos119.erdos_119.parts.ii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that $\\limsup M_n = \\infty$?\n\nThis is Problem 4.1 in [Ha74] where it is attributed to Erdős.\n\nThe weaker conjecture that $\\limsup M_n=\\infty$ was proved by Wagner [Wa80], who show that there is\nsome $c>0$ with $M_n>(\\log n)^c$ infinitely often.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«119»","statement":"True ↔ ∀ (z : ℕ → ℂ), (∀ (i : ℕ), ‖z i‖ = 1) → Filter.limsup (fun n => ↑(Erdos119.M z n)) Filter.atTop = ⊤","subjects":["30"],"theorem":"Erdos119.erdos_119.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that there exists $c > 0$ such that, for all large $n$, $\\sum_{k \\leq n} M_k > n^{1 + c}$?\n\nThe \\$100 prize was offered for the third question in [Er97f]. This was resolved by GPT 5.6 and\nKorsky (see the proof claims), who proved that $\\sum_{k\\leq n}M_k \\gg \\frac{n^{5/4}}{\\sqrt{\\log n}}$\n(and hence for infinitely many $n$ we have $M_n> n^{1/4-o(1)}$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«119»","statement":"True ↔\n  ∀ (z : ℕ → ℂ),\n    (∀ (i : ℕ), ‖z i‖ = 1) →\n      ∃ c, ∃ (_ : c > 0), ∀ᶠ (n : ℕ) in Filter.atTop, ∑ k ∈ Finset.range n, Erdos119.M z k > ↑n ^ (1 + c)","subjects":["30"],"theorem":"Erdos119.erdos_119.parts.iii"},{"answerKinds":[],"category":"research solved","docstring":"$F(n) \\le O(n^{1/2} \\ln ^ {3/4} n)$\n\nTheorem 1.4 from [AKS07]\n\n[AKS07] Alon, N. and Krivelevich, M. and Sudakov, B., Large nearly regular induced subgraphs. arXiv:0710.2106 (2007).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«82»","statement":"(fun n => ↑(Erdos82.F n)) =O[Filter.atTop] fun n => √↑n * Real.log ↑n ^ (3 / 4)","subjects":["5"],"theorem":"Erdos82.erdos_82.variants.F_upper_bound"},{"answerKinds":[],"category":"research open","docstring":"$F(n) / \\log n \\to \\infty as n \\to \\infty$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«82»","statement":"Filter.Tendsto (fun n => ↑(Erdos82.F n) / Real.log ↑n) Filter.atTop Filter.atTop","subjects":["5"],"theorem":"Erdos82.erdos_82"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there some $h(n)\\to \\infty$ such that for all $2\\leq i<j\\leq n/2$\n$$\\textrm{gcd}\\left( \\binom{n}{i},\\binom{n}{j}\\right) \\geq h(n)?$$\n\nThis was resolved by Bergman [Be11], who proved that for any $2\\leq i<j\\leq n/2$\n$$\\textrm{gcd}\\left( \\binom{n}{i},\\binom{n}{j}\\right) \\gg n^{1/2}\\frac{2^i}{i^{3/2}},$$\nwhere the implied constant is absolute.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«698»","statement":"True ↔\n  ∃ h,\n    Filter.Tendsto h Filter.atTop Filter.atTop ∧\n      ∀ (n i j : ℕ), 2 ≤ i → i < j → j ≤ n / 2 → h n ≤ (n.choose i).gcd (n.choose j)","subjects":["5","11"],"theorem":"Erdos698.erdos_698"},{"answerKinds":[],"category":"research solved","docstring":"This inequality is sharp for $i=1$, $j=p$, and $n=2p$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«698»","statement":"∀ (p : ℕ),\n  Nat.Prime p →\n    2 < p →\n      ↑(((2 * p).choose 1).gcd ((2 * p).choose p)) = ↑((2 * p).choose 1) / ↑(p.choose 1) ∧\n        ↑((2 * p).choose 1) / ↑(p.choose 1) = 2 ^ 1","subjects":["5","11"],"theorem":"Erdos698.erdos_698.variants.erdos_szekeres_sharp"},{"answerKinds":[],"category":"research solved","docstring":"A problem of Erdős and Szekeres, who observed that\n$$\\textrm{gcd}\\left( \\binom{n}{i},\\binom{n}{j}\\right) \\geq \\frac{\\binom{n}{i}}{\\binom{j}{i}}\n\\geq 2^i$$\n(in particular the greatest common divisor is always $>1$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«698»","statement":"∀ (n i j : ℕ),\n  1 ≤ i →\n    i < j →\n      j ≤ n / 2 →\n        ↑(n.choose i) / ↑(j.choose i) ≤ ↑((n.choose i).gcd (n.choose j)) ∧ 2 ^ i ≤ ↑(n.choose i) / ↑(j.choose i)","subjects":["5","11"],"theorem":"Erdos698.erdos_698.variants.erdos_szekeres"},{"answerKinds":[],"category":"research solved","docstring":"This was resolved by Bergman [Be11], who proved that for any $2\\leq i<j\\leq n/2$\n$$\\textrm{gcd}\\left( \\binom{n}{i},\\binom{n}{j}\\right) \\gg n^{1/2}\\frac{2^i}{i^{3/2}},$$\nwhere the implied constant is absolute.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos698.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«698»","statement":"∃ c,\n  0 < c ∧ ∀ (n i j : ℕ), 2 ≤ i → i < j → j ≤ n / 2 → c * (√↑n * 2 ^ i / (↑i * √↑i)) ≤ ↑((n.choose i).gcd (n.choose j))","subjects":["5","11"],"theorem":"Erdos698.erdos_698.variants.bergman"},{"answerKinds":[],"category":"research solved","docstring":"Bedert [Be25c] proved an upper bound of $-c N^{1/7}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«510»","statement":"∃ c,\n  ∃ (_ : 0 < c),\n    ∀ᶠ (N : ℕ) in Filter.atTop,\n      ∀ (A : Finset ℕ), 0 ∉ A → A.card = N → ∃ θ, ∑ n ∈ A, Real.cos (↑n * θ) < -c * ↑N ^ (1 / 7)","subjects":["11"],"theorem":"Erdos510.erdos_510.variants.bedert"},{"answerKinds":["Prop"],"category":"research open","docstring":"**Chowla's cosine problem**\n\nIf $A\\subset \\mathbb{N}$ is a finite set of positive integers of size $N > 0$ then is there some\nabsolute constant $c>0$ and $\\theta$ such that\n$$\\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/2}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«510»","statement":"sorry ↔\n  ∃ c,\n    ∃ (_ : 0 < c),\n      ∀ᶠ (N : ℕ) in Filter.atTop, ∀ (A : Finset ℕ), 0 ∉ A → A.card = N → ∃ θ, ∑ n ∈ A, Real.cos (↑n * θ) < -c * √↑N","subjects":["11"],"theorem":"Erdos510.erdos_510"},{"answerKinds":[],"category":"research solved","docstring":"Ruzsa [Ru04] proved an upper bound of $-\\exp(O(\\sqrt{\\log N})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«510»","statement":"∃ c,\n  ∃ (_ : 0 < c),\n    ∀ᶠ (N : ℕ) in Filter.atTop,\n      ∀ (A : Finset ℕ), 0 ∉ A → A.card = N → ∃ θ, ∑ n ∈ A, Real.cos (↑n * θ) < -Real.exp (c * √(Real.log ↑N))","subjects":["11"],"theorem":"Erdos510.erdos_510.variants.ruzsa"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subset \\mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$.\n\nIs the upper density of\n$$\\{ n : 1_A\\ast 1_A(n)=g(n)\\}$$\nalways $<c$ for some constant $c<1$?\n\nThe answer is trivially no to both questions: indeed if $A=\\mathbb{N}$ (assuming $0\\in\\mathbb{N}$)\nthen $1_A\\ast 1_A(n)=n+1$ for all $n$. Presumably Erdős had some additional restrictions on either\n$g$ or $A$ in mind, but these are not recorded in [Er80].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1193»","statement":"False ↔\n  ∃ c < 1,\n    ∀ (A : Set ℕ) (g : ℕ → ℕ),\n      Monotone g → (∀ (n : ℕ), 0 < g n) → {n | AdditiveCombinatorics.sumRep A n = g n}.upperDensity < c","subjects":["5","11"],"theorem":"Erdos1193.erdos_1193.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Indeed if $A=\\mathbb{N}$ (assuming $0\\in\\mathbb{N}$) then $1_A\\ast 1_A(n)=n+1$ for all $n$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1193.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1193»","statement":"∀ (n : ℕ), AdditiveCombinatorics.sumRep Set.univ n = n + 1","subjects":["5","11"],"theorem":"Erdos1193.erdos_1193.variants.sumRep_univ"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subset \\mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$.\n\nIs the lower density of\n$$\\{ n : 1_A\\ast 1_A(n)=g(n)\\}$$\nalways $0$?\n\nThe answer is trivially no to both questions: indeed if $A=\\mathbb{N}$ (assuming $0\\in\\mathbb{N}$)\nthen $1_A\\ast 1_A(n)=n+1$ for all $n$. Presumably Erdős had some additional restrictions on either\n$g$ or $A$ in mind, but these are not recorded in [Er80].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1193»","statement":"False ↔\n  ∀ (A : Set ℕ) (g : ℕ → ℕ),\n    Monotone g → (∀ (n : ℕ), 0 < g n) → {n | AdditiveCombinatorics.sumRep A n = g n}.lowerDensity = 0","subjects":["5","11"],"theorem":"Erdos1193.erdos_1193.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Erdős writes the upper density can be positive, but he believes it is bounded away from $1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1193»","statement":"∃ A g, Monotone g ∧ (∀ (n : ℕ), 0 < g n) ∧ 0 < {n | AdditiveCombinatorics.sumRep A n = g n}.upperDensity","subjects":["5","11"],"theorem":"Erdos1193.erdos_1193.variants.upper_density_pos"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $α$ be the infinite ordinal $\\omega^{\\omega^2}$. Is it true that any red/blue colouring of the\nedges of $K_α$ there is either a red $K_α$ or a blue $K_3$?\n\nThis is true and was proved independently by Schipperus [Sc10] and Darby.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«591»","statement":"True ↔ OrdinalCardinalRamsey (Ordinal.omega0 ^ Ordinal.omega0 ^ 2) (Ordinal.omega0 ^ Ordinal.omega0 ^ 2) 3","subjects":["3"],"theorem":"Erdos591.erdos_591"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the largest $k$ such that in any permutation of $\\mathbb{Z}$ there must exist a\nmonotone $k$-term arithmetic progression $x_1 < \\cdots < x_k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«195»","statement":"sorry = sSup {k | ∀ (f : ℤ ≃ ℤ), HasMonotoneAP (⇑f) k}","subjects":["5"],"theorem":"Erdos195.erdos_195"},{"answerKinds":[],"category":"research solved","docstring":"Geneson [Ge19] proved that k ≤ 5.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«195»","statement":"5 ≥ sSup {k | ∀ (f : ℤ ≃ ℤ), HasMonotoneAP (⇑f) k}","subjects":["5"],"theorem":"Erdos195.erdos_195.variants.leq_5_bound"},{"answerKinds":[],"category":"research solved","docstring":"Adenwalla [Ad22] proved that k ≤ 4.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«195»","statement":"4 ≥ sSup {k | ∀ (f : ℤ ≃ ℤ), HasMonotoneAP (⇑f) k}","subjects":["5"],"theorem":"Erdos195.erdos_195.variants.leq_4_bound"},{"answerKinds":[],"category":"textbook","docstring":"Show that\n$$\n\\sum_{n} \\frac{1}{2^n - 1} = \\sum_{n} \\frac{d(n)}{2^n},\n$$\nwhere $d(n)$ is the number of divisors of $n$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«257»","statement":"∑' (n : ℕ), 1 / (2 ^ n - 1) = ∑' (n : ℕ), ↑n.divisors.card / 2 ^ n","subjects":["11"],"theorem":"Erdos257.erdos_257.variants.tsum_top_eq"},{"answerKinds":[],"category":"research open","docstring":"Let $A\\subseteq\\mathbb{N}$ be an infinite set. Is\n$$\n\\sum_{n\\in A} \\frac{1}{2^n - 1}\n$$\nirrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«257»","statement":"True ↔ ∀ (A : Set ℕ), A.Infinite → Irrational (∑' (n : ↑A), 1 / (2 ^ ↑n - 1))","subjects":["11"],"theorem":"Erdos257.erdos_257"},{"answerKinds":[],"category":"research solved","docstring":"Show that\n$$\n\\sum_{n} \\frac{d(n)}{2^n}\n$$\nis irrational.\n\n[Er48] Erdős, P., _On arithmetical properties of Lambert series_. J. Indian Math. Soc. (N.S.) (1948), 63-66.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«257»","statement":"Irrational (∑' (n : ℕ), ↑n.divisors.card / 2 ^ n)","subjects":["11"],"theorem":"Erdos257.erdos_257.variants.tsum_top"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that if $A \\subseteq \\mathbb{N}\\setminus\\{1\\}$ is a finite set with\n$\\sum_{n \\in A} \\frac{1}{n} < 2$ then there is a partition $A=A_1 \\sqcup A_2$\nsuch that $\\sum_{n \\in A_i} \\frac{1}{n} < 1$ for $i=1,2$?\n\nThis is not true in general, as shown by Sándor [Sa97].\n\nThe minimal counterexample is $\\{2,3,4,5,6,7,10,11,13,14,15\\}$, found by Tom Stobart.\n\nThis was formalized in Lean by Mehta.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«316»","statement":"False ↔\n  ∀ (A : Finset ℕ),\n    0 ∉ A →\n      1 ∉ A → ∑ n ∈ A, 1 / ↑n < 2 → ∃ A₁ A₂, Disjoint A₁ A₂ ∧ A = A₁ ∪ A₂ ∧ ∑ n ∈ A₁, 1 / ↑n < 1 ∧ ∑ n ∈ A₂, 1 / ↑n < 1","subjects":["5","11"],"theorem":"Erdos316.erdos_316"},{"answerKinds":[],"category":"textbook","docstring":"This is not true if $A$ is a multiset, for example $2,3,3,5,5,5,5$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«316»","statement":"∃ A,\n  0 ∉ A ∧\n    1 ∉ A ∧\n      (Multiset.map (fun x => 1 / x) do\n              let a ← A\n              pure ↑a).sum <\n          2 ∧\n        ∀ (A₁ A₂ : Multiset ℕ),\n          A = A₁ + A₂ →\n            1 ≤\n                (Multiset.map (fun x => 1 / x) do\n                    let a ← A₁\n                    pure ↑a).sum ∨\n              1 ≤\n                (Multiset.map (fun x => 1 / x) do\n                    let a ← A₂\n                    pure ↑a).sum","subjects":["5","11"],"theorem":"Erdos316.erdos_316.variants.multiset"},{"answerKinds":[],"category":"research solved","docstring":"This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of\n$120$ form a counterexample. More generally, Sándor shows that for any $n\\geq 2$ there exists a\nfinite set $A\\subseteq \\mathbb{N}\\backslash\\{1\\}$ with $\\sum_{k\\in A}\\frac{1}{k} < n$ and no\npartition into $n$ parts each of which has $\\sum_{k\\in A_i}\\frac{1}{k}<1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«316»","statement":"∀ (n : ℕ),\n  2 ≤ n →\n    ∃ A,\n      A.Nonempty ∧\n        0 ∉ A ∧\n          1 ∉ A ∧ ∑ k ∈ A, 1 / ↑k < ↑n ∧ ∀ (P : Finpartition A), P.parts.card = n → ∃ p ∈ P.parts, 1 ≤ ∑ n ∈ p, 1 / ↑n","subjects":["5","11"],"theorem":"Erdos316.erdos_316.variants.generalized"},{"answerKinds":[],"category":"research open","docstring":"Let $k\\geq 3$. Is there a choice of congruence classes $a_p\\pmod{p}$ for every prime $p$\nsuch that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$\nand integer $t\\geq k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«279»","statement":"True ↔ ∀ k ≥ 3, ∃ a N, (∀ (p : ℕ), Nat.Prime p → a p < p) ∧ ∀ n ≥ N, ∃ p, ∃ t ≥ k, Nat.Prime p ∧ n = a p + t * p","subjects":["11"],"theorem":"Erdos279.erdos_279"},{"answerKinds":[],"category":"research open","docstring":"Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$\nwith $P(n+1) > P(n)$ has density $\\frac{1}{2}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«371»","statement":"{n | (n + 1).maxPrimeFac > n.maxPrimeFac}.HasDensity (1 / 2)","subjects":["11"],"theorem":"Erdos371.erdos_371"},{"answerKinds":[],"category":"research solved","docstring":"The counterexample: a nonempty family of connected bipartite graphs, none acyclic, that is not\ncompact.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/openai/ten-proofs/blob/94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6/CompactnessAndDegeneracy.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«180»","statement":"∃ family,\n  family.Nonempty ∧\n    (∀ forbidden ∈ family, forbidden.graph.Connected ∧ forbidden.graph.IsBipartite ∧ ¬forbidden.graph.IsAcyclic) ∧\n      ¬Erdos180.IsCompactFamily family","subjects":["5"],"theorem":"Erdos180.erdos_180.variants.counterexample"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $\\mathcal{F}$ is a finite set of finite graphs then $\\mathrm{ex}(n;\\mathcal{F})$ is the maximum\nnumber of edges a graph on $n$ vertices can have without containing any subgraphs from\n$\\mathcal{F}$. Note that it is trivial that $\\mathrm{ex}(n;\\mathcal{F})\\leq \\mathrm{ex}(n;G)$ for\nevery $G\\in\\mathcal{F}$. Is it true that, for every $\\mathcal{F}$, there exists $G\\in\\mathcal{F}$\nsuch that\n$$\\mathrm{ex}(n;G)\\ll_{\\mathcal{F}}\\mathrm{ex}(n;\\mathcal{F})?$$\n\nThis is the Erdős–Simonovits compactness conjecture. The answer is no: OpenAI [OpenAI26] give a\nfamily of connected bipartite graphs, none of them acyclic, for which no single member controls\nthe family extremal number. See `erdos_180.variants.counterexample`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«180»","statement":"False ↔\n  ∀ (family : Finset Erdos180.FiniteGraph),\n    family.Nonempty → Erdos180.IsCyclicFamily family → Erdos180.IsCompactFamily family","subjects":["5"],"theorem":"Erdos180.erdos_180"},{"answerKinds":[],"category":"research open","docstring":"Erdős and Szekeres conjectured that, apart from a finite exceptional set of triples `(n, i, j)`,\none can always take `p > i` in the prime divisor statement. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«699»","statement":"True ↔\n  ∃ E,\n    ∀ (n i j : ℕ),\n      1 ≤ i → i < j → j ≤ n / 2 → (n, i, j) ∉ E → ∃ p, Nat.Prime p ∧ i < p ∧ p ∣ (n.choose i).gcd (n.choose j)","subjects":["11"],"theorem":"Erdos699.erdos_szekeres_strengthening"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 699.** Is it true that for every $1 \\le i < j \\le n / 2$ there exists a prime\n$p \\ge i$ with $p \\mid \\gcd\\big(\\binom{n}{i}, \\binom{n}{j}\\big)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«699»","statement":"True ↔ ∀ (n i j : ℕ), 1 ≤ i → i < j → j ≤ n / 2 → ∃ p, Nat.Prime p ∧ i ≤ p ∧ p ∣ (n.choose i).gcd (n.choose j)","subjects":["11"],"theorem":"Erdos699.erdos_699"},{"answerKinds":[],"category":"research solved","docstring":"Sylvester and Schur: for $1 \\le i \\le n/2$ there is a prime $p > i$ dividing `n.choose i`. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/AllenGrahamHart/FormalConjectures-Bench/blob/482dacc4d9335240f26218cdc62032da3100392b/formalizations/erdos699/Erdos699Formalization.lean#L7679"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«699»","statement":"∀ (n i : ℕ), 1 ≤ i → i ≤ n / 2 → ∃ p, Nat.Prime p ∧ i < p ∧ p ∣ n.choose i","subjects":["11"],"theorem":"Erdos699.sylvester_schur"},{"answerKinds":[],"category":"research open","docstring":"Does this imply that\n$$\n\\liminf \\frac{|A \\cap [1,x]|}{x} = 0?\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«143»","statement":"True ↔\n  ∀ (A : Set ℝ), Erdos143.WellSeparatedSet A → Filter.liminf (fun x => ↑(A ∩ Set.Icc 1 x).ncard / x) Filter.atTop = 0","subjects":["11"],"theorem":"Erdos143.erdos_143.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Or\n$$\n\\sum_{x \\in A} \\frac{1}{x \\log x} < \\infty,\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«143»","statement":"∀ (A : Set ℝ), Erdos143.WellSeparatedSet A → Summable fun x => 1 / (↑x * Real.log ↑x)","subjects":["11"],"theorem":"Erdos143.erdos_143.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $\\epsilon>0$. Does there exist $A\\subseteq \\mathbb{N}$\nsuch that the lower density of $A+A$ is at least $1-\\epsilon$\nand yet $1_A\\ast 1_A(n) \\ll_\\epsilon 1$ for all $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«749»","statement":"True ↔\n  ∀ ε > 0, ∃ A, 1 - ε ≤ (A + A).lowerDensity ∧ (Nat.cast ∘ AdditiveCombinatorics.sumRep A) =O[Filter.atTop] fun n => 1","subjects":["11"],"theorem":"Erdos749.erdos_749"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ be a group, and let $A = \\{a_1G_1, \\dots, a_kG_k\\}$ be a finite system of left cosets of\nsubgroups $G_1, \\dots, G_k$ of $G$.\n\nHerzog and Schönheim conjectured that if $A$ forms a partition of $G$ with $k > 1$, then the\nindices $[G:G_1], \\dots, [G:G_k]$ cannot be distinct.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«274»","statement":"∀ {G : Type u_1} [inst : Group G],\n  1 < ENat.card G →\n    ∀ {ι : Type u_2} [inst_1 : Fintype ι],\n      1 < Fintype.card ι →\n        ∀ (P : Erdos274.Group.ExactCovering G ι), ∃ i j, i ≠ j ∧ (P.parts i).index = (P.parts j).index","subjects":["20"],"theorem":"Erdos274.herzog_schonheim"},{"answerKinds":["Prop"],"category":"research open","docstring":"If $G$ is a group, can there exist an exact covering of $G$ by more than one coset\nof different sizes? (i.e. each element is contained in exactly one of the cosets.)\n\nThe conjectured answer is no: in every such exact covering, two of the subgroups have\nthe same cardinality.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«274»","statement":"sorry ↔\n  ∀ (G : Type u_1) [inst : Group G],\n    1 < ENat.card G →\n      ∀ (ι : Type u_2) [inst_1 : Fintype ι] (P : Erdos274.Group.ExactCovering G ι),\n        1 < Fintype.card ι → ∃ i j, i ≠ j ∧ Cardinal.mk ↥(P.parts i) = Cardinal.mk ↥(P.parts j)","subjects":["20"],"theorem":"Erdos274.erdos_274"},{"answerKinds":[],"category":"research solved","docstring":"If `G` is a finite abelian group then there cannot exist an exact covering of `G` by more\nthan one cosets of different sizes? (i.e. each element is contained in exactly one\nof the cosets.)\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jostamon/erdos274-hs-abelian/blob/2ab8a2e39e7dd7836adf577b52555f069244466f/Erdos274/Main.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«274»","statement":"∀ {G : Type u_1} [inst : Fintype G] [inst_1 : CommGroup G],\n  1 < Fintype.card G →\n    ∀ {ι : Type u_2} [inst : Fintype ι] (P : Erdos274.Group.ExactCovering G ι),\n      1 < Fintype.card ι → ∃ i j, i ≠ j ∧ Cardinal.mk ↥(P.parts i) = Cardinal.mk ↥(P.parts j)","subjects":["20"],"theorem":"Erdos274.erdos_274.variants.abelian"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $k\\geq 3$. Must any ordering of $\\mathbb{R}$ contain a monotone $k$-term arithmetic progression,\nthat is, some $x_1 <\\cdots < x_k$ which forms an increasing or decreasing $k$-term arithmetic\nprogression?\n\nThe answer is no, even for $k=3$, as shown by Ardal, Brown, and Jungić [ABJ11].\n-","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://gist.githubusercontent.com/ster-oc/ffe9e4fa1b813111f40c0e417bbe8be0/raw/6f748a76e55d47e24ca319a9c00fd20ab79422bb/Erdos194.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«194»","statement":"False ↔\n  ∀ k ≥ 3,\n    ∀ (r : ℝ → ℝ → Prop),\n      IsStrictTotalOrder ℝ r → ∃ s, s.IsAPOfLength k ∧ (List.Pairwise r s ∨ List.Pairwise (flip r) s)","subjects":["5"],"theorem":"Erdos194.erdos_194"},{"answerKinds":[],"category":"research open","docstring":"Can $4$ be written as\n$$4=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«686»","statement":"True ↔ ∃ k ≥ 2, ∃ n, ∃ m ≥ n + k, 4 = ↑(∏ i ∈ Finset.Icc 1 k, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 k, (n + i))","subjects":["11"],"theorem":"Erdos686.erdos_686.variants.four"},{"answerKinds":[],"category":"research solved","docstring":"The number $4$ cannot be written as\n$$4=\\frac{\\prod_{1\\leq i\\leq 3}(m+i)}{\\prod_{1\\leq i\\leq 3}(n+i)}$$\nfor $m≥n+3$!\n\nSee [comment section on erdosproblems.com](https://www.erdosproblems.com/forum/thread/686#post-4599)\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«686»","statement":"¬∃ n, ∃ m ≥ n + 3, 4 = ↑(∏ i ∈ Finset.Icc 1 3, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 3, (n + i))","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos686.erdos_686.variants.four_three"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Can $9$ be written as\n$$9=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«686»","statement":"True ↔ ∃ k ≥ 2, ∃ n, ∃ m ≥ n + k, 9 = ↑(∏ i ∈ Finset.Icc 1 k, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 k, (n + i))","subjects":["11"],"theorem":"Erdos686.erdos_686.variants.nine"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Can every non-square $N≥2$ be written as\n$$N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«686»","statement":"True ↔\n  ∀ N ≥ 2,\n    ¬IsSquare N → ∃ k ≥ 2, ∃ n, ∃ m ≥ n + k, ↑N = ↑(∏ i ∈ Finset.Icc 1 k, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 k, (n + i))","subjects":["11"],"theorem":"Erdos686.erdos_686.variants.non_square"},{"answerKinds":[],"category":"research open","docstring":"Can $25$ be written as\n$$25=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«686»","statement":"True ↔ ∃ k ≥ 2, ∃ n, ∃ m ≥ n + k, 25 = ↑(∏ i ∈ Finset.Icc 1 k, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 k, (n + i))","subjects":["11"],"theorem":"Erdos686.erdos_686.variants.twenty_five"},{"answerKinds":[],"category":"research open","docstring":"Can every integer $N≥2$ be written as\n$$N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«686»","statement":"True ↔ ∀ N ≥ 2, ∃ k ≥ 2, ∃ n, ∃ m ≥ n + k, ↑N = ↑(∏ i ∈ Finset.Icc 1 k, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 k, (n + i))","subjects":["11"],"theorem":"Erdos686.erdos_686"},{"answerKinds":[],"category":"research solved","docstring":"The number $4$ cannot be written as\n$$4=\\frac{\\prod_{1\\leq i\\leq 2}(m+i)}{\\prod_{1\\leq i\\leq 2}(n+i)}$$\nfor $m≥n+2$!\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«686»","statement":"¬∃ n, ∃ m ≥ n + 2, 4 = ↑(∏ i ∈ Finset.Icc 1 2, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 2, (n + i))","subjects":["11"],"theorem":"Erdos686.erdos_686.variants.four_two"},{"answerKinds":[],"category":"research open","docstring":"Can every square $N≥2$ be written as\n$$N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}$$\nfor some $k≥2$ and $m≥n+k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«686»","statement":"True ↔\n  ∀ N ≥ 2,\n    IsSquare N → ∃ k ≥ 2, ∃ n, ∃ m ≥ n + k, ↑N = ↑(∏ i ∈ Finset.Icc 1 k, (m + i)) / ↑(∏ i ∈ Finset.Icc 1 k, (n + i))","subjects":["11"],"theorem":"Erdos686.erdos_686.variants.square"},{"answerKinds":[],"category":"test","docstring":"This theorem provides a sanity check, showing that the main conjecture (`erdos_89`) is strictly\nstronger than the solved Guth and Katz result. It proves that, trivially, if the lower bound\n$\\frac{n}{\\sqrt{\\log n}}$ holds, then the weaker lower bound $\\frac{n}{\\log n}$ must also hold.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«89»","statement":"((fun n => ↑n / √(Real.log ↑n)) =O[Filter.atTop] fun n => ↑(minimalDistinctDistances (EuclideanSpace ℝ (Fin 2)) n)) →\n  (fun n => ↑n / Real.log ↑n) =O[Filter.atTop] fun n => ↑(minimalDistinctDistances (EuclideanSpace ℝ (Fin 2)) n)","subjects":["52"],"theorem":"Erdos89.erdos_89.variants.implies_n_dvd_log_n"},{"answerKinds":[],"category":"research solved","docstring":"Guth and Katz [GuKa15] proved that there are always $\\gg \\frac{n}{\\log n}$\nmany distinct distances.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«89»","statement":"(fun n => ↑n / Real.log ↑n) =O[Filter.atTop] fun n => ↑(minimalDistinctDistances (EuclideanSpace ℝ (Fin 2)) n)","subjects":["52"],"theorem":"Erdos89.erdos_89.variants.n_dvd_log_n"},{"answerKinds":[],"category":"research open","docstring":"Erdős [Er46] asked whether every set of $n$ distinct points in $\\mathbb{R}^2$\ndetermines $\\gg \\frac{n}{\\sqrt{\\log n}}$ many distinct distances.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«89»","statement":"(fun n => ↑n / √(Real.log ↑n)) =O[Filter.atTop] fun n => ↑(minimalDistinctDistances (EuclideanSpace ℝ (Fin 2)) n)","subjects":["52"],"theorem":"Erdos89.erdos_89"},{"answerKinds":[],"category":"research solved","docstring":"The square grid construction, going back to Erdős and Moser, shows that\n$\\frac{n}{\\sqrt{\\log n}}$ is the correct order if the conjecture is true:\nthere are configurations whose number of distinct distances is\n$O(\\frac{n}{\\sqrt{\\log n}})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«89»","statement":"(fun n => ↑(minimalDistinctDistances (EuclideanSpace ℝ (Fin 2)) n)) =O[Filter.atTop] fun n => ↑n / √(Real.log ↑n)","subjects":["52"],"theorem":"Erdos89.erdos_89.variants.grid_upper_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $R(N)$ be the size of the largest $A\\subseteq\\{1, ..., N\\}$ such that all sums $\\sum_{n\\in S} \\frac{1}{n}$ are distinct for $S\\subseteq A$. Find the simplest $g(N)$ such that $R(N) = O(g(N))$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«321»","statement":"(fun N => ↑(Erdos321.R N)) =O[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos321.erdos_321.variants.isBigO"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $R(N)$ be the size of the largest $A\\subseteq\\{1, ..., N\\}$ such that all sums $\\sum_{n\\in S} \\frac{1}{n}$ are distinct for $S\\subseteq A$. Find the simplest $g(N)$ such that $R(N) = o(g(N))$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«321»","statement":"(fun N => ↑(Erdos321.R N)) =o[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos321.erdos_321.variants.isLittleO"},{"answerKinds":[],"category":"research solved","docstring":"Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that\n$$\n\\frac{N}{\\log N} \\prod_{i=3}^{k} \\log_i N \\le R(N),\n$$\nvalid for any $k \\ge 4$ with $\\log_k N \\ge k$ and any $r \\ge 1$ with $\\log_{2r} N \\ge 1$. (In these bounds $\\log_i n$ denotes the $i$-fold iterated logarithm.)\n\n[BlEr75] Bleicher, M. N. and Erdős, P., _The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$_. Math. Comp. (1975), 29-42.\n[BlEr76b] Bleicher, Michael N. and Erdős, Paul, _Denominators of Egyptian fractions. II_. Illinois J. Math. (1976), 598-613.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«321»","statement":"∀ (N k : ℕ), 4 ≤ k → ↑k ≤ Real.log^[k] ↑N → ↑N / Real.log ↑N * ∏ i ∈ Finset.Icc 3 k, Real.log^[i] ↑N ≤ ↑(Erdos321.R N)","subjects":["11"],"theorem":"Erdos321.erdos_321.variants.lower"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $R(N)$ be the size of the largest $A\\subseteq\\{1, ..., N\\}$ such that all sums\n$\\sum_{n\\in S} \\frac{1}{n}$ are distinct for $S\\subseteq A$. What is $R(N)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«321»","statement":"∀ (N : ℕ), Erdos321.R N = sorry","subjects":["11"],"theorem":"Erdos321.erdos_321"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $R(N)$ be the size of the largest $A\\subseteq\\{1, ..., N\\}$ such that all sums\n$\\sum_{n\\in S} \\frac{1}{n}$ are distinct for $S\\subseteq A$. What is $\\Theta(R(N))$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«321»","statement":"(fun N => ↑(Erdos321.R N)) =Θ[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos321.erdos_321.variants.isTheta"},{"answerKinds":[],"category":"research solved","docstring":"Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that\n$$\nR(N) \\le \\frac{1}{\\log 2} \\log_r N \\left( \\frac{N}{\\log N} \\prod_{i=3}^{r} \\log_i N \\right),\n$$\nvalid for any $k \\ge 4$ with $\\log_k N \\ge k$ and any $r \\ge 1$ with $\\log_{2r} N \\ge 1$. (In these bounds $\\log_i n$ denotes the $i$-fold iterated logarithm.)\n\n[BlEr75] Bleicher, M. N. and Erdős, P., _The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$_. Math. Comp. (1975), 29-42.\n[BlEr76b] Bleicher, Michael N. and Erdős, Paul, _Denominators of Egyptian fractions. II_. Illinois J. Math. (1976), 598-613.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«321»","statement":"∀ (N r : ℕ),\n  1 ≤ r →\n    1 ≤ Real.log^[2 * r] ↑N →\n      ↑(Erdos321.R N) ≤ 1 / Real.log 2 * Real.log^[r] ↑N * ↑N / Real.log ↑N * ∏ i ∈ Finset.Icc 3 r, Real.log^[i] ↑N","subjects":["11"],"theorem":"Erdos321.erdos_321.variants.upper"},{"answerKinds":[],"category":"research open","docstring":"Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\\ldots$\nsuch that $\\lvert x_{n+1}-x_n\\rvert \\ll 1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«952»","statement":"True ↔ ∃ x C, Function.Injective x ∧ ∀ (n : ℕ), Prime (x n) ∧ Zsqrtd.norm (x (n + 1) - x n) < C","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos952.erdos_952"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, if $A\\subseteq \\mathbb{N}$ is sparse enough and does not cover all residue\nclasses modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for\nall $a\\in A$?\n\nWeisenberg [We24] has shown the answer is no: $A$ can be arbitrarily sparse and missing at\nleast one residue class modulo every prime $p$, and yet $A+n$ is not contained in the primes\nfor any $n\\in \\mathbb{Z}$. (Weisenberg gives several constructions of such an $A$.)\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos429.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«429»","statement":"False ↔\n  ∃ f,\n    Filter.Tendsto f Filter.atTop Filter.atTop ∧\n      ∀ (A : Set ℕ),\n        A.Infinite →\n          (∀ (N : ℕ), (A ∩ Set.Icc 1 N).ncard ≤ f N) →\n            (∀ (p : ℕ), Nat.Prime p → ∃ b, ∀ a ∈ A, ↑a ≠ b) → ∃ n, ∀ a ∈ A, Nat.Prime (n + a)","subjects":["11"],"theorem":"Erdos429.erdos_429"},{"answerKinds":[],"category":"research open","docstring":"Let $F(n,\\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of\n$K_n$ so that every $X\\subseteq [n]$ with $\\lvert X\\rvert\\geq m$ contains more than\n$\\alpha \\binom{\\lvert X\\rvert}{2}$ many edges of each colour.\n\nProve that, for every $0\\leq \\alpha < 1/2$,\n$$F(n,\\alpha)\\sim c_\\alpha\\log n$$\nfor some constant $c_\\alpha$ depending only on $\\alpha$.\n\nThis problem is #39 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«563»","statement":"∀ (α : ℝ),\n  0 ≤ α → α < 1 / 2 → ∃ c, 0 < c ∧ Filter.Tendsto (fun n => ↑(Erdos563.F n α) / Real.log ↑n) Filter.atTop (nhds c)","subjects":["5"],"theorem":"Erdos563.erdos_563"},{"answerKinds":[],"category":"research open","docstring":"Is there a sequence $1 \\le d_1 < d_2 < \\dots$ with density 1 such that all products\n$\\prod_{u \\le i \\le v} d_i$ are distinct? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«421»","statement":"True ↔\n  ∃ d,\n    StrictMono d ∧\n      1 ≤ d 0 ∧\n        (Set.range d).HasDensity 1 ∧\n          Set.InjOn\n            (fun x =>\n              match x with\n              | (u, v) => ∏ i ∈ Finset.Icc u v, d i)\n            {(u, v) | u ≤ v}","subjects":["11"],"theorem":"Erdos421.erdos_421"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 846**\nLet `A ⊂ ℝ²` be an infinite set for which there exists some `ϵ>0` such that in any subset of `A`\nof size `n` there are always at least `ϵn` with no three on a line.\nIs it true that `A` is the union of a finite number of sets where no three are on a line?\n\nIn other words, prove or disprove the following statement: every infinite `ε`-non-trilinear subset of the\nplane is weakly non-trilinar.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/2404258180688283e5141021c75464dc2acfb798/FormalConjectures/ErdosProblems/846.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«846»","statement":"False ↔\n  ∀ (A : Set (EuclideanSpace ℝ (Fin 2))),\n    ∀ ε > 0, A.Infinite → Erdos846.NonTrilinearFor A ε → Erdos846.WeaklyNonTrilinear A","subjects":["11"],"theorem":"Erdos846.erdos_846"},{"answerKinds":[],"category":"research open","docstring":"Is the sum $\\sum\\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph?\n\nThis problem is #65 in Extremal Graph Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«65»","statement":"True ↔\n  ∀ (k : ℝ),\n    0 < k →\n      ∀ (n : ℕ) (V : Type) [inst : Fintype V] (G : SimpleGraph V),\n        0 < n →\n          Fintype.card V = n →\n            ↑G.edgeSet.ncard = k * ↑n →\n              ∀ (A B : Type) [inst : Fintype A] [inst_1 : Fintype B],\n                Fintype.card (A ⊕ B) = n →\n                  ↑(completeBipartiteGraph A B).edgeSet.ncard = k * ↑n →\n                    ∑ᶠ (a : ℕ) (_ : a ∈ (completeBipartiteGraph A B).cycleLengths), 1 / ↑a ≤\n                      ∑ᶠ (a : ℕ) (_ : a ∈ G.cycleLengths), 1 / ↑a","subjects":["5"],"theorem":"Erdos65.erdos_65.parts.ii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\\cdots$ be the lengths of\ncycles in $G$. Assume $n>0$ and $k>0$. Is it true that\n$$\\sum\\frac{1}{a_i}\\gg \\log k?$$\n\nGyárfás, Komlós, and Szemerédi [GKS84] have proved that this sum is $\\gg \\log k$, so that only\nthe second question remains.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«65»","statement":"True ↔\n  ∃ c > 0,\n    ∀ (k : ℝ),\n      0 < k →\n        ∀ (n : ℕ) (V : Type) [inst : Fintype V] (G : SimpleGraph V),\n          0 < n →\n            Fintype.card V = n →\n              ↑G.edgeSet.ncard = k * ↑n → ∑ᶠ (a : ℕ) (_ : a ∈ G.cycleLengths), 1 / ↑a ≥ c * Real.log k","subjects":["5"],"theorem":"Erdos65.erdos_65.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A$ be a finite set of integers such that $\\lvert A+A\\rvert \\ll \\lvert A\\rvert$. Is it true that\n$$\\lvert AA\\rvert \\gg \\frac{\\lvert A\\rvert^2}{(\\log \\lvert A\\rvert)^C}$$\nfor some constant $C>0$?\n\nThis was proved by Solymosi [So09d], in the strong form\n$$\\lvert AA\\rvert \\gg \\frac{\\lvert A\\rvert^2}{\\log \\lvert A\\rvert}.$$\nSee also [52].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos818.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«818»","statement":"True ↔\n  ∀ (K : ℝ),\n    0 < K →\n      ∃ C,\n        0 < C ∧\n          ∃ c,\n            0 < c ∧\n              ∀ (A : Finset ℤ),\n                2 ≤ A.card → ↑(A + A).card ≤ K * ↑A.card → c * ↑A.card ^ 2 / Real.log ↑A.card ^ C ≤ ↑(A * A).card","subjects":["5","11"],"theorem":"Erdos818.erdos_818"},{"answerKinds":[],"category":"research solved","docstring":"This was proved by Solymosi [So09d], in the strong form\n$$\\lvert AA\\rvert \\gg \\frac{\\lvert A\\rvert^2}{\\log \\lvert A\\rvert}.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«818»","statement":"∀ (K : ℝ),\n  0 < K →\n    ∃ c,\n      0 < c ∧\n        ∀ (A : Finset ℤ), 2 ≤ A.card → ↑(A + A).card ≤ K * ↑A.card → c * ↑A.card ^ 2 / Real.log ↑A.card ≤ ↑(A * A).card","subjects":["5","11"],"theorem":"Erdos818.erdos_818.variants.solymosi"},{"answerKinds":[],"category":"research open","docstring":"For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that\nthe values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c.\nThis is an open problem.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1004»","statement":"True ↔ ∀ c > 0, ∀ᶠ (x : ℕ) in Filter.atTop, ∃ n ≤ x, Erdos1004.IsDistinctTotientRun n ⌊Real.log ↑x ^ c⌋₊","subjects":["11"],"theorem":"Erdos1004.erdos_1004"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Erdős, Pomerance, and Sárközy [EPS87] proved that if φ(n+k) are all distinct for 1 ≤ k ≤ K then\nK ≤ n / exp(c (log n)^{1/3}) for some constant c > 0.\nHere we state the existence of such a constant c.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1004»","statement":"True ↔\n  ∃ c,\n    ∃ (_ : c > 0),\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ (K : ℕ), Erdos1004.IsDistinctTotientRun n K → ↑K ≤ ↑n / Real.exp (c * Real.log ↑n ^ (1 / 3))","subjects":["11"],"theorem":"Erdos1004.erdos_1004.variants.le_of_isDistinctTotientRun"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, for every $k$, there is some $f(k)$ such that if $G$ has chromatic number\n$\\geq f(k)$ then $G$ contains a triangle-free subgraph with chromatic number $\\geq k$?\n\nThis is true, as shown by Rödl [Ro77].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos923.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«923»","statement":"True ↔\n  ∀ (V : Type u_1) (n : ℕ),\n    ∃ k, ∀ (G : SimpleGraph V), ↑k ≤ G.chromaticNumber → ∃ H ≤ G, ↑n ≤ H.chromaticNumber ∧ H.CliqueFree 3","subjects":["5"],"theorem":"Erdos923.erdos_923"},{"answerKinds":[],"category":"research solved","docstring":"A question of Erdős and Selfridge [ErSe67], who observe that\n$\\liminf_{n\\to \\infty}\\sum_{0\\leq i < k}\\omega(n+i)\\geq k+\\pi(k)-1$ for every $k$. This follows from\nPólya's theorem that the set of $k$-smooth integers has unbounded gaps - indeed,\n$n(n+1)\\cdots (n+k-1)$ is divisible by all primes $\\leq k$ and, provided $n$ is large, all but at\nmost one of $n,n+1,\\ldots,n+k-1$ has a prime factor $>k$ by Pólya's theorem.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«890»","statement":"∀ (k : ℕ),\n  Filter.liminf (fun n => ∑ i ∈ Finset.range k, ↑(ArithmeticFunction.cardDistinctFactors (n + i))) Filter.atTop ≥\n    ↑k + ↑k.primeCounting - 1","subjects":["11"],"theorem":"Erdos890.erdos_890.variants.liminf_lower_bound"},{"answerKinds":[],"category":"research open","docstring":"If $\\omega_k(n)$ counts the number of distinct prime factors of $n$ which are $>k$, then is it true\nthat, for every $k\\geq 1$,\n$$\\liminf_{n\\to \\infty}\\sum_{0\\leq i < k}\\omega_k(n+i)\\leq k?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«890»","statement":"True ↔ ∀ k ≥ 1, Filter.liminf (fun n => ∑ i ∈ Finset.range k, ↑(Erdos890.omegaGt k (n + i))) Filter.atTop ≤ ↑k","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos890.erdos_890.parts.a"},{"answerKinds":[],"category":"research open","docstring":"Is it true that\n$$\\limsup_{n\\to \\infty}\\left(\\sum_{0\\leq i < k}\\omega(n+i)\\right) \\frac{\\log\\log n}{\\log n}=1,$$\nwhere $\\omega$ counts the number of distinct prime factors without restriction?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«890»","statement":"True ↔\n  ∀ k ≥ 1,\n    Filter.limsup\n        (fun n =>\n          (∑ i ∈ Finset.range k, ↑(ArithmeticFunction.cardDistinctFactors (n + i))) *\n            (↑(Real.log (Real.log ↑n)) / ↑(Real.log ↑n)))\n        Filter.atTop =\n      1","subjects":["11"],"theorem":"Erdos890.erdos_890.parts.b"},{"answerKinds":[],"category":"research solved","docstring":"It is a classical fact that $\\limsup_{n\\to \\infty}\\omega(n)\\frac{\\log\\log n}{\\log n}=1.$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«890»","statement":"Filter.limsup (fun n => ↑(ArithmeticFunction.cardDistinctFactors n) * (↑(Real.log (Real.log ↑n)) / ↑(Real.log ↑n)))\n    Filter.atTop =\n  1","subjects":["11"],"theorem":"Erdos890.erdos_890.variants.omega_limsup"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $A$ and $B$ are equal-sized Sidon sets in $\\{1,\\ldots,N\\}$ with\n$(A-A)\\cap(B-B)=\\{0\\}$, can the bound be improved to\n$$\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\n    \\leq (1-c+o(1))\\binom{f(N)}{2}$$\nfor some constant $c>0$?\n\nThe answer is no; the Erdős Problems page records a negative answer due to Barreto.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«43»","statement":"False ↔\n  ∃ c > 0,\n    ∃ o,\n      o =o[Filter.atTop] 1 ∧\n        ∀ᶠ (N : ℕ) in Filter.atTop,\n          ∀ (A B : Finset ℕ),\n            A ⊆ Finset.Icc 1 N →\n              B ⊆ Finset.Icc 1 N →\n                IsSidon ↑A →\n                  IsSidon ↑B →\n                    A.card = B.card →\n                      (A - A) ∩ (B - B) = {0} →\n                        ↑(A.card.choose 2 + B.card.choose 2) ≤ (1 - c + o N) * ↑((Erdos43.f N).choose 2)","subjects":["5","11"],"theorem":"Erdos43.erdos_43.parts.ii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $A$ and $B$ are Sidon sets in $\\{1,\\ldots,N\\}$ with\n$(A-A)\\cap(B-B)=\\{0\\}$, is it true that\n$$\\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1)?$$\n\nThe answer is no; the Erdős Problems page notes that this follows from the solution to\nErdős Problem 42.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«43»","statement":"False ↔\n  ∃ C,\n    ∀ᶠ (N : ℕ) in Filter.atTop,\n      ∀ (A B : Finset ℕ),\n        A ⊆ Finset.Icc 1 N →\n          B ⊆ Finset.Icc 1 N →\n            IsSidon ↑A →\n              IsSidon ↑B →\n                (A - A) ∩ (B - B) = {0} → ↑(A.card.choose 2 + B.card.choose 2) ≤ ↑((Erdos43.f N).choose 2) + C","subjects":["5","11"],"theorem":"Erdos43.erdos_43.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Romanoff [Ro34] proved that the answer is yes if $C$ is an integer.\n\n[Ro34] Romanoff, N. P., _Über einige Sätze der additiven Zahlentheorie_.\nMath. Ann. (1934), 668-678. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«244»","statement":"∀ {C : ℕ}, 1 < C → 0 < {x | ∃ p k, ∃ (_ : Nat.Prime p), p + ⌊C ^ k⌋₊ = x}.lowerDensity","subjects":["11"],"theorem":"Erdos244.erdos_244.variants.Romanoff"},{"answerKinds":[],"category":"research open","docstring":"Let $C > 1$. Does the set of integers of the form $p + \\lfloor C^k \\rfloor$,\nfor some prime $p$ and $k\\geq 0$, have density $>0$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«244»","statement":"True ↔ ∀ C > 1, 0 < {x | ∃ p k, ∃ (_ : Nat.Prime p), p + ⌊C ^ k⌋₊ = x}.lowerDensity","subjects":["11"],"theorem":"Erdos244.erdos_244"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n;r,k)$ be the maximal number of edges in an $r$-uniform hypergraph which contains no set of $k$ many independent edges.\n\nFor all $r\\geq 3$, $$f(n;r,k)=\\max\\left(\\binom{rk-1}{r}, \\binom{n}{r}-\\binom{n-k+1}{r}\\right).$$\n\nNote: the source states the formula with no range on `n` or `k`, but some restriction\nis needed: e.g. for `r = 3`, `k = 2`, `n = 4` no two disjoint triples fit in `4`\nvertices, so the left-hand side is `4.choose 3 = 4` while the right-hand side is\n`5.choose 3 = 10`. We require `k ≥ 1` and `n ≥ r*k - 1`: this is the smallest `n`\naccommodating the construction counted by the first term (all `r`-subsets of a fixed\n`(r*k - 1)`-set), and at `n = r*k - 1` the equality holds trivially, since the complete\n`r`-uniform hypergraph has no `k`-matching. The source's commentary likewise calls the\ncase `n < k*r` trivial.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1020»","statement":"∀ (r : ℕ),\n  3 ≤ r →\n    ∀ (n k : ℕ),\n      0 < k → r * k - 1 ≤ n → Erdos1020.f n r k = max ((r * k - 1).choose r) (n.choose r - (n - k + 1).choose r)","subjects":["5"],"theorem":"Erdos1020.erdos_1020"},{"answerKinds":[],"category":"research solved","docstring":"Suppose $n$ points in $\\mathbb{R}^2$ determine a convex polygon and the set of distances between\nthem is $\\{u_1,\\ldots,u_t\\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of\npoints. Then\n$$\\sum_i f(u_i)^2 \\ll n^3.$$\n\nIn [Er97c] Erdős claims that Fishburn solved this, but gives no reference.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos94.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«94»","statement":"∃ C > 0,\n  ∀ (P : Finset (EuclideanSpace ℝ (Fin 2))),\n    EuclideanGeometry.ConvexIndep ↑P → ∑ u ∈ distanceSet P, ↑(distanceMultiplicity P u) ^ 2 ≤ C * ↑P.card ^ 3","subjects":["5","52"],"theorem":"Erdos94.erdos_94"},{"answerKinds":[],"category":"research solved","docstring":"Lefmann and Theile [LeTh95] prove a stronger version of this question, that\n$$\\sum_i f(u_i)^2 \\ll n^3$$\nunder the weaker assumption that no three points are on a line.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«94»","statement":"∃ C > 0,\n  ∀ (P : Finset (EuclideanSpace ℝ (Fin 2))),\n    EuclideanGeometry.NonTrilinear ↑P → ∑ u ∈ distanceSet P, ↑(distanceMultiplicity P u) ^ 2 ≤ C * ↑P.card ^ 3","subjects":["5","52"],"theorem":"Erdos94.erdos_94.variants.no_three_on_a_line"},{"answerKinds":[],"category":"test","docstring":"Note it is trivial that $\\sum f(u_i)=\\binom{n}{2}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«94»","statement":"∀ (P : Finset (EuclideanSpace ℝ (Fin 2))), ∑ u ∈ distanceSet P, distanceMultiplicity P u = P.card.choose 2","subjects":["5","52"],"theorem":"Erdos94.erdos_94.variants.sum_multiplicity"},{"answerKinds":[],"category":"research open","docstring":"Erdős and Fishburn also make the stronger conjecture that $\\sum f(u_i)^2$ is maximal for the\nregular $n$-gon (for large enough $n$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«94»","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ (P : Finset (EuclideanSpace ℝ (Fin 2))),\n    P.card = n →\n      EuclideanGeometry.ConvexIndep ↑P →\n        ∑ u ∈ distanceSet P, ↑(distanceMultiplicity P u) ^ 2 ≤\n          ∑ u ∈ distanceSet (Erdos94.regularNGon n), ↑(distanceMultiplicity (Erdos94.regularNGon n) u) ^ 2","subjects":["5","52"],"theorem":"Erdos94.erdos_94.variants.regular_ngon"},{"answerKinds":[],"category":"research open","docstring":"In [Er71] Erdős suggests that only $n-1$ many cycles and edges are required if we do not\nrequire them to be edge-disjoint.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«184»","statement":"True ↔\n  ∀ {V : Type} [inst : Fintype V] [DecidableEq V] [Nonempty V] (G : SimpleGraph V),\n    ∃ D, (∀ H ∈ D, Erdos184.IsCycleOrEdge H.coe) ∧ ⋃ H ∈ D, H.edgeSet = G.edgeSet ∧ ↑D.card ≤ ↑(Fintype.card V) - 1","subjects":["5"],"theorem":"Erdos184.erdos_184.variants.covering"},{"answerKinds":[],"category":"research solved","docstring":"Conlon, Fox, and Sudakov [CFS14] proved that $O_\\epsilon(n)$ cycles and edges suffice if $G$ has\nminimum degree at least $\\epsilon n$, for any $\\epsilon>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«184»","statement":"∀ ε > 0,\n  ∃ f,\n    (f =O[Filter.atTop] fun n => ↑n) ∧\n      ∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V),\n        ↑G.minDegree ≥ ε * ↑(Fintype.card V) →\n          ∃ D, (∀ H ∈ D, Erdos184.IsCycleOrEdge H.coe) ∧ G.IsDecomposition D ∧ ↑D.card ≤ f (Fintype.card V)","subjects":["5"],"theorem":"Erdos184.erdos_184.variants.conlon_fox_sudakov"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Gallai [EGP66] proved that $O(n \\log n)$ many cycles and edges suffices.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«184»","statement":"∃ f,\n  (f =O[Filter.atTop] fun n => ↑n * Real.log ↑n) ∧\n    ∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V),\n      ∃ D, (∀ H ∈ D, Erdos184.IsCycleOrEdge H.coe) ∧ G.IsDecomposition D ∧ ↑D.card ≤ f (Fintype.card V)","subjects":["5"],"theorem":"Erdos184.erdos_184.variants.n_log_n"},{"answerKinds":[],"category":"research solved","docstring":"The best bound available is due to Bucić and Montgomery [BM22], who prove that $O(n\\log^* n)$ many\ncycles and edges suffice, where $\\log^*$ is the iterated logarithm function.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«184»","statement":"∃ f,\n  (f =O[Filter.atTop] fun n => ↑n * ↑(↑n).iteratedLog) ∧\n    ∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V),\n      ∃ D, (∀ H ∈ D, Erdos184.IsCycleOrEdge H.coe) ∧ G.IsDecomposition D ∧ ↑D.card ≤ f (Fintype.card V)","subjects":["5"],"theorem":"Erdos184.erdos_184.variants.bucic_montgomery"},{"answerKinds":[],"category":"research solved","docstring":"The graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some\nconstant $c>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«184»","statement":"∃ c > 0,\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    have G := SimpleGraph.fromRel fun i j => ↑i < 3 ∧ 3 ≤ ↑j;\n    ∀ (D : Finset G.Subgraph), (∀ H ∈ D, Erdos184.IsCycleOrEdge H.coe) → G.IsDecomposition D → (1 + c) * ↑n ≤ ↑D.card","subjects":["5"],"theorem":"Erdos184.erdos_184.variants.lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«184»","statement":"∃ f,\n  (f =O[Filter.atTop] fun n => ↑n) ∧\n    ∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V),\n      ∃ D, (∀ H ∈ D, Erdos184.IsCycleOrEdge H.coe) ∧ G.IsDecomposition D ∧ ↑D.card ≤ f (Fintype.card V)","subjects":["5"],"theorem":"Erdos184.erdos_184"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $k\\geq 2$ and let $g_k(n)$ be the largest possible size of\n$A\\subseteq \\{1,\\ldots,n\\}$ such that every $m$ has $<k$ solutions to\n$m=a_1a_2$ with $a_1<a_2\\in A$. Is it true that\n$$g_3(n)=\\frac{\\log\\log n}{\\log n}n+(c+o(1))\\frac{n}{\\log n}$$\nfor some constant $c$?\n\nThe answer is yes: the rescaled error `normalizedError` converges.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-796/Research/CanonicalTail.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«796»","statement":"True ↔ ∃ c, Filter.Tendsto Erdos796.normalizedError Filter.atTop (nhds c)","subjects":["11"],"theorem":"Erdos796.erdos_796"},{"answerKinds":[],"category":"research open","docstring":"Let $\\frac a b\\in \\mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1 < n_1 < \\dots < n_k$,\neach the product of two distinct primes, such that $\\frac{a}{b}=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«306»","statement":"True ↔\n  ∀ (q : ℚ),\n    0 < q →\n      Squarefree q.den →\n        ∃ k n,\n          n 0 = 1 ∧\n            StrictMono n ∧\n              (∀ i ∈ Finset.Icc 1 (Fin.last k),\n                  ArithmeticFunction.cardDistinctFactors (n i) = 2 ∧ ArithmeticFunction.cardFactors (n i) = 2) ∧\n                q = ∑ i ∈ Finset.Icc 1 (Fin.last k), 1 / ↑(n i)","subjects":["11"],"theorem":"Erdos306.erdos_306"},{"answerKinds":[],"category":"research solved","docstring":"Every positive integer can be expressed as an Egyptian fraction where each denominator is the\nproduct of three distinct primes.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«306»","statement":"∀ (m : ℕ),\n  0 < m →\n    ∃ k > 0,\n      ∃ n,\n        n 0 = 1 ∧\n          ∀ (i : ℕ) (hik : i < k),\n            n ⟨i, ⋯⟩ < n ⟨i + 1, ⋯⟩ ∧\n              (∀ i ∈ Finset.Icc 1 (Fin.last k),\n                  ArithmeticFunction.cardDistinctFactors (n i) = 3 ∧ ArithmeticFunction.cardFactors (n i) = 3) ∧\n                ↑m = ∑ i ∈ Finset.Icc 1 (Fin.last k), 1 / ↑(n i)","subjects":["11"],"theorem":"Erdos306.erdos_306.variants.integer_three_primes"},{"answerKinds":[],"category":"research open","docstring":"In [Er88c], Erdős asks the weaker question of whether there exists a rational $x$ with at\nleast two representations\n$$x = \\sum_{k=1}^{\\infty} \\frac{a_k}{2^{a_k}}$$\nby pairwise distinct positive integers $a_k$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«261»","statement":"True ↔ ∃ x, 2 ≤ Cardinal.mk ↑{a | Erdos261.Erdos261InfiniteRepresentation x a}","subjects":["11"],"theorem":"Erdos261.erdos_261.variants.two_representations"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there infinitely many positive integers $n$ such that there exist some $t \\ge 2$ and\ndistinct integers $a_1, \\ldots, a_t \\ge 1$ satisfying\n$$\\frac{n}{2^n} = \\sum_{1 \\le k \\le t} \\frac{a_k}{2^{a_k}}?$$\n\nIn [Er88c], Erdős notes that Cusick had a simple proof that infinitely many such $n$ exist. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«261»","statement":"True ↔ {n | 0 < n ∧ Erdos261.Erdos261Prop n}.Infinite","subjects":["11"],"theorem":"Erdos261.erdos_261.parts.i"},{"answerKinds":[],"category":"textbook","docstring":"The Borwein--Loring construction gives the required property when $m \\ge 2$. This lower\nbound ensures that the representation contains at least two terms. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«261»","statement":"∀ (m : ℕ), 2 ≤ m → Erdos261.Erdos261Prop (2 ^ (m + 1) - m - 2)","subjects":["11"],"theorem":"Erdos261.erdos_261.variants.borwein_loring_property"},{"answerKinds":[],"category":"research open","docstring":"Do all positive integers $n$ have the required property? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«261»","statement":"True ↔ ∀ n > 0, Erdos261.Erdos261Prop n","subjects":["11"],"theorem":"Erdos261.erdos_261.parts.ii"},{"answerKinds":[],"category":"textbook","docstring":"For every positive integer $m$, if $n = 2^{m+1} - m - 2$, then\n$$\\frac{n}{2^n} = \\sum_{n < k \\le n + m} \\frac{k}{2^k}.$$\n\nThis construction is due to Borwein and Loring [BoLo90]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«261»","statement":"∀ (m : ℕ),\n  0 < m →\n    have n := 2 ^ (m + 1) - m - 2;\n    ↑n / 2 ^ n = ∑ k ∈ Finset.Ioc n (n + m), ↑k / 2 ^ k","subjects":["11"],"theorem":"Erdos261.erdos_261.variants.borwein_loring"},{"answerKinds":[],"category":"research solved","docstring":"Tengely, Ulas, and Zygadlo [TUZ20] verified that every positive integer $n \\le 10000$ has\nthe required property. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«261»","statement":"∀ {n : ℕ}, 0 < n → n ≤ 10000 → Erdos261.Erdos261Prop n","subjects":["11"],"theorem":"Erdos261.erdos_261.variants.le_10000"},{"answerKinds":[],"category":"research open","docstring":"Is there a rational number $x$ such that\n$$x = \\sum_{k=1}^{\\infty} \\frac{a_k}{2^{a_k}}$$\nhas at least $2^{\\aleph_0}$ representations by pairwise distinct positive integers $a_k$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«261»","statement":"True ↔ ∃ x, Cardinal.continuum ≤ Cardinal.mk ↑{a | Erdos261.Erdos261InfiniteRepresentation x a}","subjects":["11"],"theorem":"Erdos261.erdos_261.parts.iii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"We call a graph $D$-balanced (or $D$-almost-regular) if the maximum degree is at most $D$ times the\nminimum degree.\n\nLet $ε, α > 0$ and $D$ and $n$ be sufficiently large. If $G$ is a graph on $n$ vertices with at\nleast $n^{1+α}$ edges, then must $G$ contain a $D$-balanced subgraph on $m > n^{1-α}$ vertices with\nat least $εm^{1+α}$ edges?\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos1077.lean#L265"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1077»","statement":"False ↔\n  ∀ ε > 0,\n    ε < 1 →\n      ∀ α > 0,\n        α < 1 →\n          ∀ᶠ (D : ℝ) in Filter.atTop,\n            ∀ᶠ (n : ℕ) in Filter.atTop,\n              ∀ (G : SimpleGraph (Fin n)),\n                ↑G.edgeSet.ncard > ↑n ^ (1 + α) →\n                  ∃ H,\n                    H.coe.IsBalanced D ∧ ↑H.verts.ncard > ↑n ^ (1 - α) ∧ ↑H.edgeSet.ncard > ε * ↑H.verts.ncard ^ (1 + α)","subjects":["5"],"theorem":"Erdos1077.erdos_1077"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $k\\geq 4$ and let $f_k(n)$ be the largest $m$ such that there is a graph on $n$ vertices\nwith chromatic number $k$ in which every odd cycle has length $> m$.\nThen\n$$f_k(n) \\asymp n^{\\frac{1}{k-2}}.$$\n\nA question of Erdős and Gallai.\n\nProved for all $k\\geq 4$ by Kierstead, Szemerédi, and Trotter [KST84].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«921»","statement":"True ↔\n  ∀ (k : ℕ),\n    4 ≤ k →\n      ∃ c₁ c₂,\n        0 < c₁ ∧\n          0 < c₂ ∧\n            (∀ᶠ (n : ℕ) in Filter.atTop,\n                ∃ G, G.chromaticNumber = ↑k ∧ ∀ l ∈ G.oddCycleLengths, c₁ * ↑n ^ (1 / (↑k - 2)) < ↑l) ∧\n              ∀ᶠ (n : ℕ) in Filter.atTop,\n                ∀ (G : SimpleGraph (Fin n)),\n                  G.chromaticNumber = ↑k → ∃ l ∈ G.oddCycleLengths, ↑l ≤ c₂ * ↑n ^ (1 / (↑k - 2))","subjects":["5"],"theorem":"Erdos921.erdos_921"},{"answerKinds":[],"category":"research open","docstring":"Is $\\sum_{n=1}^\\infty \\frac{p_n}{2^n}$ irrational? Here $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \\dots$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«251»","statement":"True ↔ Irrational (∑' (n : ℕ), ↑(Nat.nth Nat.Prime n) / 2 ^ n)","subjects":["11"],"theorem":"Erdos251.erdos_251"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval\nin $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where\neach $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct.\n\nIn particular is it true that $f(m)\\leq \\sqrt{m}$?\n\nGPT 5.4 Pro (prompted by He, Li, and Tang) proved $f(m)\\leq \\lceil 2\\sqrt{m}\\rceil$. A\ncorresponding lower bound was given by GPT 5.4 Pro and Aristotle; it is now known (see the paper\nof van Doorn, Li, and Tang [VLT26]) that\n$$f(m) = \\min(m, \\lceil 2\\sqrt{m}\\rceil)$$\nfor all $m$.\n\nThis was formalized in Lean by van Doorn using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem650.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«650»","statement":"False ↔ ∀ (m : ℕ), ↑(Erdos650.f m) ≤ √↑m","subjects":["5","11"],"theorem":"Erdos650.erdos_650.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval\nin $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where\neach $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct.\n\nEstimate $f(m)$.\n\nGPT 5.4 Pro (prompted by He, Li, and Tang) proved $f(m)\\leq \\lceil 2\\sqrt{m}\\rceil$. A\ncorresponding lower bound was given by GPT 5.4 Pro and Aristotle; it is now known (see the paper\nof van Doorn, Li, and Tang [VLT26]) that\n$$f(m) = \\min(m, \\lceil 2\\sqrt{m}\\rceil)$$\nfor all $m$.\n\nThis was formalized in Lean by van Doorn using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem650.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«650»","statement":"∀ (m : ℕ), Erdos650.f m = min m ⌈2 * √↑m⌉₊","subjects":["5","11"],"theorem":"Erdos650.erdos_650.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Surányi [ErSu59] proved that $f(m)\\geq\\sqrt{m}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«650»","statement":"∀ (m : ℕ), √↑m ≤ ↑(Erdos650.f m)","subjects":["5","11"],"theorem":"Erdos650.erdos_650.variants.erdos_suranyi"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Selfridge proved (see [Er78] and [Er86c]) that $f(m^2)\\leq 2m$, which implies\n$f(m)\\leq 2\\lceil \\sqrt{m}\\rceil$ for all $m$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«650»","statement":"∀ (m : ℕ), Erdos650.f (m ^ 2) ≤ 2 * m","subjects":["5","11"],"theorem":"Erdos650.erdos_650.variants.erdos_selfridge"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, for any $x$, if $A\\subset [x,\\infty)$ is a primitive set of integers (so that no distinct elements of $A$ divide each other) then$$\\sum_{a\\in A}\\frac{1}{a\\log a}&#60; 1+o(1),$$where the $o(1)$ term $\\to 0$ as $x\\to \\infty$?\n-","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/math-inc/Erdos1196/blob/02fba13be7487cc51315f68d8fa7ef277633d3c8/PrimitiveSetsAboveX/FormalConjecturesErdos1196.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1196»","statement":"True ↔\n  ∃ o,\n    o =o[Filter.atTop] 1 ∧\n      ∀ x > 0, ∀ A ⊆ Set.Ici x, Erdos1196.IsPrimitive A → ∑' (a : ↑A), 1 / (Real.log ↑↑a * ↑↑a) < 1 + o x","subjects":["11"],"theorem":"Erdos1196.erdos_1196"},{"answerKinds":[],"category":"test","docstring":"This lemma confirms that the set of possible unit-distance counts is bounded above, which\nensures that taking the supremum (`sSup`) is a well-defined operation. The trivial upper bound is\nthe total number of pairs of points, $\\binom{n}{2}$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«96»","statement":"∀ (n : ℕ), BddAbove (Erdos96.convexUnitDistanceCounts n)","subjects":["52"],"theorem":"Erdos96.convexUnitDistanceCounts_bddAbove"},{"answerKinds":[],"category":"research open","docstring":"If $n$ points in $\\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are\ndistance $1$ apart.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«96»","statement":"True ↔ (fun n => ↑(Erdos96.maxConvexUnitDistances n)) =O[Filter.atTop] fun n => ↑n","subjects":["52"],"theorem":"Erdos96.erdos_96"},{"answerKinds":[],"category":"research open","docstring":"Let $r \\geq 2$. Is it true that $\\frac{e(n,r+1)}{e(n,r)} \\to 1$ as $n \\to \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«600»","statement":"True ↔\n  ∀ (r : ℕ),\n    2 ≤ r → Filter.Tendsto (fun n => ↑(Erdos600.eFunction n (r + 1)) / ↑(Erdos600.eFunction n r)) Filter.atTop (nhds 1)","subjects":["5"],"theorem":"Erdos600.erdos_600.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Ruzsa and Szemerédi [RuSz78] proved that $e(n,r)=o(n^2)$ for any fixed $r$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«600»","statement":"∀ (r : ℕ), (fun n => ↑(Erdos600.eFunction n r)) =o[Filter.atTop] fun n => ↑n ^ 2","subjects":["5"],"theorem":"Erdos600.erdos_600.variants.ruzsa_szemeredi_upper_bound"},{"answerKinds":[],"category":"research open","docstring":"Let $r \\geq 2$. Is it true that $e(n,r+1) - e(n,r) \\to \\infty$ as $n \\to \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«600»","statement":"True ↔\n  ∀ (r : ℕ),\n    2 ≤ r →\n      Filter.Tendsto (fun n => ↑(Erdos600.eFunction n (r + 1)) - ↑(Erdos600.eFunction n r)) Filter.atTop Filter.atTop","subjects":["5"],"theorem":"Erdos600.erdos_600.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is there some $c > 0$ such that every measurable $A \\subseteq \\mathbb{R}^2$ of measure $\\geq c$\ncontains the vertices of a triangle of area 1?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«352»","statement":"True ↔\n  ∃ c > 0,\n    ∀ (A : Set (EuclideanSpace ℝ (Fin 2))),\n      MeasurableSet A →\n        ↑(MeasureTheory.volume A) ≥ ↑c →\n          ∃ t,\n            (∀ (p : Fin 3), t.points p ∈ A) ∧ EuclideanGeometry.triangle_area (t.points 0) (t.points 1) (t.points 2) = 1","subjects":["51"],"theorem":"Erdos352.erdos_352"},{"answerKinds":["Prop"],"category":"research open","docstring":"**Erdos Problem 830, Part 2**\nWe say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$.\nIf $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then is it true that\n$$A(x) > x^{1-o(1)}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«830»","statement":"sorry ↔ ∃ o, o =o[Filter.atTop] 1 ∧ ∀ᶠ (x : ℝ) in Filter.atTop, x ^ (1 - o x) < Erdos830.A x","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos830.erdos_830.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$.\nIf $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then one can show that\n$A(x) \\leq x \\exp(-(\\log x)^{1/3})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«830»","statement":"∀ᶠ (x : ℝ) in Filter.atTop, Erdos830.A x ≤ x * Real.exp (-Real.nthRoot 3 (Real.log x))","subjects":["11"],"theorem":"Erdos830.erdos_830.variants.pomerance"},{"answerKinds":["Prop"],"category":"research open","docstring":"**Erdos Problem 830, Part 1**\nWe say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$. Are there\ninfinitely many amicable pairs?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«830»","statement":"sorry ↔ {(a, b) | IsAmicable a b}.Infinite","subjects":["11"],"theorem":"Erdos830.erdos_830.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$.\nIf $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then one can show that $A(x) = o(x)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«830»","statement":"Erdos830.A =o[Filter.atTop] id","subjects":["11"],"theorem":"Erdos830.erdos_830.variants.erdos"},{"answerKinds":[],"category":"research solved","docstring":"We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$.\nIf $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then one can show that\n$A(x) \\leq x \\exp(-(\\tfrac{1}{2}+o(1))(\\log x\\log\\log x)^{1/2})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«830»","statement":"∃ o,\n  o =o[Filter.atTop] 1 ∧\n    ∀ᶠ (x : ℝ) in Filter.atTop, Erdos830.A x ≤ x * Real.exp (-(1 / 2 + o x) * √(Real.log x * Real.log (Real.log x)))","subjects":["11"],"theorem":"Erdos830.erdos_830.variants.pomerance_stronger"},{"answerKinds":[],"category":"test","docstring":"Membership in `limitPointSet` is exactly the existence of an infinite sequence of indices\nalong which the normalised prime gaps converge, as in the statement of `erdos_5`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∀ (x : ℝ),\n  x ∈ Erdos5.limitPointSet ↔\n    ∃ n, StrictMono n ∧ Filter.Tendsto (fun i => Erdos5.normalizedGap (n i)) Filter.atTop (nhds x)","subjects":["11"],"theorem":"Erdos5.mem_limitPointSet_iff"},{"answerKinds":[],"category":"research open","docstring":"Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\\log n$. This problem asks whether\n$S=[0,\\infty]$.\n\nSince $\\infty\\in S$ is known (see `erdos_5.variants.westzynthius`), the open content is the\nequality of the finite part of $S$ with $[0,\\infty)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"True ↔ Erdos5.limitPointSet = Set.Ici 0","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.limit_point_set"},{"answerKinds":[],"category":"research solved","docstring":"$\\infty\\in S$ by Westzynthius' result [We31] on large prime gaps.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∃ n, StrictMono n ∧ Filter.Tendsto (fun i => Erdos5.normalizedGap (n i)) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.westzynthius"},{"answerKinds":[],"category":"research solved","docstring":"The $1/3$ claim itself [Me20, Corollary 2]: $\\lambda([0,T]\\cap S)\\geq T/3$ for all $T>0$.\n\nUnlike the $1/8$ of [BFM16], this bound holds uniformly in $T$ with no error term.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∀ T > 0, ENNReal.ofReal (T / 3) ≤ MeasureTheory.volume (Erdos5.limitPointSet ∩ Set.Icc 0 T)","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.merikoski_measure"},{"answerKinds":[],"category":"test","docstring":"Every limit point of the normalised prime gaps is nonnegative. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«5»","statement":"Erdos5.limitPointSet ⊆ Set.Ici 0","subjects":["11"],"theorem":"Erdos5.limitPointSet_subset_Ici"},{"answerKinds":[],"category":"research solved","docstring":"The $12.5\\%$ claim itself, as deduced in [BFM16, Corollary 1.2]:\n$\\lambda([0,T]\\cap S)\\geq (1-o(1))T/8$ as $T\\to\\infty$.\n\nNote that the constant $1/8$ is only attained asymptotically and ineffectively; the bound\n[BFM16] obtain for *all* $T>0$ is the weaker $\\lambda([0,T]\\cap S)>T/22$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∀ ε > 0,\n  ∀ᶠ (T : ℝ) in Filter.atTop,\n    ENNReal.ofReal ((1 - ε) * T / 8) ≤ MeasureTheory.volume (Erdos5.limitPointSet ∩ Set.Icc 0 T)","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.banks_freiberg_maynard_measure"},{"answerKinds":[],"category":"research solved","docstring":"Merikoski [Me20] showed that at least $1/3$ of $[0,\\infty)$ belongs to $S$.\n\nThis is [Me20, Theorem 1]: for any reals $\\beta_1\\leq\\beta_2\\leq\\beta_3\\leq\\beta_4$, at least\none of the differences $\\beta_j-\\beta_i$ with $i<j$ belongs to $S$. Note that, in contrast with\n`erdos_5.variants.banks_freiberg_maynard`, the $\\beta_i$ are not required to be nonnegative.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∀ (β : Fin 4 → ℝ), Monotone β → ∃ i j, i < j ∧ β j - β i ∈ Erdos5.limitPointSet","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.merikoski"},{"answerKinds":[],"category":"research open","docstring":"Let $C\\geq 0$. Is there an infinite sequence of $n_i$ such that\n$$\\lim_{i\\to \\infty}\\frac{p_{n_i+1}-p_{n_i}}{\\log n_i}=C?$$\n\nWe formalise \"an infinite sequence of $n_i$\" as a strictly monotone sequence of indices\n`n : ℕ → ℕ`. Note that the numerator is the gap between the two *consecutive* primes\n$p_{n_i}$ and $p_{n_i+1}$, which is `primeGap (n i)`, and not the gap between the primes\nindexed by two consecutive members of the sequence.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"True ↔ ∀ (C : ℝ), 0 ≤ C → ∃ n, StrictMono n ∧ Filter.Tendsto (fun i => Erdos5.normalizedGap (n i)) Filter.atTop (nhds C)","subjects":["11"],"theorem":"Erdos5.erdos_5"},{"answerKinds":[],"category":"research solved","docstring":"Merikoski [Me20] showed that $S$ has bounded gaps.\n\nThis is [Me20, Corollary 3]: there is a (ineffective) constant $C\\geq 0$ such that\n$S\\cap[T,T+C]\\neq\\emptyset$ for all $T\\geq 0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∃ C ≥ 0, ∀ T ≥ 0, (Erdos5.limitPointSet ∩ Set.Icc T (T + C)).Nonempty","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.merikoski_bounded_gaps"},{"answerKinds":[],"category":"research solved","docstring":"$0\\in S$ by the work of Goldston, Pintz, and Yildirim [GPY09] on small prime gaps.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"0 ∈ Erdos5.limitPointSet","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.goldston_pintz_yildirim"},{"answerKinds":[],"category":"research solved","docstring":"Pintz [Pi16] showed that there exists some small constant $c>0$ such that $[0,c]\\subset S$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∃ c > 0, Set.Icc 0 c ⊆ Erdos5.limitPointSet","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.pintz"},{"answerKinds":[],"category":"research open","docstring":"In [Er65b], [Er85c], and [Er97c] Erdős asks whether $S$ is everywhere dense (but Weisenberg\nnotes that clearly $S$ is closed so this is equivalent to asking whether $S=[0,\\infty]$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"True ↔ Set.Ici 0 ⊆ closure Erdos5.limitPointSet","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.dense"},{"answerKinds":[],"category":"test","docstring":"The normalised prime gaps are nonnegative. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∀ (n : ℕ), 0 ≤ Erdos5.normalizedGap n","subjects":["11"],"theorem":"Erdos5.normalizedGap_nonneg"},{"answerKinds":[],"category":"research solved","docstring":"Banks, Freiberg, and Maynard [BFM16] showed that at least $12.5\\%$ of $[0,\\infty)$ belongs\nto $S$.\n\nThis is [BFM16, Theorem 1.1]: for any nine nonnegative reals\n$\\beta_1\\leq\\beta_2\\leq\\cdots\\leq\\beta_9$, at least one of the differences $\\beta_j-\\beta_i$\nwith $i<j$ belongs to $S$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∀ (β : Fin 9 → ℝ), (∀ (i : Fin 9), 0 ≤ β i) → Monotone β → ∃ i j, i < j ∧ β j - β i ∈ Erdos5.limitPointSet","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.banks_freiberg_maynard"},{"answerKinds":[],"category":"test","docstring":"The statement of `erdos_5` is equivalent to the description of the set of limit points in\n`erdos_5.variants.limit_point_set`; combine with `mem_limitPointSet_iff` to unfold the\nmembership into the sequence of indices $n_i$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«5»","statement":"(∀ (C : ℝ), 0 ≤ C → C ∈ Erdos5.limitPointSet) ↔ Erdos5.limitPointSet = Set.Ici 0","subjects":["11"],"theorem":"Erdos5.erdos_5_iff_limit_point_set"},{"answerKinds":[],"category":"test","docstring":"Weisenberg's remark, as reported on [erdosproblems.com/5](https://www.erdosproblems.com/5):\nsince $S$ is closed, asking that $S$ be everywhere dense in $[0,\\infty)$ is the same as asking\nthat $S=[0,\\infty)$, so `erdos_5.variants.dense` and `erdos_5.variants.limit_point_set` pose the\nsame question.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«5»","statement":"Set.Ici 0 ⊆ closure Erdos5.limitPointSet ↔ Erdos5.limitPointSet = Set.Ici 0","subjects":["11"],"theorem":"Erdos5.dense_iff_limit_point_set"},{"answerKinds":[],"category":"research solved","docstring":"Hildebrand and Maier [HiMa88] showed that $S$ contains arbitrarily large (finite) numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∀ (C : ℝ), ∃ x ∈ Erdos5.limitPointSet, C < x","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.hildebrand_maier"},{"answerKinds":[],"category":"research solved","docstring":"[HiMa88] in fact prove the stronger statement that there is a constant $c>0$ with\n$\\lambda([0,T]\\cap S)\\geq cT$ for all sufficiently large $T$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"∃ c > 0, ∀ᶠ (T : ℝ) in Filter.atTop, ENNReal.ofReal (c * T) ≤ MeasureTheory.volume (Erdos5.limitPointSet ∩ Set.Icc 0 T)","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.hildebrand_maier_measure"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er55] and Ricci [Ri56] independently showed that $S$ has positive Lebesgue measure.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«5»","statement":"0 < MeasureTheory.volume Erdos5.limitPointSet","subjects":["11"],"theorem":"Erdos5.erdos_5.variants.erdos_ricci"},{"answerKinds":[],"category":"test","docstring":"The set $S$ of limit points is closed, as Weisenberg notes in the acknowledgements to\n[erdosproblems.com/5](https://www.erdosproblems.com/5); consequently `erdos_5.variants.dense`\nand `erdos_5.variants.limit_point_set` ask the same question.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«5»","statement":"IsClosed Erdos5.limitPointSet","subjects":["11"],"theorem":"Erdos5.isClosed_limitPointSet"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there a lacunary sequence $A\\subseteq \\mathbb{N}$ (so that $A=\\{a_1 < \\cdots\\}$ and\nthere exists some $\\lambda > 1$ such that $a_{n+1}/a_n\\geq \\lambda$ for all $n\\geq 1$) such that\n$$\\left\\{ \\sum_{a\\in A'}\\frac{1}{a} : A'\\subseteq A\\textrm{ finite}\\right\\}$$\ncontain all rationals in some open interval?\n\nBleicher and Erdős conjectured the answer is no.\n\nIn fact the answer is yes, with any lacunarity constant $\\lambda\\in (1,2)$ (though not $\\lambda=2$),\nas proved by van Doorn and Kova\\v{c} [DoKo25].\n\nThis was formalized in Lean by van Doorn using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem355.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«355»","statement":"True ↔\n  ∃ A,\n    IsLacunary A ∧\n      ∃ u v, u < v ∧ ∀ (q : ℚ), ↑q ∈ Set.Ioo u v → q ∈ {x | ∃ A', ∃ (_ : ↑A' ⊆ Set.range A), ∑ a ∈ A', 1 / ↑a = x}","subjects":["11"],"theorem":"Erdos355.erdos_355"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that in any finite colouring of the integers there exists a monochromatic solution\nto $\\frac 1 a = \\frac 1 b + \\frac 1 c$ with distinct $a, b, c$?\n\nThis is true, as proved by Brown and Rödl [BrRo91].\n\nThis was formalized in Lean by Yuan using Seed-Prover.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://www.erdosproblems.com/forum/thread/303"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«303»","statement":"True ↔\n  ∀ (𝓒 : ℤ → ℤ),\n    (Set.range 𝓒).Finite → ∃ a b c, [a, b, c, 0].Nodup ∧ 1 / ↑a = 1 / ↑b + 1 / ↑c ∧ (𝓒 '' {a, b, c}).Subsingleton","subjects":["5","11"],"theorem":"Erdos303.erdos_303"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n) = \\min_{1 < k \\le n/2} \\gcd(n, \\binom{n}{k})$.\n\n**(b)** Are there infinitely many composite $n$ such that $f(n) > n^{1/2}$?\n\nErdős–Szekeres [ErSz78] could not prove this. (Since $f(n) \\ge p(n)$, the least prime factor of\n$n$, there are infinitely many $n$ — those of the form $p^2$ — with $f(n) \\ge n^{1/2}$; the\nquestion asks for the strict inequality.) Here $f(n) > n^{1/2}$ is written as `(f n) ^ 2 > n`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«700»","statement":"True ↔ {n | ¬Nat.Prime n ∧ 1 < n ∧ Erdos700.f n ^ 2 > n}.Infinite","subjects":["11"],"theorem":"Erdos700.erdos_700.parts.ii"},{"answerKinds":[],"category":"API","docstring":"Lucas (one step): for prime `P ∣ n`, if `P ∤ C(n,k)` then `P ∣ k`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«700»","statement":"∀ (P n k : ℕ), Nat.Prime P → P ∣ n → ¬P ∣ n.choose k → P ∣ k","subjects":["11"],"theorem":"Erdos700.prime_dvd_of_not_dvd_choose"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n) = \\min_{1 < k \\le n/2} \\gcd(n, \\binom{n}{k})$.\n\n**(c)** Is it true that, for every composite $n$, $f(n) \\ll_A n/(\\log n)^A$ for every $A > 0$?\n\nErdős–Szekeres [ErSz78] prove the weaker bound $f(n) \\le (1 + o(1)) n/\\log n$ (the case $A = 1$).\nHere $f(n) \\ll_A n/(\\log n)^A$ is spelled out as: for every `A > 0` there is a constant `C`\n(depending on `A`) with `f(n) ≤ C · n/(log n)^A` for every composite `n`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«700»","statement":"True ↔ ∀ (A : ℝ), 0 < A → ∃ C, 0 < C ∧ ∀ (n : ℕ), ¬Nat.Prime n → 1 < n → ↑(Erdos700.f n) ≤ C * ↑n / Real.log ↑n ^ A","subjects":["11"],"theorem":"Erdos700.erdos_700.parts.iii"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n) = \\min_{1 < k \\le n/2} \\gcd(n, \\binom{n}{k})$ and let $P(n)$ be the largest prime\ndividing $n$.\n\n**(a)** Characterise those composite $n$ such that $f(n) = n/P(n)$.\n\nErdős–Szekeres [ErSz78] note that $f(n) = n/P(n)$ when $n$ is a product of two primes\n(`erdos_700.variants.prime_mul`), with $n = 30$ a further example. The characterisation itself is\nopen; we state it as the (unknown) predicate that is equivalent to being such an `n`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«700»","statement":"∀ (n : ℕ), ¬Nat.Prime n → 1 < n → (Erdos700.f n = n / Erdos700.P n ↔ True)","subjects":["11"],"theorem":"Erdos700.erdos_700.parts.i"},{"answerKinds":[],"category":"API","docstring":"`f n` unfolds to the infimum of `fSet n`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«700»","statement":"∀ (n : ℕ), Erdos700.f n = sInf (Erdos700.fSet n)","subjects":["11"],"theorem":"Erdos700.f_eq"},{"answerKinds":[],"category":"research solved","docstring":"`f(pq) = p` for distinct primes `p < q` (recorded by Erdős–Szekeres [ErSz78]); in particular\n`pq` is always a \"hit\" (`f(pq) = pq / P(pq) = p`). Proof via Lucas' theorem:\n`p ∤ C(pq,k) ⟹ p ∣ k` (and the same for `q`),\nso any `k` with `gcd(pq, C(pq,k)) = 1` must be a multiple of `pq`, of which there are none in\n`(1, pq/2]`; the witness `k = q` gives `gcd(pq, C(pq,q)) = p`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«700»","statement":"∀ (p q : ℕ), Nat.Prime p → Nat.Prime q → p < q → Erdos700.f (p * q) = p","subjects":["11"],"theorem":"Erdos700.erdos_700.variants.prime_mul"},{"answerKinds":[],"category":"API","docstring":"Each `gcd(n, C(n,k))` with `1 < k ≤ n/2` belongs to `fSet n`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«700»","statement":"∀ (n k : ℕ), 1 < k → k ≤ n / 2 → n.gcd (n.choose k) ∈ Erdos700.fSet n","subjects":["11"],"theorem":"Erdos700.f_mem"},{"answerKinds":[],"category":"API","docstring":"`f n` is a lower bound: `f n ≤ gcd(n, C(n,k))` for every `1 < k ≤ n/2`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«700»","statement":"∀ (n k : ℕ), 1 < k → k ≤ n / 2 → Erdos700.f n ≤ n.gcd (n.choose k)","subjects":["11"],"theorem":"Erdos700.f_le"},{"answerKinds":[],"category":"research solved","docstring":"`f(p^a) = p` for a prime `p` and `a ≥ 2` (recorded by Erdős–Szekeres [ErSz78]). In particular,\nsince `(p^a) / P(p^a) = p^{a-1}`, the prime power `p^a` is a \"hit\" (`f(n) = n / P(n)`) if and only\nif `a = 2`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«700»","statement":"∀ (p a : ℕ), Nat.Prime p → 2 ≤ a → Erdos700.f (p ^ a) = p","subjects":["11"],"theorem":"Erdos700.erdos_700.variants.prime_pow"},{"answerKinds":[],"category":"research solved","docstring":"Let $\\epsilon > 0$. Is there some $r \\ll_\\epsilon 1$ such that the density of integers of the\nform $2^k+n$, where $k \\geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\\epsilon$?\n\nThis was proved affirmatively by Price and GPT-5.2 Pro [Pr26].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«851»","statement":"∀ ε ∈ Set.Ioo 0 1, ∃ r d, (Erdos851.TwoPowAddSet r).HasDensity d ∧ 1 - ε ≤ d","subjects":["11"],"theorem":"Erdos851.erdos_851"},{"answerKinds":[],"category":"research solved","docstring":"The set of integers of the form `2^k+p` (where `p` is prime) has positive lower density.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«851»","statement":"0 < Erdos851.twoPowAddPrimeSet.lowerDensity","subjects":["11"],"theorem":"Erdos851.erdos_851.variants.romanoff"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $$\\mathrm{ex}(n; K_{r,r}) \\gg n^{2-1/r}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«714»","statement":"True ↔\n  ∀ (r : ℕ),\n    2 ≤ r →\n      ∃ c,\n        0 < c ∧\n          ∀ᶠ (n : ℕ) in Filter.atTop,\n            c * ↑n ^ (2 - 1 / ↑r) ≤ ↑(SimpleGraph.extremalNumber n (completeBipartiteGraph (Fin r) (Fin r)))","subjects":["5"],"theorem":"Erdos714.erdos_714"},{"answerKinds":[],"category":"research solved","docstring":"Alon [Al92] proved that, for every $n$, there exists a graph $G$ on $n$ vertices such that\n$$\\chi_L(G)+\\chi_L(G^c)\\ll (n\\log n)^{1/2},$$\nwhere the implied constant is absolute.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«753»","statement":"∃ C,\n  0 < C ∧\n    ∀ (n : ℕ),\n      2 ≤ n →\n        ∃ G, ↑(Erdos753.listChromaticNumber G) + ↑(Erdos753.listChromaticNumber Gᶜ) ≤ C * (↑n * Real.log ↑n) ^ (1 / 2)","subjects":["5"],"theorem":"Erdos753.erdos_753.variants.alon"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any\nassignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different\nvertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent\nvertices receive distinct colours.\n\nDoes there exist some constant $c>0$ such that\n$$\\chi_L(G)+\\chi_L(G^c)> n^{1/2+c}$$\nfor every graph $G$ on $n$ vertices (where $G^c$ is the complement of $G$)?\n\nA problem of Erdős, Rubin, and Taylor.\n\nThe answer is no: Alon [Al92] proved that, for every $n$, there exists a graph $G$ on $n$ vertices\nsuch that\n$$\\chi_L(G)+\\chi_L(G^c)\\ll (n\\log n)^{1/2},$$\nwhere the implied constant is absolute.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos753.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«753»","statement":"False ↔\n  ∃ c,\n    0 < c ∧\n      ∀ (n : ℕ),\n        0 < n →\n          ∀ (G : SimpleGraph (Fin n)),\n            ↑n ^ (1 / 2 + c) < ↑(Erdos753.listChromaticNumber G) + ↑(Erdos753.listChromaticNumber Gᶜ)","subjects":["5"],"theorem":"Erdos753.erdos_753"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Folkman–Nešetřil–Rödl (finite version) [Fo70, NeRo75]**: For every `n ≥ 1`, there exists a\ngraph `G` (on a finite vertex set) that contains no $K_4$ and whose edges cannot be covered by\n`n` triangle-free graphs.\n\nMore precisely: for every `n : ℕ` with `1 ≤ n`, there exist a finite type `V` and a graph\n`G : SimpleGraph V` with:\n1. `G.CliqueFree 4` (no $K_4$), and\n2. For every family `H : Fin n → SimpleGraph V` of triangle-free graphs, `G ≠ ⨆ i, H i`.\n\nThis is the finite analogue of Problem 595. The proofs of Folkman [Fo70] and Nešetřil–Rödl\n[NeRo75] give different explicit constructions.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«595»","statement":"True ↔\n  ∀ (n : ℕ),\n    1 ≤ n → ∃ V x G, G.CliqueFree 4 ∧ ∀ (H : Fin n → SimpleGraph V), (∀ (i : Fin n), (H i).CliqueFree 3) → G ≠ ⨆ i, H i","subjects":["5"],"theorem":"Erdos595.erdos_595.variants.folkman_finite"},{"answerKinds":[],"category":"textbook","docstring":"**Triangle-free graphs are trivially countable unions of triangle-free graphs**: if `G` is\nalready triangle-free, then `G = ⨆ i : ℕ, G_i` where `G_0 = G` and `G_i = ⊥` for `i ≥ 1`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«595»","statement":"∀ {V : Type u_1} (G : SimpleGraph V), G.CliqueFree 3 → Erdos595.IsCountableUnionOfTriangleFree G","subjects":["5"],"theorem":"Erdos595.erdos_595.variants.triangle_free_is_union"},{"answerKinds":[],"category":"textbook","docstring":"**The complete graph `⊤` on `Fin 4` is not $K_4$-free**: `⊤` on `Fin 4` equals the complete\ngraph $K_4$, so it contains $K_4$ as a subgraph and is not $K_4$-free.\n\nThis sanity check confirms the $K_4$-free hypothesis of Problem 595 is non-trivial.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«595»","statement":"¬⊤.CliqueFree 4","subjects":["5"],"theorem":"Erdos595.erdos_595.variants.K4_not_cliqueFree"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 595 (\\$250)**: Is there an infinite graph $G$ which contains no $K_4$ and is\nnot the union of countably many triangle-free graphs?\n\nA problem of Erdős and Hajnal [Er87].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«595»","statement":"True ↔ ∃ V, ∃ (_ : Infinite V), ∃ G, G.CliqueFree 4 ∧ ¬Erdos595.IsCountableUnionOfTriangleFree G","subjects":["5"],"theorem":"Erdos595.erdos_595"},{"answerKinds":[],"category":"textbook","docstring":"**Monotonicity**: If `G` is a countable union of triangle-free graphs and `H ≤ G` (i.e., `H` is\na subgraph of `G`), then `H` is also a countable union of triangle-free graphs.\n\n**Proof**: If `G = ⨆ i, G_i` with each `G_i` triangle-free, then `H = ⨆ i, H ⊓ G_i`.\nEach `H ⊓ G_i` is triangle-free because it is a subgraph of `G_i`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«595»","statement":"∀ {V : Type u_1} {G H : SimpleGraph V},\n  H ≤ G → Erdos595.IsCountableUnionOfTriangleFree G → Erdos595.IsCountableUnionOfTriangleFree H","subjects":["5"],"theorem":"Erdos595.erdos_595.variants.subgraph_of_countable_union"},{"answerKinds":[],"category":"test","docstring":"**Reformulation via edge colourings**: A graph `G` is a countable union of triangle-free graphs\nif and only if there is a colouring of the edges of `G` by `ℕ` such that no monochromatic\ntriangle exists.\n\nMore precisely: `IsCountableUnionOfTriangleFree G` is equivalent to the existence of a map\n`c : G.edgeSet → ℕ` such that for each `n : ℕ`, the subgraph of edges coloured `n` is triangle-free.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«595»","statement":"∀ {V : Type u_1} (G : SimpleGraph V),\n  Erdos595.IsCountableUnionOfTriangleFree G ↔\n    ∃ c, ∀ (n : ℕ), (SimpleGraph.fromEdgeSet {e | ∃ (h : e ∈ G.edgeSet), c ⟨e, h⟩ = n}).CliqueFree 3","subjects":["5"],"theorem":"Erdos595.erdos_595.variants.reformulation_edge_colouring"},{"answerKinds":[],"category":"textbook","docstring":"**The complete graph `⊤` on `ℕ` is a countable union of triangle-free graphs**: we decompose\nit into the family of star graphs `{H_m}_{m : ℕ}`, where `H_m` is the graph with edges `{m, n}`\nfor all `n ≠ m`. Each star is triangle-free (any two non-center vertices share no edge within\nthe star), and their union covers all edges of `⊤`.\n\n**Proof sketch (star triangle-free):** If `{a, b, c}` were a triangle in `H_m`, then each of\nthe three edges `{a, b}`, `{a, c}`, `{b, c}` would pass through `m`. In particular, from\n`{a, b}` we get `a = m` or `b = m`; from `{b, c}` we get `b = m` or `c = m`. Case analysis\nshows that two vertices must equal `m`, contradicting the triangle having three distinct vertices.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«595»","statement":"Erdos595.IsCountableUnionOfTriangleFree ⊤","subjects":["5"],"theorem":"Erdos595.erdos_595.variants.complete_nat_is_union"},{"answerKinds":[],"category":"research solved","docstring":"For graphs, Spencer [Sp71] constructed a graph which contains cliques of at least\n$n-\\log_2n+O(1)$ different sizes, which Moon and Moser [MoMo65] showed to be best possible.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«775»","statement":"∃ C, ∀ (n : ℕ) (G : SimpleGraph (Fin n)), ↑G.cliqueSizes.ncard ≤ ↑n - Real.logb 2 ↑n + C","subjects":["5"],"theorem":"Erdos775.erdos_775.variants.moon_moser"},{"answerKinds":[],"category":"research solved","docstring":"For graphs, Spencer [Sp71] constructed a graph which contains cliques of at least\n$n-\\log_2n+O(1)$ different sizes.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«775»","statement":"∃ C, ∀ (n : ℕ), ∃ G, ↑n - Real.logb 2 ↑n - C ≤ ↑G.cliqueSizes.ncard","subjects":["5"],"theorem":"Erdos775.erdos_775.variants.spencer"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different\nsizes of cliques (maximal complete subgraphs)?\n\nThe answer is no, as proved by Gao [Ga25]: more generally, for any $k\\geq 3$, every $k$-uniform\nhypergraph on $n$ vertices contains at most $n-f_k(n)$ different sizes of cliques, where\n$f_k(n)\\to \\infty$ as $n\\to \\infty$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos775.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«775»","statement":"False ↔ ∃ C, ∃ᶠ (n : ℕ) in Filter.atTop, ∃ H, n - C ≤ H.cliqueSizes.ncard","subjects":["5"],"theorem":"Erdos775.erdos_775"},{"answerKinds":[],"category":"research solved","docstring":"The answer is no, as proved by Gao [Ga25]: more generally, for any $k\\geq 3$, every $k$-uniform\nhypergraph on $n$ vertices contains at most $n-f_k(n)$ different sizes of cliques, where\n$f_k(n)\\to \\infty$ as $n\\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«775»","statement":"∃ f,\n  Filter.Tendsto f Filter.atTop Filter.atTop ∧\n    ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (H : ThreeUniformHypergraph (Fin n)), H.cliqueSizes.ncard + f n ≤ n","subjects":["5"],"theorem":"Erdos775.erdos_775.variants.gao"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 1.**\nAre there infinitely many practical numbers $m$ such that $h(m) < (\\log \\log m)^{O(1)}$?\n\nMore precisely: does there exist a constant $C > 0$ such that for infinitely many\npractical numbers $m$, we have $h(m) < (\\log \\log m)^C$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«18»","statement":"True ↔ ∃ C, 0 < C ∧ ∃ᶠ (m : ℕ) in Filter.atTop, m.IsPractical ∧ ↑(Erdos18.practicalH m) < Real.log (Real.log ↑m) ^ C","subjects":["11"],"theorem":"Erdos18.erdos_18a"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 3.**\nOr perhaps even $h(n!) < (\\log n)^{O(1)}$?\n\nErdős offered \\$250 for a proof or disproof.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«18»","statement":"True ↔ ∃ C, 0 < C ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos18.practicalH n.factorial) < Real.log ↑n ^ C","subjects":["11"],"theorem":"Erdos18.erdos_18c"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 2.**\nIs it true that $h(n!) < n^{o(1)}$? That is, for all $\\varepsilon > 0$,\nis $h(n!) < n^\\varepsilon$ for sufficiently large $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«18»","statement":"True ↔ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos18.practicalH n.factorial) < ↑n ^ ε","subjects":["11"],"theorem":"Erdos18.erdos_18b"},{"answerKinds":[],"category":"textbook","docstring":"$h(n!)$ is well-defined since $n!$ is practical for $n ≥ 1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«18»","statement":"∀ (n : ℕ), n.factorial.IsPractical","subjects":["11"],"theorem":"Erdos18.factorial_isPractical"},{"answerKinds":[],"category":"test","docstring":"$h(2) = 1$: divisors are {1, 2}, each of m=1,2 needs only 1 divisor. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«18»","statement":"Erdos18.practicalH 2 = 1","subjects":["11"],"theorem":"Erdos18.practicalH_two"},{"answerKinds":[],"category":"test","docstring":"For any practical number $n$, $h(n)$ ≤ number of divisors of $n$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«18»","statement":"∀ (n : ℕ), n.IsPractical → Erdos18.practicalH n ≤ n.divisors.card","subjects":["11"],"theorem":"Erdos18.practicalH_le_divisors"},{"answerKinds":[],"category":"test","docstring":"$h(1) = 1$: we need the single divisor {1} to represent 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«18»","statement":"Erdos18.practicalH 1 = 1","subjects":["11"],"theorem":"Erdos18.practicalH_one"},{"answerKinds":[],"category":"research solved","docstring":"**Vose's Theorem.**\nVose proved the existence of infinitely many practical numbers $m$ such that\n$h(m) \\ll (\\log m)^{1/2}$. This gives a positive answer to a weaker form of Conjecture 1.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«18»","statement":"∃ C, 0 < C ∧ ∃ᶠ (m : ℕ) in Filter.atTop, m.IsPractical ∧ ↑(Erdos18.practicalH m) < C * Real.log ↑m ^ (1 / 2)","subjects":["11"],"theorem":"Erdos18.erdos_18_vose"},{"answerKinds":[],"category":"test","docstring":"$h(6) = 2$: divisors are {1, 2, 3, 6}. The hardest m to represent is\nm=4 or m=5, each requiring 2 divisors: 4=1+3, 5=2+3. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«18»","statement":"Erdos18.practicalH 6 = 2","subjects":["11"],"theorem":"Erdos18.practicalH_six"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős's Theorem.**\nErdős proved that $h(n!) < n$ for all $n \\ge 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«18»","statement":"∀ᶠ (n : ℕ) in Filter.atTop, Erdos18.practicalH n.factorial < n","subjects":["11"],"theorem":"Erdos18.erdos_18_upper_bound"},{"answerKinds":[],"category":"test","docstring":"$h(12) = 3$: divisors are {1, 2, 3, 4, 6, 12}. The hardest m is\nm=11, requiring 3 divisors: 11=1+4+6. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«18»","statement":"Erdos18.practicalH 12 = 3","subjects":["11"],"theorem":"Erdos18.practicalH_twelve"},{"answerKinds":[],"category":"research open","docstring":"Let $n_1 < n_2 < \\dots$ be an arbitrary sequence of integers, each with an associated residue class\n$a_i \\pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n < n_i$ or\n$n \\not\\equiv a_i \\pmod{n_i}$. Must the logarithmic density of $A$ exist?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«25»","statement":"True ↔\n  ∀ (seq_n : ℕ → ℕ) (seq_a : ℕ → ℤ),\n    (∀ (i : ℕ), 0 < seq_n i) →\n      StrictMono seq_n → ∃ d, {x | ∀ (i : ℕ), ↑x < ↑(seq_n i) ∨ ¬↑x ≡ seq_a i [ZMOD ↑(seq_n i)]}.HasLogDensity d","subjects":["11"],"theorem":"Erdos25.erdos_25"},{"answerKinds":[],"category":"textbook","docstring":"In fact, the answer to this question as written is easily seen to be no, since there are no\nsolutions to $2^k\\equiv -1\\pmod{7}$, and hence this fails with $p=2$ and $q=7$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«649»","statement":"¬∃ n, n.maxPrimeFac = 2 ∧ (n + 1).maxPrimeFac = 7","subjects":["11"],"theorem":"Erdos649.erdos_649.variants.no_solution_two_seven"},{"answerKinds":[],"category":"textbook","docstring":"Problem 6 in the 12th Romanian Master of Mathematics Competitions in 2020 was to prove that there\nexist infinitely many odd primes $p$ such that, for every $n$, $P(n)P(n+1)\\neq 2p$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«649»","statement":"{p | Nat.Prime p ∧ Odd p ∧ ∀ (n : ℕ), n.maxPrimeFac * (n + 1).maxPrimeFac ≠ 2 * p}.Infinite","subjects":["11"],"theorem":"Erdos649.erdos_649.variants.rmm_2020"},{"answerKinds":[],"category":"research solved","docstring":"Even with such amendments, this problem is false in a strong sense: Alan Tong has provided the\nfollowing elegant elementary proof that, for any given prime $p$, there are infinitely many\nprimes $q$ such that this statement is false: let $m$ be the product of all primes $\\leq p$, and\nchoose a prime $q$ congruent to $-1$ modulo $4m$. If $p$ is the greatest prime divisor of $n$\nthen, using quadratic reciprocity, every prime divisor of $n$ is a quadratic residue modulo $q$,\nand hence $n$ is a quadratic residue modulo $q$. On the other hand, since $q\\equiv 3\\pmod{4}$ we\nknow that $-1$ is not a quadratic residue modulo $q$, and hence $n\\not\\equiv -1\\pmod{q}$, so it\nis impossible for $q\\mid n+1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«649»","statement":"∀ (p : ℕ), Nat.Prime p → {q | Nat.Prime q ∧ ¬∃ n, n.maxPrimeFac = p ∧ (n + 1).maxPrimeFac = q}.Infinite","subjects":["11"],"theorem":"Erdos649.erdos_649.variants.tong"},{"answerKinds":[],"category":"research open","docstring":"Tong asks whether, for any given odd prime $q$, there are infinitely many primes $p$ such that\nthere is no integer $n$ with $P(n)=p$ and $P(n+1)=q$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«649»","statement":"True ↔ ∀ (q : ℕ), Nat.Prime q → Odd q → {p | Nat.Prime p ∧ ¬∃ n, n.maxPrimeFac = p ∧ (n + 1).maxPrimeFac = q}.Infinite","subjects":["11"],"theorem":"Erdos649.erdos_649.variants.tong_question"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $P(m)$ denote the greatest prime factor of $m$. Is it true that, for any two primes $p,q$,\nthere exists some integer $n$ such that $P(n)=p$ and $P(n+1)=q$?\n\nIn fact, the answer to this question as written is easily seen to be no, since there are no\nsolutions to $2^k\\equiv -1\\pmod{7}$, and hence this fails with $p=2$ and $q=7$. It is possible\nthat Erdős meant to exclude such obstructions, by amending this to 'odd primes' or 'all\nsufficiently large primes' or such.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos649.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«649»","statement":"False ↔ ∀ (p q : ℕ), Nat.Prime p → Nat.Prime q → ∃ n, n.maxPrimeFac = p ∧ (n + 1).maxPrimeFac = q","subjects":["11"],"theorem":"Erdos649.erdos_649"},{"answerKinds":[],"category":"textbook","docstring":"Sampaio independently observed that the answer to Erdős' original problem is no if one of the\nprimes can be $2$ - for example this is false with $p=19$ and $q=2$, since if $n+1=2^k$ and\n$19\\mid n$ then (since $2$ is a primitive root modulo $19$) we must have $18\\mid k$, and hence\n$73\\mid 2^{18}-1\\mid n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«649»","statement":"¬∃ n, n.maxPrimeFac = 19 ∧ (n + 1).maxPrimeFac = 2","subjects":["11"],"theorem":"Erdos649.erdos_649.variants.sampaio"},{"answerKinds":[],"category":"research solved","docstring":"If `f(z) = ∑ aₖzⁿₖ` is an entire function (with `aₖ ≠ 0` for all `k`) such that `∑ 1 / nₖ < ∞`,\nthen `f` assumes every value infinitely often. This theorem is proved in [Bi28]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«517»","statement":"∀ {f : ℂ → ℂ} {n : ℕ → ℕ},\n  HasFejerGaps n →\n    ∀ {a : ℕ → ℂ},\n      (∀ (k : ℕ), a k ≠ 0) → (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) → ∀ (z : ℂ), {x | f x = z}.Infinite","subjects":["30"],"theorem":"Erdos517.erdos_517.variants.fejer"},{"answerKinds":[],"category":"research open","docstring":"If `f(z) = ∑ aₖzⁿₖ` is an entire function (with `aₖ ≠ 0` for all `k`) such that `nₖ / k → ∞`,\nis it true that `f` assumes every value infinitely often? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«517»","statement":"True ↔\n  ∀ {f : ℂ → ℂ} {n : ℕ → ℕ},\n    HasFabryGaps n →\n      ∀ {a : ℕ → ℂ},\n        (∀ (k : ℕ), a k ≠ 0) → (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) → ∀ (z : ℂ), {x | f x = z}.Infinite","subjects":["30"],"subsets":["FC100OpenSet1"],"theorem":"Erdos517.erdos_517"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there infinitely many integers $n, m$ such that $ϕ(n) = σ(m)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«48»","statement":"True ↔ {(n, m) | n.totient = (ArithmeticFunction.sigma 1) m}.Infinite","subjects":["11"],"theorem":"Erdos48.erdos_48"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Graham [ErGr80] also say that it is not hard to construct very irregular sequences\nsatisfying the aforementioned properties. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«346»","statement":"∃ A,\n  IsAddStronglyCompleteNatSeq A ∧\n    (∀ B ⊆ Set.range A, B.Infinite → ¬IsAddComplete (Set.range A \\ B)) ∧\n      Filter.liminf (fun n => ↑(A (n + 1)) / 2) Filter.atTop = 1 ∧\n        Filter.limsup (fun n => ↑(A (n + 1)) / ↑(A n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos346.erdos_346.variants.example"},{"answerKinds":[],"category":"research solved","docstring":"The sequence `f` is not complete whenever infinitely many terms are removed from it, and this\nis proved in [Gr64d]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«346»","statement":"∀ {B : Set ℕ}, B ⊆ Set.range Erdos346.f → B.Infinite → ¬IsAddComplete (Set.range Erdos346.f \\ B)","subjects":["11"],"theorem":"Erdos346.erdos_346.variants.f_not_isAddComplete"},{"answerKinds":[],"category":"research open","docstring":"Is it true that for every lacunary, strongly complete sequence `A` that is not complete whenever\ninfinitely many terms are removed from it, `lim A (n + 1) / A n = (1 + √5) / 2`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«346»","statement":"True ↔\n  ∀ {A : ℕ → ℕ},\n    IsLacunary A →\n      IsAddStronglyCompleteNatSeq A →\n        (∀ B ⊆ Set.range A, B.Infinite → ¬IsAddComplete (Set.range A \\ B)) →\n          Filter.Tendsto (fun n => ↑(A (n + 1)) / ↑(A n)) Filter.atTop (nhds ((1 + √5) / 2))","subjects":["11"],"theorem":"Erdos346.erdos_346"},{"answerKinds":[],"category":"test","docstring":"The sequence `f` is lacunary. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«346»","statement":"IsLacunary Erdos346.f","subjects":["11"],"theorem":"Erdos346.erdos_346.variants.f_isLacunary"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Graham [ErGr80] remark that it is easy to see that if `A (n + 1) / A n > (1 + √5) / 2`\nthen the second property is automatically satisfied. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«346»","statement":"∀ {A : ℕ → ℕ},\n  (∀ (n : ℕ), (1 + √5) / 2 * ↑(A n) < ↑(A (n + 1))) →\n    ∀ {B : Set ℕ}, B ⊆ Set.range A → B.Infinite → ¬IsAddComplete (Set.range A \\ B)","subjects":["11"],"theorem":"Erdos346.erdos_346.variants.gt_goldenRatio_not_IsAddComplete"},{"answerKinds":[],"category":"research solved","docstring":"The sequence `f` is strongly complete, and this is proved in [Gr64d]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«346»","statement":"IsAddStronglyCompleteNatSeq Erdos346.f","subjects":["11"],"theorem":"Erdos346.erdos_346.variants.f_isAddStronglyCompleteNatSeq"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many 3-full $n$ such that $n+1$ is 2-full?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«366»","statement":"True ↔ {n | Nat.Full 3 n ∧ Nat.Full 2 (n + 1)}.Infinite","subjects":["11"],"theorem":"Erdos366.erdos_366.variants.three_two"},{"answerKinds":[],"category":"test","docstring":"Note that $8$ is $3$-full and $9$ is 2-full.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«366»","statement":"∃ n > 0, Nat.Full 3 n ∧ Nat.Full 2 (n + 1)","subjects":["11"],"theorem":"Erdos366.exists_three_full_then_two_full"},{"answerKinds":[],"category":"research open","docstring":"Are there any consecutive pairs of $3$-full integers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«366»","statement":"True ↔ ∃ n > 0, Nat.Full 3 n ∧ Nat.Full 3 (n + 1)","subjects":["11"],"theorem":"Erdos366.erdos_366.variants.weaker"},{"answerKinds":[],"category":"research open","docstring":"Are there any $2$-full $n$ such that $n+1$ is $3$-full?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«366»","statement":"True ↔ ∃ n > 0, Nat.Full 2 n ∧ Nat.Full 3 (n + 1)","subjects":["11"],"theorem":"Erdos366.erdos_366"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that for every infinite arithmetic progression $P$ which contains even numbers\nthere is some constant $c=c(P)$ such that every graph with average degree at least $c$\ncontains a cycle whose length is in $P$?\n\nIn [Er82e] Erdős credits this conjecture to himself and Burr. This has been proved by\nBollobás [Bo77]. The best dependence of the constant $c(P)$ is unknown.\n\nThe infinite arithmetic progression is encoded as a set $P \\subseteq \\mathbb{N}$ satisfying\n`P.IsAPOfLength ⊤` (which forces a positive common difference), and \"contains even numbers\"\nas the existence of an even element. The average degree of a finite simple graph is\n`SimpleGraph.averageDegree`, i.e. $(\\sum_v \\deg v)/|V| \\in \\mathbb{Q}$, and a cycle whose\nlength is in $P$ is a cycle walk `w` with `w.length ∈ P`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/110d489ed5c07e5b216453e092e9113127c98c9a/problems/71/Erdos71.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«71»","statement":"True ↔\n  ∀ (P : Set ℕ),\n    P.IsAPOfLength ⊤ →\n      (∃ n ∈ P, Even n) →\n        ∃ c,\n          ∀ (V : Type) [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n            c ≤ G.averageDegree → ∃ v w, w.IsCycle ∧ w.length ∈ P","subjects":["5"],"theorem":"Erdos71.erdos_71"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many binomial coefficients with deficiency 1?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1093»","statement":"True ↔\n  {x |\n      have k := x.1;\n      have n := x.2;\n      2 * k ≤ n ∧ Erdos1093.deficiency n k = 1 ∧ ∀ (p : ℕ), Nat.Prime p → p ∣ n.choose k → k < p}.Infinite","subjects":["5"],"theorem":"Erdos1093.erdos_1093.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Are there only finitely many binomial coefficients with deficiency > 1?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1093»","statement":"{x |\n    have k := x.1;\n    have n := x.2;\n    2 * k ≤ n ∧ Erdos1093.deficiency n k > 1 ∧ ∀ (p : ℕ), Nat.Prime p → p ∣ n.choose k → k < p}.Finite","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"Erdos1093.erdos_1093.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Any $A\\subseteq \\mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and $C$\nare infinite.\n\nThe Erdős sumset conjecture. Proved by Moreira, Richter, and Robertson [MRR19].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«109»","statement":"∀ (A : Set ℕ), A.upperDensity > 0 → ∃ B C, B.Infinite ∧ C.Infinite ∧ B + C ⊆ A","subjects":["5"],"theorem":"Erdos109.erdos_109"},{"answerKinds":[],"category":"research open","docstring":"Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ and let $R(x_i)=\\#\\{ \\lvert x_j-x_i\\rvert : j\\neq i\\}$,\nwhere the points are ordered such that\n$$R(x_1)\\leq \\cdots \\leq R(x_n).$$\nLet $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that\n$g(n) \\geq (1-o(1))n$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«653»","statement":"True ↔\n  ∃ o,\n    o =o[Filter.atTop] 1 ∧\n      ∀ᶠ (n : ℕ) in Filter.atTop, (1 - o n) * ↑n ≤ ↑(EuclideanGeometry.maximalDistinctDistancesFrom n)","subjects":["5","52"],"theorem":"Erdos653.erdos_653"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A_1,A_2,\\ldots$ be an infinite collection of infinite sets of integers, say\n$A_i=\\{a_{i1}<a_{i2}<\\cdots\\}$. Does there exist some $f:\\mathbb{N}\\to\\{-1,1\\}$ such that\n$$\\max_{m, 1\\leq i\\leq d} \\left\\lvert \\sum_{1\\leq j\\leq m} f(a_{ij})\\right\\rvert \\ll_d 1$$\nfor all $d\\geq 1$?\n\nErdős remarks 'it seems certain that the answer is affirmative'. This was solved by Beck [Be81]. Recently Beck [Be17] proved that one can replace $\\ll_d 1$ with $\\ll d^{4+\\epsilon}$ for any $\\epsilon>0$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos178.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«178»","statement":"True ↔\n  ∀ (a : ℕ → ℕ → ℕ),\n    (∀ (i : ℕ), StrictMono (a i)) →\n      ∃ f, (∀ (n : ℕ), f n = 1 ∨ f n = -1) ∧ ∀ (d : ℕ), ∃ C, ∀ (m i : ℕ), i < d → |∑ j ∈ Finset.range m, f (a i j)| ≤ ↑C","subjects":["11"],"theorem":"Erdos178.erdos_178"},{"answerKinds":[],"category":"research open","docstring":"Let $h(n)$ be maximal such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the\nset$$\\left\\{ \\frac{a}{(a,b)}: a,b\\in A\\right\\}$$has size at least $h(n)$.\nIs $h(n) = \\Theta(\\sqrt{n})$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«539»","statement":"(fun n => ↑(Erdos539.cofactorThreshold n)) =Θ[Filter.atTop] fun n => √↑n","subjects":["5","11"],"theorem":"Erdos539.erdos_539.variants.sq"},{"answerKinds":[],"category":"research solved","docstring":"Granville and Roesler [GR99] showed that $$h(n)\\ll n^{2/3}$$.","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«539»","statement":"(fun n => ↑(Erdos539.cofactorThreshold n)) =O[Filter.atTop] fun n => ↑n ^ (2 / 3)","subjects":["5","11"],"theorem":"Erdos539.erdos_539.variants.isBigO_sq_cube_root"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Szemerédi proved that$$n^{1/2} \\ll h(n)$$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«539»","statement":"(fun n => √↑n) =O[Filter.atTop] fun n => ↑(Erdos539.cofactorThreshold n)","subjects":["5","11"],"theorem":"Erdos539.erdos_539.variants.sq_isBigO"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $h(n)$ be maximal such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the\nset$$\\left\\{ \\frac{a}{(a,b)}: a,b\\in A\\right\\}$$has size at least $h(n)$. Estimate $h(n)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«539»","statement":"(fun n => ↑(Erdos539.cofactorThreshold n)) =Θ[Filter.atTop] sorry","subjects":["5","11"],"theorem":"Erdos539.erdos_539"},{"answerKinds":["Prop"],"category":"research solved","docstring":"This lower bound is false; see [Sc+26, Theorem A.1] and the linked Lean proof. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/erdos-539-formal-conjectures/blob/79897cf9241390eb168572f4a481e29e0e64b5f7/lean/Erdos539/FC.lean#L282-L296"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«539»","statement":"False ↔ (fun n => ↑n ^ (2 / 3)) =O[Filter.atTop] fun n => ↑(Erdos539.cofactorThreshold n)","subjects":["5","11"],"theorem":"Erdos539.erdos_539.variants.sq_cube_root_isBigO"},{"answerKinds":[],"category":"research open","docstring":"To prove `erdos_539.variants.sq` it suffices to show $$ h(n)\\ll n^{1/2}$$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«539»","statement":"(fun n => ↑(Erdos539.cofactorThreshold n)) =O[Filter.atTop] fun n => √↑n","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos539.erdos_539.variants.isBigO_sq"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $h(n)$ be maximal such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the\nset$$\\left\\{ \\frac{a}{(a,b)}: a,b\\in A\\right\\}$$has size at least $h(n)$.\nIs $h(n) = \\Theta(n^{2/3})$?\nThe answer is negative; see [Sc+26, Theorem A.1] and the linked Lean proof. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/erdos-539-formal-conjectures/blob/79897cf9241390eb168572f4a481e29e0e64b5f7/lean/Erdos539/FC.lean#L282-L296"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«539»","statement":"False ↔ (fun n => ↑(Erdos539.cofactorThreshold n)) =Θ[Filter.atTop] fun n => ↑n ^ (2 / 3)","subjects":["5","11"],"theorem":"Erdos539.erdos_539.variants.sq_cube_root"},{"answerKinds":["non-Prop"],"category":"research solved","docstring":"From [Er73]: The determination of\n$$\n  \\lim_{n\\to\\infty}\\frac{\\log(h(n))}{\\log(n)}\n$$\nwill perhaps be not too difficult.\nThe limit is $1/2$; see [Sc+26, Theorem A.1] and the linked Lean proof.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/erdos-539-formal-conjectures/blob/79897cf9241390eb168572f4a481e29e0e64b5f7/lean/Erdos539/FC.lean#L282-L296"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«539»","statement":"Filter.Tendsto (fun n => Real.log ↑(Erdos539.cofactorThreshold n) / Real.log ↑n) Filter.atTop (nhds (1 / 2))","subjects":["5","11"],"theorem":"Erdos539.erdos_539.variants.limit"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $r\\geq 2$ and let $t_r(n)$ be the Turán number (the maximal number of edges in a graph on $n$\nvertices with no $K_{r+1}$).\n\nIf $G$ is a graph with $n$ vertices and $m\\geq t_r(n)$ edges there exists a clique on $r$ vertices,\nsay $x_1,\\ldots,x_r$, such that $$d(x_1)+\\cdots+d(x_r)\\geq \\frac{2rm}{n}.$$\n\nA conjecture of Bollobás and Erdős. This was conjectured in [Er75] only in the special case $r=3$. Edwards [Ed78] proved the conjecture for $2\\leq r\\leq 8$ (under the additional assumption that $n\\geq r^2$). Faudree [Fa92] proved the conjecture for all $r\\geq 2$ provided $n>\\frac{r-1}{4}r^2$. The full conjecture was proved by Bollobás and Nikiforov [BoNi05].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos904.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«904»","statement":"True ↔\n  ∀ (V : Type u_1) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n    ∀ r ∈ Set.Icc 1 (Erdos904.n V),\n      Erdos904.turanNumber (Erdos904.n V) r ≤ G.edgeFinset.card →\n        ∃ s, G.IsNClique r s ∧ 2 * r * G.edgeFinset.card ≤ Erdos904.n V * ∑ v ∈ s, G.degree v","subjects":["5"],"theorem":"Erdos904.erdos_904"},{"answerKinds":[],"category":"textbook","docstring":"For any maximal Sidon set, the difference set contains 0.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«42»","statement":"∀ (A : Set ℕ) (N : ℕ), 1 ≤ N → A.IsMaximalSidonSetIn N → 0 ∈ A - A","subjects":["5","11"],"theorem":"Erdos42.maximal_sidon_contains_zero"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 42**: Let M ≥ 1 and N be sufficiently large in terms of M. Is it true that for every\nmaximal Sidon set `A ⊆ {1,…,N}` there is another Sidon set `B ⊆ {1,…,N}` of size M such that\n`(A - A) ∩ (B - B) = {0}`?\n\nThis was proved for all $M$ by GPT 5.5 Pro (prompted by Sandhu), see discussion thread for more details.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P42/CompactCayley/Proof.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«42»","statement":"True ↔\n  ∀ M ≥ 1,\n    ∀ᶠ (N : ℕ) in Filter.atTop,\n      ∀ (A : Set ℕ), A.IsMaximalSidonSetIn N → ∃ B ⊆ Set.Icc 1 N, IsSidon B ∧ B.ncard = M ∧ (A - A) ∩ (B - B) = {0}","subjects":["5","11"],"theorem":"Erdos42.erdos_42"},{"answerKinds":[],"category":"textbook","docstring":"The set `{1, 2, 4}` is a maximal Sidon set in `{1, ..., 4}`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«42»","statement":"{1, 2, 4}.IsMaximalSidonSetIn 4","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos42.example_maximal_sidon"},{"answerKinds":[],"category":"textbook","docstring":"The difference set of `{1, 2, 4}` is `{0, 1, 2, 3}`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«42»","statement":"{1, 2, 4} - {1, 2, 4} = {0, 1, 2, 3}","subjects":["5","11"],"theorem":"Erdos42.example_difference_set"},{"answerKinds":["Prop"],"category":"research solved","docstring":"A variant asking for explicit bounds on how large N needs to be in terms of M.\n\nThis version provides a constructive function f such that for all M ≥ 1 and N ≥ f(M),\nevery maximal Sidon set A ⊆ {1,…,N} has another Sidon set B ⊆ {1,…,N} of size M with\ndisjoint difference sets (apart from 0).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/KitaKen1/erdos-42-constructive-variant/blob/1f82c76be43cb56f22e2f7f792e392d5fb3ff78c/lean/Erdos42Constructive.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«42»","statement":"True ↔\n  ∃ f,\n    ∀ (M N : ℕ),\n      1 ≤ M →\n        f M ≤ N →\n          ∀ (A : Set ℕ), A.IsMaximalSidonSetIn N → ∃ B ⊆ Set.Icc 1 N, IsSidon B ∧ B.ncard = M ∧ (A - A) ∩ (B - B) = {0}","subjects":["5","11"],"theorem":"Erdos42.erdos_42.variants.constructive"},{"answerKinds":[],"category":"research open","docstring":"Is there a dense subset of ℝ^2 such that all pairwise distances\nare rational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«212»","statement":"True ↔ ∃ u, Dense u ∧ u.Pairwise fun c₁ c₂ => dist c₁ c₂ ∈ Set.range Rat.cast","subjects":["52"],"theorem":"Erdos212.erdos_212"},{"answerKinds":[],"category":"research open","docstring":"Let $\\alpha,\\beta \\in \\mathbb{R}$. Is it true that$$\\liminf_{n\\to \\infty} n \\| n\\alpha \\|\n\\| n\\beta\\| =0$$? This is also known as the Littlewood conjecture.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«495»","statement":"True ↔ ∀ (α β : ℝ), Filter.liminf (fun n => ↑n * distToNearestInt (↑n * α) * distToNearestInt (↑n * β)) Filter.atTop = 0","subjects":["11"],"theorem":"Erdos495.erdos_495"},{"answerKinds":[],"category":"research solved","docstring":"Given any infinite set $A\\subset \\mathbb{N}$ there is a set $B$ of density $0$ such that $A+B$ contains all except finitely many integers.\n\nConjectured by Erdős and Straus. Proved by Lorentz [Lo54].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos31.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«31»","statement":"∀ (A : Set ℕ), A.Infinite → ∃ B, B.HasDensity 0 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, n ∈ A + B","subjects":["11"],"theorem":"Erdos31.erdos_31"},{"answerKinds":[],"category":"textbook","docstring":"A trivial upper bound: a play can claim at most the $n - 1$ elements of $\\{2, \\dots, n\\}$, so\n$L(n) \\leq n - 1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«872»","statement":"∀ (n : ℕ), 2 ≤ n → Erdos872.L n ≤ n - 1","subjects":["5","11","91"],"theorem":"Erdos872.erdos_872.trivial_upper_bound"},{"answerKinds":[],"category":"research open","docstring":"Erdős Problem 872, part (i) (weak form): there exists a constant $\\epsilon > 0$ such that the\ngame length is at least $\\epsilon \\cdot n$ for all sufficiently large $n$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«872»","statement":"True ↔ ∃ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos872.L n) ≥ ε * ↑n","subjects":["5","11","91"],"theorem":"Erdos872.erdos_872.parts.i"},{"answerKinds":[],"category":"API","docstring":"Each move claims exactly one pool element, so the minimax value never exceeds the number of\nalready claimed elements plus the number of still unclaimed elements. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«872»","statement":"∀ {n : ℕ} (fuel : ℕ) (turn : Bool) (p : Erdos872.GamePos n),\n  Erdos872.gameValueAux fuel turn p ≤ p.claimed.card + p.pool.card","subjects":["5"],"theorem":"Erdos872.gameValueAux_le"},{"answerKinds":[],"category":"research open","docstring":"Forum-related variant: how small can a maximal primitive subset of $\\{2, \\dots, n\\}$ be?\nThe set of primes in $\\{2, \\dots, n\\}$ is a maximal primitive subset of size $\\pi(n)$, and the forum\nthread asks whether this is the smallest possible for all $n \\geq 2$. Equivalently: must every\ncompleted play of the saturation game, by both players and regardless of strategy, claim at least\n$\\pi(n)$ elements? (Terminal positions of the game are exactly the maximal primitive subsets.) ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«872»","statement":"True ↔\n  ∀ n ≥ 2,\n    ∀ (A : Finset ℕ), Maximal (Erdos872.IsPrimitive n) A → (Finset.filter Nat.Prime (Finset.Icc 2 n)).card ≤ A.card","subjects":["5","11","91"],"theorem":"Erdos872.erdos_872.variants.prime_question"},{"answerKinds":[],"category":"API","docstring":"Membership in `legalMoves`: a legal move is a pool element whose insertion preserves\nprimitiveness. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«872»","statement":"∀ {n : ℕ} {p : Erdos872.GamePos n} {x : ℕ},\n  x ∈ Erdos872.legalMoves p ↔ x ∈ p.pool ∧ Erdos872.IsPrimitive n (insert x p.claimed)","subjects":["5"],"theorem":"Erdos872.mem_legalMoves"},{"answerKinds":[],"category":"research open","docstring":"Erdős Problem 872, part (ii) (strong form): for every $\\epsilon > 0$, the game length is at\nleast $(1-\\epsilon) \\cdot n / 2$ for all sufficiently large $n$.\n\nStatus note: the forum thread (April-May 2026) records Shortener strategies giving\n$L(n) \\leq (23/48 + o(1)) \\cdot n$ (described in the thread as accepted as correct, with a Lean\nformalization in progress) and a claimed $L(n) \\leq 0.19 \\cdot n$, either of which would answer this\nquestion negatively under the Prolonger-first convention. Neither is published, so the statement\nis recorded here as the original Erdős question. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«872»","statement":"True ↔ ∀ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos872.L n) ≥ (1 - ε) * ↑n / 2","subjects":["5","11","91"],"theorem":"Erdos872.erdos_872.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $A \\subset \\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct\nfor $a,b,c \\in A$ (aside from the trivial coincidences). Is it true that\n$$\\liminf_{N \\to \\infty} \\frac{\\lvert A \\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/3}}=0?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«41»","statement":"∀ (A : Set ℕ),\n  Erdos41.NtupleCondition A 3 →\n    A.Infinite → Filter.liminf (fun N => ↑(A ∩ Set.Icc 1 N).ncard / ↑N ^ (1 / 3)) Filter.atTop = 0","subjects":["11"],"theorem":"Erdos41.erdos_41"},{"answerKinds":[],"category":"research solved","docstring":"Erdős proved that if the pairwise sums $a+b$ are all distinct aside from the trivial\ncoincidences, then\n$$\\liminf_{N \\to \\infty} \\frac{\\lvert A \\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«41»","statement":"∀ (A : Set ℕ),\n  Erdos41.NtupleCondition A 2 → A.Infinite → Filter.liminf (fun N => ↑(A ∩ Set.Icc 1 N).ncard / √↑N) Filter.atTop = 0","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos41.erdos_41.variants.pairwise"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points. Can there be $\\gg n$ many distinct distances\neach of which occurs for more than $n$ many pairs from $A$?\n\nThe answer is yes: Bhowmick [Bh24] constructs a set of $n$ points in $\\mathbb{R}^2$ such that\n$\\lfloor\\frac{n}{4}\\rfloor$ distances occur at least $n+1$ times.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«756»","statement":"True ↔ (fun n => ↑n) =O[Filter.atTop] fun n => ↑(Erdos756.maxRichDistances n)","subjects":["52"],"theorem":"Erdos756.erdos_756"},{"answerKinds":[],"category":"research solved","docstring":"More generally, they construct, for any $m$ and large $n$, a set of $n$ points such that\n$\\lfloor \\frac{n}{2(m+1)}\\rfloor$ distances occur at least $n+m$ times.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«756»","statement":"∀ (m : ℕ), ∀ᶠ (n : ℕ) in Filter.atTop, ∃ A, A.card = n ∧ n / (2 * (m + 1)) ≤ (Erdos756.richDistances A (n + m)).card","subjects":["52"],"theorem":"Erdos756.erdos_756.variants.bhowmick_general"},{"answerKinds":[],"category":"research solved","docstring":"Bhowmick [Bh24] constructs a set of $n$ points in $\\mathbb{R}^2$ such that\n$\\lfloor\\frac{n}{4}\\rfloor$ distances occur at least $n+1$ times.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos756.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«756»","statement":"∀ (n : ℕ), ∃ A, A.card = n ∧ n / 4 ≤ (Erdos756.richDistances A (n + 1)).card","subjects":["52"],"theorem":"Erdos756.erdos_756.variants.bhowmick"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots\n$z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all\nintervals $I\\subseteq [0,2\\pi]$\n$$\n\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll\n\\left(n\\log M\\right)^{1/2},\n$$\nwhere $n$ is the number of non-zero coefficients of $f$ and\n$$\nM=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}.\n$$\n\nAn internal OpenAI model (see [APSSV26b]) has disproved the conjecture, constructing, for every\n$n\\geq 1$, a polynomial $f$ with $n$ non-zero coefficients such that $M<3$ and with a positive real\nzero of multiplicity $n-1$, whence letting $I=[0,c/d]$ for a suitably small $c>0$,\n$$\n\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\geq n-1.\n$$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos990.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«990»","statement":"False ↔\n  ∃ C,\n    ∀ (f : Polynomial ℂ),\n      f.coeff 0 ≠ 0 →\n        ∀ (α β : ℝ),\n          0 ≤ α →\n            α ≤ β →\n              β ≤ 2 * Real.pi →\n                |↑(Erdos990.rootArgCount f (Set.Icc α β)) - (β - α) / (2 * Real.pi) * ↑f.natDegree| ≤\n                  C * √(↑f.support.card * Real.log (Erdos990.M f))","subjects":["12","30"],"theorem":"Erdos990.erdos_990"},{"answerKinds":[],"category":"research solved","docstring":"Hayman [Ha72b] proved\n$$\n\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\leq n-1,\n$$\nand noted this is essentially sharp since $f(x)=(x^{p}-1)^{n-1}$ has $n$ non-zero coefficients and\nhas $1$ as a positive real zero of multiplicity $n-1$ (although for this $f$ the parameter $M$\nbecomes very large).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«990»","statement":"∀ (f : Polynomial ℂ),\n  f.coeff 0 ≠ 0 →\n    ∀ (α β : ℝ),\n      0 ≤ α →\n        α ≤ β →\n          β ≤ 2 * Real.pi →\n            |↑(Erdos990.rootArgCount f (Set.Icc α β)) - (β - α) / (2 * Real.pi) * ↑f.natDegree| ≤ ↑f.support.card - 1","subjects":["12","30"],"theorem":"Erdos990.erdos_990.variants.hayman"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Turán [ErTu50] proved such an upper bound with $n$ replaced by $d$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«990»","statement":"∃ C,\n  ∀ (f : Polynomial ℂ),\n    f.coeff 0 ≠ 0 →\n      ∀ (α β : ℝ),\n        0 ≤ α →\n          α ≤ β →\n            β ≤ 2 * Real.pi →\n              |↑(Erdos990.rootArgCount f (Set.Icc α β)) - (β - α) / (2 * Real.pi) * ↑f.natDegree| ≤\n                C * √(↑f.natDegree * Real.log (Erdos990.M f))","subjects":["12","30"],"theorem":"Erdos990.erdos_990.variants.erdos_turan"},{"answerKinds":[],"category":"research solved","docstring":"An internal OpenAI model (see [APSSV26b]) has disproved the conjecture, constructing, for every\n$n\\geq 1$, a polynomial $f$ with $n$ non-zero coefficients such that $M<3$ and with a positive real\nzero of multiplicity $n-1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«990»","statement":"∀ (n : ℕ),\n  1 ≤ n →\n    ∃ f, f.coeff 0 ≠ 0 ∧ f.support.card = n ∧ Erdos990.M f < 3 ∧ ∃ x, 0 < x ∧ Polynomial.rootMultiplicity (↑x) f = n - 1","subjects":["12","30"],"theorem":"Erdos990.erdos_990.variants.counterexample"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a constant `c > 0` such that, for any `K > 1`, whenever `A` is a sufficiently large\nfinite multiset of integers with $\\sum_{n \\in A} 1/n > K$ there exists some $S \\subseteq A$ such that\n$1 - \\exp(-(c*K)) < \\sum_{n \\in S} 1/n \\le 1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«312»","statement":"True ↔\n  ∃ c,\n    0 < c ∧\n      ∀ (K : ℝ),\n        1 < K →\n          ∃ N₀,\n            ∀ (n : ℕ) (a : Fin n → ℕ),\n              n ≥ N₀ ∧ ∑ i, (↑(a i))⁻¹ > K →\n                ∃ S, 1 - Real.exp (-(c * K)) < ∑ i ∈ S, (↑(a i))⁻¹ ∧ ∑ i ∈ S, (↑(a i))⁻¹ ≤ 1","subjects":["5","11"],"theorem":"Erdos312.erdos_312"},{"answerKinds":[],"category":"research open","docstring":"If $n(n+1)=2^k3^lm$, where $(m,6)=1$, then is it true that\n$\\limsup_{n\\to \\infty} \\frac{2^k3^l}{n\\log n}=\\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«933»","statement":"True ↔ Filter.limsup (fun n => ↑↑(2 ^ Erdos933.k n * 3 ^ Erdos933.l n) / (↑↑n * ↑(Real.log ↑n))) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos933.erdos_933"},{"answerKinds":[],"category":"research solved","docstring":"Mahler proved (a more general result that implies in particular) that $2^k3^l<n^{1+o(1)}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«933»","statement":"∃ c, c =o[Filter.atTop] 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑(2 ^ Erdos933.k n * 3 ^ Erdos933.l n) < ↑n ^ (1 + c n)","subjects":["11"],"theorem":"Erdos933.erdos_933.variants.mahler"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er76d] wrote 'it is easy to see' that for infinitely many $n$, $2^k 3^l > n\\log n$.\n\nSteinerberger has noted a simple proof of this fact follows from taking $n=2^{3^r}$ for any\ninteger $r\\geq 1$, when $k=3^r$ and $l=r+1$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«933»","statement":"{n | ↑(2 ^ Erdos933.k n * 3 ^ Erdos933.l n) > ↑n * Real.log ↑n}.Infinite","subjects":["11"],"theorem":"Erdos933.erdos_933.variants.lower_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $k\\geq 1$. What is the best possible $c_k$ such that\n$$R(C_{2k+1},H)\\leq c_k m$$\nfor any graph $H$ on $m$ edges without isolated vertices?\n\nThis problem is #34 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«569»","statement":"have c := sorry;\n∀ (k : ℕ),\n  1 ≤ k →\n    sInf\n        {C |\n          0 < C ∧\n            ∀ (m : ℕ) (W : Type) [inst : Fintype W] (H : SimpleGraph W) [inst_1 : DecidableRel H.Adj],\n              (∀ (v : W), 0 < H.degree v) →\n                H.edgeSet.ncard = m → ↑((SimpleGraph.cycleGraph (2 * k + 1)).graphRamsey H) ≤ C * ↑m} =\n      c k","subjects":["5"],"theorem":"Erdos569.erdos_569"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does the set of integers of the form $n + \\varphi(n)$ have positive (lower) density?\n\n[GIL24] proved this was true.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«822»","statement":"True ↔ 0 < (Set.range fun n => n + n.totient).lowerDensity","subjects":["11"],"theorem":"Erdos822.erdos_822"},{"answerKinds":[],"category":"research solved","docstring":"Let $P(n)$ denote the largest prime factor of $n$. There are infinitely many $n$ such that\n$P(n)>P(n+1)>P(n+2)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«372»","statement":"{n | n.maxPrimeFac > (n + 1).maxPrimeFac ∧ (n + 1).maxPrimeFac > (n + 2).maxPrimeFac}.Infinite","subjects":["11"],"theorem":"Erdos372.erdos_372"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq \\mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the\nvertices of an isosceles trapezoid of area $1$? What about an isosceles triangle, or a\nright-angled triangle, or a cyclic quadrilateral, or a convex polygon with congruent sides?\n\nKoizumi [Ko25] has resolved this question, proving that any set with infinite measure must\ncontain the vertices of an isosceles trapezoid, an isosceles triangle, and a right-angled\ntriangle, all of area $1$.\n\nThis statement formalizes the leading question, for isosceles trapezoids; the remaining\nconfigurations are given as variants below. The area of a polygon is taken to be the Lebesgue\nmeasure of the convex hull of its vertices.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/110d489ed5c07e5b216453e092e9113127c98c9a/problems/353/Erdos353.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«353»","statement":"True ↔\n  ∀ (A : Set (EuclideanSpace ℝ (Fin 2))),\n    MeasurableSet A →\n      MeasureTheory.volume A = ⊤ →\n        ∃ a ∈ A,\n          ∃ b ∈ A,\n            ∃ c ∈ A,\n              ∃ d ∈ A,\n                EuclideanGeometry.IsIsoscelesTrapezoid a b c d ∧ MeasureTheory.volume ((convexHull ℝ) {a, b, c, d}) = 1","subjects":["28","51"],"theorem":"Erdos353.erdos_353"},{"answerKinds":[],"category":"research solved","docstring":"The answer is negative for convex polygons with congruent sides: Kovač and Predojević\n[KoPr24] prove that there exists a set of infinite measure such that every convex polygon\nwith congruent sides and all vertices in the set has area $<1$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/110d489ed5c07e5b216453e092e9113127c98c9a/problems/353/Erdos353.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«353»","statement":"∃ A,\n  MeasurableSet A ∧\n    MeasureTheory.volume A = ⊤ ∧\n      ∀ (n : ℕ) (v : Fin (n + 3) → EuclideanSpace ℝ (Fin 2)),\n        EuclideanGeometry.IsCcwConvexPolygon v →\n          (∀ (i : Fin (n + 3)), v i ∈ A) →\n            (∀ (i : Fin (n + 3)), dist (v i) (v (i + 1)) = dist (v 0) (v 1)) →\n              MeasureTheory.volume ((convexHull ℝ) (Set.range v)) < 1","subjects":["28","51"],"theorem":"Erdos353.erdos_353.variants.congruent_sides"},{"answerKinds":[],"category":"research solved","docstring":"Every measurable $A\\subseteq \\mathbb{R}^2$ with infinite measure contains the vertices of a\ncyclic quadrilateral of area $1$.\n\nKovač and Predojević [KoPr24] have proved that this is true for cyclic quadrilaterals - that\nis, every set with infinite measure contains four distinct points on a circle such that the\nquadrilateral determined by these four points has area $1$. The quadrilateral determined by\nfour distinct concyclic points is their convex hull.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/110d489ed5c07e5b216453e092e9113127c98c9a/problems/353/Erdos353.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«353»","statement":"∀ (A : Set (EuclideanSpace ℝ (Fin 2))),\n  MeasurableSet A →\n    MeasureTheory.volume A = ⊤ →\n      ∃ Q ⊆ A, Q.ncard = 4 ∧ EuclideanGeometry.Cospherical Q ∧ MeasureTheory.volume ((convexHull ℝ) Q) = 1","subjects":["28","51"],"theorem":"Erdos353.erdos_353.variants.cyclic_quadrilateral"},{"answerKinds":[],"category":"research solved","docstring":"Every measurable $A\\subseteq \\mathbb{R}^2$ with infinite measure contains the vertices of a\nright-angled triangle of area $1$.\n\nKoizumi [Ko25] has resolved this question, proving that any set with infinite measure must\ncontain the vertices of an isosceles trapezoid, an isosceles triangle, and a right-angled\ntriangle, all of area $1$.\n\nNote the area condition forces `a`, `b`, `c` to be affinely independent, so no separate\nnon-degeneracy hypothesis is needed.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/110d489ed5c07e5b216453e092e9113127c98c9a/problems/353/Erdos353.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«353»","statement":"∀ (A : Set (EuclideanSpace ℝ (Fin 2))),\n  MeasurableSet A →\n    MeasureTheory.volume A = ⊤ →\n      ∃ a ∈ A,\n        ∃ b ∈ A, ∃ c ∈ A, EuclideanGeometry.IsRightAngled a b c ∧ MeasureTheory.volume ((convexHull ℝ) {a, b, c}) = 1","subjects":["28","51"],"theorem":"Erdos353.erdos_353.variants.right_angled_triangle"},{"answerKinds":[],"category":"research solved","docstring":"Every measurable $A\\subseteq \\mathbb{R}^2$ with infinite measure contains the vertices of an\nisosceles triangle of area $1$.\n\nKoizumi [Ko25] has resolved this question, proving that any set with infinite measure must\ncontain the vertices of an isosceles trapezoid, an isosceles triangle, and a right-angled\ntriangle, all of area $1$.\n\nNote the area condition forces `a`, `b`, `c` to be affinely independent, so no separate\nnon-degeneracy hypothesis is needed.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/110d489ed5c07e5b216453e092e9113127c98c9a/problems/353/Erdos353.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«353»","statement":"∀ (A : Set (EuclideanSpace ℝ (Fin 2))),\n  MeasurableSet A →\n    MeasureTheory.volume A = ⊤ →\n      ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, IsIsosceles a b c ∧ MeasureTheory.volume ((convexHull ℝ) {a, b, c}) = 1","subjects":["28","51"],"theorem":"Erdos353.erdos_353.variants.isosceles_triangle"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Determine the largest length of an interval in $[x,2x]$ on which\n$\\omega(n) > \\log\\log n$ everywhere. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«452»","statement":"Erdos452.largeOmegaIntervalLength = sorry","subjects":["11"],"theorem":"Erdos452.erdos_452"},{"answerKinds":[],"category":"test","docstring":"This lemma confirms that the set of possible unit distance counts is bounded above, which\nensures that taking the supremum (`sSup`) is a well-defined operation. The trivial upper bound is\nthe total number of pairs of points, $\\binom{n}{2}$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«90»","statement":"∀ (n : ℕ), BddAbove (Erdos90.unitDistanceCounts n)","subjects":["52"],"theorem":"Erdos90.unitDistanceCounts_BddAbove"},{"answerKinds":[],"category":"research solved","docstring":"**Sawin's explicit exponent.** The constructive disproof can be realised with $c \\ge 0.014114$\n(absorbing the implicit constant $C$ of Sawin's Theorem 1 into a slightly smaller exponent for\nall large enough $n$). Reference: Theorem 1 of Sawin, [arXiv:2605.20579](https://arxiv.org/abs/2605.20579)\n(2026).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«90»","statement":"{n | ↑n ^ 1.014114 ≤ ↑(Erdos90.maxUnitDistances n)}.Infinite","subjects":["52"],"theorem":"Erdos90.erdos_90.variants.sawin_explicit"},{"answerKinds":[],"category":"research solved","docstring":"**Sawin's Lemmas 11–12 / Remarks Proposition 2.3: the totally real tower.**\n\nThere exist $rdBound : \\mathbb{R}$ and a *single* infinite set $Q$ of rational primes\n$q \\equiv 1 \\pmod 4$ such that for every $N$ one can find a totally real number field $F/\\mathbb{Q}$\nof degree $\\ge N$ with bounded root discriminant $|disc F|^{1/[F:\\mathbb{Q}]} \\le rdBound$ in which\n*every* prime $q \\in Q$ splits completely.\n\nThe load-bearing feature is the quantifier order: $Q$ is fixed *before* $F$, so the same primes\nsplit completely in fields of *unbounded* degree. (For a single fixed $F$, Chebotarev already\ngives infinitely many completely split primes $\\equiv 1 \\pmod 4$, so a per-field statement would\nbe vacuous.) This uniform splitting in an unbounded tower is the key arithmetic input to the\ndisproof. It is proved as Lemmas 11–12 of Sawin, [arXiv:2605.20579](https://arxiv.org/abs/2605.20579),\nand as Proposition 2.3 of the [Remarks](https://arxiv.org/abs/2605.20695) paper, via the\nGolod–Shafarevich inequality for pro-$2$ groups together with the Hajir–Maire–Ramakrishna (2003)\ntower construction.\n\nA \"completely split\" rational prime $q$ in $F$ is one for which $(q)$ is the product of exactly\n$[F:\\mathbb{Q}]$ distinct maximal ideals.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/n-yamaguchi-0729/SawinTotallyRealTowers/blob/3a455e1aa9140dbbe7b7d68f508392a69c86d0f4/Lean4/SawinTotallyRealTowers/SawinTotallyRealTower.lean#L31"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«90»","statement":"∃ rdBound Q,\n  Q.Infinite ∧\n    (∀ q ∈ Q, Nat.Prime q ∧ q % 4 = 1) ∧\n      ∀ (N : ℕ),\n        ∃ F x,\n          ∃ (x_1 : CharZero F) (x_2 : NumberField F) (_ : NumberField.IsTotallyReal F),\n            N ≤ Module.finrank ℚ F ∧\n              |↑(NumberField.discr F)| ^ (1 / ↑(Module.finrank ℚ F)) ≤ rdBound ∧\n                ∀ q ∈ Q, ∃ factors, factors.card = Module.finrank ℚ F ∧ ∀ p ∈ factors, p.IsMaximal ∧ ↑q ∈ p","subjects":["11"],"theorem":"Erdos90.sawin_totally_real_tower"},{"answerKinds":[],"category":"research solved","docstring":"**Constructive form of the disproof.** There is an absolute constant $c > 0$ such that\ninfinitely many $n$ admit a configuration realising at least $n^{1+c}$ unit distances.\n\nThis is the qualitative content of Theorem 1.1 of Alon–Bloom–Gowers–Litt–Sawin–Shankar–\nTsimerman–Wang–Matchett Wood, [*Remarks on the disproof of the unit distance conjecture*](https://arxiv.org/abs/2605.20695)\n(2026). An explicit bound $c \\ge 0.014114$ is given by Sawin, [*An explicit lower bound for the\nunit distance problem*](https://arxiv.org/abs/2605.20579) (2026); see\n`erdos_90.variants.sawin_explicit` below.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«90»","statement":"∃ c > 0, {n | ↑n ^ (1 + c) ≤ ↑(Erdos90.maxUnitDistances n)}.Infinite","subjects":["52"],"theorem":"Erdos90.erdos_90.variants.polynomial_lower_bound"},{"answerKinds":[],"category":"test","docstring":"Sawin's explicit bound implies the qualitative polynomial lower bound, by taking\n$c = 0.014114$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«90»","statement":"{n | ↑n ^ 1.014114 ≤ ↑(Erdos90.maxUnitDistances n)}.Infinite →\n  ∃ c > 0, {n | ↑n ^ (1 + c) ≤ ↑(Erdos90.maxUnitDistances n)}.Infinite","subjects":["52"],"theorem":"Erdos90.erdos_90.variants.sawin_explicit_implies_polynomial_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"**Sawin's Lemma 2: lattice geometry of unit distances** (Sawin, [arXiv:2605.20579](https://arxiv.org/abs/2605.20579)).\n\nLet $d \\ge 1$, $R \\ge 2$, and suppose $\\Lambda \\subset \\mathbb{R}^{2d}$ is a lattice equipped with\nan additive embedding $\\pi : \\Lambda \\to \\mathbb{R}^2$. Suppose $S \\subseteq \\Lambda$ is a finite\nset of \"matching\" vectors satisfying $\\|v\\| \\le 1$ and $\\|\\pi v\\| = 1$ for every $v \\in S$. Then\nthere is a finite point set $U \\subset \\mathbb{R}^2$ with unit-distance density at least\n$(1 - 1/R)^{2d}\\,\\#S$, i.e. $(1-1/R)^{2d}\\,\\#S\\,\\#U \\le \\#\\{\\text{unit pairs in } U\\}$.\n\nThis pure geometry-of-numbers reduction is the elementary heart of the disproof.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«90»","statement":"∀ (d : ℕ),\n  1 ≤ d →\n    ∀ (R : ℝ),\n      2 ≤ R →\n        ∀ (Λ : Submodule ℤ (EuclideanSpace ℝ (Fin (2 * d)))) (π : ↥Λ →+ EuclideanSpace ℝ (Fin 2)),\n          Function.Injective ⇑π →\n            ∀ (S : Finset ↥Λ),\n              (∀ v ∈ S, ‖↑v‖ ≤ 1) →\n                (∀ v ∈ S, ‖π v‖ = 1) → ∃ U, 0 < U.card ∧ (1 - 1 / R) ^ (2 * d) * ↑S.card * ↑U.card ≤ ↑(unitDistNum U)","subjects":["52"],"theorem":"Erdos90.sawin_lattice_reduction"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does every set of $n$ distinct points in $\\mathbb{R}^2$ contain at most\n$n^{1+O(\\frac{1}{\\log\\log n})}$ many pairs which are distance $1$ apart?\n\nThis was\n[disproved](https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf)\nby an internal model at OpenAI, which constructed (for infinitely many $n$) a set $P$ of $n$ points\nin $\\mathbb{R}^2$ such that the number of unit distance pairs in $P$ is at least $n^{1+c}$, where\n$c > 0$ is an absolute constant.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«90»","statement":"False ↔\n  ∃ O,\n    ∃ (_ : O =O[Filter.atTop] fun n => 1 / Real.log (Real.log ↑n)),\n      (fun n => ↑(Erdos90.maxUnitDistances n)) =ᶠ[Filter.atTop] fun n => ↑n ^ (1 + O n)","subjects":["52"],"theorem":"Erdos90.erdos_90"},{"answerKinds":["Prop"],"category":"test","docstring":"The polynomial lower bound implies the answer to Erdős 90 is `False`: a fixed positive\nexponent $c$ is incompatible with the conjectured $O(1 / \\log \\log n)$ growth. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«90»","statement":"(∃ c > 0, {n | ↑n ^ (1 + c) ≤ ↑(Erdos90.maxUnitDistances n)}.Infinite) →\n  (False ↔\n    ∃ O,\n      ∃ (_ : O =O[Filter.atTop] fun n => 1 / Real.log (Real.log ↑n)),\n        (fun n => ↑(Erdos90.maxUnitDistances n)) =ᶠ[Filter.atTop] fun n => ↑n ^ (1 + O n))","subjects":["52"],"theorem":"Erdos90.erdos_90.variants.polynomial_lower_bound_implies_erdos_90"},{"answerKinds":[],"category":"research open","docstring":"Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are\ncoloured red and blue then there exists either a red $K_k$ or a blue $K_l$.\n\nProve the existence of some $c>0$ such that\n$$\\lim_{k\\to \\infty}\\frac{R(k+1,k)}{R(k,k)}> 1+c.$$\n\nA problem of Erdős and Sós.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1030»","statement":"∃ c > 0,\n  ∃ L,\n    Filter.Tendsto (fun k => ↑(SimpleGraph.classicalRamsey (k + 1) k) / ↑(SimpleGraph.classicalRamsey k k)) Filter.atTop\n        (nhds L) ∧\n      L > 1 + c","subjects":["5"],"theorem":"Erdos1030.erdos_1030"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A,B\\subseteq \\mathbb{N}$ such that for all large $N$$$\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg\nN^{1/2}$$and$$\\lvert B\\cap \\{1,\\ldots,N\\}\\rvert \\gg N^{1/2}.$$\nIs it true that there are infinitely many solutions to $a_1-a_2=b_1-b_2\\neq 0$ with $a_1,a_2\\in A$\nand $b_1,b_2\\in B$?\n\nRuzsa has observed that there is a simple counterexample: take $A$ to be the set of numbers whose\nbinary representation has only non-zero digits in even places, and $B$ similarly but with non-zero\ndigits only in odd places. It is easy to see $A$ and $B$ both grow like $\\gg N^{1/2}$ and yet for\nany $n\\geq 1$ there is exactly one solution to $n=a+b$ with $a\\in A$ and $b\\in B$.\n\nThis was formalized in Lean by van Doorn using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/d30552f64c55686d40b928a0a3b8e2396357a4ee/ErdosProblem331.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«331»","statement":"False ↔\n  ∀ (A B : Set ℕ),\n    ((fun n => ↑n ^ (1 / 2)) =O[Filter.atTop] fun n => ↑(Nat.count (fun x => x ∈ A) n)) →\n      ((fun n => ↑n ^ (1 / 2)) =O[Filter.atTop] fun n => ↑(Nat.count (fun x => x ∈ B) n)) →\n        {(a₁, a₂, b₁, b₂) | a₁ ∈ A ∧ a₂ ∈ A ∧ b₁ ∈ B ∧ b₂ ∈ B ∧ a₁ ≠ a₂ ∧ a₁ + b₂ = a₂ + b₁}.Infinite","subjects":["11"],"theorem":"Erdos331.erdos_331"},{"answerKinds":[],"category":"research open","docstring":"Ruzsa suggests that a non-trivial variant of this problem arises if one imposes the stronger\ncondition that $|A \\cap \\{1,\\dots,N\\}| \\sim c_A N^{1/2}$ for some constant $c_A>0$, and similarly\nfor $B$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«331»","statement":"True ↔\n  ∀ (A B : Set ℕ),\n    (∃ c_A > 0,\n        Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Nat.count (fun x => x ∈ A) n)) fun n => c_A * ↑n ^ (1 / 2)) →\n      (∃ c_B > 0,\n          Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Nat.count (fun x => x ∈ B) n)) fun n =>\n            c_B * ↑n ^ (1 / 2)) →\n        {(a₁, a₂, b₁, b₂) | a₁ ∈ A ∧ a₂ ∈ A ∧ b₁ ∈ B ∧ b₂ ∈ B ∧ a₁ ≠ a₂ ∧ a₁ + b₂ = a₂ + b₁}.Infinite","subjects":["11"],"theorem":"Erdos331.erdos_331.variants.ruzsa"},{"answerKinds":[],"category":"research open","docstring":"Let $A \\subset \\mathbb{N}$ be an additive basis of order 2.\n\nMust there exist $B = \\{b_1 < b_2 < \\dots\\} \\subseteq A$ which is also a basis such that\n$\\lim_{k\\to\\infty} \\frac{b_k}{k^2}$ does not exist?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«326»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    A.IsAddBasisOfOrder 2 →\n      ∃ b,\n        StrictMono b ∧\n          ∀ (n : ℕ),\n            b n ∈ A ∧\n              (Set.range b).IsAddBasis ∧ ∀ (x : ℝ), ¬Filter.Tendsto (fun n => ↑(b n) / ↑n ^ 2) Filter.atTop (nhds x)","subjects":["5","11"],"theorem":"Erdos326.erdos_326"},{"answerKinds":[],"category":"research solved","docstring":"Erdős originally asked whether this was true with `A = B`, but this was disproved by Cassels.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«326»","statement":"∃ a,\n  ∃ (_ : StrictMono a) (_ : (Set.range a).IsAddBasisOfOrder 2),\n    ∃ x, ∃ (_ : 0 < x), Filter.Tendsto (fun n => ↑(a n) / ↑n ^ 2) Filter.atTop (nhds x)","subjects":["5","11"],"theorem":"Erdos326.erdos_326.variants.eq"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Estimate `m(n,k)`, or better give an asymptotic formula.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«857»","statement":"have f := sorry;\n∀ (k : ℕ), 3 ≤ k → Filter.Tendsto (fun n => ↑(Erdos857.m n k) / f k n) Filter.atTop (nhds 1)","subjects":["5"],"theorem":"Erdos857.erdos_857"},{"answerKinds":[],"category":"research solved","docstring":"The set is known to be infinite. In [Er77c] Erdős credits Schinzel with proving that there are\ninfinitely many odd integers not of this form, but gives no reference.\n\n[Er77c] Erdős, P., _Problems and results on combinatorial number theory. III._.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«9»","statement":"Erdos9.Erdos9A.Infinite","subjects":["5","11"],"theorem":"Erdos9.erdos_9.variants.infinite"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«9»","statement":"1 ∈ Erdos9.Erdos9A","subjects":["5","11"],"theorem":"Erdos9.erdos9A_contains_one"},{"answerKinds":[],"category":"research open","docstring":"Is the upper density of the set of odd numbers that cannot be expressed as a prime plus\ntwo powers of 2 positive?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«9»","statement":"True ↔ 0 < Erdos9.Erdos9A.upperDensity","subjects":["5","11"],"theorem":"Erdos9.erdos_9"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«9»","statement":"3 ∈ Erdos9.Erdos9A","subjects":["5","11"],"theorem":"Erdos9.erdos9A_contains_three"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«9»","statement":"5 ∉ Erdos9.Erdos9A","subjects":["5","11"],"theorem":"Erdos9.erdos9A_not_contains_five"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Determine which countable ordinals $β$ have the property that, if $α = \\omega^β$, then in any\nred/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«592»","statement":"∀ (β : Ordinal.{u}),\n  β.card ≤ Cardinal.aleph0 → OrdinalCardinalRamsey (Ordinal.omega0 ^ β) (Ordinal.omega0 ^ β) 3 ↔ sorry β","subjects":["3"],"theorem":"Erdos592.erdos_592"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1059»","statement":"∀ (d : ℕ), Erdos1059.DecidableAllFactorialSubtractionsComposite d ↔ Erdos1059.AllFactorialSubtractionsComposite d","subjects":["11"],"theorem":"Erdos1059.allFactorialSubtractionsComposite_equivalent"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1059»","statement":"¬Erdos1059.AllFactorialSubtractionsComposite 89","subjects":["11"],"theorem":"Erdos1059.notAllFactorialSubtractionsComposite_89"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1059»","statement":"∀ (d : ℕ), Erdos1059.IsFactorial d ↔ Erdos1059.DecidableIsFactorial d","subjects":["11"],"theorem":"Erdos1059.isFactorial_equivalent"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1059»","statement":"Erdos1059.AllFactorialSubtractionsComposite 211","subjects":["11"],"theorem":"Erdos1059.allFactorialSubtractionsComposite_211"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1059»","statement":"∀ (n : ℕ), Erdos1059.factorialsLessThanN n = ↑(Erdos1059.decidableFactorialsLessThanN n)","subjects":["11"],"theorem":"Erdos1059.factorialsLessThanN_equivalent"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1059»","statement":"True ↔ {p | Nat.Prime p ∧ Erdos1059.AllFactorialSubtractionsComposite p}.Infinite","subjects":["11"],"theorem":"Erdos1059.erdos_1059"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1059»","statement":"Erdos1059.AllFactorialSubtractionsComposite 101","subjects":["11"],"theorem":"Erdos1059.allFactorialSubtractionsComposite_101"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1059»","statement":"Erdos1059.factorialsLessThanN 100 = {1, 2, 6, 24}","subjects":["11"],"theorem":"Erdos1059.testFactorialsLessThanN"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, for any $C > 0$, there infinitely many $n$ such that:\n$$\n  p_{n + 1} - p_n > C \\frac{\\log\\log n\\log\\log\\log\\log n}{(\\log\\log\\log n) ^ 2}\\log n\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«4»","statement":"True ↔ ∀ C > 0, Erdos4.Erdos4For C","subjects":["11"],"theorem":"Erdos4.erdos_4"},{"answerKinds":[],"category":"research solved","docstring":"Rankin's theorem: there exists a positive constant $C$ such that `Erdos4For C` holds. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«4»","statement":"∃ C > 0, Erdos4.Erdos4For C","subjects":["11"],"theorem":"Erdos4.erdos_4.variants.rankin"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges. Must there be a triangle $T$ in $G$\nand vertices $y_1,\\ldots,y_t$, where $t>(\\frac{1}{2}-o(1))n$, such that every $y_i$ is joined to\nat least two vertices of $T$?\n\nA conjecture of Erdős and Faudree; a stronger version of [905].\n\nThis has been solved in the negative by Ma and Tang [MaTa25], who construct a graph with $n$\nvertices and $>n^2/4$ edges in which every triangle has at most $(2-(5/2)^{1/2}+o(1))n$ vertices\nadjacent to at least two of its vertices (note that $2-(5/2)^{1/2}\\approx 0.4189$).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1034.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1034»","statement":"False ↔\n  ∀ (ε : ℝ),\n    0 < ε →\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ (G : SimpleGraph (Fin n)),\n          ↑n ^ 2 / 4 < ↑G.edgeSet.ncard →\n            ∃ T, G.IsNClique 3 T ∧ ∃ Y, Erdos1034.JoinedToTwo G T Y ∧ (1 / 2 - ε) * ↑n < ↑Y.card","subjects":["5"],"theorem":"Erdos1034.erdos_1034"},{"answerKinds":[],"category":"research solved","docstring":"The construction of Ma and Tang [MaTa25] of a graph with $n$ vertices and $>n^2/4$ edges in which\nevery triangle has at most $(2-(5/2)^{1/2}+o(1))n$ vertices adjacent to at least two of its\nvertices shows that, for the threshold $h(n)$ of `erdos_1034.variants.lower_bound`,\n$$h(n) \\leq (2-(5/2)^{1/2}+o(1))n.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1034»","statement":"∀ (ε : ℝ),\n  0 < ε →\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∃ G,\n        ↑n ^ 2 / 4 < ↑G.edgeSet.ncard ∧\n          ∀ (T : Finset (Fin n)),\n            G.IsNClique 3 T → ∀ (Y : Finset (Fin n)), Erdos1034.JoinedToTwo G T Y → ↑Y.card ≤ (2 - √(5 / 2) + ε) * ↑n","subjects":["5"],"theorem":"Erdos1034.erdos_1034.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Erdős suggested that the answer is different if $G$ has no $K_4$. In the comments Ma and Tang\nsketch a proof that the conjecture remains false even if we assume that $G$ contains no $K_4$,\nconstructing a graph with $n$ vertices, $>n^2/4$ edges, and no $K_4$, in which every triangle has\nat most $(2\\sqrt{3}-3+o(1))n$ vertices adjacent to at least two of its vertices (note that\n$2\\sqrt{3}-3\\approx 0.464$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1034»","statement":"∀ (ε : ℝ),\n  0 < ε →\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∃ G,\n        G.CliqueFree 4 ∧\n          ↑n ^ 2 / 4 < ↑G.edgeSet.ncard ∧\n            ∀ (T : Finset (Fin n)),\n              G.IsNClique 3 T → ∀ (Y : Finset (Fin n)), Erdos1034.JoinedToTwo G T Y → ↑Y.card ≤ (2 * √3 - 3 + ε) * ↑n","subjects":["5"],"theorem":"Erdos1034.erdos_1034.variants.k4_free"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Faudree asked about the threshold $h(n)$ such that every graph with $n$ vertices and\n$>n^2/4$ edges contained a triangle and $h(n)$ other vertices which are connected to at least two\nvertices of the triangle. The fact that every graph with $>n^2/4$ edges contains a book of size\n$n/6$ shows that\n$$(1/6-o(1))n \\leq h(n).$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1034»","statement":"∀ (ε : ℝ),\n  0 < ε →\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (G : SimpleGraph (Fin n)),\n        ↑n ^ 2 / 4 < ↑G.edgeSet.ncard →\n          ∃ T, G.IsNClique 3 T ∧ ∃ Y, Erdos1034.JoinedToTwo G T Y ∧ (1 / 6 - ε) * ↑n ≤ ↑Y.card","subjects":["5"],"theorem":"Erdos1034.erdos_1034.variants.lower_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $n_1<n_2<\\cdots $ be an infinite sequence of integers with associated $a_k\\pmod{n_k}$, such that for some $\\epsilon>0$ we have $n_k>(1+\\epsilon)k\\log k$ for all $k$. Then\n$$\n\\#\\{ m<n_k : m\\not\\equiv a_i\\pmod{n_i} \\textrm{ for }1\\leq i\\leq k\\}\\neq o(k).\n$$\n\nCambie observed that this is false.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos280.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«280»","statement":"False ↔\n  ∀ (n a : ℕ → ℕ),\n    StrictMono n →\n      (∀ (i : ℕ), 1 ≤ i → a i < n i) →\n        (∃ ε, 0 < ε ∧ ∀ (k : ℕ), 1 ≤ k → ↑(n k) > (1 + ε) * ↑k * Real.log ↑k) →\n          ¬Filter.Tendsto (fun k => ↑(Erdos280.uncoveredCount n a k) / ↑k) Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Erdos280.erdos_280"},{"answerKinds":[],"category":"research open","docstring":"Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we\n$2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some\nmonochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.\n\nProve that, for $r \\ge 3$,\n$$ \\log_{r-1} R_r(n) \\asymp_r n, $$\nwhere $\\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«562»","statement":"True ↔ ∀ r ≥ 3, (fun n => Real.log^[r - 1] ↑(Combinatorics.hypergraphRamsey r n)) =Θ[Filter.atTop] fun n => ↑n","subjects":["5"],"theorem":"Erdos562.erdos_562"},{"answerKinds":[],"category":"research solved","docstring":"For all transcendental entire function `f`, `liminf (fun r : ℝ => ratio r f) atTop > 1 / 2`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«513»","statement":"⨆ f, Filter.liminf (fun r => Erdos513.ratio r ↑f) Filter.atTop > 1 / 2","subjects":["30"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos513.erdos_513.variants.lower_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let `f` be a transcendental entire function. What is the greatest possible value of\n`liminf (fun r : ℝ => ratio r f) atTop`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«513»","statement":"sorry = ⨆ f, Filter.liminf (fun r => Erdos513.ratio r ↑f) Filter.atTop","subjects":["30"],"theorem":"Erdos513.erdos_513"},{"answerKinds":[],"category":"research solved","docstring":"For all transcendental entire function `f`, `liminf (fun r : ℝ => ratio r f) atTop ≤ 2 / π - c`\nfor some `c > 0`. This is proved in [ClHa64]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«513»","statement":"∃ c > 0, ⨆ f, Filter.liminf (fun r => Erdos513.ratio r ↑f) Filter.atTop ≤ 2 / Real.pi - c","subjects":["30"],"theorem":"Erdos513.erdos_513.variants.upper_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 613:**\nLet $n \\geq 3$ and $G$ be a graph with $\\binom{2n+1}{2} - \\binom{n}{2} - 1$ edges.\nMust $G$ be the union of a bipartite graph and a graph with maximum degree less than $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«613»","statement":"False ↔\n  ∀ n ≥ 3,\n    ∀ (V : Type u_1) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n      G.edgeFinset.card = (2 * n + 1).choose 2 - n.choose 2 - 1 →\n        ∃ B D,\n          ∀ [DecidableRel B.Adj] [inst_3 : DecidableRel D.Adj], G = B ⊔ D ∧ B.IsBipartite ∧ ∀ (v : V), D.degree v < n","subjects":["5"],"theorem":"Erdos613.erdos_613"},{"answerKinds":[],"category":"research open","docstring":"What is the size of the largest Sidon subset $A\\subseteq\\{1,2^2,\\ldots,N^2\\}$? Is it $N^{1-o(1)}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«773»","statement":"True ↔\n  ∀ ε > 0,\n    ∀ᶠ (N : ℕ) in Filter.atTop, ↑N ^ (1 - ε) ≤ ↑(Finset.image (fun n => n ^ 2) (Finset.Icc 1 N)).maxSidonSubsetCard","subjects":["11"],"theorem":"Erdos773.erdos_773"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Stein conjectured that $f(N)=o(N)$, which was proved by Erdős and Szemerédi [ErSz68].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«202»","statement":"(fun N => ↑(Erdos202.f N)) =o[Filter.atTop] fun N => ↑N","subjects":["5","11"],"theorem":"Erdos202.erdos_202.variants.erdos_szemeredi"},{"answerKinds":[],"category":"research solved","docstring":"Let $n_1<\\cdots < n_r\\leq N$ with associated $a_i\\pmod{n_i}$ such that the congruence classes are\ndisjoint (that is, every integer is $\\equiv a_i\\pmod{n_i}$ for at most one $1\\leq i\\leq r$). How\nlarge can $r$ be in terms of $N$?\n\nLet $f(N)$ be the maximum possible $r$, and let $L(N)=\\exp(\\sqrt{\\log N\\log\\log N})$.\n\nThis was proved by GPT-5.4 Pro (prompted by Ho Boon Suan), using the argument of [BFV13] together\nwith the resolution of the Kahn-Kalai conjecture by Park and Pham [PaPh24], so that\n$$f(N)= N L(N)^{-1+o(1)}.$$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos202.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«202»","statement":"∃ o, o =o[Filter.atTop] 1 ∧ ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos202.f N) = ↑N * scaleL N ^ (-1 + o N)","subjects":["5","11"],"theorem":"Erdos202.erdos_202"},{"answerKinds":[],"category":"research solved","docstring":"In fact, Kovač and Tao proved in [KoTa24] that there exists a strictly increasing\nsequence $a_n$ of positive integers such that $\\sum \\frac{1}{a_n + t}$ converges to a rational\nnumber for all $t \\in \\mathbb{Q}$ such that $t \\ne -a_n$ for any $n$.\n\n[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series.\n         [arXiv:2406.17593](https://arxiv.org/abs/2406.17593) (2024).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«266»","statement":"∃ a, StrictMono a ∧ a 0 ≥ 1 ∧ ∀ (t : ℚ), (¬∃ n, t = -↑(a n)) → ∃ q, HasSum (fun n => 1 / (↑(a n) + ↑t)) ↑q","subjects":["11"],"theorem":"Erdos266.erdos_266.variants.all_rationals"},{"answerKinds":[],"category":"research solved","docstring":"Let $a_n$ be an infinite sequence of positive integers such that $\\sum \\frac{1}{a_n}$ converges.\nThere exists some integer $t \\ge 1$ such that $\\sum \\frac{1}{a_n + t}$ is irrational.\n\nThis was disproven by Kovač and Tao in [KoTa24].\n\n[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series.\n         [arXiv:2406.17593](https://arxiv.org/abs/2406.17593) (2024).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«266»","statement":"¬∀ (a : ℕ → ℕ),\n    ((∀ (n : ℕ), a n ≥ 1) ∧ Summable fun x => 1 / ↑(a x)) → ∃ t ≥ 1, Irrational (∑' (n : ℕ), 1 / (↑(a n) + ↑t))","subjects":["11"],"theorem":"Erdos266.erdos_266"},{"answerKinds":[],"category":"research open","docstring":"Is it true that there is a constant $c_k$ such that for almost all $n < x$ we have\n$g_k(n)=c_k\\log x+o(\\log x)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«400»","statement":"True ↔\n  ∀ k ≥ 2,\n    ∃ c,\n      ∀ ε > 0,\n        Filter.Tendsto\n          (fun x => ↑{n ∈ Finset.Icc 1 x | |↑(Erdos400.g k n) - c * Real.log ↑x| ≤ ε * Real.log ↑x}.card / ↑x)\n          Filter.atTop (nhds 1)","subjects":["11"],"theorem":"Erdos400.erdos_400.parts.ii"},{"answerKinds":[],"category":"test","docstring":"For $k \\ge 2$, $g_k(n) > 0$. We show this by choosing $a = (n, 1, 0, \\ldots, 0)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«400»","statement":"∀ (k n : ℕ), k ≥ 2 → 0 < Erdos400.g k n","subjects":["11"],"theorem":"Erdos400.erdos_400.variants.g_pos"},{"answerKinds":[],"category":"research open","docstring":"Can one show that $\\sum_{n\\leq x}g_k(n) \\sim c_k x\\log x$ for some constant $c_k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«400»","statement":"True ↔\n  ∀ k ≥ 2,\n    ∃ c,\n      Asymptotics.IsEquivalent Filter.atTop (fun x => ∑ n ∈ Finset.Icc 1 x, ↑(Erdos400.g k n)) fun x =>\n        c * ↑x * Real.log ↑x","subjects":["11"],"theorem":"Erdos400.erdos_400.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Graham write that it is easy to show that $g_k(n) \\ll_k \\log n$ always, but the best\npossible constant is unknown.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«400»","statement":"∀ k ≥ 2, (fun n => ↑(Erdos400.g k n)) =O[Filter.atTop] fun n => Real.log ↑n","subjects":["11"],"theorem":"Erdos400.erdos_400.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"If ℕ is $2$-coloured then there must exist a monochromatic three-term arithmetic progression\n$x,x+d,x+2d$ such that $d>x$.\n\nThis was first proved by Brown and Landman [BrLa99], who in fact show that this is always possible\nwith $d>f(x)$ for any increasing function $f$.\n\nThis was formalized in Lean by Alexeev using Aristotle and ChatGPT.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos645.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«645»","statement":"∀ (c : ℕ → Bool), ∃ x d, 0 < x ∧ x < d ∧ ∃ C, c x = C ∧ c (x + d) = C ∧ c (x + 2 * d) = C","subjects":["5","11"],"theorem":"Erdos645.erdos_645"},{"answerKinds":[],"category":"research open","docstring":"Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\\log g(k) \\asymp \\frac{k}{\\log k}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1095»","statement":"Asymptotics.IsEquivalent Filter.atTop (fun k => Real.log ↑(Erdos1095.g k)) fun k => ↑k / Real.log ↑k","subjects":["11"],"theorem":"Erdos1095.erdos_1095.variants.log_equivalent"},{"answerKinds":[],"category":"research open","docstring":"Ecklund, Erdős, and Selfridge [EES74] conjectured $g(k)\\leq \\exp((1+o(1))k)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1095»","statement":"∃ f, Filter.Tendsto f Filter.atTop (nhds 0) ∧ ∀ᶠ (k : ℕ) in Filter.atTop, ↑(Erdos1095.g k) ≤ Real.exp (↑k * (1 + f k))","subjects":["11"],"theorem":"Erdos1095.erdos_1095.variants.upper_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that\n$g(k)\\geq\\exp(c\\frac{k}{\\log k})$ for some constant $c>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1095»","statement":"∃ c > 0, ∀ᶠ (k : ℕ) in Filter.atTop, ↑(Erdos1095.g k) ≥ Real.exp (c * ↑k / Real.log ↑k)","subjects":["11"],"theorem":"Erdos1095.erdos_1095.variants.lower_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"The current record is $g(k) \\gg \\exp(c(\\log k)^2)$ for some $c>0$, due to Konyagin [Ko99b]. -","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1095»","statement":"∃ c > 0, (fun k => Real.exp (c * Real.log ↑k ^ 2)) =O[Filter.atTop] fun k => ↑(Erdos1095.g k)","subjects":["11"],"theorem":"Erdos1095.erdos_1095.variants.lower_solved"},{"answerKinds":[],"category":"research solved","docstring":"Since $t_2(p)=p-1$ for prime $p$ it is trivial that $\\sum_{n\\leq x}t_2(n)\\gg \\frac{x^2}{\\log x}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«394»","statement":"(fun x => ↑x ^ 2 / Real.log ↑x) =O[Filter.atTop] fun x => ∑ n ∈ Finset.Icc 1 ⌊x⌋₊, ↑(Erdos394.t 2 n)","subjects":["11"],"theorem":"Erdos394.erdos_394.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"They proved (with Selfridge) that this holds for $n=10$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«394»","statement":"∀ (k : ℕ), 2 ≤ k → k < 10 → Erdos394.t k (Nat.factorial 10) < Erdos394.t (k - 1) (Nat.factorial 10) - 1","subjects":["11"],"theorem":"Erdos394.erdos_394.variants.factorial_gap_10"},{"answerKinds":[],"category":"research solved","docstring":"In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$. This was proved by Erdős\nand Hall [ErHa78], who proved that in fact\n$\\sum_{n\\leq x}t_2(n)\\ll \\frac{\\log\\log\\log x}{\\log\\log x}x^2.$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«394»","statement":"(fun x => ∑ n ∈ Finset.Icc 1 ⌊x⌋₊, ↑(Erdos394.t 2 n)) =O[Filter.atTop] fun x =>\n  ↑x ^ 2 * (Real.log (Real.log (Real.log ↑x)) / Real.log (Real.log ↑x))","subjects":["11"],"theorem":"Erdos394.erdos_394.variants.hall_bound"},{"answerKinds":[],"category":"API","docstring":"The least positive multiple of `n` is `n`, so `t 1 n = n`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«394»","statement":"∀ {n : ℕ}, 0 < n → Erdos394.t 1 n = n","subjects":["11"],"theorem":"Erdos394.t_one"},{"answerKinds":[],"category":"research open","docstring":"They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$,\n$t_k(n!)< t_{k-1}(n!)-1$ for all $1\\leq k < n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«394»","statement":"True ↔ {n | ∀ (k : ℕ), 2 ≤ k → k < n → Erdos394.t k n.factorial < Erdos394.t (k - 1) n.factorial - 1}.Infinite","subjects":["11"],"theorem":"Erdos394.erdos_394.variants.factorial_gap_conjecture"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that $\\sum_{n\\leq x}t_2(n)\\ll \\frac{x^2}{(\\log x)^c}$ for some $c>0$?\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-394/Research/FirstQuestion.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«394»","statement":"True ↔ ∃ c > 0, (fun x => ∑ n ∈ Finset.Icc 1 ⌊x⌋₊, ↑(Erdos394.t 2 n)) =O[Filter.atTop] fun x => ↑x ^ 2 / Real.log ↑x ^ c","subjects":["11"],"theorem":"Erdos394.erdos_394.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Erdős and Hall conjecture that the sum is $o(x^2/(\\log x)^c)$ for any $c<\\log 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«394»","statement":"∀ c < Real.log 2, (fun x => ∑ n ∈ Finset.Icc 1 ⌊x⌋₊, ↑(Erdos394.t 2 n)) =o[Filter.atTop] fun x => x ^ 2 / Real.log x ^ c","subjects":["11"],"theorem":"Erdos394.erdos_394.variants.hall_conjecture"},{"answerKinds":[],"category":"API","docstring":"`t k n = v` when `v` works and nothing positive below it does. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«394»","statement":"∀ {n k v : ℕ},\n  0 < v →\n    n ∣ ∏ i ∈ Finset.range k, (v + i) →\n      (∀ m ∈ Finset.range v, 0 < m → ¬n ∣ ∏ i ∈ Finset.range k, (m + i)) → Erdos394.t k n = v","subjects":["11"],"theorem":"Erdos394.t_eq_of"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, for $k\\geq 2$, $\\sum_{n\\leq x}t_{k+1}(n) =o\\left(\\sum_{n\\leq x}t_k(n)\\right)?$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-394/Research/DenseHierarchyLittleO.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«394»","statement":"True ↔\n  ∀ k ≥ 2,\n    (fun x => ∑ n ∈ Finset.Icc 1 ⌊x⌋₊, ↑(Erdos394.t (k + 1) n)) =o[Filter.atTop] fun x =>\n      ∑ n ∈ Finset.Icc 1 ⌊x⌋₊, ↑(Erdos394.t k n)","subjects":["11"],"theorem":"Erdos394.erdos_394.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Prove that, for any finite set $A\\subset\\mathbb{N}$, there exist $a, b\\in A$ such\nthat\n$$\n  \\gcd(a, b)\\leq a/|A|.\n$$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«402»","statement":"∀ (A : Finset ℕ), 0 ∉ A → A.Nonempty → ∃ a ∈ A, ∃ b ∈ A, ↑(a.gcd b) ≤ ↑a / ↑A.card","subjects":["11"],"theorem":"Erdos402.erdos_402"},{"answerKinds":[],"category":"research solved","docstring":"Proved for all sufficiently large sets (including the sharper version which\ncharacterises the case of equality) independently by Szegedy [Sz86] and\nZaharescu [Za87]. The following is taken from [Sz86].\n\nThere exists an effectively computable $n_0$ with the following properties:\n(i) if $n \\ge n_0$ and $a_1, a_2, \\dots, a_n$ are distinct natural numbers then\n$\\max_{i, j} \\frac{a_i}{(a_i, a_j)} \\ge n$.\n(ii) If equality holds then the system $\\{a_1, a_2, \\dots, a_n\\}$ is either of the\ntype $\\{k, 2k, \\dots, nk\\}$ or of the type\n$\\left\\{\\frac{k}{1}, \\frac{k}{2}, \\dots, \\frac{k}{n}\\right\\}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«402»","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ (A : Finset ℕ),\n    A.card = n →\n      0 ∉ A →\n        (n ≤ (A ×ˢ A).sup fun x => x.1 / x.1.gcd x.2) ∧\n          ((n = (A ×ˢ A).sup fun x => x.1 / x.1.gcd x.2) ↔\n            ∃ k > 0,\n              A = Finset.image (fun x => k * x) (Finset.Icc 1 n) ∨\n                A = Finset.image (fun x => k * (Finset.Icc 1 n).lcm id / x) (Finset.Icc 1 n))","subjects":["11"],"theorem":"Erdos402.erdos_402.variants.szegedy_zaharescu_weak"},{"answerKinds":[],"category":"research solved","docstring":"A conjecture of Graham [Gr70], who also conjectured that (assuming $A$ itself\nhas no common divisor) the only cases where equality is achieved are when\n$A = \\{1, \\dots, n\\}$ or $A = \\{L/1, \\dots, L/n\\}$ (where $L = \\operatorname{lcm}(1, \\dots, n)$) or\n$A = \\{2, 3, 4, 6\\}$.\nNote: The source [BaSo96] mentioned on the Erdős page makes it clear what\nquantifiers to use for \"where equality is achieved\". See Theorem 1.1 there.\n\nTODO(firsching): Consider if we should have the other direction here as well or\nan iff statement.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«402»","statement":"∀ (A : Finset ℕ),\n  0 ∉ A →\n    A.Nonempty →\n      A.gcd id = 1 →\n        (∀ a ∈ A, ∀ b ∈ A, ↑a / ↑A.card ≤ ↑(a.gcd b)) →\n          A = Finset.Icc 1 A.card ∨\n            A = Finset.image (fun x => (Finset.Icc 1 A.card).lcm id / x) (Finset.Icc 1 A.card) ∨ A = {2, 3, 4, 6}","subjects":["11"],"theorem":"Erdos402.erdos_402.variants.equality"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«509»","statement":"∀ {M : Type u} [inst : MetricSpace M] (S : Set M),\n  S.Nonempty → ∀ (r : ℝ) (ι : Type v) (bdc : Erdos509.BoundedDiscCover S r ι), 0 < r","subjects":["54"],"theorem":"Erdos509.BoundedDiscCover.bound_nonneg_of_nonempty"},{"answerKinds":[],"category":"research open","docstring":"Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set\n$\\{z ∈ ℂ : |f(z)| ≤ 1\\}$\nbe covered by a set of closed discs the sum of whose radii is $≤ 2$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«509»","statement":"True ↔\n  ∀ (f : Polynomial ℂ),\n    f.Monic → f.natDegree ≠ 0 → ∃ ι, Nonempty (Erdos509.BoundedDiscCover {z | ‖Polynomial.eval z f‖ ≤ 1} 2 ι)","subjects":["30"],"theorem":"Erdos509.erdos_509"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set\n$\\{z ∈ ℂ : |f(z)| ≤ 1\\}$\nbe covered by a set of closed discs the sum of whose radii is $≤ 2e$?\nSolution: True. This is due to Cartan.\nSee *Sur les systèmes de fonctions holomorphes à variétés linéaires\nlacunaires et leurs applications*, Henri Cartan,\nhttp://www.numdam.org/article/ASENS_1928_3_45__255_0.pdf\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«509»","statement":"True ↔\n  ∀ (f : Polynomial ℂ),\n    f.Monic →\n      f.natDegree ≠ 0 → ∃ ι, Nonempty (Erdos509.BoundedDiscCover {z | ‖Polynomial.eval z f‖ ≤ 1} (2 * Real.exp 1) ι)","subjects":["30"],"theorem":"Erdos509.erdos_509.variants.Cartan_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set\n$\\{z ∈ ℂ : |f(z)| ≤ 1\\}$\nbe covered by a set of closed discs the sum of whose radii is $≤ 2.59$?\nSolution: True. This is due to Pommerenke.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«509»","statement":"True ↔\n  ∀ (f : Polynomial ℂ),\n    f.Monic → f.natDegree ≠ 0 → ∃ ι, Nonempty (Erdos509.BoundedDiscCover {z | ‖Polynomial.eval z f‖ ≤ 1} 2.59 ι)","subjects":["30"],"theorem":"Erdos509.erdos_509.variants.Pommerenke_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial.\nIf it is connected, can the set $\\{z ∈ ℂ : |f(z)| ≤ 1\\}$\nbe covered by a set of circles the sum of whose radii is $≤ 2$?\nSolution: True. This is due to Pommerenke.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«509»","statement":"True ↔\n  ∀ (f : Polynomial ℂ),\n    f.Monic →\n      f.natDegree ≠ 0 →\n        IsConnected {z | ‖Polynomial.eval z f‖ ≤ 1} →\n          ∃ ι, Nonempty (Erdos509.BoundedDiscCover {z | ‖Polynomial.eval z f‖ ≤ 1} 2 ι)","subjects":["30"],"theorem":"Erdos509.erdos_509.variants.Pommerenke_connected"},{"answerKinds":[],"category":"test","docstring":"Sanity check: $\\tau^+(12) = 4$. Divisors $1, 2, 3, 4, 6, 12$ lie in dyadic blocks\n$k = 0, 1, 1, 2, 2, 3$, so the distinct blocks are $\\{0, 1, 2, 3\\}$. ($\\tau(12) = 6$.) ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«448»","statement":"Erdos448.tauPlus 12 = 4","subjects":["11"],"theorem":"Erdos448.tauPlus_twelve"},{"answerKinds":[],"category":"research solved","docstring":"Hall and Tenenbaum [HaTe88] further prove that $\\tau^+(n)/\\tau(n)$ has a distribution function:\nthere is a function `F` such that, for every $z$, the set $\\{n : \\tau^+(n)/\\tau(n) \\le z\\}$ has\ndensity `F z`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«448»","statement":"∃ F, ∀ (z : ℝ), {n | ↑(Erdos448.tauPlus n) / ↑n.divisors.card ≤ z}.HasDensity (F z)","subjects":["11"],"theorem":"Erdos448.erdos_448.variants.hall_tenenbaum_distribution"},{"answerKinds":[],"category":"test","docstring":"Sanity check: $\\tau^+(6) = 3$. Divisors $1, 2, 3, 6$ lie in dyadic blocks $k = 0, 1, 1, 2$,\nso the distinct blocks are $\\{0, 1, 2\\}$. ($\\tau(6) = 4$.) ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«448»","statement":"Erdos448.tauPlus 6 = 3","subjects":["11"],"theorem":"Erdos448.tauPlus_six"},{"answerKinds":[],"category":"research solved","docstring":"A more precise result of Hall and Tenenbaum [HaTe88, §4.6]: the upper density of\n$\\{n : \\tau^+(n) < \\epsilon \\cdot \\tau(n)\\}$ is $\\ll \\epsilon \\cdot \\log(2/\\epsilon)$ as\n$\\epsilon \\to 0^+$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«448»","statement":"(fun ε => {n | ↑(Erdos448.tauPlus n) < ε * ↑n.divisors.card}.upperDensity) =O[nhdsWithin 0 (Set.Ioi 0)] fun ε =>\n  ε * Real.log (2 / ε)","subjects":["11"],"theorem":"Erdos448.erdos_448.variants.hall_tenenbaum_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Graham asked whether there is a good inequality for $\\sum_{n \\le x} \\tau^+(n)$. This was\nanswered by Ford [Fo08], who proved\n$$ \\sum_{n \\le x} \\tau^+(n) \\asymp x \\cdot \\frac{(\\log x)^{1 - \\alpha}}{(\\log\\log x)^{3/2}}, $$\nwhere $\\alpha = 1 - \\frac{1 + \\log\\log 2}{\\log 2} = 0.08607\\ldots$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«448»","statement":"(fun x => ∑ n ∈ Finset.Icc 1 x, ↑(Erdos448.tauPlus n)) =Θ[Filter.atTop] fun x =>\n  have α := 1 - (1 + Real.log (Real.log 2)) / Real.log 2;\n  ↑x * Real.log ↑x ^ (1 - α) / Real.log (Real.log ↑x) ^ (3 / 2)","subjects":["11"],"theorem":"Erdos448.erdos_448.variants.ford"},{"answerKinds":[],"category":"research solved","docstring":"Quantitative form of the (negative) answer to `erdos_448`. Erdős and Tenenbaum [ErTe81] showed that\nthe upper density of $\\{n : \\tau^+(n) < \\epsilon \\cdot \\tau(n)\\}$ is in fact\n$\\asymp \\epsilon^{1 - o(1)}$, where the $o(1)$ in the exponent tends to $0$ as $\\epsilon \\to 0$.\nEquivalently, $\\log(\\text{upper density}) / \\log \\epsilon \\to 1$ as $\\epsilon \\to 0^+$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«448»","statement":"Filter.Tendsto (fun ε => Real.log {n | ↑(Erdos448.tauPlus n) < ε * ↑n.divisors.card}.upperDensity / Real.log ε)\n  (nhdsWithin 0 (Set.Ioi 0)) (nhds 1)","subjects":["11"],"theorem":"Erdos448.erdos_448.variants.erdos_tenenbaum"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $\\tau(n)$ count the divisors of $n$ and $\\tau^+(n)$ count the number of $k$ such that $n$ has a\ndivisor in $[2^k, 2^{k+1})$. Is it true that, for all $\\epsilon > 0$,\n$$ \\tau^+(n) < \\epsilon \\cdot \\tau(n) $$\nfor almost all $n$?\n\nThis is false, and was disproved by Erdős and Tenenbaum [ErTe81], who showed that in fact the upper\ndensity of the set of such $n$ is $\\asymp \\epsilon^{1-o(1)}$ (where the $o(1)$ in the exponent\n$\\to 0$ as $\\epsilon \\to 0$). A more precise result was proved by Hall and Tenenbaum [HaTe88]\n(see Section 4.6), who showed that the upper density is $\\ll \\epsilon \\log(2/\\epsilon)$. Hall and\nTenenbaum further prove that $\\tau^+(n)/\\tau(n)$ has a distribution function. Erdős and Graham also\nasked whether there is a good inequality known for $\\sum_{n \\leq x} \\tau^+(n)$. This was provided by\nFord [Fo08] who proved\n$$ \\sum_{n \\leq x} \\tau^+(n) \\asymp x\\frac{(\\log x)^{1-\\alpha}}{(\\log\\log x)^{3/2}} $$\nwhere\n$$ \\alpha = 1-\\frac{1+\\log\\log 2}{\\log 2} = 0.08607\\cdots. $$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«448»","statement":"False ↔ ∀ (ε : ℝ), 0 < ε → {n | ↑(Erdos448.tauPlus n) < ε * ↑n.divisors.card}.HasDensity 1","subjects":["11"],"theorem":"Erdos448.erdos_448"},{"answerKinds":[],"category":"test","docstring":"Always $\\tau^+(n) \\le \\tau(n)$: the occupied dyadic blocks are the image of the divisor set\nunder `Nat.log 2`, and an image has at most as many elements as its source. This is what makes the\n$\\epsilon < 1$ comparison in the problem meaningful. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«448»","statement":"∀ (n : ℕ), Erdos448.tauPlus n ≤ n.divisors.card","subjects":["11"],"theorem":"Erdos448.tauPlus_le_tau"},{"answerKinds":[],"category":"research solved","docstring":"A folklore result states that any $a_n$ satisfying $\\lim_{n \\to \\infty} a_n^{\\frac{1}{2^n}} = \\infty$\nhas $\\sum \\frac{1}{a_n}$ converging to an irrational number.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«263»","statement":"∀ (a : ℕ → ℕ),\n  Filter.Tendsto (fun n => ↑(a n) ^ (1 / 2 ^ n)) Filter.atTop Filter.atTop → Irrational (∑' (n : ℕ), 1 / ↑(a n))","subjects":["11"],"theorem":"Erdos263.erdos_263.variants.folklore"},{"answerKinds":[],"category":"research open","docstring":"Must every irrationality sequence $a_n$ in the above sense\nsatisfy $a_n^{1/n} \\to \\infty$ as $n \\to \\infty$?\n\nNote: this was answered **false** for the *pre-correction* statement, which did not\nrequire monotonicity — the counterexample sequence is not increasing. The problem\nwas corrected on erdosproblems.com on 2026-04-02 to require increasing sequences;\nfor the corrected statement this question is **open**. The earlier formal proof\n(for the pre-correction definition) is preserved at\nhttps://github.com/google-deepmind/formal-conjectures/blob/c8cf651906abe91051cf835d4232ad5648412113/FormalConjectures/ErdosProblems/263.lean#L298\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«263»","statement":"True ↔\n  ∀ (a : ℕ → ℕ),\n    Erdos263.IsIrrationalitySequence a → Filter.Tendsto (fun n => ↑(a n) ^ (1 / ↑n)) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos263.erdos_263.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Kovač and Tao [KoTa24] proved that any strictly increasing sequence $a_n$ such that\n$\\sum \\frac{1}{a_n}$ converges and $\\lim \\frac{a_{n+1}}{a_n^2} = 0$ is not\nan irrationality sequence in the above sense.\n\n[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series.\n         arXiv:2406.17593 (2024).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«263»","statement":"∀ (a : ℕ → ℕ),\n  StrictMono a →\n    (Summable fun n => 1 / ↑(a n)) →\n      Filter.Tendsto (fun n => ↑(a (n + 1)) / ↑(a n) ^ 2) Filter.atTop (nhds 0) → ¬Erdos263.IsIrrationalitySequence a","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos263.erdos_263.variants.sub_doubly_exponential"},{"answerKinds":[],"category":"research solved","docstring":"On the other hand, if there exists some $\\varepsilon > 0$ such that $a_n$ satisfies\n$\\liminf \\frac{a_{n+1}}{a_n^{2+\\varepsilon}} > 0$, then $a_n$ is an irrationality sequence\nby the above folklore result `erdos_263.variants.folklore`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/arex1337/erdos-263-lean/blob/95de79a5cd49050df80e95be6cfc161580830799/Erdos263/Folklore.lean#L700"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«263»","statement":"∀ (a : ℕ → ℕ),\n  (∀ (n : ℕ), a n > 0) →\n    StrictMono a →\n      (∃ ε > 0, Filter.liminf (fun n => ↑(a (n + 1)) / ↑(a n) ^ (2 + ε)) Filter.atTop > 0) →\n        Erdos263.IsIrrationalitySequence a","subjects":["11"],"theorem":"Erdos263.erdos_263.variants.super_doubly_exponential"},{"answerKinds":[],"category":"research solved","docstring":"Koizumi [Ko25] showed that $a_n = \\lfloor \\alpha^{2^n} \\rfloor$ is an irrationality sequence\nfor all but countably many $\\alpha > 1$.\n\n[Ko25] Koizumi, J., Irrationality of the reciprocal sum of doubly exponential sequences,\n       arXiv:2504.05933 (2025).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«263»","statement":"∀ᶠ (α : ℝ) in Filter.cocountable, α > 1 → Erdos263.IsIrrationalitySequence fun n => ⌊α ^ 2 ^ n⌋₊","subjects":["11"],"theorem":"Erdos263.erdos_263.variants.doubly_exponential_all_but_countable"},{"answerKinds":[],"category":"research open","docstring":"Is $a_n = 2^{2^n}$ an irrationality sequence in the above sense?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«263»","statement":"True ↔ Erdos263.IsIrrationalitySequence fun n => 2 ^ 2 ^ n","subjects":["11"],"theorem":"Erdos263.erdos_263.parts.i"},{"answerKinds":[],"category":"research open","docstring":"In [Er79] Erdős says perhaps $s_{n+1} - s_n \\ll \\log s_n$, but he is 'very doubtful'.\n\n[Er79] Erdős, Paul, __Some unconventional problems in number theory__. Math. Mag. (1979), 67-70.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«208»","statement":"(fun n => ↑(Erdos208.erdos208.s (n + 1)) - ↑(Erdos208.erdos208.s n)) =O[Filter.atTop] fun n =>\n  Real.log ↑(Erdos208.erdos208.s n)","subjects":["11"],"theorem":"Erdos208.erdos_208.variants.log_bound"},{"answerKinds":[],"category":"research open","docstring":"Let $s_1 < s_2 < \\dots$ be the sequence of squarefree numbers. Is it true that\nfor any $\\epsilon > 0$ and large $n$, $s_{n+1} - s_n \\ll_\\epsilon s_n^\\epsilon$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«208»","statement":"True ↔\n  ∀ ε > 0,\n    (fun n => ↑(Erdos208.erdos208.s (n + 1)) - ↑(Erdos208.erdos208.s n)) =O[Filter.atTop] fun n =>\n      ↑(Erdos208.erdos208.s n) ^ ε","subjects":["11"],"theorem":"Erdos208.erdos_208.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Let $s_1 < s_2 < \\dots$ be the sequence of squarefree numbers. Is it true that\n$s_{n + 1} - s_n \\le (1 + o(1)) \\cdot (\\pi^2 / 6) \\cdot \\log (s_n) / \\log (\\log (s_n))$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«208»","statement":"True ↔\n  ∃ c,\n    c =o[Filter.atTop] 1 ∧\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ↑(Erdos208.erdos208.s (n + 1)) - ↑(Erdos208.erdos208.s n) ≤\n          (1 + c n) * (Real.pi ^ 2 / 6) * Real.log ↑(Erdos208.erdos208.s n) /\n            Real.log (Real.log ↑(Erdos208.erdos208.s n))","subjects":["11"],"theorem":"Erdos208.erdos_208.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $\\limsup f(n)=\\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«950»","statement":"True ↔ Filter.limsup (fun n => ↑(Erdos950.f n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos950.erdos_950.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Erdős writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that,\nfor every $\\epsilon>0$, if $x$ is sufficiently large there exists $y<x$ such that\n$\\pi(x)< \\pi(y)+\\epsilon \\pi(x-y)$. Compare this to [855].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«950»","statement":"True ↔ ∀ ε > 0, ∀ᶠ (x : ℕ) in Filter.atTop, ∃ y < x, ↑x.primeCounting < ↑y.primeCounting + ε * ↑(x - y).primeCounting","subjects":["11"],"theorem":"Erdos950.erdos_950.variants.weaker_pi"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $\\liminf f(n)=1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«950»","statement":"True ↔ Filter.liminf (fun n => ↑(Erdos950.f n)) Filter.atTop = 1","subjects":["11"],"theorem":"Erdos950.erdos_950.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $f(n)=o(\\log\\log n)$ for all $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«950»","statement":"True ↔ Erdos950.f =o[Filter.atTop] fun n => Real.log (Real.log ↑n)","subjects":["11"],"theorem":"Erdos950.erdos_950.parts.iii"},{"answerKinds":[],"category":"research solved","docstring":"He notes that if $\\pi(x)< \\pi(y)+O\\left(\\frac{x-y}{\\log x}\\right)$ for all $y<x-(\\log x)^C$ for\nsome constant $C>0$ then $f(n)\\ll \\log\\log\\log n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«950»","statement":"(∃ C > 0,\n    ∃ K > 0,\n      ∀ᶠ (x : ℕ) in Filter.atTop,\n        ∀ (y : ℕ), ↑y < ↑x - Real.log ↑x ^ C → ↑x.primeCounting < ↑y.primeCounting + K * ((↑x - ↑y) / Real.log ↑x)) →\n  Erdos950.f =O[Filter.atTop] fun n => Real.log (Real.log (Real.log ↑n))","subjects":["11"],"theorem":"Erdos950.erdos_950.variants.weaker_pi_implies_f"},{"answerKinds":[],"category":"research solved","docstring":"This function was considered by de Bruijn, Erdős, and Turán, who showed that\n$\\sum_{n<x}f(n)\\sim \\sum_{n<x}f(n)^2\\sim x$. They gave no proofs, but a proof of the (harder) second\nclaim is given by Gorodetsky here [mathoverflow/508491].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«950»","statement":"(Asymptotics.IsEquivalent Filter.atTop (fun x => ∑ n ∈ Finset.range x, Erdos950.f n) fun x => ↑x) ∧\n  Asymptotics.IsEquivalent Filter.atTop (fun x => ∑ n ∈ Finset.range x, Erdos950.f n ^ 2) fun x => ↑x","subjects":["11"],"theorem":"Erdos950.erdos_950.variants.debruijn_erdos_turan"},{"answerKinds":[],"category":"research open","docstring":"The study of $f(p)$ is even harder, and Erdős could not prove that\n$\\sum_{p<x}f(p)^2\\sim \\pi(x)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«950»","statement":"True ↔\n  Asymptotics.IsEquivalent Filter.atTop (fun x => ∑ p ∈ Finset.range x with Prime p, Erdos950.f p ^ 2) fun x =>\n    ↑x.primeCounting","subjects":["11"],"theorem":"Erdos950.erdos_950.variants.sum_primes"},{"answerKinds":[],"category":"research solved","docstring":"Straus [Str66] proved that $h(n) \\ll \\sqrt{n}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«789»","statement":"(fun n => ↑(Erdos789.subsetSumThreshold n)) =O[Filter.atTop] fun n => √↑n","subjects":["5"],"theorem":"Erdos789.erdos_789.variants.isBigO_sq"},{"answerKinds":[],"category":"research open","docstring":"By the solved variant `erdos_789.variants.isBigO_sq`, in order to prove\n`erdos_789.variants.sq` it suffices to show $\\sqrt{n}=O(h(n))$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«789»","statement":"(fun n => √↑n) =O[Filter.atTop] fun n => ↑(Erdos789.subsetSumThreshold n)","subjects":["5"],"theorem":"Erdos789.erdos_789.variants.sq_isBigO"},{"answerKinds":[],"category":"research open","docstring":"By the solved variant `erdos_789.variants.cube_root_linearithmic_isBigO`, in order to prove\n`erdos_789.variants.cube_root_linarithmic` it suffices to show $h(n) = O((n\\log(n))^{1/3})$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«789»","statement":"(fun n => ↑(Erdos789.subsetSumThreshold n)) =O[Filter.atTop] fun n => (↑n * Real.log ↑n) ^ (1 / 3)","subjects":["5"],"theorem":"Erdos789.erdos_789.variants.isBigO_cube_root_linearithmic"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er62c] and Choi [Ch74b] proved that $(n\\log(n))^{1/3}\\ll h(n)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«789»","statement":"(fun n => (↑n * Real.log ↑n) ^ (1 / 3)) =O[Filter.atTop] fun n => ↑(Erdos789.subsetSumThreshold n)","subjects":["5"],"theorem":"Erdos789.erdos_789.variants.cube_root_linearithmic_isBigO"},{"answerKinds":[],"category":"research open","docstring":"Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$\nthen there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if\n$a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.\n\nIs $h(n) = \\Theta(\\sqrt{n})$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«789»","statement":"(fun n => ↑(Erdos789.subsetSumThreshold n)) =Θ[Filter.atTop] fun n => √↑n","subjects":["5"],"theorem":"Erdos789.erdos_789.variants.sq"},{"answerKinds":[],"category":"research open","docstring":"Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$\nthen there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if\n$a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.\n\nIs $h(n) = \\Theta((n\\log(n)))^{1/3})$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«789»","statement":"(fun n => ↑(Erdos789.subsetSumThreshold n)) =Θ[Filter.atTop] fun n => (↑n * Real.log ↑n) ^ (1 / 3)","subjects":["5"],"theorem":"Erdos789.erdos_789.variants.cube_root_linearithmic"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$\nthen there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if\n$a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.\n\nEstimate $h(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«789»","statement":"(fun n => ↑(Erdos789.subsetSumThreshold n)) =Θ[Filter.atTop] sorry","subjects":["5"],"theorem":"Erdos789.erdos_789"},{"answerKinds":[],"category":"research solved","docstring":"Steinhaus [St20] has proved Erdős 120 to be false whenever $A$ is a finite set.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«120»","statement":"∀ {A : Set ℝ}, A.Finite → ¬Erdos120.Erdos120For A","subjects":["5","28"],"theorem":"Erdos120.erdos_120.variants.finite_set"},{"answerKinds":[],"category":"research open","docstring":"Let $A \\subseteq \\mathbb{R}$ be an infinite set. Must there be a set $E \\subseteq \\mathbb{R}$\nof positive measure which does not contain any set of the shape $a * A + b$\nfor some $a,b \\in \\mathbb{R}$ and $a \\neq 0$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«120»","statement":"True ↔ ∀ (A : Set ℝ), A.Infinite → Erdos120.Erdos120For A","subjects":["5","28"],"theorem":"Erdos120.erdos_120"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $1<q<1+\\epsilon$ and consider the set of numbers of the shape\n$\\sum_{i\\in S}q^i$ (for all finite $S$), ordered by size as\n$0=x_1<x_2<\\cdots$.\n\nIs it true that, provided $\\epsilon>0$ is sufficiently small, $x_{k+1}-x_k \\to 0$?\n\nThis was solved affirmatively by Erdős and Komornik [ErKo98], who proved the conclusion\nwhenever $1<q<\\sqrt{q_1}$, where $q_1$ is the second Pisot-Vijayaraghavan number.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos1096.lean#L44"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1096»","statement":"True ↔\n  ∃ ε > 0,\n    ∀ (q : ℝ),\n      1 < q →\n        q < 1 + ε →\n          ∀ (x : ℕ → ℝ),\n            StrictMono x →\n              Set.range x = {x | ∃ S, ∑ i ∈ S, q ^ i = x} →\n                Filter.Tendsto (fun k => x (k + 1) - x k) Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Erdos1096.erdos_1096"},{"answerKinds":[],"category":"research open","docstring":"Is it true that for every $k$ there exists $n$ such that\n$$\\prod_{0\\leq i\\leq k}(n-i) \\mid \\binom{2n}{n}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«396»","statement":"True ↔ ∀ (k : ℕ), ∃ n, n.descFactorial (k + 1) ∣ n.centralBinom","subjects":["11"],"theorem":"Erdos396.erdos_396"},{"answerKinds":[],"category":"research solved","docstring":"Nguyen and Vu proved that $|A| \\ll N^{1/3} (\\log N)^{O(1)}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«587»","statement":"∃ O > 0, ∃ O' > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos587.MaxNotSqSum N) ≤ O' * Real.nthRoot 3 ↑N * Real.log ↑N ^ O","subjects":["11"],"theorem":"Erdos587.erdos_587.variants.nguyen_vu"},{"answerKinds":[],"category":"research solved","docstring":"Let `f = ∑ aₖzⁿₖ` be an entire function of finite order such that `nₖ / k → ∞`.\nThen `limsup (fun r => ratio r f) atTop = 1`. This is proved in [Fu63]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«516»","statement":"∀ {f : ℂ → ℂ} {n : ℕ → ℕ},\n  HasFabryGaps n →\n    ∀ {a : ℕ → ℂ},\n      (∀ (n : ℕ), a n ≠ 0) →\n        (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) →\n          Erdos516.OfFiniteOrder f → Filter.limsup (fun r => Erdos516.ratio r f) Filter.atTop = 1","subjects":["30"],"theorem":"Erdos516.erdos_516"},{"answerKinds":[],"category":"research open","docstring":"Is it true that for all entire functions `f = ∑ aₖzⁿₖ` such that `∑' 1 / nₖ < ∞`,\n`limsup (fun r => ratio r f) atTop = 1`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«516»","statement":"True ↔\n  ∀ {f : ℂ → ℂ} {n : ℕ → ℕ},\n    HasFejerGaps n →\n      ∀ {a : ℕ → ℂ},\n        (∀ (n : ℕ), a n ≠ 0) →\n          (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) →\n            Filter.limsup (fun r => Erdos516.ratio r f) Filter.atTop = 1","subjects":["30"],"theorem":"Erdos516.erdos_516.variants.limsup_ratio_eq_one_of_hasFejerGaps"},{"answerKinds":[],"category":"research solved","docstring":"Let `f = ∑ aₖzⁿₖ` be an entire function such that `nₖ > k (log k) ^ (2 + c)`.\nThen `limsup (fun r => ratio r f) atTop = 1`. This is proved in [Ko65]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«516»","statement":"∀ {f : ℂ → ℂ} {n : ℕ → ℕ},\n  (∃ c > 0, ∀ (k : ℕ), ↑(n k) > ↑k * Real.log ↑k ^ (2 + c)) →\n    ∀ {a : ℕ → ℂ},\n      (∀ (n : ℕ), a n ≠ 0) →\n        (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) →\n          Filter.limsup (fun r => Erdos516.ratio r f) Filter.atTop = 1","subjects":["30"],"theorem":"Erdos516.erdos_516.variants.limsup_ratio_eq_one"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Erdős asks in [Er75b] if for every 2-coloring of ℝ, there is an uncountable set $A ⊆ ℝ$ such that\nall sums $a + b$ for $a, b ∈ A, a ≠ b$ have the same colour.\n\nIn [Ko16] Péter Komjáth constructed a counterexample.\nThe same result was proven independently in [SWCol] by Sokoup and Weiss.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos965.lean#L42"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«965»","statement":"False ↔ ∀ (f : ℝ → Fin 2), ∃ A, ¬A.Countable ∧ ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, ∀ d ∈ A, a ≠ b → c ≠ d → f (a + b) = f (c + d)","subjects":["3","5"],"theorem":"Erdos965.erdos_965"},{"answerKinds":["Prop"],"category":"research solved","docstring":"In fact, in both [Ko16] and [SWCol] a generalized example for $k$-sums is constructed.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«965»","statement":"False ↔\n  ∀ k ≥ 2,\n    ∀ (f : ℝ → Fin 2),\n      ∃ A, ¬A.Countable ∧ ∀ (s t : Finset ℝ), ↑s ⊆ A → ↑t ⊆ A → s.card = k → t.card = k → f (s.sum id) = f (t.sum id)","subjects":["3","5"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos965.erdos_965.variants.generalization"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Estimate $\\epsilon_n$ - lower bound.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«688»","statement":"sorry =O[Filter.atTop] Erdos688.epsilonFunction","subjects":["11"],"theorem":"Erdos688.erdos_688.parts.i.lower_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Estimate $\\epsilon_n$ - upper bound.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«688»","statement":"Erdos688.epsilonFunction =O[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos688.erdos_688.parts.i.upper_bound"},{"answerKinds":[],"category":"research open","docstring":"In particular, is it true that $\\epsilon_n = o(1)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«688»","statement":"True ↔ Erdos688.epsilonFunction =o[Filter.atTop] fun n => 1","subjects":["11"],"theorem":"Erdos688.erdos_688.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Erdős claims in [Er80] (p. 106) that it is not difficult to prove\n$\\epsilon_n \\gg \\frac{\\log\\log\\log n}{\\log\\log n}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«688»","statement":"(fun n => Real.log (Real.log (Real.log ↑n)) / Real.log (Real.log ↑n)) =O[Filter.atTop] Erdos688.epsilonFunction","subjects":["11"],"theorem":"Erdos688.erdos_688.variants.lglglg_over_lglg_is_big_o"},{"answerKinds":[],"category":"research solved","docstring":"Achieving density $1/3$ is trivial, taking $A$ to be all multiples of $3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1136»","statement":"Erdos1136.AvoidsPowersOfTwo {n | 3 ∣ n} ∧ {n | 3 ∣ n}.HasDensity (1 / 3)","subjects":["11"],"theorem":"Erdos1136.erdos_1136.variants.multiples_of_three"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does there exist $A\\subset \\mathbb{N}$ with lower density $>1/3$ such that $a+b\\neq 2^k$ for\nany $a,b\\in A$ and $k\\geq 0$?\n\nMüller [Mu11] settled this question in the affirmative: in fact one can take $A$ to be\nthe set of all integers congruent to $3\\cdot 2^i\\pmod{2^{i+2}}$ for any $i\\geq 0$, which has\ndensity $1/2$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1136.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1136»","statement":"True ↔ ∃ A, 1 / 3 < A.lowerDensity ∧ Erdos1136.AvoidsPowersOfTwo A","subjects":["11"],"theorem":"Erdos1136.erdos_1136"},{"answerKinds":[],"category":"research solved","docstring":"Müller [Mu11] settled this question in the affirmative: in fact one can take $A$ to be\nthe set of all integers congruent to $3\\cdot 2^i\\pmod{2^{i+2}}$ for any $i\\geq 0$, which has\ndensity $1/2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1136»","statement":"Erdos1136.AvoidsPowersOfTwo Erdos1136.muellerSet ∧ Erdos1136.muellerSet.HasDensity (1 / 2)","subjects":["11"],"theorem":"Erdos1136.erdos_1136.variants.mueller"},{"answerKinds":[],"category":"research solved","docstring":"Müller also proved this is best possible, in that $A$ with the property in the question has\nlower density at most $1/2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1136»","statement":"∀ (A : Set ℕ), Erdos1136.AvoidsPowersOfTwo A → A.lowerDensity ≤ 1 / 2","subjects":["11"],"theorem":"Erdos1136.erdos_1136.variants.upper_bound"},{"answerKinds":[],"category":"research open","docstring":"Is every odd $n > 1$ the sum of a squarefree number and a power of 2?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«11»","statement":"∀ (n : ℕ), Odd n → 1 < n → ∃ k l, Squarefree k ∧ n = k + 2 ^ l","subjects":["11"],"theorem":"Erdos11.erdos_11"},{"answerKinds":[],"category":"research open","docstring":"Erdős often asked this under the weaker assumption that $n > 1$\nis not divisible by 4.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«11»","statement":"∀ (n : ℕ), ¬4 ∣ n → 1 < n → ∃ k l, Squarefree k ∧ n = k + 2 ^ l","subjects":["11"],"theorem":"Erdos11.erdos_11.variants.not_four_dvd"},{"answerKinds":[],"category":"research solved","docstring":"Every odd $1 < n < 10^7$ is the sum of a squarefree number and a power of 2.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«11»","statement":"∀ (n : ℕ), Odd n → n < 10 ^ 7 → 1 < n → ∃ k l, Squarefree k ∧ n = k + 2 ^ l","subjects":["11"],"theorem":"Erdos11.erdos_11.variants.finite_bound1"},{"answerKinds":[],"category":"research solved","docstring":"Every odd $1 < n < 2^50$ is the sum of a squarefree number and a power of 2.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«11»","statement":"∀ (n : ℕ), Odd n → n < 2 ^ 50 → 1 < n → ∃ k l, Squarefree k ∧ n = k + 2 ^ l","subjects":["11"],"theorem":"Erdos11.erdos_11.variants.finite_bound2"},{"answerKinds":[],"category":"research solved","docstring":"Suppose that every odd $n$ is the sum of a squarefree number and a power of 2. Then the set of primes\n$p$ such that $2 ^ p ≡ 2 \\mod p ^ 2$ is infinite. This is Theorem 1 in [GrSo98].\n[GrSo98] Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«11»","statement":"(∀ (n : ℕ), Odd n → 1 < n → ∃ k l, Squarefree k ∧ n = k + 2 ^ l) → {p | Nat.Prime p ∧ 2 ^ p ≡ 2 [MOD p ^ 2]}.Infinite","subjects":["11"],"theorem":"Erdos11.erdos_11.variants.granville_soundararajan"},{"answerKinds":[],"category":"research open","docstring":"Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«11»","statement":"∀ (n : ℕ), Odd n → 1 < n → ∃ k l m, Squarefree k ∧ n = k + 2 ^ l + 2 ^ m","subjects":["11"],"theorem":"Erdos11.erdos_11.variants.two_pow_two"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for every $\\varepsilon > 0$, $h(N) = \\sqrt N + O_{\\varepsilon}(N^\\varepsilon)$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«30»","statement":"True ↔ ∀ ε > 0, (fun N => ↑(Erdos30.h N) - √↑N) =O[Filter.atTop] fun N => ↑N ^ ε","subjects":["11"],"theorem":"Erdos30.erdos_30"},{"answerKinds":[],"category":"research solved","docstring":"Freud [Fr93] constructed a sequence with density $\\geq 19/36$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«867»","statement":"∀ (ε : ℝ),\n  0 < ε → ∀ᶠ (N : ℕ) in Filter.atTop, ∃ A ⊆ Finset.Icc 1 N, Erdos867.ConsecutiveSumFree A ∧ (19 / 36 - ε) * ↑N ≤ ↑A.card","subjects":["5","11"],"theorem":"Erdos867.erdos_867.variants.freud"},{"answerKinds":[],"category":"research solved","docstring":"Adenwalla has observed that\n$$\\lvert A\\rvert \\leq (\\tfrac{2}{3}+o(1))N.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«867»","statement":"∀ (ε : ℝ),\n  0 < ε → ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, Erdos867.ConsecutiveSumFree A → ↑A.card ≤ (2 / 3 + ε) * ↑N","subjects":["5","11"],"theorem":"Erdos867.erdos_867.variants.adenwalla"},{"answerKinds":[],"category":"research solved","docstring":"The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the\nmaximal size of such an $A$ satisfies\n$$\\frac{13}{24}N -O(1)\\leq \\lvert A\\rvert.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«867»","statement":"∃ C, ∀ (N : ℕ), ∃ A ⊆ Finset.Icc 1 N, Erdos867.ConsecutiveSumFree A ∧ 13 / 24 * ↑N - C ≤ ↑A.card","subjects":["5","11"],"theorem":"Erdos867.erdos_867.variants.coppersmith_phillips_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Taking $A=(N/2,N]\\cap \\mathbb{N}$ shows $\\lvert A\\rvert \\geq N/2-O(1)$ is possible.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«867»","statement":"∃ C, ∀ (N : ℕ), ∃ A ⊆ Finset.Icc 1 N, Erdos867.ConsecutiveSumFree A ∧ ↑N / 2 - C ≤ ↑A.card","subjects":["5","11"],"theorem":"Erdos867.erdos_867.variants.lower_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that if $A=\\{a_1<\\cdots <a_t\\}\\subseteq \\{1,\\ldots,N\\}$ has no solutions to\n$$a_i+a_{i+1}+\\cdots+a_j\\in A$$\nthen\n$$\\lvert A\\rvert \\leq \\frac{N}{2}+O(1)?$$\n\nIn fact this problem is false. Freud [Fr93] constructed a sequence with density $\\geq 19/36$.\nThe current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the\nmaximal size of such an $A$ satisfies\n$$\\frac{13}{24}N -O(1)\\leq \\lvert A\\rvert \\leq \\left(\\frac{2}{3}-\\frac{1}{512}\\right)N+\\log N.$$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos867.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«867»","statement":"False ↔ ∃ C, ∀ (N : ℕ), ∀ A ⊆ Finset.Icc 1 N, Erdos867.ConsecutiveSumFree A → ↑A.card ≤ ↑N / 2 + C","subjects":["5","11"],"theorem":"Erdos867.erdos_867"},{"answerKinds":[],"category":"research solved","docstring":"The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the\nmaximal size of such an $A$ satisfies\n$$\\lvert A\\rvert \\leq \\left(\\frac{2}{3}-\\frac{1}{512}\\right)N+\\log N.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«867»","statement":"∀ᶠ (N : ℕ) in Filter.atTop,\n  ∀ A ⊆ Finset.Icc 1 N, Erdos867.ConsecutiveSumFree A → ↑A.card ≤ (2 / 3 - 1 / 512) * ↑N + Real.log ↑N","subjects":["5","11"],"theorem":"Erdos867.erdos_867.variants.coppersmith_phillips_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Must every additive complement $A$ to the primes satisfy\n$\\liminf_{N \\to \\infty} \\frac{|A \\cap \\{1, \\ldots, N\\}|}{\\log N} > 1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«32»","statement":"∀ (A : Set ℕ),\n  Erdos32.IsAdditiveComplementToPrimes A →\n    1 < Filter.liminf (fun N => ↑{x ∈ Finset.Icc 1 N | x ∈ A}.card / ↑(Real.log ↑N)) Filter.atTop","subjects":["11"],"theorem":"Erdos32.erdos_32.variants.liminf_gt_one"},{"answerKinds":[],"category":"research open","docstring":"Can the bound $O(\\log N)$ be achieved for an additive complement to the primes? [Guy04] writes\nthat Erdős offered \\$50 for the solution.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«32»","statement":"True ↔\n  ∃ A,\n    Erdos32.IsAdditiveComplementToPrimes A ∧\n      (fun N => ↑{x ∈ Finset.Icc 1 N | x ∈ A}.card) =O[Filter.atTop] fun N => Real.log ↑N","subjects":["11"],"theorem":"Erdos32.erdos_32.variants.log_bound"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a set $A \\subseteq \\mathbb{N}$ such that $|A \\cap \\{1, \\ldots, N\\}| = o((\\log N)^2)$\nand every sufficiently large integer can be written as $p + a$ for some prime $p$ and $a \\in A$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«32»","statement":"True ↔\n  ∃ A,\n    Erdos32.IsAdditiveComplementToPrimes A ∧\n      (fun N => ↑{x ∈ Finset.Icc 1 N | x ∈ A}.card) =o[Filter.atTop] fun N => Real.log ↑N ^ 2","subjects":["11"],"theorem":"Erdos32.erdos_32"},{"answerKinds":[],"category":"research solved","docstring":"Ruzsa proved that any additive complement $A$ to the primes must satisfy\n$\\liminf_{N \\to \\infty} \\frac{|A \\cap \\{1, \\ldots, N\\}|}{\\log N} \\geq e^\\gamma$,\nwhere $\\gamma$ is the Euler-Mascheroni constant.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«32»","statement":"∀ (A : Set ℕ),\n  Erdos32.IsAdditiveComplementToPrimes A →\n    ↑(Real.exp Real.eulerMascheroniConstant) ≤\n      Filter.liminf (fun N => ↑{x ∈ Finset.Icc 1 N | x ∈ A}.card / ↑(Real.log ↑N)) Filter.atTop","subjects":["11"],"theorem":"Erdos32.erdos_32.variants.ruzsa"},{"answerKinds":[],"category":"research solved","docstring":"Erdős proved in [Erd54] that there exists an additive complement $A$ to the primes with\n$|A \\cap \\{1, \\ldots, N\\}| = O((\\log N)^2)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«32»","statement":"∃ A,\n  Erdos32.IsAdditiveComplementToPrimes A ∧\n    (fun N => ↑{x ∈ Finset.Icc 1 N | x ∈ A}.card) =O[Filter.atTop] fun N => Real.log ↑N ^ 2","subjects":["11"],"theorem":"Erdos32.erdos_32.variants.log_squared"},{"answerKinds":[],"category":"research open","docstring":"Let $a_1 < a_2 < \\dots$ be a sequence of integers such that\n$\\lim_{n\\to\\infty} \\frac{a_n}{a_{n-1}^2} = 1$ and $\\sum \\frac{1}{a_n} \\in \\mathbb{Q}$.\n\nThen, for all sufficiently large $n \\ge 1$, $a_n = a_{n-1}^2 - a_{n-1} + 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«243»","statement":"∀ (a : ℕ → ℕ),\n  StrictMono a →\n    Filter.Tendsto (fun n => ↑(a n) / ↑(a (n - 1)) ^ 2) Filter.atTop (nhds 1) →\n      (Summable fun x => 1 / ↑(a x)) → ∀ᶠ (n : ℕ) in Filter.atTop, a n = a (n - 1) ^ 2 - a (n - 1) + 1","subjects":["40"],"theorem":"Erdos243.erdos_243"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {a : ℕ}, Erdos304.unitFractionExpressible a 0 = {0}","subjects":["11"],"theorem":"Erdos304.unitFractionExpressible_zero_right"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {a b : ℕ}, 0 ∉ Erdos304.unitFractionExpressible a b ↔ a ≠ 0 ∧ b ≠ 0","subjects":["11"],"theorem":"Erdos304.zero_notMem_unitFractionExpressible"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"Erdos304.smallestCollection 2 15 = 2","subjects":["11"],"theorem":"Erdos304.smallestCollection_two_fifteen"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {a b : ℕ}, Erdos304.smallestCollection a b = 1 → a ∣ b","subjects":["11"],"theorem":"Erdos304.dvd_of_smallestCollection_eq_one"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {a b : ℕ}, Erdos304.smallestCollection a b = 1 → ∃ m, 1 < m ∧ ↑a / ↑b = (↑m)⁻¹","subjects":["11"],"theorem":"Erdos304.eq_one_of_smallestCollection_eq_one"},{"answerKinds":[],"category":"research solved","docstring":"In 1950, Erdős [Er50c] proved the lower bound $$\\log \\log b \\ll N(b)$$.\n[Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \\ldots + {1}/{x_n} =A/B$ egyenlet eg\\'{E}sz sz\\'{A}m\\'{u} megold\\'{A}sairól. Mat. Lapok (1950), 192-210.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«304»","statement":"(fun b => Real.log (Real.log ↑b)) =O[Filter.atTop] fun b => ↑(Erdos304.smallestCollectionTo b)","subjects":["11"],"theorem":"Erdos304.erdos_304.variants.lower_1950"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {a b : ℕ}, a = 0 ∨ b = 0 → Erdos304.unitFractionExpressible a b = {0}","subjects":["11"],"theorem":"Erdos304.unitFractionExpressible_of_zero"},{"answerKinds":[],"category":"research solved","docstring":"In 1985 Vose [Vo85] proved the upper bound $$N(b) \\ll \\sqrt{\\log b}$$.\n[Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«304»","statement":"(fun b => ↑(Erdos304.smallestCollectionTo b)) =O[Filter.atTop] fun b => √(Real.log ↑b)","subjects":["11"],"theorem":"Erdos304.erdos_304.variants.upper_1985"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {a b : ℕ}, 1 ∈ Erdos304.unitFractionExpressible a b → a ∣ b","subjects":["11"],"theorem":"Erdos304.dvd_of_one_mem_unitFractionExpressible"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {a b : ℕ}, a ≠ 0 → b ≠ 0 → (Erdos304.unitFractionExpressible a b).Nonempty → 0 < Erdos304.smallestCollection a b","subjects":["11"],"theorem":"Erdos304.smallestCollection_pos"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {a b : ℕ}, 0 ∈ Erdos304.unitFractionExpressible a b ↔ a = 0 ∨ b = 0","subjects":["11"],"theorem":"Erdos304.zero_mem_unitFractionExpressible_iff"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ (b : ℕ), 1 < b → Erdos304.smallestCollection 1 b = 1","subjects":["11"],"theorem":"Erdos304.smallestCollection_left_one"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $$N(b) \\ll \\log \\log b$$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«304»","statement":"True ↔ (fun b => ↑(Erdos304.smallestCollectionTo b)) =O[Filter.atTop] fun b => Real.log (Real.log ↑b)","subjects":["11"],"theorem":"Erdos304.upper_bound"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {a b : ℕ}, 1 ∈ Erdos304.unitFractionExpressible a b → ∃ m, 1 < m ∧ ↑a / ↑b = (↑m)⁻¹","subjects":["11"],"theorem":"Erdos304.eq_inv_of_one_mem_unitFractionExpressible"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«304»","statement":"∀ {b : ℕ}, Erdos304.unitFractionExpressible 0 b = {0}","subjects":["11"],"theorem":"Erdos304.unitFractionExpressible_zero_left"},{"answerKinds":[],"category":"research solved","docstring":"In 1950, Erdős [Er50c] proved the upper bound $$N(b) \\ll \\log b / \\log \\log b$$.\n[Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \\ldots + {1}/{x_n} =A/B$ egyenlet eg\\'{E}sz sz\\'{A}m\\'{u} megold\\'{A}sairól. Mat. Lapok (1950), 192-210.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«304»","statement":"(fun b => ↑(Erdos304.smallestCollectionTo b)) =O[Filter.atTop] fun b => Real.log ↑b / Real.log (Real.log ↑b)","subjects":["11"],"theorem":"Erdos304.erdos_304.variants.upper_1950"},{"answerKinds":[],"category":"research solved","docstring":"The partial result of Erdős, Hajnal, Sós, and Szemerédi [EHSS83]: the statement of\n`erdos_579` holds whenever the edge-density coefficient exceeds $1/8$. That is, for every\n$\\delta > 1/8$ there is a $c > 0$ such that for all sufficiently large $n$, every\n$K_{2,2,2}$-free graph $G$ on $n$ vertices with at least $\\delta n^2$ edges has an independent\nset of size at least $c n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«579»","statement":"∀ (δ : ℝ),\n  1 / 8 < δ →\n    ∃ c,\n      0 < c ∧\n        ∀ᶠ (n : ℕ) in Filter.atTop,\n          ∀ (G : SimpleGraph (Fin n)),\n            Erdos579.octahedron.Free G → δ * ↑n ^ 2 ≤ ↑G.edgeFinset.card → c * ↑n ≤ ↑G.indepNum","subjects":["5"],"theorem":"Erdos579.erdos_579.variants.ehss_large_delta"},{"answerKinds":[],"category":"research open","docstring":"Let $\\delta > 0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no\n$K_{2,2,2}$ (the octahedron) and at least $\\delta n^2$ edges, must $G$ contain an independent\nset of size $\\gg_\\delta n$?\n\nThis is a problem of Erdős, Hajnal, Sós, and Szemerédi [EHSS83]. It is **open**; they proved\nthe statement for $\\delta > 1/8$ (see `erdos_579.variants.ehss_large_delta`), and the\ndifficulty is to push the edge-density threshold down to an arbitrary $\\delta > 0$.\n\nHere $K_{2,2,2}$ is the complete tripartite graph with all parts of size $2$, encoded as\n`completeMultipartiteGraph (fun _ : Fin 3 => Fin 2)`; \"contains no $K_{2,2,2}$\" is expressed\nvia `SimpleGraph.Free`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«579»","statement":"True ↔\n  ∀ (δ : ℝ),\n    0 < δ →\n      ∃ c,\n        0 < c ∧\n          ∀ᶠ (n : ℕ) in Filter.atTop,\n            ∀ (G : SimpleGraph (Fin n)),\n              Erdos579.octahedron.Free G → δ * ↑n ^ 2 ≤ ↑G.edgeFinset.card → c * ↑n ≤ ↑G.indepNum","subjects":["5"],"theorem":"Erdos579.erdos_579"},{"answerKinds":[],"category":"test","docstring":"Sanity check that the forbidden structure is non-trivial: the octahedron is of course not\noctahedron-free, since it contains a copy of itself.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«579»","statement":"¬Erdos579.octahedron.Free Erdos579.octahedron","subjects":["5"],"theorem":"Erdos579.erdos_579.variants.octahedron_not_free"},{"answerKinds":[],"category":"research open","docstring":"Let $\\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if\n$B\\subseteq A\\in\\mathcal{F}$ then $B\\in \\mathcal{F}$). There exists some element $x$ such that\nwhenever $\\mathcal{F}'\\subseteq \\mathcal{F}$ is an intersecting subfamily we have\n$$\\lvert \\mathcal{F}'\\rvert \\leq \\lvert \\{ A\\in \\mathcal{F} : x\\in A\\}\\rvert.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«701»","statement":"True ↔\n  ∀ {X : Type} [Nonempty X] [Fintype X] (F : Set (Set X)),\n    IsLowerSet F → ∃ x, ∀ F' ⊆ F, F'.Intersecting → Cardinal.mk ↑F' ≤ Cardinal.mk ↑{A | A ∈ F ∧ x ∈ A}","subjects":["5"],"theorem":"Erdos701.erdos_701"},{"answerKinds":[],"category":"research open","docstring":"In particular, there should be infinitely many $n$, but the set of such $n$ should have\ndensity zero. Unfortunately this heuristic is difficult to turn into a proof.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«291»","statement":"Filter.Tendsto (fun N => ↑{n ∈ Finset.Icc 1 N | (Erdos291.a n).gcd (Erdos291.L n) = 1}.card / ↑N) Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Erdos291.erdos_291.variants.shiu_heuristic_density_zero"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that $(a_n,L_n)>1$ occurs for infinitely many $n$?\n\nSteinerberger has observed that the answer to the second question is trivially yes: for example, any\n$n$ which begins with a $2$ in base $3$ has $3\\mid (a_n,L_n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«291»","statement":"True ↔ {n | (Erdos291.a n).gcd (Erdos291.L n) > 1}.Infinite","subjects":["11"],"theorem":"Erdos291.erdos_291.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"This leads to a heuristic prediction (see for example a preprint of Shiu [Sh16]) of\n$\\asymp\\frac{x}{\\log x}$ for the number of $n\\in [1,x]$ such that $(a_n,L_n)=1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«291»","statement":"(fun x => ↑{n ∈ Finset.Icc 1 x | (Erdos291.a n).gcd (Erdos291.L n) = 1}.card) =Θ[Filter.atTop] fun x => ↑x / Real.log ↑x","subjects":["11"],"theorem":"Erdos291.erdos_291.variants.shiu_heuristic_asymptotic"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«291»","statement":"Erdos291.a 1 = 1 ∧ Erdos291.a 2 = 3 ∧ Erdos291.a 3 = 11 ∧ Erdos291.a 4 = 25","subjects":["11"],"theorem":"Erdos291.a_eval"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«291»","statement":"Erdos291.L 1 = 1 ∧ Erdos291.L 2 = 2 ∧ Erdos291.L 3 = 6 ∧ Erdos291.L 4 = 12","subjects":["11"],"theorem":"Erdos291.L_eval"},{"answerKinds":[],"category":"research solved","docstring":"More generally, if the leading digit of $n$ in base $p$ is $p-1$ then $p\\mid (a_n,L_n)$. There is\nin fact a necessary and sufficient condition: a prime $p\\leq n$ divides $(a_n,L_n)$ if and only if\n$p$ divides the numerator of $1+\\cdots+\\frac{1}{k}$, where $k$ is the leading digit of $n$ in base\n$p$. This can be seen by writing $a_n = \\frac{L_n}{1}+\\cdots+\\frac{L_n}{n}$ and observing that the\nright-hand side is congruent to $1+\\cdots+1/k$ modulo $p$. (The previous claim about $p-1$ follows\nimmediately from Wolstenholme's theorem.)\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«291»","statement":"∀ (n p : ℕ),\n  Nat.Prime p →\n    p ≤ n →\n      have k := n / p ^ Nat.log p n;\n      p ∣ (Erdos291.a n).gcd (Erdos291.L n) ↔ ↑p ∣ (∑ i ∈ Finset.Icc 1 k, 1 / ↑i).num","subjects":["11"],"theorem":"Erdos291.erdos_291.variants.steinerberger_generalization"},{"answerKinds":[],"category":"research open","docstring":"Let $n\\geq 1$ and define $L_n$ to be the least common multiple of $\\{1,\\ldots,n\\}$ and $a_n$ by\n$\\sum_{1\\leq k\\leq n}\\frac{1}{k}=\\frac{a_n}{L_n}$.\n\nIs it true that $(a_n,L_n)=1$ occurs for infinitely many $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«291»","statement":"True ↔ {n | (Erdos291.a n).gcd (Erdos291.L n) = 1}.Infinite","subjects":["11"],"theorem":"Erdos291.erdos_291.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Wu and Yan [WuYa22] have proved, conditional on $\\frac{1}{\\log p}$ being linearly independent\nover $\\mathbb{Q}$ for any finite collection of primes $p$ (itself a consequence of Schanuel's\nconjecture), that the set of $n$ for which $(a_n,L_n)>1$ has upper density $1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«291»","statement":"(LinearIndependent ℚ fun p => 1 / Real.log ↑↑p) →\n  Filter.limsup (fun N => ↑↑{n ∈ Finset.Icc 1 N | (Erdos291.a n).gcd (Erdos291.L n) > 1}.card / ↑↑N) Filter.atTop = 1","subjects":["11"],"theorem":"Erdos291.erdos_291.variants.wu_yan"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq \\mathbb{N}$ have positive density. Must there exist distinct $a,b,c\\in A$ such\nthat $[a,b]=c$ (where $[a,b]$ is the least common multiple of $a$ and $b$)?\n\nThis is true, a consequence of the positive solution to [447] by Kleitman [Kl71].\n\nDavenport and Erdős [DaEr36] showed that there must exist an infinite sequence $a_1<a_2\\cdots$\nin $A$ such that $a_i\\mid a_j$ for all $i\\leq j$, under the assumption that the upper logarithmic\ndensity of $A$ is positive.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos487.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«487»","statement":"True ↔ ∀ (A : Set ℕ), A.HasPosDensity → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, a ≠ b ∧ b ≠ c ∧ a ≠ c ∧ a.lcm b = c","subjects":["11"],"theorem":"Erdos487.erdos_487"},{"answerKinds":[],"category":"research open","docstring":"Erdős and Rosenfeld, ask whether $4$ is the best possible $K$ for the infinitude of $n$\nwith (at least) $K$ divisors in $(n^{\\frac{1}{2}}, n^{\\frac{1}{2}} + n^{\\frac{1}{4}})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«887»","statement":"IsGreatest {K | ∃ C > 0, Infinite ↑{n | K ≤ {d ∈ Finset.Ioo ⌊√↑n⌋₊ ⌈√↑n + C * ↑n ^ (1 / 4)⌉₊ | d ∣ n}.card}} 4","subjects":["11"],"theorem":"Erdos887.erdos_887.variants.rosenfeld_4"},{"answerKinds":[],"category":"research open","docstring":"Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then\n$n$ has at most $K$ divisors in $(n^{\\frac{1}{2}}, n^{\\frac{1}{2}} + C n^{\\frac{1}{4}})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«887»","statement":"∃ K, ∀ C > 0, ∀ᶠ (n : ℕ) in Filter.atTop, {d ∈ Finset.Ioo ⌊√↑n⌋₊ ⌈√↑n + C * ↑n ^ (1 / 4)⌉₊ | d ∣ n}.card ≤ K","subjects":["11"],"theorem":"Erdos887.erdos_887.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"A question of Erdős and Rosenfeld, who proved that there are infinitely many $n$ with (at least)\n$4$ divisors in $(n^{\\frac{1}{2}}, n^{\\frac{1}{2}} + cn^{\\frac{1}{4}})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«887»","statement":"∃ C > 0, Infinite ↑{n | 4 ≤ {d ∈ Finset.Ioo ⌊√↑n⌋₊ ⌈√↑n + C * ↑n ^ (1 / 4)⌉₊ | d ∣ n}.card}","subjects":["11"],"theorem":"Erdos887.erdos_887.variants.rosenfeld_infinite"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then\n$n$ has at most $K$ divisors in $(n^{\\frac{1}{2}}, n^{\\frac{1}{2}} + C n^{\\frac{1}{4}})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«887»","statement":"∀ C > 0, ∀ᶠ (n : ℕ) in Filter.atTop, {d ∈ Finset.Ioo ⌊√↑n⌋₊ ⌈√↑n + C * ↑n ^ (1 / 4)⌉₊ | d ∣ n}.card ≤ sorry","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos887.erdos_887.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Let m be a finite cardinal $< \\omega$. Let $α$ be the infinite ordinal $\\omega^{\\omega}$.\nIt was proved by Milnor that any red/blue colouring of the edges of $K_α$ there is either a\nred $K_α$ or a blue $K_3$. A shorter proof was found by Larson [La73]\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«590»","statement":"∀ (m : ℕ), OrdinalCardinalRamsey (Ordinal.omega0 ^ Ordinal.omega0) (Ordinal.omega0 ^ Ordinal.omega0) ↑m","subjects":["3"],"theorem":"Erdos590.erdos_590.variants.finite_cardinal"},{"answerKinds":[],"category":"research solved","docstring":"Let $α$ be the infinite ordinal $\\omega^{\\omega}$. It was proved by Chang [Ch72] that any red/blue\ncolouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«590»","statement":"OrdinalCardinalRamsey (Ordinal.omega0 ^ Ordinal.omega0) (Ordinal.omega0 ^ Ordinal.omega0) 3","subjects":["3"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos590.erdos_590"},{"answerKinds":[],"category":"research solved","docstring":"Specker [Sp57] proved that when $α=ω^n$ for $3≤ n < \\omega$ then it is not the case that any\nred/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«590»","statement":"∀ {n : ℕ}, 3 ≤ n → ¬OrdinalCardinalRamsey (Ordinal.omega0 ^ n) (Ordinal.omega0 ^ n) 3","subjects":["3"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos590.erdos_590.variants.ge_three_false"},{"answerKinds":[],"category":"research solved","docstring":"Specker [Sp57] proved that when $α=ω^2$ any red/blue\ncolouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«590»","statement":"OrdinalCardinalRamsey (Ordinal.omega0 ^ 2) (Ordinal.omega0 ^ 2) 3","subjects":["3"],"theorem":"Erdos590.erdos_590.variants.two"},{"answerKinds":[],"category":"research solved","docstring":"The infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that\nall of its roots are real and contained in `[-1,1]` is `< 1.835`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1038»","statement":"∀ (n : ℕ), ⨅ f, MeasureTheory.volume {x | |Polynomial.eval x ↑f| < 1} < 1.835","subjects":["28"],"theorem":"Erdos1038.erdos_1038.variants.inf_upperBound"},{"answerKinds":[],"category":"research solved","docstring":"The infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that\nall of its roots are real and contained in `[-1,1]` is `≥ 2 ^ (4 / 3) - 1`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1038»","statement":"∀ (n : ℕ), 2 ^ (4 / 3) - 1 ≤ ⨅ f, MeasureTheory.volume {x | |Polynomial.eval x ↑f| < 1}","subjects":["28"],"theorem":"Erdos1038.erdos_1038.varaints.inf_lowerBound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such\nthat all of its roots are real and contained in `[-1,1]`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1038»","statement":"∀ (n : ℕ), sorry = ⨅ f, MeasureTheory.volume {x | |Polynomial.eval x ↑f| < 1}","subjects":["28"],"theorem":"Erdos1038.erdos_1038.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"The supremum of `|{x ∈ ℝ : |f x| < 1}|` over all monic polynomials `f` such that\nall of its roots are real and contained in `[-1,1]` is `2 * 2 ^ (1 / 2)`. This is proved in\n[Tao25]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1038»","statement":"∀ (n : ℕ), 2 * 2 ^ (1 / 2) = ⨆ f, MeasureTheory.volume {x | |Polynomial.eval x ↑f| < 1}","subjects":["28"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos1038.erdos_1038.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"If $0\\leq r\\leq 1/2$ then the component which contains $0$ must have diameter $\\geq 2$, which\n$f(z)=z^n$ shows is best possible.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1048»","statement":"∀ (r : ℝ),\n  0 ≤ r →\n    r ≤ 1 / 2 →\n      ∀ (f : Polynomial ℂ),\n        f.Monic →\n          f.degree ≥ 1 → (∀ z ∈ f.roots, ‖z‖ ≤ r) → 2 ≤ Metric.ediam (connectedComponentIn (Erdos1043.levelSet f) 0)","subjects":["30"],"theorem":"Erdos1048.erdos_1048.variants.diam_ge_two"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $f\\in \\mathbb{C}[x]$ is a monic polynomial with all roots satisfying\n$\\lvert z\\rvert \\leq r$ for some $r<2$, then must\n$$\\{ z: \\lvert f(z)\\rvert <1\\}$$\nhave a connected component with diameter $>2-r$?\n\nA problem of Erdős, Herzog, and Piranian [EHP58].\n\nPommerenke [Po61] proved the answer is no for $r>1$, showing that if $f(z)=z^n-r^n$ then\n$\\{ z: \\lvert f(z)\\rvert \\leq 1\\}$ has $n$ connected components, all with diameter\n$\\to 0$ as $n\\to \\infty$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1048.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1048»","statement":"False ↔\n  ∀ (r : ℝ) (f : Polynomial ℂ),\n    r < 2 →\n      f.Monic →\n        f.degree ≥ 1 →\n          (∀ z ∈ f.roots, ‖z‖ ≤ r) →\n            ∃ z ∈ Erdos1048.openLevelSet f,\n              ENNReal.ofReal (2 - r) < Metric.ediam (connectedComponentIn (Erdos1048.openLevelSet f) z)","subjects":["30"],"theorem":"Erdos1048.erdos_1048"},{"answerKinds":[],"category":"research solved","docstring":"If $1/2<r\\leq \\frac{\\sqrt{5}-1}{2}$ then the component which contains $0$ must have\ndiameter $>1/r$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1048»","statement":"∀ (r : ℝ),\n  1 / 2 < r →\n    r ≤ (√5 - 1) / 2 →\n      ∀ (f : Polynomial ℂ),\n        f.Monic →\n          f.degree ≥ 1 →\n            (∀ z ∈ f.roots, ‖z‖ ≤ r) →\n              ENNReal.ofReal (1 / r) < Metric.ediam (connectedComponentIn (Erdos1043.levelSet f) 0)","subjects":["30"],"theorem":"Erdos1048.erdos_1048.variants.diam_gt_inv_r"},{"answerKinds":[],"category":"research solved","docstring":"Pommerenke [Po61] proved the answer is no for $r>1$, showing that if $f(z)=z^n-r^n$ then\n$\\{ z: \\lvert f(z)\\rvert \\leq 1\\}$ has $n$ connected components, all with diameter\n$\\to 0$ as $n\\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1048»","statement":"∀ (r : ℝ),\n  1 < r →\n    ∀ (n : ℕ),\n      1 ≤ n →\n        ∀ (f : Polynomial ℂ),\n          f = Polynomial.X ^ n - Polynomial.C (↑r ^ n) →\n            {C | ∃ z ∈ Erdos1043.levelSet f, C = connectedComponentIn (Erdos1043.levelSet f) z}.ncard = n","subjects":["30"],"theorem":"Erdos1048.erdos_1048.variants.pommerenke_ncard_components"},{"answerKinds":[],"category":"research solved","docstring":"If $0\\leq r\\leq 1/2$ then the component which contains $0$ must have diameter $\\geq 2$, which\n$f(z)=z^n$ shows is best possible.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1048»","statement":"∀ (n : ℕ), 1 ≤ n → Metric.ediam (connectedComponentIn (Erdos1043.levelSet (Polynomial.X ^ n)) 0) = 2","subjects":["30"],"theorem":"Erdos1048.erdos_1048.variants.diam_ge_two_is_best"},{"answerKinds":[],"category":"research solved","docstring":"On the other hand, if $0<r\\leq 1$, then the answer is yes, as also shown by Pommerenke [Po61].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1048»","statement":"∀ (r : ℝ),\n  0 < r →\n    r ≤ 1 →\n      ∀ (f : Polynomial ℂ),\n        f.Monic →\n          f.degree ≥ 1 →\n            (∀ z ∈ f.roots, ‖z‖ ≤ r) →\n              ∃ z ∈ Erdos1048.openLevelSet f,\n                ENNReal.ofReal (2 - r) < Metric.ediam (connectedComponentIn (Erdos1048.openLevelSet f) z)","subjects":["30"],"theorem":"Erdos1048.erdos_1048.variants.r_le_one"},{"answerKinds":[],"category":"research solved","docstring":"If $\\frac{\\sqrt{5}-1}{2}\\leq r\\leq 1$ then the component which contains $0$ must have\ndiameter $>2-r^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1048»","statement":"∀ (r : ℝ),\n  (√5 - 1) / 2 ≤ r →\n    r ≤ 1 →\n      ∀ (f : Polynomial ℂ),\n        f.Monic →\n          f.degree ≥ 1 →\n            (∀ z ∈ f.roots, ‖z‖ ≤ r) →\n              ENNReal.ofReal (2 - r ^ 2) < Metric.ediam (connectedComponentIn (Erdos1043.levelSet f) 0)","subjects":["30"],"theorem":"Erdos1048.erdos_1048.variants.diam_gt_two_sub_sq"},{"answerKinds":[],"category":"research solved","docstring":"Pommerenke [Po61] proved the answer is no for $r>1$, showing that if $f(z)=z^n-r^n$ then\n$\\{ z: \\lvert f(z)\\rvert \\leq 1\\}$ has $n$ connected components, all with diameter\n$\\to 0$ as $n\\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1048»","statement":"∀ (r : ℝ),\n  1 < r →\n    ∀ (ε : ℝ),\n      0 < ε →\n        ∀ᶠ (n : ℕ) in Filter.atTop,\n          ∀ (f : Polynomial ℂ),\n            f = Polynomial.X ^ n - Polynomial.C (↑r ^ n) →\n              ∀ z ∈ Erdos1043.levelSet f,\n                Metric.ediam (connectedComponentIn (Erdos1043.levelSet f) z) < ENNReal.ofReal ε","subjects":["30"],"theorem":"Erdos1048.erdos_1048.variants.pommerenke_diam_tendsto_zero"},{"answerKinds":[],"category":"research open","docstring":"When $A =\\{a_1 < \\cdots\\}$ corresponds to the set of primes, it is conjectured that the set of\nnumbers $n$ that have representations $$n=\\sum_{u\\leq i\\leq v}a_i$$ has positive upper density.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«358»","statement":"0 < {n | (Erdos358.intervalRepresentations (Nat.nth Nat.Prime) n).Nonempty}.upperDensity","subjects":["5","11"],"theorem":"Erdos358.erdos_358.variants.prime_set_density_representation"},{"answerKinds":[],"category":"research open","docstring":"When $A =\\{a_1 < \\cdots\\}$ corresponds to the set of primes, it is conjectured that the\n$\\limsup$ of the number of representations $$n=\\sum_{u\\leq i\\leq v}a_i$$ is infinite.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«358»","statement":"Filter.limsup (fun n => ↑(Erdos358.f (Nat.nth Nat.Prime) n)) Filter.atTop = ⊤","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos358.erdos_358.variants.prime_set"},{"answerKinds":[],"category":"textbook","docstring":"When $A_n = n$, the function $f$ defined above counts the number of odd divisors of $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«358»","statement":"Erdos358.f id = fun n => {d ∈ n.divisors | Odd d}.card","subjects":["5","11"],"theorem":"Erdos358.f_id"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A=\\{a_1 < \\cdots\\}$ be an infinite sequence of integers. Let $f(n)$ count the number of\nsolutions to $$n=\\sum_{u\\leq i\\leq v}a_i.$$\nIs there such an $A$ for which $f(n)\\to \\infty$ as $n\\to \\infty$?\n\nTao [Ta26] constructed such a sequence with $f(n) \\gg \\log n$ for all sufficiently large $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«358»","statement":"True ↔ ∃ A, StrictMono A ∧ Filter.Tendsto (Erdos358.f A) Filter.atTop Filter.atTop","subjects":["5","11"],"theorem":"Erdos358.erdos_358.parts.i"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that if $A =\\{a_1 < \\cdots\\}$ and $g$ counts the number of representations\n$$n=\\sum_{u\\leq i\\leq v}a_i$$ such that the sum has at least two terms, then for all $n$ we have\n$1 \\leq g(n)$ for sufficiently large $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«358»","statement":"∃ A, StrictMono A ∧ ∀ᶠ (n : ℕ) in Filter.atTop, 1 ≤ Erdos358.g A n","subjects":["5","11"],"theorem":"Erdos358.erdos_358.variants.one_le"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A=\\{a_1 < \\cdots\\}$ be an infinite sequence of integers. Let $f(n)$ count the number of\nsolutions to $$n=\\sum_{u\\leq i\\leq v}a_i.$$\nIs there an $A$ such that $f(n)\\geq 2$ for all large $n$?\n\nThis also follows from Tao's construction with $f(n) \\gg \\log n$ [Ta26].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«358»","statement":"True ↔ ∃ A, StrictMono A ∧ ∀ᶠ (n : ℕ) in Filter.atTop, 2 ≤ Erdos358.f A n","subjects":["5","11"],"theorem":"Erdos358.erdos_358.parts.ii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"This is false, and $c_t<2$ for all $t$: a counterexample is provided by Wood [Wo13b], who\nconstructs, for any $r\\geq 2$, a triangle-free $2$-degenerate $r$-uniform hypergraph with\nchromatic number $3$. A similar counterexample was found independently by KoishiChan in the\ncomments.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1022»","statement":"True ↔ ∀ (t : ℕ) (c : ℝ), Erdos1022.SparseImpliesPropertyB t c → c < 2","subjects":["5"],"theorem":"Erdos1022.erdos_1022.variants.lt_two"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there a constant $c_t$, where $c_t\\to \\infty$ as $t\\to \\infty$, such that if $\\mathcal{F}$\nis a finite family of finite sets, all of size at least $t$, and for every set $X$ there are\n$<c_t\\lvert X\\rvert$ many $A\\in \\mathcal{F}$ with $A\\subseteq X$, then $\\mathcal{F}$ has\nchromatic number $2$ (in other words, has property B)?\n\nThis is false, and $c_t<2$ for all $t$: a counterexample is provided by Wood [Wo13b], who\nconstructs, for any $r\\geq 2$, a triangle-free $2$-degenerate $r$-uniform hypergraph with\nchromatic number $3$. A similar counterexample was found independently by KoishiChan in the\ncomments.\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/latest/ErdosProblems/Erdos1022.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1022»","statement":"False ↔ ∃ c, Filter.Tendsto c Filter.atTop Filter.atTop ∧ ∀ (t : ℕ), Erdos1022.SparseImpliesPropertyB t (c t)","subjects":["5"],"theorem":"Erdos1022.erdos_1022"},{"answerKinds":[],"category":"research solved","docstring":"Erdős originally conjectured, in this language, that $c_2=1$, which he reports in [Er71] was\nproved by Lovász.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1022»","statement":"IsGreatest {c | Erdos1022.SparseImpliesPropertyB 2 c} 1","subjects":["5"],"theorem":"Erdos1022.erdos_1022.variants.lovasz"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subset \\mathbb{R}^2$ be a finite set of size $n$, and let $\\{d_1,\\ldots,d_k\\}$ be the set of\ndistances determined by $A$. Let $f(d)$ be the multiplicity of $d$, that is, the number of\nunordered pairs from $A$ of distance $d$ apart.\n\nIs it true that $k=n-1$ and $\\{f(d_i)\\}=\\{n-1,\\ldots,1\\}$ if and only if $A$ is a set of\nequidistant points on a line or a circle?\n\nErdős conjectured that the answer is no, and other such configurations exist.\n\nThis was proved by Clemen, Dumitrescu, and Liu [CDL25], who observed that equidistant points on a\nshort circular arc on a circle of radius $1$, together with the centre, are also an example.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos958.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«958»","statement":"False ↔\n  ∀ (n : ℕ) (A : Finset (EuclideanSpace ℝ (Fin 2))),\n    A.card = n →\n      (distanceSet A).card = n - 1 ∧ Finset.image (distanceMultiplicity A) (distanceSet A) = Finset.Icc 1 (n - 1) →\n        Erdos958.IsEquidistantOnLine A ∨ Erdos958.IsEquidistantOnCircle A","subjects":["5","52"],"theorem":"Erdos958.erdos_958"},{"answerKinds":[],"category":"research open","docstring":"Erdős Problem 855 (Segal's conjecture): $\\pi(x + y) \\le \\pi(x) + \\pi(y)$\nfor sufficiently large $x, y$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«855»","statement":"True ↔ ∀ᶠ (x : ℕ) (y : ℕ) in Filter.atTop, (x + y).primeCounting ≤ x.primeCounting + y.primeCounting","subjects":["11"],"theorem":"Erdos855.erdos_855"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős Problem 707**: It is false that any finite Sidon set can be embedded in a perfect\ndifferent set modulo some $n$.\n\nAs described in [arxiv/2510.19804], a counterexample is provided in [Ha47], see below.\nThe proof of this has been formalized.\n\nThis was formalized in Lean by Alexeev using ChatGPT.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos707.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«707»","statement":"(∀ (A : Set ℕ), A.Finite → IsSidon A → ∃ B, ∃ n > 0, A ⊆ B ∧ IsPerfectDifferenceSet B n) ↔ False","subjects":["5","11"],"theorem":"Erdos707.erdos_707"},{"answerKinds":[],"category":"research solved","docstring":"Alexeev and Mixon [arxiv/2510.19804] have disproved this conjecture,\nshowing that $\\{1, 2, 4, 8, 13\\}$ cannot be extended to any perfect difference set.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«707»","statement":"∀ (A : Set ℕ), A = {1, 2, 4, 8, 13} → Finite ↑A ∧ IsSidon A ∧ ∀ (B : Set ℕ) (n : ℕ), A ⊆ B → ¬IsPerfectDifferenceSet B n","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.counterexample_mian_chowla"},{"answerKinds":[],"category":"textbook","docstring":"The Singer construction gives perfect difference sets for `n = p^2 + p + 1` where `p` is a\nprime power.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«707»","statement":"∀ (p : ℕ), IsPrimePow p → ∃ B, IsPerfectDifferenceSet B (p ^ 2 + p + 1) ∧ B.ncard = p + 1","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.singer_construction"},{"answerKinds":[],"category":"textbook","docstring":"The set `{1, 2, 4}` is a Sidon set.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«707»","statement":"IsSidon {1, 2, 4}","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.example_sidon_set"},{"answerKinds":[],"category":"textbook","docstring":"For small Sidon sets, we can check the conjecture directly.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«707»","statement":"∀ (A : Set ℕ),\n  A.Finite → A.ncard ≤ 3 → IsSidon A → ∃ B p, IsPrimePow p ∧ A ⊆ B ∧ IsPerfectDifferenceSet B (p ^ 2 + p + 1)","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.small_sidon_sets"},{"answerKinds":[],"category":"textbook","docstring":"The set `{1, 2, 4}` can be embedded in a perfect difference set modulo 7.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«707»","statement":"∃ B, {1, 2, 4} ⊆ B ∧ IsPerfectDifferenceSet B 7","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.example_embedding"},{"answerKinds":[],"category":"research solved","docstring":"Alexeev and Mixon [arxiv/2510.19804] have disproved this conjecture, proving that $\\{1,2,4,8\\}$\ncannot be extended to a perfect difference set modulo $p^2+p+1$\nfor any prime $p$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«707»","statement":"∀ (A : Set ℕ),\n  A = {1, 2, 4, 8} →\n    Finite ↑A ∧ IsSidon A ∧ ∀ (B : Set ℕ) (p : ℕ), Prime p → A ⊆ B → ¬IsPerfectDifferenceSet B (p ^ 2 + p + 1)","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.counterexample_prime"},{"answerKinds":[],"category":"research solved","docstring":"It is false that any finite Sidon set can be embedded in a perfect\ndifference set modulo `p^2 + p + 1` for some prime `p`.\n\nAs described in [arxiv/2510.19804], a counterexample is provided in [Ha47], see below.\nThe proof of this has been formalized.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«707»","statement":"(∀ (A : Set ℕ), A.Finite → IsSidon A → ∃ B p, Nat.Prime p ∧ A ⊆ B ∧ IsPerfectDifferenceSet B (p ^ 2 + p + 1)) ↔ False","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.prime"},{"answerKinds":[],"category":"research solved","docstring":"It is false that any finite Sidon set can be embedded in a perfect\ndifference set modulo `p^2 + p + 1` for some prime power `p`.\n\nAs described in [arxiv/2510.19804], a counterexample is provided in [Ha47], see below.\nThe proof of this has been formalized.\n-","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«707»","statement":"(∀ (A : Set ℕ), A.Finite → IsSidon A → ∃ B p, IsPrimePow p ∧ A ⊆ B ∧ IsPerfectDifferenceSet B (p ^ 2 + p + 1)) ↔ False","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.prime_power"},{"answerKinds":[],"category":"research solved","docstring":"This conjecture was actually first disproved by Hall in 1947 [Ha47], long before Erdős asked\nthis question.\nA counterexample for any modulus from from [Ha47] in the paragraph following Theorem 4.3, where it\nwas given as $\\{-8, -6, 0, 1, 4\\}$, but this can be shifted to natural numbers\nas pointed out in [arxiv/2510.19804].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«707»","statement":"∀ (A : Set ℕ),\n  A = {1, 3, 9, 10, 13} → Finite ↑A ∧ IsSidon A ∧ ∀ (B : Set ℕ) (n : ℕ), A ⊆ B → ¬IsPerfectDifferenceSet B n","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.counterexample_hall"},{"answerKinds":[],"category":"textbook","docstring":"A perfect difference set modulo `n` must have size `≤ √n + 1`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«707»","statement":"∀ (B : Set ℕ) (n : ℕ), IsPerfectDifferenceSet B n → B.ncard ≤ n.sqrt + 1","subjects":["5","11"],"theorem":"Erdos707.erdos_707.variants.perfect_difference_set_size_bound"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Gyárfás and Ruszinkó [EGR98]: there is a constant $c > 0$ such that for every\n$n$ there exist connected triangle-free graphs $G$ on $n$ vertices with\n$h_3(G) \\geq n - c$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«619»","statement":"∃ c > 0,\n  ∀ (n : ℕ), 0 < n → ∃ V x G, Fintype.card V = n ∧ G.Connected ∧ G.CliqueFree 3 ∧ ↑n - c ≤ ↑(Erdos619.minNewEdges 3 G)","subjects":["5"],"theorem":"Erdos619.erdos_619.variants.h_three_lower"},{"answerKinds":[],"category":"test","docstring":"The graph with a single vertex is triangle-free and has diameter `0`, so it needs no\nnew edges to reach diameter at most `4`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«619»","statement":"Erdos619.minNewEdges 4 ⊥ = 0","subjects":["5"],"theorem":"Erdos619.erdos_619.test.minNewEdges_singleton"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Gyárfás and Ruszinkó [EGR98]: $h_3(G) \\leq n$ for every connected triangle-free\ngraph $G$ on $n$ vertices.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«619»","statement":"∀ {V : Type u_1} [inst : Fintype V] (G : SimpleGraph V),\n  G.Connected → G.CliqueFree 3 → Erdos619.minNewEdges 3 G ≤ Fintype.card V","subjects":["5"],"theorem":"Erdos619.erdos_619.variants.h_three_le"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Gyárfás and Ruszinkó [EGR98]: $h_5(G) \\leq \\frac{n-1}{2}$ for every connected\ntriangle-free graph $G$ on $n$ vertices.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«619»","statement":"∀ {V : Type u_1} [inst : Fintype V] (G : SimpleGraph V),\n  G.Connected → G.CliqueFree 3 → ↑(Erdos619.minNewEdges 5 G) ≤ (↑(Fintype.card V) - 1) / 2","subjects":["5"],"theorem":"Erdos619.erdos_619.variants.h_five_le"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Gyárfás and Ruszinkó [EGR98]: every connected triangle-free graph on a finite\nvertex set can be extended, by adding edges, to a triangle-free graph of diameter at\nmost $3$. This shows that the infimum defining `minNewEdges r G` ranges over a nonempty\nset for every `3 ≤ r`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«619»","statement":"∀ {V : Type u_1} [Fintype V] (G : SimpleGraph V),\n  G.Connected → G.CliqueFree 3 → ∃ H, G ≤ H ∧ H.CliqueFree 3 ∧ H.ediam ≤ 3","subjects":["5"],"theorem":"Erdos619.erdos_619.variants.add_edges_diam_three"},{"answerKinds":[],"category":"test","docstring":"A triangle-free graph that already has diameter at most `r` needs no new edges. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«619»","statement":"∀ {V : Type u_1} {r : ℕ} {G : SimpleGraph V}, G.CliqueFree 3 → G.ediam ≤ ↑r → Erdos619.minNewEdges r G = 0","subjects":["5"],"theorem":"Erdos619.erdos_619.test.minNewEdges_eq_zero"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 619** [EGR98, Er99]: For a triangle-free graph $G$ let $h_r(G)$ be the\nsmallest number of edges that need to be added to $G$ so that it has diameter $r$ (while\npreserving the property of being triangle-free). Is it true that there exists a constant\n$c>0$ such that if $G$ is a connected graph on $n$ vertices then $h_4(G)<(1-c)n$?\n\nThe answer is **no**: for every $\\eta>0$ there exist connected triangle-free graphs on\n$n$ vertices with $h_4(G)\\geq(1-\\eta)n$, so no such constant $c$ exists. The original proof\nwas generated by Claude Fable 5; the Lean formalization was sketched by Fable and\nimplemented by GPT 5.5 with Codex (see the linked `formal_proof`).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/b8c7a76f267c29eaa41d1212c211a920be8b05ea/FormalConjectures/ErdosProblems/619.lean#L6009"},{"conditions":[],"kind":"lean4","link":"https://github.com/nick-kuhn/erdos-619/blob/7f65718b8c1019ecc24e6c9a6b04ec4c66a4e26f/Solution.lean#L5869"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«619»","statement":"False ↔\n  ∃ c > 0,\n    ∀ (V : Type) [inst : Fintype V] (G : SimpleGraph V),\n      G.Connected → G.CliqueFree 3 → ↑(Erdos619.minNewEdges 4 G) < (1 - c) * ↑(Fintype.card V)","subjects":["5"],"theorem":"Erdos619.erdos_619"},{"answerKinds":[],"category":"research solved","docstring":"The following theorem is proved in [Ma66]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«996»","statement":"∀ (C : ℝ),\n  0.5 < C →\n    ∀ (f : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)) (n : ℕ → ℕ),\n      IsLacunary n →\n        ((fun k =>\n              (MeasureTheory.eLpNorm (↑↑f - Erdos996.fourierPartial f k) 2\n                  AddCircle.haarAddCircle).toReal) =O[Filter.atTop]\n            fun k => 1 / Real.log (Real.log ↑k) ^ C) →\n          ∀ᵐ (x : AddCircle 1),\n            Filter.Tendsto (fun N => (∑ k ∈ Finset.range N, ↑↑f (n k • x)) / ↑N) Filter.atTop\n              (nhds (∫ (t : AddCircle 1), ↑↑f t ∂AddCircle.haarAddCircle))","subjects":["42"],"theorem":"Erdos996.erdos_996.variants.log2"},{"answerKinds":[],"category":"research open","docstring":"Does there exists a positive constant `C` such that for all `f ∈ L²[0,1]` and all lacunary\nsequences `n`, if `‖f - fₖ‖₂ = O(1 / log log log k ^ C)`, then for almost every `x`,\n`lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«996»","statement":"True ↔\n  ∃ C,\n    0 < C ∧\n      ∀ (f : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)) (n : ℕ → ℕ),\n        IsLacunary n →\n          ((fun k =>\n                (MeasureTheory.eLpNorm (↑↑f - Erdos996.fourierPartial f k) 2\n                    AddCircle.haarAddCircle).toReal) =O[Filter.atTop]\n              fun k => 1 / Real.log (Real.log (Real.log ↑k)) ^ C) →\n            ∀ᵐ (x : AddCircle 1),\n              Filter.Tendsto (fun N => (∑ k ∈ Finset.range N, ↑↑f (n k • x)) / ↑N) Filter.atTop\n                (nhds (∫ (t : AddCircle 1), ↑↑f t ∂AddCircle.haarAddCircle))","subjects":["42"],"subsets":["FC100OpenSet1"],"theorem":"Erdos996.erdos_996"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A=\\{n_1<n_2<\\cdots\\}$ be an infinite sequence of integers, and let $\\phi_A(k)$ count the\nnumber of $1\\leq m\\leq n_k$ such that the fraction $\\frac{m}{n_k}$ does not have denominator $n_j$\nfor $j<k$ when written in lowest form; equivalently,\n$$\n\\frac{n_k}{(m,n_k)}\\neq n_j\n$$\nfor all $1\\leq j<k$.\n\nIs there a sequence $A$ such that\n$$\n\\lim_{N\\to \\infty}\\frac{1}{N}\\sum_{k\\leq N}\\frac{\\phi_A(k)}{n_k}=0?\n$$\n\nThis was solved by Haight [Ha] who proved that such a sequence does exist (contrary to Erdős'\nexpectations).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1000.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1000»","statement":"True ↔ ∃ n, StrictMono n ∧ 0 < n 0 ∧ Filter.Tendsto (Erdos1000.phiAvg n) Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Erdos1000.erdos_1000"},{"answerKinds":[],"category":"research solved","docstring":"The study of $\\phi_A$ was introduced by Cassels [Ca50b], who proved that there exist sequences\nsuch that\n$$\n\\liminf_{N\\to \\infty}\\frac{1}{N}\\sum_{k\\leq N}\\frac{\\phi_A(k)}{n_k}=0.\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1000»","statement":"∃ n, StrictMono n ∧ 0 < n 0 ∧ Filter.liminf (Erdos1000.phiAvg n) Filter.atTop = 0","subjects":["11"],"theorem":"Erdos1000.erdos_1000.variants.liminf_eq_zero"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er64b] proved that the limit of $\\frac{\\phi_A(k)}{n_k}$ as $k\\to \\infty$ cannot be $0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1000»","statement":"∀ (n : ℕ → ℕ),\n  StrictMono n → 0 < n 0 → ¬Filter.Tendsto (fun k => ↑(Erdos1000.phiSeq n k) / ↑(n k)) Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Erdos1000.erdos_1000.variants.not_tendsto_zero"},{"answerKinds":[],"category":"research solved","docstring":"In fact he proved that if $\\liminf \\frac{\\phi_A(k)}{n_k}=0$ then\n$\\limsup \\frac{\\phi_A(k)}{n_k}=1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1000»","statement":"∀ (n : ℕ → ℕ),\n  StrictMono n →\n    0 < n 0 →\n      Filter.liminf (fun k => ↑(Erdos1000.phiSeq n k) / ↑(n k)) Filter.atTop = 0 →\n        Filter.limsup (fun k => ↑(Erdos1000.phiSeq n k) / ↑(n k)) Filter.atTop = 1","subjects":["11"],"theorem":"Erdos1000.erdos_1000.variants.limsup_eq_one"},{"answerKinds":[],"category":"research solved","docstring":"It is trivial that $\\phi_A(k)\\geq \\phi(n_k)$, where $\\phi$ is the Euler totient function.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1000»","statement":"∀ (n : ℕ → ℕ), StrictMono n → ∀ (k : ℕ), (n k).totient ≤ Erdos1000.phiSeq n k","subjects":["11"],"theorem":"Erdos1000.erdos_1000.variants.totient_le"},{"answerKinds":[],"category":"research open","docstring":"Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set\n$A\\subset \\mathbb{Z}$ such that for any $n\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and\n$b\\in \\{ f(k) : k\\in\\mathbb{Z}\\}$ such that $n=a+b$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«477»","statement":"True ↔ ∃ f, 2 ≤ f.degree ∧ ∃ A, ∀ (z : ℤ), ∃! ab, (ab ∈ A ×ˢ Set.range fun x => Polynomial.eval x f) ∧ z = ab.1 + ab.2","subjects":["12"],"theorem":"Erdos477.erdos_477"},{"answerKinds":[],"category":"research open","docstring":"Probably there is no such $A$ for the polynomial $X^k$ for any $k \\ge 2$. This is asked in [Sek59].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«477»","statement":"∀ (k : ℕ),\n  2 ≤ k → ∀ (A : Set ℤ), ∃ z, ¬∃! a, (a ∈ A ×ˢ Set.range fun x => Polynomial.eval x (Polynomial.X ^ k)) ∧ z = a.1 + a.2","subjects":["12"],"theorem":"Erdos477.erdos_477.variants.monomial"},{"answerKinds":[],"category":"research solved","docstring":"There is no such $A$ for any polynomial $f(x) = aX^2 + bX + c$, if $a | b$\nwith $a \\ne 0$ and $b \\ne 0$.\nThis was found be AlphaProof for the specific instance $X^2 - X + 1$ and then generalised.\n ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«477»","statement":"∀ {a b c : ℤ},\n  a ≠ 0 →\n    b ≠ 0 →\n      a ∣ b →\n        have f := a • Polynomial.X ^ 2 + b • Polynomial.X + Polynomial.C c;\n        ∀ (A : Set ℤ), ∃ z, ¬∃! a, (a ∈ A ×ˢ Set.range fun x => Polynomial.eval x f) ∧ z = a.1 + a.2","subjects":["12"],"theorem":"Erdos477.erdos_477.variants.degree_two_dvd_condition_b_ne_zero"},{"answerKinds":[],"category":"research open","docstring":"Probably there is no such $A$ for the polynomial $X^3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«477»","statement":"∀ (A : Set ℤ), ∃ z, ¬∃! a, (a ∈ A ×ˢ Set.range fun x => Polynomial.eval x (Polynomial.X ^ 3)) ∧ z = a.1 + a.2","subjects":["12"],"theorem":"Erdos477.erdos_477.variants.X_pow_three"},{"answerKinds":[],"category":"research solved","docstring":"There is no such $A$ for the polynomial $f(x) = X^2$.\n\nThis is shown in [Sek59].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«477»","statement":"∀ (A : Set ℤ), ∃ z, ¬∃! a, (a ∈ A ×ˢ Set.range fun x => Polynomial.eval x (Polynomial.X ^ 2)) ∧ z = a.1 + a.2","subjects":["12"],"theorem":"Erdos477.erdos_477.variants.S_sq"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $k\\ge 0$. Let $G$ be a graph such that every subgraph $H$ contains an independent set of size\n$\\ge (n-k)/2$, where $n$ is the number of vertices of $H$. Must $G$ be the union of a bipartite\ngraph and $O_k(1)$ many vertices?\n\nProved by Reed [Re99].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«73»","statement":"True ↔\n  ∀ (k : ℕ),\n    ∃ C,\n      ∀ (V : Type) [Fintype V] (G : SimpleGraph V),\n        (∀ (S : Finset V), ∃ I ⊆ S, (SimpleGraph.induce (↑I) G).edgeSet = ∅ ∧ ↑I.card ≥ (↑S.card - ↑k) / 2) →\n          ∃ D, D.card ≤ C ∧ (SimpleGraph.induce (↑D)ᶜ G).Colorable 2","subjects":["5"],"theorem":"Erdos73.erdos_73"},{"answerKinds":["Prop"],"category":"research solved","docstring":"For every fixed $k \\geq 2$, Erdős and Graham conjectured that\n$$g(k,n) \\sim \\frac{n^2}{k-1}.$$\nThe conjecture was proved by Dixmier [Di90]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«433»","statement":"True ↔ ∀ (k : ℕ), 2 ≤ k → Filter.Tendsto (fun n => ↑(Erdos433.g k n) / (↑n ^ 2 / (↑k - 1))) Filter.atTop (nhds 1)","subjects":["11"],"theorem":"Erdos433.erdos_433"},{"answerKinds":[],"category":"research open","docstring":"Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of\n$∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1106»","statement":"True ↔ Filter.Tendsto (fun n => (∏ i ∈ Finset.Icc 1 n, Erdos1106.p i).primeFactors.card) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos1106.erdos_1106.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of\n$∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1106»","statement":"True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, (∏ i ∈ Finset.Icc 1 n, Erdos1106.p i).primeFactors.card > n","subjects":["11"],"theorem":"Erdos1106.erdos_1106.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"`f n - 2 * n = θ (n / log n)`. This is proved in [EGS82]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«390»","statement":"(fun n => ↑(Erdos390.f n) - 2 * ↑n) =Θ[Filter.atTop] fun n => ↑n / Real.log ↑n","subjects":["11"],"theorem":"Erdos390.erdos_390.variants.theta"},{"answerKinds":[],"category":"research open","docstring":"Does there exists a constant `c` such that `f n - 2 * n ~ c * (n / log n)`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«390»","statement":"True ↔ ∃ c, Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Erdos390.f n) - 2 * ↑n) fun n => c * ↑n / Real.log ↑n","subjects":["11"],"theorem":"Erdos390.erdos_390"},{"answerKinds":[],"category":"research solved","docstring":"A solution to erdos_418 was shown by Browkin and Schinzel [BrSc95] by showing that any integer of\nthe form $2^(k + 1)\\cdot 509203$ is not of the form $n - \\phi(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«418»","statement":"{x | ∃ k, 2 ^ (k + 1) * 509203 = x} ⊆ {x | ∃ n, n - n.totient = x}ᶜ","subjects":["11"],"theorem":"Erdos418.erdos_418.variants.soln"},{"answerKinds":[],"category":"research open","docstring":"It is open whether the set of non-cototients has positive density.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«418»","statement":"True ↔ ∃ S, ∃ (_ : S.HasPosDensity), S ⊆ {x | ∃ n, n - n.totient = x}ᶜ","subjects":["11"],"theorem":"Erdos418.erdos_418.variants.density"},{"answerKinds":[],"category":"test","docstring":"A sanity check for the definition: $7$ is a cototient, since $7 = 15 - \\phi(15)$ and $\\phi(15) = 8$.\nIn particular $7$ is not a non-cototient.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«418»","statement":"7 ∈ {x | ∃ n, n - n.totient = x}","subjects":["11"],"theorem":"Erdos418.erdos_418.variants.seven_mem_cototient"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there infinitely many integers not of the form $n - \\phi(n)$?\n\nAsked by Erdős and Sierpiński. Numbers not of the form we call non-cototients.\n\nBrowkin and Schinzel [BrSc95] provided an affirmative answer to this question, proving that any\ninteger of the shape $2^{k}\\cdot 509203$ for $k\\geq 1$ is a non-cototient.\n\nThis is discussed in problem B36 of Guy's collection [Gu04].\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos418.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«418»","statement":"True ↔ {x | ∃ n, n - n.totient = x}ᶜ.Infinite","subjects":["11"],"theorem":"Erdos418.erdos_418"},{"answerKinds":[],"category":"research solved","docstring":"It follows from a slight strengthening of the Goldbach conjecture that every odd number can be\nwritten as $n - \\phi(n)$.\nIn particular, we assume that every even number greater than 6 can be written as the sum of two\n*distinct* primes, in contrast to the usual Goldbach conjecture that every even number greater than\n2 can be written as the sum of two primes.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«418»","statement":"(∀ (n : ℕ), 6 < n → Even n → ∃ p q, p ≠ q ∧ Nat.Prime p ∧ Nat.Prime q ∧ n = p + q) →\n  ∀ (m : ℕ), Odd m → ∃ n, m + n.totient = n","subjects":["11"],"theorem":"Erdos418.erdos_418.variants.conditional"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er73b] has shown that a positive density set of natural numbers cannot be written as\n$\\sigma(n)-n$ (numbers not of this form are called nonaliquot, or sometimes untouchable).\n\nThe density sits in an existential, so `HasPosDensity` is the *stronger* reading: the witness\n`S` is ours to choose, and weakening it to positive lower density would claim less rather than\nmore. That is the opposite of the usual situation for Erdős' \"positive density\", where the\ndensity is a hypothesis or a claim about a fixed set. Whether the nonaliquot numbers themselves\nhave a density is a separate question and is not what this states.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«418»","statement":"∃ S, ∃ (_ : S.HasPosDensity), S ⊆ {x | ∃ n, (ArithmeticFunction.sigma 1) n - n = x}ᶜ","subjects":["11"],"theorem":"Erdos418.erdos_418.variants.sigma"},{"answerKinds":[],"category":"research open","docstring":"The **Odd Noncototient Conjecture**: every non-cototient is even. Equivalently, every odd natural\nnumber is of the form $n - \\phi(n)$ for some $n$.\n\nThis is the unconditional form of `erdos_418.variants.conditional`, which derives the odd case from a\nstrengthening of the Goldbach conjecture. See [Wikipedia: Noncototient].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«418»","statement":"{x | ∃ n, n - n.totient = x}ᶜ ⊆ {k | Even k}","subjects":["11"],"theorem":"Erdos418.erdos_418.variants.odd_noncototient"},{"answerKinds":[],"category":"research open","docstring":"For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r)\ncontains a subgraph of girth ≥ r and chromatic number ≥ k?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«108»","statement":"True ↔\n  ∀ r ≥ 4,\n    ∀ k ≥ 2,\n      ∃ f,\n        ∀ (V : Type u) (G : SimpleGraph V),\n          Nonempty V → ↑f ≤ G.chromaticNumber → ∃ H, H.coe.girth ≥ r ∧ H.coe.chromaticNumber ≥ ↑k","subjects":["5"],"theorem":"Erdos108.erdos_108"},{"answerKinds":[],"category":"research open","docstring":"Does `liminf h n = ∞`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«770»","statement":"True ↔ Filter.liminf Erdos770.h Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos770.erdos_770.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"It is probably true that `h n = 3` for infinitely many `n`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«770»","statement":"{n | Erdos770.h n = 3}.Infinite","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos770.erdos_770.variants.three"},{"answerKinds":[],"category":"textbook","docstring":"For odd `n`, the values of `h n` form an unbounded set.\nThis is described as 'easy to see' in [Er74b]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«770»","statement":"Set.Unbounded (fun x1 x2 => x1 ≤ x2) (ENat.toNat '' Erdos770.h '' Odd)","subjects":["11"],"theorem":"Erdos770.erdos_770.variants.odd_h_unbounded"},{"answerKinds":[],"category":"research open","docstring":"For every prime `p`, does the density of integers with `h n = p` exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«770»","statement":"True ↔ ∀ (p : ℕ), Nat.Prime p → ∃ a, {n | Erdos770.h n = ↑p}.HasDensity a","subjects":["11"],"theorem":"Erdos770.erdos_770.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is it true that if `p` is the greatest prime such that `p - 1 ∣ n` and `p > n ^ ε`, then\n`h n = p`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«770»","statement":"True ↔\n  ∀ ε > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      have p := sSup {m | Nat.Prime m ∧ m - 1 ∣ n};\n      ↑p > ↑n ^ ε → Erdos770.h n = ↑p","subjects":["11"],"theorem":"Erdos770.erdos_770.parts.iii"},{"answerKinds":[],"category":"textbook","docstring":"`n + 1` is prime iff `h n = n + 1`. This is described as 'easy to see' in [Er74b]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«770»","statement":"∀ {n : ℕ}, 2 < n → (Erdos770.h n = ↑n + 1 ↔ Nat.Prime (n + 1))","subjects":["11"],"theorem":"Erdos770.Nat.Prime.h_eq_add_one"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er65b] proved that, for every infinite sequence $x_1, x_2, \\ldots \\in (0, 1)$,\n$A_k \\gg \\log k$ for infinitely many $k$. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"∀ (x : ℕ → ℝ), (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) → ∃ c > 0, ∃ᶠ (k : ℕ) in Filter.atTop, ↑(c * Real.log ↑k) ≤ Erdos987.A x k","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.variants.log_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"**Linear upper bound (tight via Clunie phase tracking).** Tighter than\n`linear_upper_bound_weak`: there exists a sequence $x \\in (0,1)$ with $A_k \\le k + 1$ for all\n$k \\ge 1$, via the (shifted) van der Corput sequence. Whether the $+1$ can be eliminated to\nrecover Clunie's exact $A_k \\le k$ under the strict $\\mathrm{Ioo}\\,0\\,1$ hypothesis is open in\nthis formalization (see `linear_upper_bound`). ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"∃ x, ∃ (_ : ∀ (j : ℕ), x j ∈ Set.Ioo 0 1), ∀ (k : ℕ), 1 ≤ k → Erdos987.A x k ≤ ↑↑k + 1","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.variants.linear_upper_bound_clunie"},{"answerKinds":[],"category":"research solved","docstring":"Liu [Li69] showed that, for any $\\epsilon > 0$, $A_k \\gg k^{1 - \\epsilon}$ infinitely often\nunder the additional assumption that there are only a finite number of distinct points. Clunie\nobserved in the Mathscinet review of [Li69] that under this assumption in fact $A_k = \\infty$\ninfinitely often (the version stated here). ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"∀ (x : ℕ → ℝ), (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) → (Set.range x).Finite → ∃ᶠ (k : ℕ) in Filter.atTop, Erdos987.A x k = ⊤","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.variants.finite_distinct_points"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Question 1:\n\nIs it true that $\\limsup_{k \\to \\infty} A_k = \\infty$?\n\nErdős [Er64b] remarks it is \"easy to see\" that $\\limsup_k \\sup_n |\\sum_{j \\le n} e(k x_j)| = \\infty$.\nErdős [Er65b] later found a \"very easy\" proof that $A_k \\gg \\log k$ for infinitely many $k$.\nClunie [Cl67] proved that $A_k \\gg k^{1/2}$ for infinitely many $k$, which implies the answer is\nyes (Tao independently found a proof). This is Problem 7.21 in [Ha74]. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"True ↔ ∀ (x : ℕ → ℝ), (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) → Filter.limsup (fun k => Erdos987.A x k) Filter.atTop = ⊤","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er64b] remarks it is \"easy to see\" that for every infinite sequence\n$x_1, x_2, \\ldots \\in (0, 1)$,\n$$\\limsup_{k \\to \\infty} \\sup_n \\left\\lvert \\sum_{j \\le n} e(k x_j) \\right\\rvert = \\infty.$$ ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"∀ (x : ℕ → ℝ),\n  (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) →\n    Filter.limsup (fun k => ⨆ n, ↑‖∑ j ∈ Finset.range n, additiveChar (↑k * x j)‖) Filter.atTop = ⊤","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.variants.sup_limsup_infty"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Question 2 (parts.ii)**: Is it possible for $A_k = o(k)$? Yes — there exists a sequence\n$(x_n) \\in (0, 1)$ and a bound $b(k) = o(k)$ with $A x k \\le b k$ eventually. A corollary of\n`sqrt_log_upper_bound` (which gives a $\\sqrt{k \\log k}$ bound) plus the asymptotic\n$\\sqrt{k \\log k} = o(k)$. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"True ↔\n  ∃ x,\n    ∃ (_ : ∀ (j : ℕ), x j ∈ Set.Ioo 0 1),\n      ∃ b, (b =o[Filter.atTop] fun k => ↑k) ∧ ∀ᶠ (k : ℕ) in Filter.atTop, Erdos987.A x k ≤ ↑(b k)","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Clunie [Cl67] proved that, for every infinite sequence $x_1, x_2, \\ldots \\in (0, 1)$,\n$A_k \\gg k^{1/2}$ for infinitely many $k$. (Tao independently found a proof.) ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"∀ (x : ℕ → ℝ), (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) → ∃ c > 0, ∃ᶠ (k : ℕ) in Filter.atTop, ↑(c * √↑k) ≤ Erdos987.A x k","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.variants.sqrt_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"**Linear upper bound (weakened).** A first weakened version of Clunie's `A_k ≤ k`: there\nexists a sequence $x \\in (0,1)$ with $A_k \\le 2k$ for all $k \\ge 1$. The witness is the\n(shifted) van der Corput sequence. For the tighter $A_k \\le k + 1$ bound see\n`linear_upper_bound_clunie`. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"∃ x, ∃ (_ : ∀ (j : ℕ), x j ∈ Set.Ioo 0 1), ∀ (k : ℕ), 1 ≤ k → Erdos987.A x k ≤ ↑(2 * ↑k)","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.variants.linear_upper_bound_weak"},{"answerKinds":[],"category":"research solved","docstring":"Clunie [Cl67] proved that there exists an infinite sequence $\\{z_\\nu\\}$ on the unit circle\nwith $A_\\nu \\le \\nu$ for all $\\nu \\ge 1$. Translating $z_\\nu = e(x_\\nu)$, the natural domain of\n$x_\\nu$ is the half-open unit interval $\\mathrm{Ico}\\,0\\,1$, matching the original\n[Er64b]/[Cl67] statement (any unit complex number is allowed, including $z = 1$, i.e. $x = 0$).\n\n**Note**: erdosproblems.com/987 phrases the problem with the *open* interval\n$x_\\nu \\in (0, 1)$, which excludes $z = 1$ and is strictly stronger than what [Er64b]/[Cl67]\nstate; we align with the original papers here. The shifted-vdc $\\le k + 1$ variant under the\nopen interval is preserved as `linear_upper_bound_clunie`. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"∃ x, ∃ (_ : ∀ (j : ℕ), x j ∈ Set.Ico 0 1), ∀ (k : ℕ), 1 ≤ k → Erdos987.A x k ≤ ↑↑k","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.variants.linear_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"An internal OpenAI model (see [APSSV26b, §3]) proved that there exists an infinite sequence\n$x_1, x_2, \\ldots \\in (0, 1)$ such that\n$\\sup_n \\left\\lvert \\sum_{j \\le n} e(k x_j) \\right\\rvert \\ll (k \\log k)^{1/2}$ for all $k \\ge 1$\n(in particular $A_k \\ll (k \\log k)^{1/2}$).\n\nNote: the bound is restricted to $k \\ge 2$ since $\\log 1 = 0$ would make the RHS vanish at\n$k = 1$, while the LHS $\\|\\sum_{j < n} e(x_j)\\|$ can equal $1$ (e.g. for $n = 1$). ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Marti2203/formal-conjectures/blob/19c63d48acce3099c242b059518c49bf8dc0eab8/FormalConjectures/ErdosProblems/987.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«987»","statement":"∃ x,\n  ∃ (_ : ∀ (j : ℕ), x j ∈ Set.Ioo 0 1),\n    ∃ C, ∃ (_ : 0 < C), ∀ (k n : ℕ), 2 ≤ k → ‖∑ j ∈ Finset.range n, additiveChar (↑k * x j)‖ ≤ C * √(↑k * Real.log ↑k)","subjects":["11","40","42"],"theorem":"Erdos987.erdos_987.variants.sqrt_log_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Plesník [Pl75] proved the bound $|E(G)| < 3n(n-1)/8$ for any diameter-$2$-critical graph\non $n$ vertices.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«742»","statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  Erdos742.IsDiameter2Critical G → ↑G.edgeFinset.card < 3 * ↑(Fintype.card V) * (↑(Fintype.card V) - 1) / 8","subjects":["5"],"theorem":"Erdos742.variants.plesnik_bound"},{"answerKinds":[],"category":"test","docstring":"The complete bipartite graph $K_{a, b}$ has exactly $a \\cdot b$ edges. The bound\n$\\lfloor n^2 / 4 \\rfloor$ in the Murty-Simon conjecture is attained by the balanced\ncase $K_{\\lceil n/2 \\rceil, \\lfloor n/2 \\rfloor}$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«742»","statement":"∀ (a b : ℕ), (completeBipartiteGraph (Fin a) (Fin b)).edgeSet.ncard = a * b","subjects":["5"],"theorem":"Erdos742.complete_bipartite_edge_count"},{"answerKinds":[],"category":"research solved","docstring":"Füredi [Fü92] proved the Murty-Simon conjecture for all sufficiently large $n$, that is, there\nexists $n_0$ such that every diameter-$2$-critical graph on $n \\geq n_0$ vertices has at most\n$\\lfloor n^2 / 4 \\rfloor$ edges.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«742»","statement":"∃ n₀,\n  ∀ (V : Type u_2) [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n    n₀ ≤ Fintype.card V → Erdos742.IsDiameter2Critical G → G.edgeFinset.card ≤ Fintype.card V ^ 2 / 4","subjects":["5"],"theorem":"Erdos742.variants.furedi_bound"},{"answerKinds":[],"category":"research open","docstring":"**Murty-Simon Conjecture**\n\nLet $G$ be a graph on $n$ vertices with diameter $2$ such that deleting any edge increases the\ndiameter. Is it true that $G$ has at most $\\lfloor n^2 / 4 \\rfloor$ edges? Equality is conjectured\nto hold for the complete balanced bipartite graph $K_{\\lceil n/2 \\rceil, \\lfloor n/2 \\rfloor}$.\n\nThe conjecture is resolved up to a finite check: Fan [Fa87] verified it for $n \\leq 24$ and\n$n = 26$, and Füredi [Fü92] proved it for all sufficiently large $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«742»","statement":"True ↔\n  ∀ (V : Type u_2) [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n    Erdos742.IsDiameter2Critical G → G.edgeFinset.card ≤ Fintype.card V ^ 2 / 4","subjects":["5"],"theorem":"Erdos742.erdos_742"},{"answerKinds":[],"category":"research solved","docstring":"Fan [Fa87] verified the Murty-Simon conjecture for all $n \\leq 24$ and for $n = 26$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«742»","statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  Fintype.card V ≤ 24 ∨ Fintype.card V = 26 →\n    Erdos742.IsDiameter2Critical G → G.edgeFinset.card ≤ Fintype.card V ^ 2 / 4","subjects":["5"],"theorem":"Erdos742.variants.fan_bound"},{"answerKinds":[],"category":"research solved","docstring":"Erdős proved the answer is yes under the stronger condition that\n$\\limsup \\frac{n_k}{k^t} = \\infty$ for all $t\\geq 1$.\n\n[ErGr80] Erdős, P. and Graham, R.,\n_Old and new problems and results in combinatorial number theory_.\nMonographies de L'Enseignement Mathematique (1980).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«247»","statement":"∀ (n : ℕ → ℕ),\n  StrictMono n →\n    (∀ t ≥ 1, Filter.limsup (fun k => ↑(↑(n k) / ↑k.succ ^ t)) Filter.atTop = ⊤) →\n      Transcendental ℚ (∑' (k : ℕ), 1 / 2 ^ n k)","subjects":["11"],"theorem":"Erdos247.erdos_247.variants.strong_condition"},{"answerKinds":[],"category":"research open","docstring":"Let $n_1 < n_2 < \\cdots$ be a sequence of integers such that\n$$\n  \\limsup \\frac{n_k}{k} = \\infty.\n$$\nIs\n$$\n  \\sum_{k=1}^{\\infty} \\frac{1}{2^{n_k}}\n$$\ntranscendental?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«247»","statement":"True ↔\n  ∀ (n : ℕ → ℕ),\n    StrictMono n →\n      Filter.limsup (fun k => ↑(n k) / ↑k.succ) Filter.atTop = ⊤ → Transcendental ℚ (∑' (k : ℕ), 1 / 2 ^ n k)","subjects":["11"],"theorem":"Erdos247.erdos_247"},{"answerKinds":[],"category":"research open","docstring":"Let $\\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}$ has\nsize at least $\\epsilon N$ then there must be distinct $a,b,c\\in A$ such that\n$$[a, b]=[b, c]=[a, c],$$\nwhere $[\\cdot, \\cdot]$ denotes the least common multiple?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«536»","statement":"True ↔\n  ∀ ε > 0,\n    ∀ᶠ (N : ℕ) in Filter.atTop,\n      ∀ A ⊆ Finset.Icc 1 N,\n        ε * ↑N ≤ ↑A.card → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, {a, b, c}.card = 3 ∧ a.lcm b = b.lcm c ∧ b.lcm c = a.lcm c","subjects":["11"],"theorem":"Erdos536.erdos_536"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for all sufficiently large $n$, there exists some $k$ such that\n$$\np(n+k)>k^2+1,\n$$\nwhere $p(m)$ denotes the least prime factor of $m$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«680»","statement":"True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k, k ≠ 0 ∧ (n + k).minFac > k ^ 2 + 1","subjects":["11"],"theorem":"Erdos680.erdos_680.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Can one prove this is false if we replace $k^2+1$ by $e^{(1+\\epsilon)\\sqrt{k}}+C_\\epsilon$, for all\n$\\epsilon>0$, where $C_\\epsilon>0$ is some constant?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«680»","statement":"True ↔ ∀ ε > 0, ∃ C > 0, ¬∀ᶠ (n : ℕ) in Filter.atTop, ∃ k, k ≠ 0 ∧ ↑(n + k).minFac > Real.exp ((1 + ε) * √↑k) + C","subjects":["11"],"theorem":"Erdos680.erdos_680.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ be a graph such that $R(G,T_n)\\ll n$ for any tree $T_n$ on $n$ vertices and\n$R(G,K_n)\\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices,\n$$R(G,H)\\ll m?$$\n\nIn other words, is $G$ Ramsey size linear?\n\nThis problem is #33 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«568»","statement":"True ↔\n  ∀ (V : Type) [inst : Fintype V] (G : SimpleGraph V),\n    (∃ c₁ > 0, ∀ (n : ℕ) (T : SimpleGraph (Fin n)), T.IsTree → ↑(G.graphRamsey T) ≤ c₁ * ↑n) →\n      (∃ c₂ > 0, ∀ (n : ℕ), ↑(G.graphRamsey (SimpleGraph.completeGraph (Fin n))) ≤ c₂ * ↑n ^ 2) → G.IsRamseySizeLinear","subjects":["5"],"theorem":"Erdos568.erdos_568"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«350»","statement":"Erdos350.DistinctSubsetSums {1, 2}","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos350.distinctSubsetSums_1_2"},{"answerKinds":[],"category":"API","docstring":"Small sanity check: the two predicates are saying the same thing. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«350»","statement":"∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : DecidableEq M] (A : Finset M),\n  Erdos350.DistinctSubsetSums ↑A ↔ Erdos350.DecidableDistinctSubsetSums A","subjects":["5","11"],"theorem":"Erdos350.DistinctSubsetSums_iff_DecidableDistinctSubsetSums"},{"answerKinds":[],"category":"research solved","docstring":"If `A ⊂ ℕ` is a finite set of integers all of whose subset sums are distinct then `∑ n ∈ A, 1/n^s < 1/(1 - 2^(-s))`, for any `s > 0`.\nProved by Hanson, Steele, and Stenger [HSS77].\n\nWe exclude here the case `s = 0`, because in the informal formulation then the right hand side is to be interpreted as `∞`, while the left hand side counts the elements in `A`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«350»","statement":"∀ (A : Finset ℕ), Erdos350.DecidableDistinctSubsetSums A → ∀ (s : ℝ), 0 < s → ∑ n ∈ A, (1 / ↑n) ^ s < 1 / (1 - 2 ^ (-s))","subjects":["5","11"],"theorem":"Erdos350.erdos_350.variants.strengthening"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«350»","statement":"Erdos350.DecidableDistinctSubsetSums {1, 2}","subjects":["5","11"],"theorem":"Erdos350.decidableDistinctSubsetSums_1_2"},{"answerKinds":[],"category":"research solved","docstring":"If `A ⊂ ℕ` is a finite set of integers all of whose subset sums are distinct then `∑ n ∈ A, 1/n < 2`.\nProved by Ryavec.\n\nThis was proved by Ryavec, who did not appear to ever publish the proof. Ryavec's proof is\nreproduced in [BeEr74]. More generally, Ryavec's proof delivers that\n$\\sum_{n\\in A}\\frac{1}{n}\\leq 2-2^{1-\\lvert A\\rvert},$ with equality if and only if\n$A=\\{1,2,\\ldots,2^k\\}$.\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/ba788c9124b563bce98a3413d474b3a2731fd0af/FormalConjectures/ErdosProblems/350.lean#L226"},{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos350.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«350»","statement":"∀ (A : Finset ℕ), Erdos350.DecidableDistinctSubsetSums A → ∑ n ∈ A, 1 / ↑n < 2","subjects":["5","11"],"theorem":"Erdos350.erdos_350"},{"answerKinds":[],"category":"research open","docstring":"Prove that there exists some $c>0$ such that\n$$h(n) \\sim c \\left(\\frac{n}{\\log n}\\right)^{1/2}$$\nas $n\\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«912»","statement":"∃ c > 0, Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Erdos912.h n)) fun n => c * (↑n / Real.log ↑n) ^ (1 / 2)","subjects":["11"],"theorem":"Erdos912.erdos_912"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Selfridge prove in [Er82c] that $h(n) \\asymp \\left(\\frac{n}{\\log n}\\right)^{1/2}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«912»","statement":"(fun n => ↑(Erdos912.h n)) =Θ[Filter.atTop] fun n => (↑n / Real.log ↑n) ^ (1 / 2)","subjects":["11"],"theorem":"Erdos912.erdos_912.variants.selfridge"},{"answerKinds":[],"category":"research open","docstring":"A heuristic of Tao using the Cramér model for the primes suggests this is true with\n$c=\\sqrt{2\\pi}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«912»","statement":"Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Erdos912.h n)) fun n => √(2 * Real.pi) * (↑n / Real.log ↑n) ^ (1 / 2)","subjects":["11"],"theorem":"Erdos912.erdos_912.variants.tao"},{"answerKinds":[],"category":"API","docstring":"Roughness criterion (sufficiency for the anchor conditions): if $a < s$ for all $s$ in\nthe leg and the leg stays below $a + P^-(a)$, then $a$ is coprime to the whole leg. Stated\nvia divisibility: no prime factor of $a$ divides any $s$ with $a < s < a + p$ for all\nprime factors $p$ of $a$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1212»","statement":"∀ {a s : ℕ}, a < s → (∀ (p : ℕ), Nat.Prime p → p ∣ a → s < a + p) → a.gcd s = 1","subjects":["11"],"theorem":"Erdos1212.anchor_coprime_of_short_leg"},{"answerKinds":[],"category":"API","docstring":"Isolation lemma, right neighbour (core of the no-periodic-certificate theorem): if every\nprime in $P$ divides $x$ and none divides $y$, then no prime of $P$ divides either\ncoordinate of $(x+1, y)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1212»","statement":"∀ {P : Finset ℕ} {x y : ℕ}, (∀ p ∈ P, Nat.Prime p) → (∀ p ∈ P, p ∣ x) → (∀ p ∈ P, ¬p ∣ y) → ∀ p ∈ P, ¬p ∣ x + 1 ∧ ¬p ∣ y","subjects":["11"],"theorem":"Erdos1212.right_neighbor_witness_free"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ be the graph with vertex set those pairs $(x,y)\\in \\mathbb{N}^2$ with\n$\\mathrm{gcd}(x,y)=1$, in which we join two vertices if the differ in only one coordinate, and\nthere by $\\pm 1$.\n\nIs there a path going to infinity on $G$, say $P$, such that for all $(x,y)\\in P$ both\n$\\min(x,y)>1$ and at least one of $x$ or $y$ is composite?\n\nThe weaker version (only $\\min(x,y) > 1$) was solved by C. Stewart via the prime-pair path\n$(p_k, p_{k+1}) \\to (p_{k+1}, p_{k+2})$, as recounted in [Er80]; the compositeness condition\nforbids those anchors and the question is open.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1212»","statement":"True ↔\n  ∃ f,\n    Function.Injective f ∧\n      (∀ (n : ℕ), Erdos1212.Adj (f n) (f (n + 1))) ∧\n        (∀ (n : ℕ), Erdos1212.Valid (f n)) ∧ Filter.Tendsto (fun n => (f n).1 + (f n).2) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos1212.erdos_1212"},{"answerKinds":[],"category":"API","docstring":"Core of the composite-anchor reduction: horizontal-leg vertices $(s, c)$ for\n$a \\le s \\le b$ are valid, given the anchor $c$ is composite and coprime to the whole leg. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1212»","statement":"∀ {a b c : ℕ},\n  c.Composite → 2 ≤ a → (∀ (s : ℕ), a ≤ s → s ≤ b → s.gcd c = 1) → ∀ (s : ℕ), a ≤ s → s ≤ b → Erdos1212.Valid (s, c)","subjects":["11"],"theorem":"Erdos1212.horizontal_leg_valid"},{"answerKinds":[],"category":"API","docstring":"Isolation lemma, left neighbour. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1212»","statement":"∀ {P : Finset ℕ} {x y : ℕ},\n  (∀ p ∈ P, Nat.Prime p) → 1 ≤ x → (∀ p ∈ P, p ∣ x) → (∀ p ∈ P, ¬p ∣ y) → ∀ p ∈ P, ¬p ∣ x - 1 ∧ ¬p ∣ y","subjects":["11"],"theorem":"Erdos1212.left_neighbor_witness_free"},{"answerKinds":[],"category":"API","docstring":"Core of the composite-anchor reduction: vertical-leg vertices $(a, s)$ for\n$b \\le s \\le c$ are valid vertices of the strengthened problem, given the anchor $a$ is\ncomposite and coprime to the whole leg. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1212»","statement":"∀ {a b c : ℕ},\n  a.Composite → 2 ≤ b → (∀ (s : ℕ), b ≤ s → s ≤ c → a.gcd s = 1) → ∀ (s : ℕ), b ≤ s → s ≤ c → Erdos1212.Valid (a, s)","subjects":["11"],"theorem":"Erdos1212.vertical_leg_valid"},{"answerKinds":[],"category":"API","docstring":"Isolation lemma, vertical neighbours: both coordinates even. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1212»","statement":"∀ {x y : ℕ}, 2 ∣ x → ¬2 ∣ y → (2 ∣ x ∧ 2 ∣ y + 1) ∧ (1 ≤ y → 2 ∣ x ∧ 2 ∣ y - 1)","subjects":["11"],"theorem":"Erdos1212.vertical_neighbor_both_even"},{"answerKinds":[],"category":"test","docstring":"Sanity check for `Valid`: the vertex $(4, 3)$ is valid — both coordinates exceed $1$,\nthey are coprime, and $4$ is composite. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1212»","statement":"Erdos1212.Valid (4, 3)","subjects":["11"],"theorem":"Erdos1212.valid_four_three"},{"answerKinds":[],"category":"research solved","docstring":"Is $\\sum_{n} \\mu(n)^2\\frac{n}{2^n}$ irrational?\n\nThis is true, and was proved by Chen and Ruzsa.\n\n[ChRu99] Chen, Yong-Gao and Ruzsa, Imre Z., On the irrationality of certain series. Period. Math. Hungar. (1999), 31--37.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://gist.githubusercontent.com/ster-oc/c7429943f6b3a634797dc8b2a3b01f2d/raw/8c6b5b7f08021f0aed2312542dd2e9ee7beaa6d6/Erdos259.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«259»","statement":"Irrational (∑' (n : ℕ), ↑(ArithmeticFunction.moebius n) ^ 2 * ↑n / 2 ^ n)","subjects":["11"],"theorem":"Erdos259.erdos_259"},{"answerKinds":[],"category":"research solved","docstring":"There are examples where $(n, m) ∈ S$ with $m ≠ n + 1$.\n\n(Found by AlphaProof, although it was implicit already in [A129515])\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«730»","statement":"∃ n m, (n, m) ∈ Erdos730.S ∧ m ≠ n + 1","subjects":["11"],"theorem":"Erdos730.erdos_730.variants.delta_ne_one"},{"answerKinds":[],"category":"textbook","docstring":"For example, $(87,88)$ and $(607,608)$ are such pairs.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«730»","statement":"{(87, 88), (607, 608)} ⊆ Erdos730.S","subjects":["11"],"theorem":"Erdos730.erdos_730.variants.explicit_pairs"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many pairs of integers $n < m$ such that $\\binom{2n}{n}$\nand $\\binom{2m}{m}$ have the same set of prime divisors?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«730»","statement":"True ↔ Erdos730.S.Infinite","subjects":["11"],"theorem":"Erdos730.erdos_730"},{"answerKinds":[],"category":"research open","docstring":"Is it true that in any finite colouring of $\\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums\nand products of distinct elements in $A$ are the same colour?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«172»","statement":"True ↔\n  ∀ (n : ℕ) (color : ℕ → Fin n) (m : ℕ),\n    ∃ A, A.card ≥ m ∧ ∃ c, ∀ (S : Finset ↥A), S.Nonempty → color (∑ x ∈ S, ↑x) = c ∧ color (∏ x ∈ S, ↑x) = c","subjects":["5"],"theorem":"Erdos172.erdos_172"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1 < n_2 < \\dots < n_k$ with\n$$\n  1 = \\frac{1}{n_1} + \\cdots + \\frac{1}{n_k}.\n$$\nIs it true that\n$$\n  f(k) = (1 + o(1)) \\frac{e}{e - 1} k ?\n$$\n\nProved by Martin [Ma00].\n\n[Ma00] Martin, Greg, _Denser Egyptian fractions_. Acta Arith. (2000), 231-260.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«285»","statement":"True ↔\n  ∀ (f : ℕ → ℕ) (S : Set ℕ),\n    S = {k | ∃ n, StrictMono n ∧ 0 ∉ Set.range n ∧ 1 = ∑ i, 1 / ↑(n i)} →\n      (∀ k ∈ S,\n          IsLeast {x | ∃ n, ∃ (_ : StrictMono n) (_ : 0 ∉ Set.range n) (_ : 1 = ∑ i, 1 / ↑(n i)), n (Fin.last k) = x}\n            (f k)) →\n        ∃ o, ∃ (_ : o =o[Filter.atTop] 1), ∀ k ∈ S, ↑(f k) = (1 + o k) * Real.exp 1 / (Real.exp 1 - 1) * (↑k + 1)","subjects":["5","11"],"theorem":"Erdos285.erdos_285"},{"answerKinds":[],"category":"research solved","docstring":"It is trivial that $f(k)\\geq (1 + o(1)) \\frac{e}{e - 1}k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«285»","statement":"∀ (f : ℕ → ℕ) (S : Set ℕ),\n  S = {k | ∃ n, StrictMono n ∧ 0 ∉ Set.range n ∧ 1 = ∑ i, 1 / ↑(n i)} →\n    (∀ k ∈ S,\n        IsLeast {x | ∃ n, ∃ (_ : StrictMono n) (_ : 0 ∉ Set.range n) (_ : 1 = ∑ i, 1 / ↑(n i)), n (Fin.last k) = x}\n          (f k)) →\n      ∃ o, ∃ (_ : o =o[Filter.atTop] 1), ∀ k ∈ S, (1 + o k) * Real.exp 1 / (Real.exp 1 - 1) * (↑k + 1) ≤ ↑(f k)","subjects":["5","11"],"theorem":"Erdos285.erdos_285.variants.lb"},{"answerKinds":[],"category":"test","docstring":"$97$ is the smallest prime that is not a cluster prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«17»","statement":"IsLeast {p | Nat.Prime p ∧ ¬Erdos17.IsClusterPrime p} 97","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos17.isClusterPrime_97_isLeast_non_cluster"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 17.** Are there infinitely many cluster primes? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«17»","statement":"True ↔ {p | Erdos17.IsClusterPrime p}.Infinite","subjects":["11"],"theorem":"Erdos17.erdos_17"},{"answerKinds":[],"category":"research solved","docstring":"In 2003, Elsholtz [El03] refined the upper bound to\n$$\\pi^{\\mathcal{C}}(x) \\ll x\\,\\exp\\!\\bigl(-c(\\log\\log x)^2\\bigr)$$\nfor every real $0 < c < 1/8$.\n\n[El03] Elsholtz, Christian, On cluster primes. Acta Arith. (2003), 281--284.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«17»","statement":"∃ C,\n  0 < C ∧\n    ∀ c ∈ Set.Ioo 0 (1 / 8),\n      Asymptotics.IsBigOWith C Filter.atTop (fun x => ↑(Erdos17.clusterPrimeCount x)) fun x =>\n        ↑x * Real.exp (-c * Real.log (Real.log ↑x) ^ 2)","subjects":["11"],"theorem":"Erdos17.erdos_17.variants.upper_Elsholtz"},{"answerKinds":[],"category":"research solved","docstring":"In 1999 Blecksmith, Erdős, and Selfridge [BES99] proved the upper bound\n$$\\pi^{\\mathcal{C}}(x) \\ll_A x(\\log x)^{-A}$$ for every real $A > 0$.\n\n[BES99] Blecksmith, Richard and Erd\\H os, Paul and Selfridge, J. L., Cluster primes. Amer. Math. Monthly (1999), 43--48.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«17»","statement":"∀ {A : ℝ}, 0 < A → (fun x => ↑(Erdos17.clusterPrimeCount x)) =O[Filter.atTop] fun x => ↑x / Real.log ↑x ^ A","subjects":["11"],"theorem":"Erdos17.erdos_17.variants.upper_BES"},{"answerKinds":[],"category":"research open","docstring":"Does $\\{1,2^3,\\ldots,N^3\\}$ contain a Sidon set of size $\\gg N$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1206»","statement":"True ↔\n  ∃ c,\n    0 < c ∧\n      ∀ᶠ (N : ℕ) in Filter.atTop, ∃ S ⊆ Finset.image (fun n => n ^ 3) (Finset.Icc 1 N), IsSidon ↑S ∧ c * ↑N ≤ ↑S.card","subjects":["5","11"],"theorem":"Erdos1206.erdos_1206.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is there an infinite set $A\\subset \\mathbb{N}$ of positive density such that $\\{a^3 : a\\in A\\}$ is a Sidon set?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1206»","statement":"True ↔ ∃ A, A.Infinite ∧ 0 < A.lowerDensity ∧ IsSidon ((fun a => a ^ 3) '' A)","subjects":["5","11"],"theorem":"Erdos1206.erdos_1206.parts.ii"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset\nof $\\{1, \\dots, N\\}$ that does not contain any non-trivial $k$-term arithmetic progression.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«142»","statement":"∀ (k : ℕ), (fun N => ↑(Erdos142.r k N)) =Θ[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos142.erdos_142"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Find functions $f_k$, such that $r_k(N) = O_k(f_k)$, where $r_k(N)$ the largest possible size of a\nsubset of $\\{1, \\dots, N\\}$ that does not contain any non-trivial $k$-term arithmetic progression.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«142»","statement":"∀ (k : ℕ), (fun N => ↑(Erdos142.r k N)) =O[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos142.erdos_142.variants.upper"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset\nof $\\{1, \\dots, N\\}$ that does not contain any non-trivial $3$-term arithmetic progression.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«142»","statement":"(fun N => ↑(Erdos142.r 3 N)) =Θ[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos142.erdos_142.variants.three"},{"answerKinds":[],"category":"research open","docstring":"Show that $r_k(N) = o_k(N / \\log N)$, where $r_k(N)$ the largest possible size of a subset\nof $\\{1, \\dots, N\\}$ that does not contain any non-trivial $k$-term arithmetic progression.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«142»","statement":"∀ (k : ℕ), 1 < k → (fun N => ↑(Erdos142.r k N)) =o[Filter.atTop] fun N => ↑N / Real.log ↑N","subjects":["11"],"theorem":"Erdos142.erdos_142.variants.lower"},{"answerKinds":[],"category":"test","docstring":"The empty graph on `Fin 0` is `Free` of any nontrivial subgraph (vacuous). This is the\nsimplest non-trivial witness to `G₁.Free H` appearing in `HasFiniteRamseyProperty`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«596»","statement":"(SimpleGraph.cycleGraph 4).Free ⊥","subjects":["5"],"theorem":"Erdos596.erdos_596.test.empty_is_free"},{"answerKinds":[],"category":"research solved","docstring":"Erdős–Hajnal exceptional pairs exist — recorded as a known direction of `erdos_596`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«596»","statement":"∃ U₁ U₂ G₁ G₂, G₁.IsErdosHajnalExceptional G₂","subjects":["5"],"theorem":"Erdos596.erdos_596.variants.exists_exceptional"},{"answerKinds":[],"category":"research open","docstring":"Whether $(K_4, K_3)$ is Erdős–Hajnal exceptional is precisely the content of\nErdős Problem 595. The finite Ramsey property holds (Folkman 1970, Nešetřil–Rödl\n[NeRo75]); the open part is whether every $K_4$-free graph is a countable union of\ntriangle-free graphs. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«596»","statement":"True ↔ (SimpleGraph.completeGraph (Fin 4)).IsErdosHajnalExceptional (SimpleGraph.completeGraph (Fin 3))","subjects":["5"],"theorem":"Erdos596.erdos_596.variants.K4_K3_exceptional_iff"},{"answerKinds":[],"category":"research solved","docstring":"Folkman 1970 / Nešetřil–Rödl [NeRo75]: for every $n \\geq 1$ there is a $K_4$-free\ngraph whose edges cannot be $n$-coloured without a monochromatic triangle. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«596»","statement":"(SimpleGraph.completeGraph (Fin 4)).HasFiniteRamseyProperty (SimpleGraph.completeGraph (Fin 3))","subjects":["5"],"theorem":"Erdos596.erdos_596.variants.K4_K3_finite_ramsey"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The original Erdős–Hajnal conjecture (that no exceptional pair exists) is **false** —\nwitnessed by $(C_4, C_6)$ via `C4_C6_is_exceptional`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«596»","statement":"False ↔ ∀ {U₁ U₂ : Type} (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂), ¬G₁.IsErdosHajnalExceptional G₂","subjects":["5"],"theorem":"Erdos596.erdos_596.variants.original_conjecture_is_false"},{"answerKinds":[],"category":"research solved","docstring":"Nešetřil–Rödl [NeRo75]: for every $n \\geq 1$ there is a $C_4$-free graph whose edges\ncannot be $n$-coloured without a monochromatic $C_6$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«596»","statement":"(SimpleGraph.cycleGraph 4).HasFiniteRamseyProperty (SimpleGraph.cycleGraph 6)","subjects":["5"],"theorem":"Erdos596.erdos_596.variants.C4_C6_finite_ramsey"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"**Erdős Problem 596** (Erdős–Hajnal, [Er87]). For which graph pairs $(G_1, G_2)$ is it\ntrue that\n\n  (1) for every $n \\geq 1$ there is a graph $H$ without a $G_1$ such that any\n      $n$-colouring of $H$'s edges contains a monochromatic $G_2$, and yet\n  (2) for every graph $H$ without a $G_1$ there is an $\\aleph_0$-colouring of $H$'s edges\n      with no monochromatic $G_2$?\n\nErdős and Hajnal originally conjectured that no such pair exists; but $(C_4, C_6)$\nwitnesses it (Nešetřil–Rödl + Erdős–Hajnal). The full question is to characterise the\nclass of all such pairs, recorded here as `answer(sorry)`.\n\nSee Problem 595 for the specific case $(G_1, G_2) = (K_4, K_3)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«596»","statement":"∀ {U₁ U₂ : Type} (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂),\n  G₁.IsErdosHajnalExceptional G₂ ↔ (fun {U₁ U₂} => sorry) G₁ G₂","subjects":["5"],"theorem":"Erdos596.erdos_596"},{"answerKinds":[],"category":"research solved","docstring":"Every $C_4$-free graph is a countable union of trees (Erdős–Hajnal [Er87]); trees are\nacyclic, hence $C_6$-free, giving the countable Ramsey escape for $(C_4, C_6)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«596»","statement":"(SimpleGraph.cycleGraph 4).HasCountableRamseyEscape (SimpleGraph.cycleGraph 6)","subjects":["5"],"theorem":"Erdos596.erdos_596.variants.C4_free_countable_escape"},{"answerKinds":[],"category":"research solved","docstring":"The pair $(C_4, C_6)$ is Erdős–Hajnal exceptional; combines `C4_C6_finite_ramsey` and\n`C4_free_countable_escape`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«596»","statement":"(SimpleGraph.cycleGraph 4).IsErdosHajnalExceptional (SimpleGraph.cycleGraph 6)","subjects":["5"],"theorem":"Erdos596.erdos_596.variants.C4_C6_is_exceptional"},{"answerKinds":[],"category":"research open","docstring":"Let $P_d(n)$ be such that in any set of $n$ points in $\\mathbb{R}^d$ there exist at least $P_d(n)$ many points which do not contain an isosceles triangle. Estimate $P_d(n)$ - in particular, is it true that $$P_2(n)<n^{1-c}$$ for some constant $c>0$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1207»","statement":"True ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos1207.P 2 n) < ↑n ^ (1 - c)","subjects":["52"],"theorem":"Erdos1207.erdos_1207"},{"answerKinds":[],"category":"research open","docstring":"Let $σ_1(n)=σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$.\nIs it true that, for every $m, n ≥ 2$, there exist some $i, j$ such that $σ_i(m) = σ_j(n)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«412»","statement":"True ↔ ∀ m ≥ 2, ∀ n ≥ 2, ∃ i j, (⇑(ArithmeticFunction.sigma 1))^[i] m = (⇑(ArithmeticFunction.sigma 1))^[j] n","subjects":["11"],"theorem":"Erdos412.erdos_412"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a graph on $n$ vertices, $\\alpha_1(G)$ be the maximum number of edges that contain\nat most one edge from every triangle, and $\\tau_1(G)$ be the minimum number of edges that\ncontain at least one edge from every triangle.\n\nIs it true that$$\\alpha_1(G)+\\tau_1(G) \\leq \\frac{n^2}{4}?$$\n\nA problem of Erdős, Gallai, and Tuza [EGT96], who observe that this is probably quite difficult\nsince there are different examples where equality hold: the complete graph, the complete\nbipartite graph, and the graph obtained from $K_{m,m}$ by adding one vertex joined to every\nother.\n\nThis is true, and was proved by Norin and Sun [NoSu16], who in fact proved\nthat$$\\alpha_1(G)+\\tau_B(G) \\leq \\frac{n^2}{4},$$where $\\tau_B(G)$ is the minimum number of\nedges that need to be removed to make the graph bipartite.\n\nHere $\\alpha_1(G)$ and $\\tau_1(G)$ are taken over subsets of the edge set of $G$, and the\ninequality is stated multiplied through by $4$ so that it lives in the natural numbers.\n\nThe linked file states $\\tau_1$ as the least number of edges whose deletion leaves $G$\ntriangle-free, which is the same as meeting every triangle of $G$, and quantifies over an\narbitrary `Fintype V` rather than `Fin n`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos621.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«621»","statement":"True ↔\n  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (a t : ℕ),\n    IsGreatest\n        {k |\n          ∃ A ⊆ G.edgeFinset,\n            A.card = k ∧\n              ∀ (x y z : Fin n), G.Adj x y → G.Adj y z → G.Adj x z → ({s(x, y), s(y, z), s(x, z)} ∩ A).card ≤ 1}\n        a →\n      IsLeast\n          {k |\n            ∃ T ⊆ G.edgeFinset,\n              T.card = k ∧\n                ∀ (x y z : Fin n), G.Adj x y → G.Adj y z → G.Adj x z → (T ∩ {s(x, y), s(y, z), s(x, z)}).Nonempty}\n          t →\n        4 * (a + t) ≤ n ^ 2","subjects":["5"],"theorem":"Erdos621.erdos_621"},{"answerKinds":[],"category":"research open","docstring":"Is\n$$\\sum_{n} \\frac{\\phi(n)}{2^n}$$\nirrational? Here $\\phi$ is the Euler totient function.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«249»","statement":"True ↔ Irrational (∑' (n : ℕ), ↑n.totient / 2 ^ n)","subjects":["11"],"theorem":"Erdos249.erdos_249"},{"answerKinds":[],"category":"research solved","docstring":"Nguyen, Scott, and Seymour [NSS23] proved the conjecture for $H = P_5$, the path on five\nvertices: every $P_5$-free graph on $n$ vertices has a clique or independent set of\npolynomial size.\n\n[NSS23] Nguyen, T., Scott, A. and Seymour, P., Induced subgraph density. VII. The\nfive-vertex path. [arXiv:2312.15333](https://arxiv.org/abs/2312.15333)\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«61»","statement":"∃ c > 0, Erdos61.IsErdosHajnalLowerBound (SimpleGraph.pathGraph 5) fun n => ↑n ^ c","subjects":["5"],"theorem":"Erdos61.erdos_61.variants.p5"},{"answerKinds":[],"category":"research open","docstring":"The Erdős–Hajnal Conjecture states that there is a constant $c(H) > 0$ for each\n$H$ such that we can take $f(n) = n^{c(H)}$ in the above formulation.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«61»","statement":"True ↔\n  ∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] (H : SimpleGraph α),\n    ∃ c > 0, Erdos61.IsErdosHajnalLowerBound H fun n => ↑n ^ c","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"Erdos61.erdos_61"},{"answerKinds":[],"category":"research solved","docstring":"Bucić, Nguyen, Scott, and Seymour [BNSS23] improved this to\n$f(n) = \\exp(c_H \\sqrt{\\log n \\log \\log n})$ for some constant $c_H > 0$ depending on $H$.\n\n[BNSS23] Bucić, M. and Nguyen, T. and Scott, A. and Seymour, P., A loglog step towards Erdos-Hajnal\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«61»","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] (H : SimpleGraph α),\n  ∃ c > 0, Erdos61.IsErdosHajnalLowerBound H fun n => Real.exp (c * √(Real.log ↑n * Real.log (Real.log ↑n)))","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos61.erdos_61.variants.bnss23"},{"answerKinds":[],"category":"research solved","docstring":"Chudnovsky, Scott, Seymour, and Spirkl [CSSS23] proved the conjecture for $H = C_5$, the\ncycle on five vertices: every graph with no induced five-cycle has a clique or independent\nset of polynomial size.\n\n[CSSS23] Chudnovsky, M., Scott, A., Seymour, P. and Spirkl, S., Erdős–Hajnal for graphs with\nno 5-hole. Proc. Lond. Math. Soc. (3) 126 (2023), 997–1014.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«61»","statement":"∃ c > 0, Erdos61.IsErdosHajnalLowerBound (SimpleGraph.cycleGraph 5) fun n => ↑n ^ c","subjects":["5"],"theorem":"Erdos61.erdos_61.variants.c5"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Hajnal [ErHa89] proved that we can take $f(n) = \\exp(c_H \\sqrt{\\log n})$\nfor some constant $c_H > 0$ depending on $H$.\n\n[ErHa89] Erdős, P. and Hajnal, A., Ramsey-type theorems. Discrete Appl. Math. (1989), 37-52.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«61»","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] (H : SimpleGraph α),\n  ∃ c > 0, Erdos61.IsErdosHajnalLowerBound H fun n => Real.exp (c * √(Real.log ↑n))","subjects":["5"],"theorem":"Erdos61.erdos_61.variants.erha89"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there infinitely many $n$ such that the largest prime factor of $n$ is $< n^{\\frac{1}{2}}$ and\nthe largest prime factor of $n + 1$ is $< (n + 1)^{\\frac{1}{2}}$.\n\nSteinerberger has pointed out this problem has a trivial solution.\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/f58dea7d2cc5c9da2e050ec80a73e838b54a6dd2/FormalConjectures/ErdosProblems/370.lean#L73"},{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos370.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«370»","statement":"True ↔ {n | ↑n.maxPrimeFac < √↑n ∧ ↑(n + 1).maxPrimeFac < √(↑n + 1)}.Infinite","subjects":["11"],"theorem":"Erdos370.erdos_370"},{"answerKinds":[],"category":"research solved","docstring":"**The Erdős discrepancy problem**\n\nIf $f\\colon \\mathbb N \\rightarrow \\{-1, +1\\}$ then is it true that for every $C>0$ there\nexist $d, m \\ge 1$ such that $$\\left\\lvert \\sum_{1\\leq k\\leq m}f(kd)\\right\\rvert > C?$$\nThis is true, and was proved by Tao [Ta16]\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«67»","statement":"∀ (f : ℕ → ↥{-1, 1}) (C : ℝ), 0 < C → ∃ d ≥ 1, ∃ m ≥ 1, C < |∑ k ∈ Finset.Icc 1 m, ↑(f (k * d))|","subjects":["11"],"theorem":"Erdos67.erdos_67"},{"answerKinds":[],"category":"research solved","docstring":"**The Erdős discrepancy problem (complex variant)**\n\nIf $f\\colon \\mathbb N \\rightarrow S^1 ⊆ ℂ$ then is it true that for every $C>0$ there\nexist $d, m \\ge 1$ such that $$\\left\\lvert \\sum_{1\\leq k\\leq m}f(kd)\\right\\rvert > C?$$\nThis is true, and was proved by Tao [Ta16]\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«67»","statement":"∀ (f : ℕ → ↑(Metric.sphere 0 1)) (C : ℝ), 0 < C → ∃ d ≥ 1, ∃ m ≥ 1, C < ‖∑ k ∈ Finset.Icc 1 m, ↑(f (k * d))‖","subjects":["11"],"theorem":"Erdos67.erdos_67.variants.complex"},{"answerKinds":[],"category":"research open","docstring":"Is there a constant $c > 0$ such that every graph on $2^n$ vertices with minimum degree\n$> (1-c) \\cdot 2^n$ contains the $n$-dimensional hypercube $Q_n$?\n\nThis is Erdős's question [Er93, p. 345].\n\nSee also [576] for the extremal number of edges that guarantee a $Q_n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1035»","statement":"True ↔\n  ∃ c > 0,\n    ∀ (n : ℕ) (G : SimpleGraph (Fin (2 ^ n))) [inst : DecidableRel G.Adj],\n      (∀ (v : Fin (2 ^ n)), ↑(G.degree v) > (1 - c) * 2 ^ n) → (SimpleGraph.hypercube n).IsContained G","subjects":["5"],"theorem":"Erdos1035.erdos_1035"},{"answerKinds":[],"category":"research open","docstring":"The set of indices $n$ for which a prime gap is followed by a larger or equal prime gap has a\nnatural density of $\\frac 1 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«218»","statement":"{n | primeGap n ≤ primeGap (n + 1)}.HasDensity (1 / 2)","subjects":["11"],"theorem":"Erdos218.erdos_218.variants.le"},{"answerKinds":[],"category":"research open","docstring":"The set of indices $n$ for which a prime gap is preceded by a larger or equal prime gap has a\nnatural density of $\\frac 1 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«218»","statement":"{n | primeGap (n + 1) ≤ primeGap n}.HasDensity (1 / 2)","subjects":["11"],"theorem":"Erdos218.erdos_218.variants.ge"},{"answerKinds":[],"category":"research open","docstring":"There are infinitely many indices $n$ such that the prime gap at $n$ is equal to the prime gap\nat $n+1$. This is equivalent to the existence of infinitely many arithmetic progressions of\nlength $3$, see `erdos_141.variants.infinite_three`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«218»","statement":"{n | primeGap n = primeGap (n + 1)}.Infinite","subjects":["11"],"theorem":"Erdos218.erdos_218.variants.infinite_equal_prime_gap"},{"answerKinds":[],"category":"research solved","docstring":"The smallest number of edges in a graph of dimension $4$ is achieved solely by $K_{3,3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1007»","statement":"∀ (n : ℕ) (G : SimpleGraph (Fin n)),\n  G.HasDimension 4 →\n    G.edgeSet.ncard = 9 → (∀ (v : Fin n), ∃ w, G.Adj v w) → Nonempty (G ≃g completeBipartiteGraph (Fin 3) (Fin 3))","subjects":["5","52"],"theorem":"Erdos1007.erdos_1007.variants.dimension_four_extremal"},{"answerKinds":[],"category":"research solved","docstring":"The dimension of a graph $G$ is the minimal $n$ such that $G$ can be embedded in $\\mathbb{R}^n$\nsuch that every edge of $G$ is a unit line segment.\n\nWhat is the smallest number of edges in a graph with dimension $4$?\n\nAnswer: The smallest number of edges is $9$, achieved solely by $K_{3,3}$, proved by House [Ho13]. An\nalternative proof was given by Chaffee and Noble [ChNo16], who also prove that the smallest\nnumber of edges in a graph of dimension $5$ is $15$ (achieved by $K_6$ and $K_{1,3,3}$).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1007.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1007»","statement":"IsLeast {m | ∃ n G, G.HasDimension 4 ∧ G.edgeSet.ncard = m} 9","subjects":["5","52"],"theorem":"Erdos1007.erdos_1007"},{"answerKinds":[],"category":"research solved","docstring":"The smallest number of edges in a graph of dimension $5$ is $15$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1007»","statement":"IsLeast {m | ∃ n G, G.HasDimension 5 ∧ G.edgeSet.ncard = m} 15","subjects":["5","52"],"theorem":"Erdos1007.erdos_1007.variants.dimension_five"},{"answerKinds":[],"category":"research solved","docstring":"The smallest number of edges in a graph of dimension $5$ is achieved by $K_6$ and $K_{1,3,3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1007»","statement":"((SimpleGraph.completeGraph (Fin 6)).HasDimension 5 ∧ (SimpleGraph.completeGraph (Fin 6)).edgeSet.ncard = 15) ∧\n  Erdos1007.K133.HasDimension 5 ∧ Erdos1007.K133.edgeSet.ncard = 15","subjects":["5","52"],"theorem":"Erdos1007.erdos_1007.variants.dimension_five_extremal"},{"answerKinds":["Prop"],"category":"research open","docstring":"Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \\ldots, a_n$ all possible\nvalues of $a_i a_j - 1$ with $i \\neq j$.\nIs it true that the set of integers which eventually appear has positive density?\n\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«424»","statement":"sorry ↔ 0 < Erdos424.generatedSet.lowerDensity","subjects":["11"],"theorem":"Erdos424.erdos_424"},{"answerKinds":["Prop"],"category":"research open","docstring":"A literal interpretation of \"positive density\": the natural density of `generatedSet` exists\n(i.e. the lower and upper density agree) and is positive.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«424»","statement":"sorry ↔ Erdos424.generatedSet.HasPosDensity","subjects":["11"],"theorem":"Erdos424.erdos_424.variants.exact_density"},{"answerKinds":[],"category":"research open","docstring":"Erdős was unable to prove that if the two products have the same factors\nthen there must exist a prime between $n_1$ and $n_2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«931»","statement":"∀ (k₁ k₂ n₁ n₂ : ℕ),\n  k₂ ≤ k₁ →\n    3 ≤ k₂ →\n      n₁ + k₁ ≤ n₂ →\n        (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors = (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors →\n          ∃ p, Nat.Prime p ∧ n₁ ≤ p ∧ p ≤ n₂","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos931.erdos_931.variants.exists_prime"},{"answerKinds":[],"category":"research solved","docstring":"In fact there exist counterexamples, like this one found by AlphaProof.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«931»","statement":"∃ k₁ k₂,\n  ∃ (_ : k₂ ≤ k₁) (_ : 3 ≤ k₂),\n    {(n₁, n₂) |\n        n₁ + k₁ ≤ n₂ ∧\n          n₂ ≤ 2 * (n₁ + k₁) ∧\n            (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors = (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors}.Nonempty","subjects":["11"],"theorem":"Erdos931.erdos_931.variants.additional_condition_nonempty"},{"answerKinds":[],"category":"research open","docstring":"Let $k_1 \\geq k_2 \\geq 3$. Are there only finitely many $n_2\\geq n_1 + k_1$\nsuch that\n$$\n  \\prod_{1\\leq i\\leq k_1}(n_1 + i)\\ \\text{and}\\ \\prod_{1\\leq j\\leq k_2} (n_2 + j)\n$$\nhave the same prime factors?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«931»","statement":"True ↔\n  ∀ (k₁ k₂ : ℕ),\n    k₂ ≥ 3 →\n      k₂ ≤ k₁ →\n        {(n₁, n₂) |\n            n₁ + k₁ ≤ n₂ ∧\n              (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors = (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors}.Finite","subjects":["11"],"theorem":"Erdos931.erdos_931"},{"answerKinds":[],"category":"research open","docstring":"Erdős thought perhaps if the two products have the same factors then\n$n_2 > 2(n_1 + k_1)$.\nIt is an open question whether this is true when allowing a finite number of counterexamples.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«931»","statement":"True ↔\n  ∀ (k₁ k₂ : ℕ),\n    k₂ ≥ 3 →\n      k₂ ≤ k₁ →\n        {(n₁, n₂) |\n            n₁ + k₁ ≤ n₂ ∧\n              n₂ ≤ 2 * (n₁ + k₁) ∧\n                (∏ i ∈ Finset.Icc 1 k₁, (n₁ + i)).primeFactors = (∏ j ∈ Finset.Icc 1 k₂, (n₂ + j)).primeFactors}.Finite","subjects":["11"],"theorem":"Erdos931.erdos_931.variants.additional_condition"},{"answerKinds":[],"category":"textbook","docstring":"$$\\sum_{n=2}^\\infty \\frac{1}{n!-1} = \\sum_{n=2}^\\infty \\sum_{k=1}^\\infty \\frac{1}{(n!)^k}$$\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«68»","statement":"have f := fun n k => 1 / ↑(n + 2).factorial ^ (k + 1);\n∑' (n : ℕ), 1 / (↑(n + 2).factorial - 1) = ∑' (n : ℕ) (k : ℕ), f n k","subjects":["11"],"theorem":"Erdos68.sum_factorial_inv_eq_geometric"},{"answerKinds":[],"category":"research open","docstring":"Is\n$$\\sum_{n=2}^\\infty \\frac{1}{n!-1}$$\nirrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«68»","statement":"True ↔ Irrational (∑' (n : ℕ), 1 / (↑(n + 2).factorial - 1))","subjects":["11"],"theorem":"Erdos68.erdos_68"},{"answerKinds":[],"category":"research open","docstring":"Part (ii) of Erdős Problem 1060: bound on the number of $k \\le n$ with $k \\sigma_1(k) = n$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1060»","statement":"∃ C,\n  (fun n => ↑{k ∈ Finset.Iic n | k * (ArithmeticFunction.sigma 1) k = n}.card) =O[Filter.atTop] fun n => Real.log ↑n ^ C","subjects":["11"],"theorem":"Erdos1060.erdos_1060.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"The conjecture is about the function $f(n)$ which counts the number of solutions to\n$k\\sigma(k)=n$, where $\\sigma(k)$ is the sum of divisors of $k$. The first bound is that $f(n)$ grows slower\nthan any power of $n^(\\frac{1}{\\log\\log n})$. The second bound is that $f(n)$ is at most a power of\n$\\log n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1060»","statement":"∃ h,\n  (h =o[Filter.atTop] fun n => 1 / Real.log (Real.log ↑n)) ∧\n    ∀ᶠ (n : ℕ) in Filter.atTop, ↑{k ∈ Finset.Iic n | k * (ArithmeticFunction.sigma 1) k = n}.card ≤ ↑n ^ h n","subjects":["11"],"theorem":"Erdos1060.erdos_1060.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"The primes show that $\\lvert A\\rvert \\gg n/\\log n$ is possible.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«888»","statement":"(fun n => ↑n / Real.log ↑n) =O[Filter.atTop] fun n => ↑(Nat.findGreatest (Erdos888.p n) n)","subjects":["11"],"theorem":"Erdos888.erdos_888.variants.primes"},{"answerKinds":[],"category":"research solved","docstring":"Erdős claims that Sárközy proved that $\\lvert A\\rvert =o(n)$ (a proof of this\nbound is provided by Tao in the comments).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«888»","statement":"(fun n => ↑(Nat.findGreatest (Erdos888.p n) n)) =o[Filter.atTop] Nat.cast","subjects":["11"],"theorem":"Erdos888.erdos_888.variants.sarkozy"},{"answerKinds":[],"category":"research solved","docstring":"Cambie and Weisenberg have noted in the comments that the set of semiprimes\nalso works, showing $(1+o(1))\\frac{\\log\\log n}{\\log n}n \\leq \\lvert A\\rvert$ is achievable.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«888»","statement":"(fun n => ↑n * Real.log (Real.log ↑n) / Real.log ↑n) =O[Filter.atTop] fun n => ↑(Nat.findGreatest (Erdos888.p n) n)","subjects":["11"],"theorem":"Erdos888.erdos_888.variants.semiprimes"},{"answerKinds":[],"category":"research solved","docstring":"What is the size of the largest $A\\subseteq \\{1,\\ldots,n\\}$ such that if\n$a\\leq b\\leq c\\leq d\\in A$ are such that $abcd$ is a square then $ad=bc$?\n\nThis was proved by GPT-5.5 Pro (prompted by Chojecki).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos888.lean#L47"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«888»","statement":"(fun n => ↑(Nat.findGreatest (Erdos888.p n) n)) =Θ[Filter.atTop] fun n => ↑n * Real.log (Real.log ↑n) / Real.log ↑n","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos888.erdos_888"},{"answerKinds":[],"category":"research solved","docstring":"If $G$ is a graph with infinite chromatic number and $a_1 < a_2 < \\cdots$ are lengths of the odd\ncycles of $G$ then $\\sum \\frac{1}{a_i} = \\infty$.\n\nConjectured by Erdős and Hajnal [ErHa66], and solved by Liu and Montgomery [LiMo20].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«57»","statement":"∀ {V : Type u_1} (G : SimpleGraph V), G.chromaticNumber = ⊤ → ¬Summable fun a => 1 / ↑↑a","subjects":["5"],"theorem":"Erdos57.erdos_57"},{"answerKinds":[],"category":"research solved","docstring":"The classical estimate of Mertens states that $\\sum_{p\\leq n}\\frac{1}{p}\\sim \\log\\log n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«726»","statement":"Asymptotics.IsEquivalent Filter.atTop (fun n => ∑ p ∈ Finset.range (n + 1) with Nat.Prime p, 1 / ↑p) fun n =>\n  Real.log (Real.log ↑n)","subjects":["11"],"theorem":"Erdos726.erdos_726.variants.mertens_estimate"},{"answerKinds":[],"category":"research open","docstring":"As $n\\to \\infty$ ranges over integers\n$\\sum_{p\\leq n}1_{n\\in (p/2,p)\\pmod{p}}\\frac{1}{p}\\sim \\frac{\\log\\log n}{2}$?\n\nA conjecture of Erdős, Graham, Ruzsa, and Straus [EGRS75].\n\nBy $n\\in (p/2,p)\\pmod{p}$ we mean $n\\equiv r\\pmod{p}$ for some integer $r$ with $p/2<r<p$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«726»","statement":"True ↔\n  Asymptotics.IsEquivalent Filter.atTop\n    (fun n => ∑ p ∈ Finset.range (n + 1) with Nat.Prime p ∧ ↑p / 2 < ↑n % ↑p, 1 / ↑p) fun n =>\n    Real.log (Real.log ↑n) / 2","subjects":["11"],"theorem":"Erdos726.erdos_726"},{"answerKinds":[],"category":"research open","docstring":"Let $h(n)$ be such that any $n$ points in $\\mathbb{R}^2$, with no three on a line\nand no four on a circle, determine at least $h(n)$ distinct distances. Does\n$h(n)/n\\to \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«98»","statement":"True ↔ Filter.Tendsto (fun n => ↑(Erdos98.h n) / ↑n) Filter.atTop Filter.atTop","subjects":["52"],"theorem":"Erdos98.erdos_98"},{"answerKinds":[],"category":"research solved","docstring":"Erdős could not even prove $h(n)\\geq n$. Pach has shown $h(n) < n^{\\log_2 3}$.\nErdős, Füredi, and Pach [EFPR93] have improved this to\n$h(n) < n\\exp(c\\sqrt{\\log n})$ for some constant $c>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«98»","statement":"∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos98.h n) < ↑n * Real.exp (c * √(Real.log ↑n))","subjects":["52"],"theorem":"Erdos98.erdos_98.variants.upper_bound"},{"answerKinds":["non-Prop"],"category":"research solved","docstring":"Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable.\n`V(x)=x/logx * e^((C+o(1))(log log log x)^2)`, for some explicit constant `C>0`.\nRef:Maier, Helmut and Pomerance, Carl, _On the number of distinct values of Euler's $\\phi$-function_.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«416»","statement":"have C := sorry;\n0 < C ∧\n  ∃ f,\n    f =o[Filter.atTop] 1 ∧\n      ∀ᶠ (x : ℝ) in Filter.atTop,\n        Erdos416.V x = x / Real.log x * Real.exp ((C + f x) * Real.log (Real.log (Real.log x)) ^ 2)","subjects":["11"],"theorem":"Erdos416.erdos_416.variants.Maier_Pomerance"},{"answerKinds":[],"category":"research solved","docstring":"Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable.\nPillai proved `V(x)=o(x)`.\nRef: S. Sivasankaranarayana Pillai, _On some functions connected with $\\phi(n)$_\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«416»","statement":"Erdos416.V =o[Filter.atTop] id","subjects":["11"],"theorem":"Erdos416.erdos_416.variants.Pillai"},{"answerKinds":[],"category":"research open","docstring":"Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable. Does `V(2x)/V(x)→2` ?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«416»","statement":"Filter.Tendsto (fun x => Erdos416.V (2 * x) / Erdos416.V x) Filter.atTop (nhds 2)","subjects":["11"],"theorem":"Erdos416.erdos_416.parts.i"},{"answerKinds":["non-Prop"],"category":"research solved","docstring":"Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable.\n`V(x) ≍ x/log x*e^(C_1*(log log log x − log log log log x)^2+C_2 log log log x − C_3 log log log log x)`\nRef: Ford, Kevin, _The distribution of totients_.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«416»","statement":"match sorry with\n| (C₁, C₂, C₃) =>\n  0 < C₁ ∧\n    0 < C₂ ∧\n      0 < C₃ ∧\n        have G := fun x =>\n          x / Real.log x *\n            Real.exp\n              (C₁ * (Real.log (Real.log (Real.log x)) - Real.log (Real.log (Real.log (Real.log x)))) ^ 2 +\n                  C₂ * Real.log (Real.log (Real.log x)) -\n                C₃ * Real.log (Real.log (Real.log (Real.log x))));\n        Erdos416.V =Θ[Filter.atTop] G","subjects":["11"],"theorem":"Erdos416.erdos_416.variants.Ford"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable.\nIs there an asymptotic formula for `V(x)`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«416»","statement":"have f := sorry;\nFilter.Tendsto (fun x => Erdos416.V x / f x) Filter.atTop (nhds 1)","subjects":["11"],"theorem":"Erdos416.erdos_416.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Let `V(x)` count the number of `n≤x` such that `ϕ(m)=n` is solvable.\nErdős proved V(x)=x(logx)^(−1+o(1)).\nRef: Erdős, P., _On the normal number of prime factors of $p-1$ and some related problems concerning Euler's $\\varphi$-function._\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«416»","statement":"∃ f, f =o[Filter.atTop] 1 ∧ ∀ᶠ (x : ℝ) in Filter.atTop, Erdos416.V x = x * Real.log x ^ (-1 + f x)","subjects":["11"],"theorem":"Erdos416.erdos_416.variants.Erdos"},{"answerKinds":[],"category":"research open","docstring":"Probably $f(x) = x^5$ has the property that the sums $f(a)+f(b)$ with\n$a < b$ nonnegative integers are distinct.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«324»","statement":"Set.InjOn\n  (fun x =>\n    match x with\n    | (a, b) => a ^ 5 + b ^ 5)\n  {(a, b) | a < b}","subjects":["11"],"theorem":"Erdos324.erdos_324.variants.quintic"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a polynomial $f(x)\\in\\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with\n$a < b$ nonnegative integers are distinct?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«324»","statement":"True ↔\n  ∃ f,\n    Set.InjOn\n      (fun x =>\n        match x with\n        | (a, b) => Polynomial.eval (↑a) f + Polynomial.eval (↑b) f)\n      {(a, b) | a < b}","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos324.erdos_324"},{"answerKinds":[],"category":"research open","docstring":"Let $\\operatorname{lcm}(1, \\dots, n)$ denote the least common multiple of $\\{1, \\dots, n\\}$.\nLet $p_k$ be the $k$-th prime.\nIs it true that for all $k \\geq 1$, $\\operatorname{lcm}(1, \\dots, p_{k+1}-1) < p_k \\cdot \\operatorname{lcm}(1, \\dots, p_k)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«458»","statement":"True ↔ ∀ (k : ℕ), Erdos458.lcm_upto (Nat.nth Prime (k + 1) - 1) < Nat.nth Prime k * Erdos458.lcm_upto (Nat.nth Prime k)","subjects":["11"],"theorem":"Erdos458.erdos_458"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for all sufficiently large $k$, there exists finite intervals\n$I_1, \\dotsc, I_k \\subset \\mathbb{N}$ with $|I_i| \\geq 2$ for $1 \\leq i \\leq k$ such that\n$$\n1 = \\sum_{i=1}^k \\sum_{n \\in I_i} \\frac{1}{n}.\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«289»","statement":"True ↔\n  ∀ᶠ (k : ℕ) in Filter.atTop,\n    ∃ I,\n      (∀ (i : Fin k), (I i).1 < (I i).2) ∧\n        (∀ (i j : Fin k), i ≠ j → (I i).2 < (I j).1 ∨ (I j).2 < (I i).1) ∧\n          ∑ i, ∑ n ∈ Finset.Icc (I i).1 (I i).2, (↑n)⁻¹ = 1","subjects":["11"],"theorem":"Erdos289.erdos_289"},{"answerKinds":[],"category":"test","docstring":"A singleton `{0}` is an independent set for any family `A : ℝ → Set ℝ`,\nas witnessed by `erdos_501.variants.singleton_independent`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«501»","statement":"∀ (A : ℝ → Set ℝ), {0}.Pairwise fun x y => x ∉ A y","subjects":["5","28"],"theorem":"Erdos501.erdos_501.tests.singleton_zero_independent"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Newelski–Pawlikowski–Seredyński (1987) [NPS87]: infinite independent set in the closed case.**\n\nIf all the sets `A x` are closed with Lebesgue measure `< 1`, then there **is** an\ninfinite independent set. This gives a strong affirmative answer to the second\nquestion of Problem 501. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«501»","statement":"True ↔\n  ∀ (A : ℝ → Set ℝ),\n    (∀ (x : ℝ), IsClosed (A x)) →\n      (∀ (x : ℝ), MeasureTheory.volume (A x) < 1) → ∃ X, X.Infinite ∧ X.Pairwise fun x y => x ∉ A y","subjects":["5","28"],"theorem":"Erdos501.erdos_501.variants.newelski_pawlikowski_seredynski"},{"answerKinds":[],"category":"research open","docstring":"For every $x \\in \\mathbb{R}$ let $A_x \\subset \\mathbb{R}$ be a bounded set with outer measure\n$< 1$. Must there exist an infinite independent set, that is, some infinite $X \\subseteq\n\\mathbb{R}$ such that $x \\notin A_y$ for all $x \\neq y \\in X$?\n\nIf the sets $A_x$ are closed and have measure $< 1$, then must there exist an independent set\nof size $3$?\n\nKnown results: Erdős–Hajnal [ErHa60] proved the existence of arbitrarily large finite\nindependent sets. Hechler [He72] showed the answer is **no** assuming the continuum\nhypothesis. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«501»","statement":"True ↔\n  ∀ (A : ℝ → Set ℝ),\n    (∀ (x : ℝ), Bornology.IsBounded (A x)) →\n      (∀ (x : ℝ), MeasureTheory.volume.toOuterMeasure (A x) < 1) → ∃ X, X.Infinite ∧ X.Pairwise fun x y => x ∉ A y","subjects":["5","28"],"theorem":"Erdos501.erdos_501"},{"answerKinds":[],"category":"textbook","docstring":"**Two-element sets: independent iff mutual non-membership.**\n\nA two-element set `{x, y}` (with `x ≠ y`) is independent for `A` if and only if\n`x ∉ A y` and `y ∉ A x`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«501»","statement":"∀ (A : ℝ → Set ℝ) {x y : ℝ}, x ≠ y → (({x, y}.Pairwise fun x y => x ∉ A y) ↔ x ∉ A y ∧ y ∉ A x)","subjects":["5","28"],"theorem":"Erdos501.erdos_501.variants.pair_independent_iff"},{"answerKinds":[],"category":"test","docstring":"The boundary case: the measure condition `< 1` is sharp. An interval of length ≥ 1\nhas Lebesgue measure ≥ 1, so it would fail the hypothesis. Here `[0, 1]` has measure exactly 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«501»","statement":"MeasureTheory.volume.toOuterMeasure (Set.Icc 0 1) = 1","subjects":["5","28"],"theorem":"Erdos501.erdos_501.tests.unit_interval_measure"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Closed sets case: existence of an independent set of size 3.**\n\nIf the sets `A x` are closed with Lebesgue measure `< 1`, must there exist an\nindependent set of size 3?\n\nThis is implied by the stronger theorem of Newelski–Pawlikowski–Seredyński [NPS87] below;\nGladysz [Gl62] earlier proved the existence of an independent set of size 2. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«501»","statement":"True ↔\n  ∀ (A : ℝ → Set ℝ),\n    (∀ (x : ℝ), IsClosed (A x)) →\n      (∀ (x : ℝ), MeasureTheory.volume (A x) < 1) → ∃ X, 3 ≤ X.ncard ∧ X.Pairwise fun x y => x ∉ A y","subjects":["5","28"],"theorem":"Erdos501.erdos_501.variants.closed_size3"},{"answerKinds":[],"category":"test","docstring":"Two reals form an independent set for the empty family `A _ = ∅`:\nneither 0 nor 1 belongs to ∅, so both conditions of `pair_independent_iff` hold. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«501»","statement":"{0, 1}.Pairwise fun x y => x ∉ (fun x => ∅) y","subjects":["5","28"],"theorem":"Erdos501.erdos_501.tests.pair_independent_empty"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős–Hajnal (1960): arbitrarily large finite independent sets exist.**\n\nFor every `n : ℕ` and every family `A : ℝ → Set ℝ` of bounded sets with Lebesgue\nouter measure `< 1`, there exists a finite independent set of size at least `n`.\n\nThis was proved by Erdős and Hajnal [ErHa60]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«501»","statement":"True ↔\n  ∀ (n : ℕ) (A : ℝ → Set ℝ),\n    (∀ (x : ℝ), Bornology.IsBounded (A x)) →\n      (∀ (x : ℝ), MeasureTheory.volume.toOuterMeasure (A x) < 1) → ∃ X, n ≤ X.card ∧ (↑X).Pairwise fun x y => x ∉ A y","subjects":["5","28"],"theorem":"Erdos501.erdos_501.variants.erdosHajnal_finite"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Gladysz (1962) [Gl62]: independent set of size 2 in the closed case.**\n\nIf all the sets `A x` are closed with Lebesgue measure `< 1`, then there exist two\ndistinct reals `x y` such that `x ∉ A y` and `y ∉ A x`.\n\nThis is a weaker result proved by Gladysz before the full Newelski–Pawlikowski–\nSeredyński theorem [NPS87]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«501»","statement":"True ↔\n  ∀ (A : ℝ → Set ℝ),\n    (∀ (x : ℝ), IsClosed (A x)) →\n      (∀ (x : ℝ), MeasureTheory.volume (A x) < 1) → ∃ X, 2 ≤ X.ncard ∧ X.Pairwise fun x y => x ∉ A y","subjects":["5","28"],"theorem":"Erdos501.erdos_501.variants.gladysz_size2"},{"answerKinds":[],"category":"test","docstring":"The constant family `A x = ∅` satisfies all hypotheses of the main problem:\neach `A x` is bounded (the empty set is bounded) and has Lebesgue outer measure 0 < 1.\nMoreover, all of ℝ is an independent set, showing the conclusion holds trivially.\n\nThis demonstrates that the hypotheses are non-vacuous: the family `A x = ∅` is a valid\ninput to the theorem, and `ℝ` (which is infinite) witnesses the conclusion. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«501»","statement":"have A := fun x => ∅;\n(∀ (x : ℝ), Bornology.IsBounded (A x)) ∧\n  (∀ (x : ℝ), MeasureTheory.volume.toOuterMeasure (A x) < 1) ∧ ∃ X, X.Infinite ∧ X.Pairwise fun x y => x ∉ A y","subjects":["5","28"],"theorem":"Erdos501.erdos_501.tests.empty_family_is_valid"},{"answerKinds":[],"category":"test","docstring":"The hypothesis `volume.toOuterMeasure (A x) < 1` is strictly satisfied when\n`A x = {x}` (a singleton), since Lebesgue measure of a singleton is 0. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«501»","statement":"∀ (x : ℝ), MeasureTheory.volume.toOuterMeasure {x} < 1","subjects":["5","28"],"theorem":"Erdos501.erdos_501.tests.singleton_outer_measure_lt_one"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Hechler (1972) [He72]: the answer to the main question is NO, assuming the continuum\nhypothesis.**\n\nAssuming CH (`ℵ₁ = 𝔠`), there exists a family `A : ℝ → Set ℝ` of bounded sets with\nLebesgue outer measure `< 1` for which no infinite independent set exists. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«501»","statement":"True ↔\n  Cardinal.aleph 1 = Cardinal.continuum →\n    ∃ A,\n      (∀ (x : ℝ), Bornology.IsBounded (A x)) ∧\n        (∀ (x : ℝ), MeasureTheory.volume.toOuterMeasure (A x) < 1) ∧ ¬∃ X, X.Infinite ∧ X.Pairwise fun x y => x ∉ A y","subjects":["5","28"],"theorem":"Erdos501.erdos_501.variants.hechler_CH"},{"answerKinds":[],"category":"textbook","docstring":"**Trivial lower bound: a single-element set is always independent.**\n\nFor any family `A`, any singleton `{x}` is vacuously independent: there are no two\ndistinct elements. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«501»","statement":"∀ (A : ℝ → Set ℝ) (x : ℝ), {x}.Pairwise fun x y => x ∉ A y","subjects":["5","28"],"theorem":"Erdos501.erdos_501.variants.singleton_independent"},{"answerKinds":[],"category":"test","docstring":"On a graph with no edges there is no book. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«80»","statement":"∀ {n : ℕ}, Erdos80.bookNumber ⊥ = 0","subjects":["5"],"theorem":"Erdos80.bookNumber_bot"},{"answerKinds":[],"category":"API","docstring":"A book is counted by triangles, so `bookNumber` is `0` exactly when no edge lies in one. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«80»","statement":"∀ {α : Type u_1} [inst : Fintype α] (G : SimpleGraph α) [inst_1 : DecidableRel G.Adj],\n  Erdos80.bookNumber G = 0 ↔ ∀ e ∈ G.edgeFinset, G.trianglesContaining e = ∅","subjects":["5"],"theorem":"Erdos80.bookNumber_eq_zero_iff"},{"answerKinds":[],"category":"research open","docstring":"The weaker question from the same problem: is $f_c(n) \\gg \\log n$?\n\nSame feasibility bound on `c` as above.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«80»","statement":"True ↔ ∀ (c : ℝ), 0 < c → c < 1 / 2 → (fun n => Real.log ↑n) =O[Filter.atTop] fun n => ↑(Erdos80.f c n)","subjects":["5"],"theorem":"Erdos80.erdos_80.variants.log"},{"answerKinds":[],"category":"API","docstring":"`EveryEdgeInTriangle` holds vacuously on a graph with no edges, so the admissible set is\nnonempty only through the edge count. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«80»","statement":"∀ {n : ℕ}, Erdos80.EveryEdgeInTriangle ⊥","subjects":["5"],"theorem":"Erdos80.everyEdgeInTriangle_bot"},{"answerKinds":[],"category":"research open","docstring":"Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and\nat least $cn^2$ edges, where each edge is contained in at least one triangle, must contain a\nbook of size $m$, that is, an edge shared by at least $m$ different triangles. Estimate\n$f_c(n)$. In particular, is it true that $f_c(n)>n^\\epsilon$ for some $\\epsilon>0$?\n\nThe bound $c < 1/2$ is what makes the hypothesis satisfiable: a simple graph on $n$ vertices\nhas at most $n(n-1)/2$ edges, so no graph has $cn^2$ of them once $c \\geq 1/2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«80»","statement":"True ↔ ∀ (c : ℝ), 0 < c → c < 1 / 2 → ∃ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑n ^ ε < ↑(Erdos80.f c n)","subjects":["5"],"theorem":"Erdos80.erdos_80"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Offord showed that the number of real roots of a random degree `n` polynomial with `±1`\ncoefficients is `(2/π+o(1))log n`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«522»","statement":"∃ p o,\n  Filter.Tendsto o Filter.atTop (nhds 0) ∧\n    Filter.Tendsto p Filter.atTop (nhds 0) ∧\n      ∀ (Ω : Type u_3) [inst : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume]\n        (n : ℕ),\n        2 ≤ n →\n          ∀ (f : Erdos522.KacCoefficients {-1, 1} Ω MeasureTheory.volume),\n            (MeasureTheory.volume {ω | |↑(f.roots n ω).card / Real.log ↑n - 2 / Real.pi| ≥ o n}).toReal ≤ p n","subjects":["12","60"],"theorem":"Erdos522.erdos_522.variants.number_real_roots"},{"answerKinds":[],"category":"research solved","docstring":"Yakir proved that almost all Kac polynomials have `n/2+O(n^(9/10))` many roots in `{z∈C:|z|≤1}`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«522»","statement":"∃ p,\n  Filter.Tendsto p Filter.atTop (nhds 0) ∧\n    ∀ (Ω : Type u_3) [inst : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume]\n      (n : ℕ),\n      2 ≤ n →\n        ∀ (f : Erdos522.KacCoefficients {-1, 1} Ω MeasureTheory.volume),\n          (MeasureTheory.volume\n                {ω |\n                  |↑(Multiset.countP (fun x => x ∈ Metric.closedBall 0 1) (f.roots n ω)) - ↑n / 2| ≥\n                    ↑n ^ (9 / 10)}).toReal ≤\n            p n","subjects":["12","60"],"theorem":"Erdos522.erdos_522.variants.yakir_solution"},{"answerKinds":[],"category":"research open","docstring":"Let $f(z)=\\sum_{0\\leq k\\leq n} \\epsilon_k z^k$ be a random polynomial, where\n$\\epsilon_k\\in \\{0,1\\}$ independently uniformly at random for $0\\leq k\\leq n$.\n\nIs it true that, if $R_n$ is the number of roots of $f(z)$ in\n$\\{ z\\in \\mathbb{C} : \\lvert z\\rvert \\leq 1\\}$, then\n$$\n  \\frac{R_n}{n/2}\\to 1\n$$\nalmost surely?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«522»","statement":"True ↔\n  ∀ {Ω : Type u_3} [inst : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume]\n    {n : ℕ},\n    1 ≤ n →\n      ∀ (f : Erdos522.KacCoefficients {0, 1} Ω MeasureTheory.volume),\n        MeasureTheory.volume\n            {ω | Filter.Tendsto (fun n => 2 * ↑(f.numRootsInUnitDisk n ω) / ↑n) Filter.atTop (nhds 1)} =\n          1","subjects":["12","60"],"theorem":"Erdos522.erdos_522.variants.zero_one"},{"answerKinds":[],"category":"research open","docstring":"Let $f(z)=\\sum_{0\\leq k\\leq n} \\epsilon_k z^k$ be a random polynomial, where\n$\\epsilon_k\\in \\{-1,1\\}$ independently uniformly at random for $0\\leq k\\leq n$.\n\nIs it true that, if $R_n$ is the number of roots of $f(z)$ in\n$\\{ z\\in \\mathbb{C} : \\lvert z\\rvert \\leq 1\\}$, then\n$$\n  \\frac{R_n}{n/2}\\to 1\n$$\nalmost surely?\n\nThere is some ambiguity as to whether the intended coefficient set is $\\{-1, 1\\}$ or $\\{0, 1\\}$,\nsee `erdos_522.variants.zero_one` for the alternate version.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«522»","statement":"True ↔\n  ∀ {Ω : Type u_3} [inst : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume]\n    (c : Erdos522.KacCoefficients {-1, 1} Ω MeasureTheory.volume),\n    MeasureTheory.volume {ω | Filter.Tendsto (fun n => 2 * ↑(c.numRootsInUnitDisk n ω) / ↑n) Filter.atTop (nhds 1)} = 1","subjects":["12","60"],"theorem":"Erdos522.erdos_522"},{"answerKinds":[],"category":"research solved","docstring":"The 'easy' case of Erdős Problem 1119: if moreover $\\mathfrak{m}^+ < \\mathfrak{c}$, then\nany family of entire functions taking at most $\\mathfrak{m}$ distinct values at each\npoint has cardinality at most $\\mathfrak{m}$. In [Ha74] it is written that this is\n'easy to see'.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1119»","statement":"∀ (m : Cardinal.{0}),\n  Cardinal.aleph0 < m →\n    Order.succ m < Cardinal.continuum →\n      ∀ (F : Set (ℂ → ℂ)),\n        (∀ f ∈ F, Differentiable ℂ f) → (∀ (z₀ : ℂ), Cardinal.mk ↑{y | ∃ f ∈ F, f z₀ = y} ≤ m) → Cardinal.mk ↑F ≤ m","subjects":["3","30"],"theorem":"Erdos1119.erdos_1119.variants.easy_case"},{"answerKinds":[],"category":"research solved","docstring":"Erdős's theorem [Er64g], answering a question of Wetzel: if $\\mathfrak{c} > \\aleph_1$,\nthen every family of entire functions taking only countably many distinct values at each\npoint $z_0 \\in \\mathbb{C}$ is itself countable.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1119»","statement":"Cardinal.aleph 1 < Cardinal.continuum →\n  ∀ (F : Set (ℂ → ℂ)), (∀ f ∈ F, Differentiable ℂ f) → (∀ (z₀ : ℂ), {y | ∃ f ∈ F, f z₀ = y}.Countable) → F.Countable","subjects":["3","30"],"theorem":"Erdos1119.erdos_1119.variants.erdos_wetzel"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er64g] also showed that the previous statement fails under the continuum\nhypothesis: if $\\mathfrak{c} = \\aleph_1$, then there is an **uncountable** family of\nentire functions taking only countably many distinct values at each point\n$z_0 \\in \\mathbb{C}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1119»","statement":"Cardinal.continuum = Cardinal.aleph 1 →\n  ∃ F, (∀ f ∈ F, Differentiable ℂ f) ∧ (∀ (z₀ : ℂ), {y | ∃ f ∈ F, f z₀ = y}.Countable) ∧ ¬F.Countable","subjects":["3","30"],"theorem":"Erdos1119.erdos_1119.variants.erdos_wetzel_ch"},{"answerKinds":[],"category":"research solved","docstring":"Let $\\mathfrak{m}$ be an infinite cardinal with $\\aleph_0 < \\mathfrak{m} < \\mathfrak{c} =\n2^{\\aleph_0}$. Let $\\{f_\\alpha\\}$ be a family of entire functions such that, for every\n$z_0 \\in \\mathbb{C}$, there are at most $\\mathfrak{m}$ distinct values of $f_\\alpha(z_0)$.\nMust $\\{f_\\alpha\\}$ have cardinality at most $\\mathfrak{m}$?\n\nThis is Problem 2.46 in [Ha74], where it is attributed to Erdős. The question is\n**independent of ZFC**, so the headline statement carries `answer(sorry)`: it is neither\nprovable nor refutable from the usual axioms of set theory.\n\nThe answer is yes if $\\mathfrak{m}^+ < \\mathfrak{c}$ (see\n`erdos_1119.variants.easy_case`), so the question reduces to the case\n$\\mathfrak{m}^+ = \\mathfrak{c}$, where it is undecidable: Kumar and Shelah [KuSh17]\nproduced a model of $\\mathfrak{c} = \\aleph_2$ in which the answer is yes (with\n$\\mathfrak{m} = \\aleph_1$), while Schilhan and Weinert [ScWe24] produced a different\nmodel of $\\mathfrak{c} = \\aleph_2$ in which the answer is no.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1119»","statement":"True ↔\n  ∀ (m : Cardinal.{0}),\n    Cardinal.aleph0 < m →\n      m < Cardinal.continuum →\n        ∀ (F : Set (ℂ → ℂ)),\n          (∀ f ∈ F, Differentiable ℂ f) → (∀ (z₀ : ℂ), Cardinal.mk ↑{y | ∃ f ∈ F, f z₀ = y} ≤ m) → Cardinal.mk ↑F ≤ m","subjects":["3","30"],"theorem":"Erdos1119.erdos_1119"},{"answerKinds":[],"category":"test","docstring":"Sanity check: the empty family of entire functions satisfies both the value-bound\nhypothesis and the cardinality conclusion of `erdos_1119`, for any infinite\n$\\mathfrak{m}$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1119»","statement":"∀ (m : Cardinal.{0}),\n  Cardinal.aleph0 < m →\n    (∀ f ∈ ∅, Differentiable ℂ f) ∧ (∀ (z₀ : ℂ), Cardinal.mk ↑{y | ∃ f ∈ ∅, f z₀ = y} ≤ m) ∧ Cardinal.mk ↑∅ ≤ m","subjects":["3","30"],"theorem":"Erdos1119.erdos_1119.test.empty_family"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 70**: Let $\\mathfrak{c}$ be the cardinality of the continuum,\nlet $\\beta$ be a countable ordinal, and let $2 \\le n < \\omega$.\nIs it true that $\\mathfrak{c} \\to (\\beta, n)^3_2$?\n\nNote: The cases $n \\le 3$ are trivially true (see `omega_three`), so the\ngenuine content of the conjecture begins at $n = 4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«70»","statement":"True ↔\n  ∀ (β : Ordinal.{0}) (n : ℕ),\n    β.card ≤ Cardinal.aleph0 → 2 ≤ n → Erdos70.OrdinalCardinalRamsey3 Cardinal.continuum.ord β ↑n","subjects":["3"],"theorem":"Erdos70.erdos_70"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős–Rado partial result**: $\\mathfrak{c} \\to (\\omega + n, 4)^3_2$ for any\n$2 \\le n < \\omega$. Positive partial answer to Problem 70 with $\\beta = \\omega + n$\nand the blue side fixed at $4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«70»","statement":"∀ (n : ℕ), 2 ≤ n → Erdos70.OrdinalCardinalRamsey3 Cardinal.continuum.ord (Ordinal.omega0 + ↑n) 4","subjects":["3"],"theorem":"Erdos70.erdos_70.variants.erdos_rado"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Trivial boundary case**: $\\mathfrak{c} \\to (\\omega, 3)^3_2$.\n\nThis is trivially true because in a 3-uniform hypergraph, a \\\"blue clique of size 3\\\"\nconsists of a single 3-element subset ($\\binom{3}{3} = 1$), so the blue alternative\nmerely asks for one blue triple to exist. The proof splits into two cases:\n- If any blue triple exists, it is itself a blue-monochromatic set of cardinality 3.\n- If no blue triple exists, all triples are red, and since $\\omega \\le \\mathfrak{c}$,\n  any subset of order type $\\omega$ is red-monochromatic.\n\nThe problem becomes non-trivial only for $n \\ge 4$; see `omega_times_two_four` for\nthe simplest genuinely open case.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/c024db0fa3ac32c6dddcd6c28d7b0cd994dad580/FormalConjectures/ErdosProblems/70.lean#L126"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«70»","statement":"True ↔ Erdos70.OrdinalCardinalRamsey3 Cardinal.continuum.ord Ordinal.omega0 3","subjects":["3"],"theorem":"Erdos70.erdos_70.variants.omega_three"},{"answerKinds":[],"category":"research open","docstring":"**The relation at $\\omega_1$**: $\\mathfrak{c} \\to (\\omega_1, n)^3_2$ for finite $n \\ge 2$,\nwhere $\\omega_1 = \\aleph_1$ is the first uncountable ordinal.\n\nNote that $\\omega_1$ is *not* a countable ordinal, so this is not directly an instance of the\nmain Erdős problem (which asks for *countable* $\\beta$). Under CH, $\\omega_1 = \\mathfrak{c}.\\mathrm{ord}$,\nmaking this a self-referential question about $\\mathfrak{c}.\\mathrm{ord} \\to\n(\\mathfrak{c}.\\mathrm{ord}, n)^3_2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«70»","statement":"True ↔ ∀ (n : ℕ), 2 ≤ n → Erdos70.OrdinalCardinalRamsey3 Cardinal.continuum.ord (Cardinal.aleph 1).ord ↑n","subjects":["3"],"theorem":"Erdos70.erdos_70.variants.omega_one"},{"answerKinds":[],"category":"test","docstring":"**Monotonicity of `OrdinalCardinalRamsey3`**:\nIf `OrdinalCardinalRamsey3 α β c` holds and $\\beta' \\le \\beta$, $c' \\le c$, then\n`OrdinalCardinalRamsey3 α β' c'` also holds.\n\nThis allows us to deduce weaker partition results from stronger ones.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«70»","statement":"∀ {α β β' : Ordinal.{u}} {c c' : Cardinal.{u}},\n  Erdos70.OrdinalCardinalRamsey3 α β c → β' ≤ β → c' ≤ c → Erdos70.OrdinalCardinalRamsey3 α β' c'","subjects":["3"],"theorem":"Erdos70.erdos_70.variants.ordinalCardinalRamsey3_mono"},{"answerKinds":[],"category":"research open","docstring":"**First open case beyond Erdős–Rado**: $\\mathfrak{c} \\to (\\omega \\cdot 2, 4)^3_2$.\n\nErdős and Rado proved $\\mathfrak{c} \\to (\\omega + n, 4)^3_2$ for every finite $n \\ge 2$\n(see `erdos_rado`), which covers all red ordinals below $\\omega \\cdot 2 = \\omega + \\omega$.\nThis variant asks whether the result extends to $\\beta = \\omega \\cdot 2$, the simplest\ncountable ordinal not covered by their theorem.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«70»","statement":"True ↔ Erdos70.OrdinalCardinalRamsey3 Cardinal.continuum.ord (Ordinal.omega0 * 2) 4","subjects":["3"],"theorem":"Erdos70.erdos_70.variants.omega_times_two_four"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, for every $c$, there exists an $n$ such that $\\sigma(n)>cn$ but there is no\ncovering system whose moduli all divide $n$?\n\nThis was answered affirmatively by Haight [Ha79].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«277»","statement":"True ↔ ∀ (c : ℝ), ∃ n, ↑((ArithmeticFunction.sigma 1) n) > c * ↑n ∧ ∀ (m : StrictCoveringSystem ℤ), ∃ i, ↑n ∉ m.moduli i","subjects":["11"],"theorem":"Erdos277.erdos_277"},{"answerKinds":[],"category":"research open","docstring":"Does the limit $\\lim_{n\\to\\infty} \\frac{f(2n)}{f(n)}$ tend to infinity?\n\n(Other finite limits have been ruled out by [KoLu25], see below)\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«893»","statement":"True ↔ Filter.Tendsto (fun n => ↑(Erdos893.f (2 * n)) / ↑(Erdos893.f n)) Filter.atTop Filter.atTop","subjects":["5"],"theorem":"Erdos893.erdos_893"},{"answerKinds":[],"category":"research solved","docstring":"Kovač and Luca [KoLu25] (building on a heuristic independently found by\nCambie (personal communication)) have shown that there is no finite limit, in that\n$\\lim_{n\\to\\infty} \\frac{f(2n)}{f(n)}$ is unbounded.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«893»","statement":"¬BddAbove (Set.range fun n => ↑(Erdos893.f (2 * n)) / ↑(Erdos893.f n))","subjects":["5"],"theorem":"Erdos893.erdos_893.variants.unbounded"},{"answerKinds":[],"category":"research solved","docstring":"Schinzel deduced from Pólya's theorem [Po18] (that the sequence of $k$-smooth integers has unbounded\ngaps) that this is true with $p_1\\cdots p_k$ replaced by $p_1\\cdots p_{k-1}p_{k+1}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«891»","statement":"∀ k ≥ 2,\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∃ m ∈ Finset.Ico n (n + (∏ i ∈ Finset.range (k - 1), Nat.nth Nat.Prime i) * Nat.nth Nat.Prime k),\n      k < ArithmeticFunction.cardDistinctFactors m","subjects":["11"],"theorem":"Erdos891.erdos_891.variants.schinzel"},{"answerKinds":[],"category":"research open","docstring":"Let $2=p_1 < p_2 < \\cdots$ be the primes and $k\\geq 2$. Is it true that, for all sufficiently large\n$n$, there must exist an integer in $[n,n+p_1\\cdots p_k)$ with $>k$ many prime factors?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«891»","statement":"True ↔\n  ∀ k ≥ 2,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∃ m ∈ Finset.Ico n (n + ∏ i ∈ Finset.range k, Nat.nth Nat.Prime i), k < ArithmeticFunction.cardDistinctFactors m","subjects":["11"],"theorem":"Erdos891.erdos_891"},{"answerKinds":[],"category":"research open","docstring":"Weisenberg has observed that Dickson's conjecture implies the answer is no if we replace\n$p_1\\cdots p_k$ with $p_1\\cdots p_k-1$. Indeed, let $L_k$ be the lowest common multiple of all\nintegers at most $p_1\\cdots p_k$. By Dickson's conjecture [Wikipedia], there are infinitely many\n$n'$ such that $\\frac{L_k}{m}n'+1$ is prime for all $1\\leq m < p_1\\cdots p_k$. It follows that,\nif $n=L_kn'+1$, then all integers in $[n,n+p_1\\cdots p_k-1)$ have at most $k$ prime factors.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«891»","statement":"∀ k ≥ 2,\n  ∃ᶠ (n : ℕ) in Filter.atTop,\n    ∀ m ∈ Finset.Ico n (n + ∏ i ∈ Finset.range k, Nat.nth Nat.Prime i - 1), ArithmeticFunction.cardDistinctFactors m ≤ k","subjects":["11"],"theorem":"Erdos891.erdos_891.variants.weisenberg"},{"answerKinds":[],"category":"research open","docstring":"This is unknown even for $k=2$ - that is, is it true that in every interval of $6$\n(sufficiently large) consecutive integers there must exist one with at least $3$ prime factors?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«891»","statement":"True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ m ∈ Finset.Ico n (n + 6), 3 ≤ ArithmeticFunction.cardDistinctFactors m","subjects":["11"],"theorem":"Erdos891.erdos_891.variants.case_k_2"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ be a graph with chromatic number $\\aleph_1$. Is it true that there is a colouring of the\nedges with $\\aleph_1$ many colours such that, in any countable colouring of the vertices, there\nexists a vertex colour containing all edge colours?\n\nA problem of Erdős, Galvin, and Hajnal. The consistency of this was proved by Hajnal and Komjáth.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1176»","statement":"True ↔\n  ∀ {V : Type u_1} (G : SimpleGraph V),\n    G.chromaticCardinal = Cardinal.aleph 1 →\n      ∃ EColor,\n        ∃ (_ : Cardinal.mk EColor = Cardinal.aleph 1),\n          ∃ c_edge,\n            ∀ (VColor : Type),\n              Cardinal.mk VColor ≤ Cardinal.aleph 0 →\n                ∀ (c_vert : V → VColor),\n                  ∃ vc,\n                    ∀ (ec : EColor), ∃ u v, ∃ (h : G.Adj u v), c_vert u = vc ∧ c_vert v = vc ∧ c_edge ⟨s(u, v), h⟩ = ec","subjects":["3","5"],"theorem":"Erdos1176.erdos_1176"},{"answerKinds":[],"category":"research solved","docstring":"Cambie makes the stronger conjecture that if $x_1,\\ldots,x_{k^2}$ are distinct positive real\nnumbers with $\\sum x_i=1$ then there is always a monotonic subsequence with sum at least $1/k$.\nThis is a weighted-form of the Erdős-Szekeres theorem, and is also mentioned (as an open\nquestion) in a survey on the latter by Steele [St95].\n\nThis stronger conjecture appears to have been first proved by Tidor, Wang, and Yang [TWY16],\nand is also implicit in work of Wagner [Wa17]. A proof was given and formalised by Aristotle\n(see the comments), with an alternative proof provided by Chan.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1026.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1026»","statement":"∀ (k : ℕ),\n  0 < k →\n    ∀ (x : Fin (k ^ 2) → ℝ),\n      Function.Injective x →\n        (∀ (i : Fin (k ^ 2)), 0 < x i) → ∑ i, x i = 1 → ∃ S ∈ Erdos1026.monotonicSubsequenceSums x, 1 / ↑k ≤ S","subjects":["5"],"theorem":"Erdos1026.erdos_1026.variants.weighted_erdos_szekeres"},{"answerKinds":[],"category":"research solved","docstring":"A construction of Cambie in the comments shows that $c\\leq 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1026»","statement":"1 ∈ upperBounds Erdos1026.admissibleConstants","subjects":["5"],"theorem":"Erdos1026.erdos_1026.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Hanani [Ha57] showed that every sequence is the disjoint union of at most\n$(\\sqrt{2}+o(1))\\sqrt{n}$ many monotonic subsequences.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1026»","statement":"∀ (ε : ℝ),\n  0 < ε →\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (x : Fin n → ℝ),\n        Function.Injective x →\n          ∃ m I,\n            (∀ (j : Fin m), MonotoneOn x ↑(I j) ∨ AntitoneOn x ↑(I j)) ∧\n              (Pairwise fun j k => Disjoint (I j) (I k)) ∧ Finset.univ.biUnion I = Finset.univ ∧ ↑m ≤ (√2 + ε) * √↑n","subjects":["5"],"theorem":"Erdos1026.erdos_1026.variants.hanani"},{"answerKinds":[],"category":"research solved","docstring":"Hanani [Ha57] showed that every sequence is the disjoint union of at most\n$(\\sqrt{2}+o(1))\\sqrt{n}$ many monotonic subsequences, whence $c\\geq 1/\\sqrt{2}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1026»","statement":"1 / √2 ∈ Erdos1026.admissibleConstants","subjects":["5"],"theorem":"Erdos1026.erdos_1026.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Let $x_1,\\ldots,x_n$ be a sequence of distinct real numbers. Determine\n$$\n\\max\\left(\\sum x_{i_r}\\right),\n$$\nwhere the maximum is taken over all monotonic subsequences.\n\nThis is as Erdős posed the problem in [Er71], which is rather ambiguous. Discussion between\nseveral users in the comments section has led to the following precise possible question, as\nposed by van Doorn:\n\nWhat is the largest constant $c$ such that, for all sequences of $n$ real numbers\n$x_1,\\ldots,x_n$,\n$$\n\\max\\left(\\sum x_{i_r}\\right) > (c-o(1))\\frac{1}{\\sqrt{n}}\\sum x_i\n$$\n(where again the maximum is taken over all monotonic subsequences)?\n\nCambie makes the stronger conjecture that if $x_1,\\ldots,x_{k^2}$ are distinct positive real\nnumbers with $\\sum x_i=1$ then there is always a monotonic subsequence with sum at least $1/k$.\n\nThis stronger conjecture appears to have been first proved by Tidor, Wang, and Yang [TWY16],\nand is also implicit in work of Wagner [Wa17]. A proof was given and formalised by Aristotle\n(see the comments), with an alternative proof provided by Chan. In particular, this shows that\n$c=1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1026»","statement":"IsGreatest Erdos1026.admissibleConstants 1","subjects":["5"],"theorem":"Erdos1026.erdos_1026"},{"answerKinds":[],"category":"research open","docstring":"For all $n\\ge 2k$ the least prime factor of $\\binom{n}{k}$ is $\\le\\max(n/k,k)$, with only\nfinitely many exceptions.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1094»","statement":"{(n, k) | 0 < k ∧ 2 * k ≤ n ∧ (n.choose k).minFac > max (n / k) k}.Finite","subjects":["11"],"theorem":"Erdos1094.erdos_1094"},{"answerKinds":[],"category":"research open","docstring":"Let $p(n)$ denote the least prime factor of $n$. Is there a constant $C>0$ such that\n$$\\sum_{x\\leq n\\leq x+C\\sqrt{x}(\\log x)^2}\\frac{p(n)}{n}\\gg 1$$\nfor all sufficiently large $x$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«462»","statement":"True ↔\n  ∃ C c,\n    0 < C ∧ 0 < c ∧ ∀ᶠ (x : ℕ) in Filter.atTop, c ≤ ∑ n ∈ Finset.Icc x ⌊↑x + C * √↑x * Real.log ↑x ^ 2⌋₊, ↑n.minFac / ↑n","subjects":["11"],"theorem":"Erdos462.erdos_462"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for any $a \\in \\mathbb{Z}$, there are infinitely many $n$ such that\n$$\\phi(n) | n + a$$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«828»","statement":"True ↔ ∀ (a : ℤ), {n | ↑n.totient ∣ ↑n + a}.Infinite","subjects":["11"],"theorem":"Erdos828.erdos_828"},{"answerKinds":[],"category":"research open","docstring":"When $n > 1$, Lehmer conjectured that $\\phi(n) | n - 1$ if and only if $n$ is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«828»","statement":"True ↔ ∀ n > 1, n.totient ∣ n - 1 ↔ Prime n","subjects":["11"],"theorem":"Erdos828.erdos_828.variants.lehmer_conjecture"},{"answerKinds":[],"category":"textbook","docstring":"It is an easy exercise to show that $\\phi(n) | n$ if and only if $n = 0, 1$ or $n = 2^a 3^b$ for\nsome $a > 0$.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/03e00cf8d44098d0fb06e891fca30c29769df619/FormalConjectures/ErdosProblems/828.lean#L49"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«828»","statement":"∀ {n : ℕ}, n.totient ∣ n ↔ n ≤ 1 ∨ ∃ a > 0, ∃ b, n = 2 ^ a * 3 ^ b","subjects":["11"],"theorem":"Erdos828.erdos_828.variants.phi_dvd_self_iff_pow2_pow3"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to\ncolour the vertices of $G$ such that each colour class induces either a complete graph or empty\ngraph.\n\nIs it true that if $G$ has no $K_5$ and $\\zeta(G)\\geq 4$ then $\\chi(G) \\leq \\zeta(G)+2$?\n\nThis has been disproved by Steiner [St24b], who constructed a graph $G$ with $\\omega(G)=4$,\n$\\zeta(G)=4$, and $\\chi(G)=7$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos762.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«762»","statement":"False ↔\n  ∀ (V : Type u_1) [Fintype V] (G : SimpleGraph V),\n    G.CliqueFree 5 → 4 ≤ G.cochromaticNumber → G.chromaticNumber ≤ G.cochromaticNumber + 2","subjects":["5"],"theorem":"Erdos762.erdos_762"},{"answerKinds":[],"category":"research solved","docstring":"This has been disproved by Steiner [St24b], who constructed a graph $G$ with $\\omega(G)=4$,\n$\\zeta(G)=4$, and $\\chi(G)=7$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«762»","statement":"∃ n G, G.cliqueNum = 4 ∧ G.cochromaticNumber = 4 ∧ G.chromaticNumber = 7","subjects":["5"],"theorem":"Erdos762.erdos_762.variants.steiner"},{"answerKinds":[],"category":"research solved","docstring":"A conjecture of Erdős, Gimbel, and Straight [EGS90], who proved that for every $n>2$ there exists\nsome $f(n)$ such that if $G$ contains no clique on $n$ vertices then $\\chi(G)\\leq \\zeta(G)+f(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«762»","statement":"∀ (n : ℕ),\n  2 < n →\n    ∃ f, ∀ (V : Type u_1) [Fintype V] (G : SimpleGraph V), G.CliqueFree n → G.chromaticNumber ≤ G.cochromaticNumber + ↑f","subjects":["5"],"theorem":"Erdos762.erdos_762.variants.bounded_clique_number"},{"answerKinds":[],"category":"research solved","docstring":"A number of improvements of the constant $\\frac{1}{4}$ have been given, with the current\nrecord $\\sqrt{2 / \\pi}$ first provided in unpublished work of Elkies and Gleason.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1»","statement":"∃ o,\n  ∃ (_ : o =o[Filter.atTop] 1),\n    ∀ (N : ℕ) (A : Finset ℕ), Erdos1.IsSumDistinctSet A N → (√(2 / Real.pi) - o A.card) * 2 ^ A.card / √↑A.card ≤ ↑N","subjects":["5","11"],"theorem":"Erdos1.erdos_1.variants.lb_strong"},{"answerKinds":[],"category":"research solved","docstring":"The minimal value of $N$ such that there exists a sum-distinct set with nine\nelements is $161$.\n\nhttps://oeis.org/A276661\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1»","statement":"IsLeast {N | ∃ A, Erdos1.IsSumDistinctSet A N ∧ A.card = 9} 161","subjects":["5","11"],"theorem":"Erdos1.erdos_1.variants.least_N_9"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Moser [Er56] proved\n$$\n  N \\geq (\\tfrac{1}{4} - o(1)) \\frac{2^n}{\\sqrt{n}}.\n$$\n\n[Er56] Erdős, P., _Problems and results in additive number theory_. Colloque sur la Th\\'{E}orie des Nombres, Bruxelles, 1955 (1956), 127-137.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1»","statement":"∃ o,\n  ∃ (_ : o =o[Filter.atTop] 1),\n    ∀ (N : ℕ) (A : Finset ℕ), Erdos1.IsSumDistinctSet A N → (1 / 4 - o A.card) * 2 ^ A.card / √↑A.card ≤ ↑N","subjects":["5","11"],"theorem":"Erdos1.erdos_1.variants.lb"},{"answerKinds":[],"category":"research solved","docstring":"The minimal value of $N$ such that there exists a sum-distinct set with five\nelements is $13$.\n\nhttps://oeis.org/A276661\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1»","statement":"IsLeast {N | ∃ A, Erdos1.IsSumDistinctSet A N ∧ A.card = 5} 13","subjects":["5","11"],"theorem":"Erdos1.erdos_1.variants.least_N_5"},{"answerKinds":[],"category":"textbook","docstring":"The trivial lower bound is $N \\gg 2^n / n$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1»","statement":"∃ C > 0, ∀ (N : ℕ) (A : Finset ℕ), Erdos1.IsSumDistinctSet A N → N ≠ 0 → C * 2 ^ A.card / ↑A.card < ↑N","subjects":["5","11"],"theorem":"Erdos1.erdos_1.variants.weaker"},{"answerKinds":[],"category":"research open","docstring":"A generalisation of the problem to sets $A \\subseteq (0, N]$ of real numbers, such that the subset\nsums all differ by at least $1$ is proposed in [Er73] and [ErGr80].\n\n[Er73] Erdős, P., _Problems and results on combinatorial number theory_. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[ErGr80] Erdős, P. and Graham, R., _Old and new problems and results in combinatorial number theory_. Monographies de L'Enseignement Mathematique (1980).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1»","statement":"∃ C > 0, ∀ (N : ℕ) (A : Finset ℝ), Erdos1.IsSumDistinctRealSet A N → N ≠ 0 → C * 2 ^ A.card < ↑N","subjects":["5","11"],"theorem":"Erdos1.erdos_1.variants.real"},{"answerKinds":[],"category":"textbook","docstring":"The minimal value of $N$ such that there exists a sum-distinct set with three\nelements is $4$.\n\nhttps://oeis.org/A276661\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1»","statement":"IsLeast {N | ∃ A, Erdos1.IsSumDistinctSet A N ∧ A.card = 3} 4","subjects":["5","11"],"theorem":"Erdos1.erdos_1.variants.least_N_3"},{"answerKinds":[],"category":"research open","docstring":"If $A\\subseteq\\{1, ..., N\\}$ with $|A| = n$ is such that the subset sums $\\sum_{a\\in S}a$ are\ndistinct for all $S\\subseteq A$ then\n$$\n  N \\gg 2 ^ n.\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1»","statement":"∃ C > 0, ∀ (N : ℕ) (A : Finset ℕ), Erdos1.IsSumDistinctSet A N → N ≠ 0 → C * 2 ^ A.card < ↑N","subjects":["5","11"],"theorem":"Erdos1.erdos_1"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $F(n)$ be the maximal size of a family of subsets of $\\{1,\\ldots,n\\}$ such that no set in\nthis family is the union of other members of the family. Is it true that there is a constant\n$c>0$ such that\n$$F(n)\\sim c \\frac{2^n}{n^{1/2}}?$$\n\nHunter observes in the comments that this follows from the solution to [447], which implies\n$F(n)\\sim \\binom{n}{n/2}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1023.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1023»","statement":"True ↔ ∃ c, 0 < c ∧ Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Erdos1023.F n)) fun n => c * 2 ^ n / ↑n ^ (1 / 2)","subjects":["5"],"theorem":"Erdos1023.erdos_1023"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Kleitman proved in unpublished work that\n$$F(n)\\asymp \\frac{2^n}{n^{1/2}}.$$\n([Er71] has an exponent of $3/2$, but this is presumably a typo.)\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1023»","statement":"(fun n => ↑(Erdos1023.F n)) =Θ[Filter.atTop] fun n => 2 ^ n / ↑n ^ (1 / 2)","subjects":["5"],"theorem":"Erdos1023.erdos_1023.variants.erdos_kleitman"},{"answerKinds":[],"category":"research solved","docstring":"Hunter observes in the comments that this follows from the solution to [447], which implies\n$F(n)\\sim \\binom{n}{n/2}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1023»","statement":"Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Erdos1023.F n)) fun n => ↑(n.choose (n / 2))","subjects":["5"],"theorem":"Erdos1023.erdos_1023.variants.hunter"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq \\mathbb{N}$ be a set of density zero. Does there exist a $B$ such that\n$A\\subseteq B+B$ and\n$$\\lvert B\\cap \\{1,\\ldots,N\\}\\rvert =o(N^{1/2})$$\nfor all large $N$?\n\nThe answer is no. Erdős and Newman [ErNe77] have proved this is true when $A$ is the set of\nsquares. In fact, Theorem 2 of [ErNe77] already implies a negative answer to this problem, but\nthis seems to have been overlooked by Erdős and Graham.\n\nSee also [806].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos333.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«333»","statement":"False ↔\n  ∀ (A : Set ℕ), A.HasDensity 0 → ∃ B, A ⊆ B + B ∧ (fun N => ↑(B ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => √↑N","subjects":["11"],"theorem":"Erdos333.erdos_333"},{"answerKinds":[],"category":"research open","docstring":"Is there a graph of chromatic number `ℵ_ 1` with `ℵ_ 1` vertices such that for all\n`ε > 0`, if `n` is sufficiently large and `H` is a subgraph on `n` vertices,\nthen `H` contains an independent set of size `> n ^ (1 - ε)`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«75»","statement":"True ↔\n  ∃ V G,\n    G.chromaticCardinal = Cardinal.aleph 1 ∧\n      Cardinal.mk V = Cardinal.aleph 1 ∧\n        ∀ ε > 0,\n          ∀ᶠ (n : ℕ) in Filter.atTop,\n            ∀ (H : G.Subgraph), H.verts.ncard = n → ∃ I, ↑I ⊆ H.verts ∧ G.IsIndepSet ↑I ∧ ↑I.card > ↑n ^ (1 - ε)","subjects":["5"],"theorem":"Erdos75.erdos_75"},{"answerKinds":[],"category":"research open","docstring":"Let `q : ℕ → ℕ` be a strictly increasing sequence of primes such that\n`q (n + 2) - q (n + 1) ≥ q (n + 1) - q n`. Must `lim q n / (n ^ 2) = ∞`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«455»","statement":"True ↔\n  ∀ (q : ℕ → ℕ),\n    StrictMono q →\n      (∀ (n : ℕ), Nat.Prime (q n) ∧ q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) →\n        Filter.Tendsto (fun n => ↑(q n) / ↑n ^ 2) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos455.erdos_455"},{"answerKinds":[],"category":"research solved","docstring":"Let `q : ℕ → ℕ` be a strictly increasing sequence of primes such that\n`q (n + 2) - q (n + 1) ≥ q (n + 1) - q n`. Then `liminf q n / (n ^ 2) > 0.352`, and this is proved in\n[Ri76]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«455»","statement":"∀ (q : ℕ → ℕ),\n  StrictMono q →\n    (∀ (n : ℕ), Nat.Prime (q n) ∧ q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) →\n      Filter.liminf (fun n => ↑(q n) / ↑n ^ 2) Filter.atTop > 0.352","subjects":["11"],"theorem":"Erdos455.erdos_455.variants.liminf"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a bipartite graph on $n$ vertices such that one part has $\\lfloor n^{2/3}\\rfloor$\nvertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must\ncontain a $C_6$?\n\nThe answer is no, as shown by De Caen and Székely [DeSz92], who in fact show a stronger result.\nLet $f(n,m)$ be the maximum number of edges of a bipartite graph between $n$ and $m$ vertices which\ndoes not contain either a $C_4$ or $C_6$. A positive answer to this question would then imply\n$f(n,\\lfloor n^{2/3}\\rfloor)\\ll n$. De Caen and Székely prove\n$n^{10/9}\\gg f(n,\\lfloor n^{2/3}\\rfloor) \\gg n^{58/57+o(1)}$ for $m\\sim n^{2/3}$. They also prove\nmore generally that, for $n^{1/2}\\leq m\\leq n$, $f(n,m) \\ll (nm)^{2/3},$ which was also proved by\nFaudree and Simonovits.\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos1080.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1080»","statement":"False ↔\n  ∃ c > 0,\n    ∀ (V : Type) [inst : Fintype V] [Nonempty V] (G : SimpleGraph V) (X Y : Set V),\n      Erdos1080.IsBipartition G X Y →\n        X.ncard = ⌊↑(Fintype.card V) ^ (2 / 3)⌋₊ →\n          ↑G.edgeSet.ncard ≥ c * ↑(Fintype.card V) → ∃ v walk, walk.IsCycle ∧ walk.length = 6","subjects":["5"],"theorem":"Erdos1080.erdos_1080"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«91»","statement":"Erdos91.IsOptimal Erdos91.equiTriangle 3","subjects":["52"],"theorem":"Erdos91.erdos_91.test.equiTriangle_optimal"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"Erdos91.IsOptimal Erdos91.unitSquare 4","subjects":["52"],"theorem":"Erdos91.erdos_91.test.unitSquare_optimal"},{"answerKinds":[],"category":"research solved","docstring":"For $n = 3$ the equilateral triangle is the only such set.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"Erdos91.UniqueMinimizer 3","subjects":["52"],"theorem":"Erdos91.erdos_91.variants.three"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"Erdos91.IsOptimal Erdos91.wheelSeven 7","subjects":["52"],"theorem":"Erdos91.erdos_91.test.wheelSeven_optimal"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"¬Erdos91.DilationEquivSimilar Erdos91.circleSeven Erdos91.wheelSeven","subjects":["52"],"theorem":"Erdos91.erdos_91.test.dissimilar_circleSeven_wheelSeven"},{"answerKinds":[],"category":"research open","docstring":"Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct\ndistances between points in $A$. Prove that for large $n$ there are at least two\n(and probably many) such $A$ which are non-similar.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"(∀ᶠ (n : ℕ) in Filter.atTop, ¬Erdos91.UniqueMinimizer n) ↔ True","subjects":["52"],"theorem":"Erdos91.erdos_91"},{"answerKinds":[],"category":"research solved","docstring":"In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 9$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"¬Erdos91.UniqueMinimizer 9","subjects":["52"],"theorem":"Erdos91.erdos_91.variants.nine"},{"answerKinds":[],"category":"research solved","docstring":"In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 7$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"¬Erdos91.UniqueMinimizer 7","subjects":["52"],"theorem":"Erdos91.erdos_91.variants.seven"},{"answerKinds":[],"category":"research solved","docstring":"For $n=4$ the square or two equilateral triangles sharing an edge give two\nnon-similar examples.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"¬Erdos91.UniqueMinimizer 4","subjects":["52"],"theorem":"Erdos91.erdos_91.variants.four"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"Erdos91.IsOptimal Erdos91.circleSeven 7","subjects":["52"],"theorem":"Erdos91.erdos_91.test.circleSeven_optimal"},{"answerKinds":[],"category":"research solved","docstring":"In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 6$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"¬Erdos91.UniqueMinimizer 6","subjects":["52"],"theorem":"Erdos91.erdos_91.variants.six"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), Erdos91.IsOptimal A 3 → Erdos91.DilationEquivSimilar A Erdos91.equiTriangle","subjects":["52"],"theorem":"Erdos91.erdos_91.test.equiTriangle_unique_optimal"},{"answerKinds":[],"category":"research solved","docstring":"For $n = 5$ the regular pentagon is the unique such set (which has two distinct distances).\nErdős mysteriously remarks in [Er90] this was proved by 'a colleague'. (In [Er87b] this is\ndescribed as 'a colleague from Zagreb (unfortunately I do not have his letter)'.)\nA published proof of this fact is provided by Kovács [Ko24c].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"Erdos91.UniqueMinimizer 5","subjects":["52"],"theorem":"Erdos91.erdos_91.variants.five"},{"answerKinds":[],"category":"research solved","docstring":"In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 8$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«91»","statement":"¬Erdos91.UniqueMinimizer 8","subjects":["52"],"theorem":"Erdos91.erdos_91.variants.eight"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as\ndisjoint as possible'?\n\nThat is, for all $d\\mid n$ with $d>1$ there is an associated $a_d$ such that every integer is\ncongruent to some $a_d\\pmod{d}$, and if there is some integer $x$ with\n$$x\\equiv a_d\\pmod{d}\\textrm{ and }x\\equiv a_{d'}\\pmod{d'}$$then $(d,d')=1$.\n\nThe density of such $n$ is zero. Erdős and Graham believed that no such $n$ exist.\n\nAdenwalla [Ad25] has proved there are no such $n$.\n\nThis was formalized by van Doorn in Lean using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem204.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«204»","statement":"False ↔\n  ∃ n a,\n    have D := {d | d ∣ n ∧ d > 1};\n    (∀ (x : ℤ), ∃ d ∈ D, x ≡ a d [ZMOD ↑d]) ∧\n      ∀ d ∈ D, ∀ d' ∈ D, d ≠ d' → (∃ x, x ≡ a d [ZMOD ↑d] ∧ x ≡ a d' [ZMOD ↑d']) → d.gcd d' = 1","subjects":["5"],"theorem":"Erdos204.erdos_204"},{"answerKinds":[],"category":"research solved","docstring":"Alweiss has observed a lower bound of $\\binom{n + 1}{2}$ follows from considering the subset of\n$\\mathbb{R}^{n + 1}$ formed of all vectors $e_i + e_j$ where $e_i$, $e_j$ are distinct coordinate\nvectors. This set can be viewed as a subset of some $\\mathbb{R}^n$, and is easily checked to have\nthe required property.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«503»","statement":"∀ (n : ℕ), (n + 1).choose 2 ≤ sSup {x | ∃ A, ∃ (_ : A.IsIsosceles), A.ncard = x}","subjects":["51"],"theorem":"Erdos503.erdos_503.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"When $n = 3$, the answer is 8 (due to Croft [Cr62]).\n\n[Cr62] Croft, H. T., $9$-point and $7$-point configurations in $3$-space. Proc. London Math. Soc. (3) (1962), 400-424.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«503»","statement":"IsGreatest {x | ∃ A, ∃ (_ : A.IsIsosceles), A.ncard = x} 8","subjects":["51"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos503.erdos_503.variants.R3"},{"answerKinds":[],"category":"research solved","docstring":"When $n = 2$, the answer is 6 (due to Kelly [ErKe47] - an alternative proof is given by Kovács [Ko24c]).\n\n[ErKe47] Erdős, Paul and Kelly, L. M., Elementary Problems and Solutions: Solutions: E735. Amer. Math. Monthly (1947), 227-229.\n[Ko24c] Z. Kovács, A note on Erdős's mysterious remark. arXiv:2412.05190 (2024).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«503»","statement":"IsGreatest {x | ∃ A, ∃ (_ : A.IsIsosceles), A.ncard = x} 6","subjects":["51"],"theorem":"Erdos503.erdos_503.variants.R2"},{"answerKinds":[],"category":"research solved","docstring":"The best upper bound known in general is due to Blokhius [Bl84] who showed that\n$$\n|A| \\leq \\binom{n + 2}{2}\n$$\n\n[Bl84] Blokhuis, A., Few-distance sets. (1984), iv+70.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«503»","statement":"∀ (n m : ℕ), m ∈ {x | ∃ A, ∃ (_ : A.IsIsosceles), A.ncard = x} → m ≤ (n + 2).choose 2","subjects":["51"],"theorem":"Erdos503.erdos_503.variants.upper_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the size of the largest $A \\subseteq \\mathbb{R}^n$ such that every three points from $A$\ndetermine an isosceles triangle? That is, for any three points $x$, $y$, $z$ from $A$, at least two\nof the distances $|x - y|$, $|y - z|$, $|x - z|$ are equal.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«503»","statement":"∀ (n : ℕ), IsGreatest {x | ∃ A, ∃ (_ : A.IsIsosceles), A.ncard = x} sorry","subjects":["51"],"theorem":"Erdos503.erdos_503"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such\nthat\n$$\n\\max_{1\\leq k\\leq n}\\left\\lvert \\sum_{i}z_i^k\\right\\rvert>c?\n$$\n\nAtkinson proved that $c=1/6$ suffices.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos519.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«519»","statement":"True ↔ ∃ c, 0 < c ∧ ∀ (n : ℕ) (hn : 0 < n) (z : Fin n → ℂ), z ⟨0, hn⟩ = 1 → ∃ k, c < ‖Erdos519.powerSum z (↑k + 1)‖","subjects":["30"],"theorem":"Erdos519.erdos_519"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er48] proved that this is true if $t\\geq 2$ is an integer.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1049»","statement":"∀ t ≥ 2, Irrational (∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1))","subjects":["11"],"theorem":"Erdos1049.erdos_1049.variants.geq_2_integer"},{"answerKinds":[],"category":"research open","docstring":"Let $t>1$ be a rational number. Is\n$\\sum_{n=1}^\\infty\\frac{1}{t^n-1}=\\sum_{n=1}^\\infty \\frac{\\tau(n)}{t^n}$ irrational, where\n$\\tau(n)$ counts the divisors of $n$?\n\nA conjecture of Chowla.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1049»","statement":"True ↔ ∀ t > 1, Irrational (∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1))","subjects":["11"],"theorem":"Erdos1049.erdos_1049"},{"answerKinds":[],"category":"textbook","docstring":"The classical Lambert series identity: $\\sum_{n=1}^\\infty \\frac{1}{t^n - 1} =\n\\sum_{n=1}^\\infty \\frac{\\tau(n)}{t^n}$, where $\\tau(n)$ counts the divisors of $n$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1049»","statement":"∀ (t : ℚ), ∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / ↑t ^ ↑n","subjects":["11"],"theorem":"Erdos1049.lambert_series_eq_num_divisor_sum"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős Problem 139**:\nLet $r_k(N)$ be the size of the largest subset of ${1,...,N}$ which does not contain a non-trivial\n$k$-term arithmetic progression. Prove that $r_k(N) = o(N)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«139»","statement":"∀ (k : ℕ), 1 < k → Filter.Tendsto (fun N => ↑(Erdos139.r k N) / ↑N) Filter.atTop (nhds 0)","subjects":["5","11"],"theorem":"Erdos139.erdos_139"},{"answerKinds":[],"category":"research open","docstring":"Let $f$ be a Rademacher multiplicative function.\nDoes there exist some constant $c > 0$ such that, almost surely,\n$$\n  \\limsup_{N \\to \\infty} \\frac{\\sum_{m \\leq N} f(m)}{\\sqrt{N \\log \\log N}} = c?\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«520»","statement":"True ↔\n  ∃ c > 0,\n    ∀ (Ω : Type) [inst : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume]\n      (f : ℕ → Ω → ℝ),\n      Erdos520.IsRademacherMultiplicative f →\n        ∀ᵐ (ω : Ω), Filter.limsup (fun N => ∑ m ≤ N, f m ω / √(↑N * Real.log (Real.log ↑N))) Filter.atTop = c","subjects":["11","60"],"theorem":"Erdos520.erdos_520"},{"answerKinds":[],"category":"research solved","docstring":"Selfridge and Straus [SeSt58] showed that the conjecture is true\nwhen $k = 4$ and $|A| > 12$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«494»","statement":"∀ card > 12, Erdos494.Erdos494Unique 4 card","subjects":["5"],"theorem":"Erdos494.erdos_494.variants.k_eq_4_card_gt_12"},{"answerKinds":[],"category":"research solved","docstring":"A counterexample to the product version of the conjecture (by Steinerberger). ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/hjyuh/formal-conjectures/blob/e0da6ec78953b17618895a093d4bee90fd3f6f67/FormalConjectures/ErdosProblems/494.lean#L951"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«494»","statement":"∃ A B, A.card = B.card ∧ Erdos494.prodMultiset A 3 = Erdos494.prodMultiset B 3 ∧ A ≠ B","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos494.erdos_494.variants.product"},{"answerKinds":[],"category":"research solved","docstring":"Gordon, Fraenkel, and Straus [GRS62] proved that the claim is true for all $k > 2$ when\n$|A|$ is sufficiently large.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«494»","statement":"∀ k > 2, ∀ᶠ (card : ℕ) in Filter.atTop, Erdos494.Erdos494Unique k card","subjects":["5"],"theorem":"Erdos494.erdos_494.variants.gordon_fraenkel_straus"},{"answerKinds":[],"category":"research solved","docstring":"Kruyt noted that the conjecture fails when $|A| = k$, by rotating $A$ around an appropriate point.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/hjyuh/formal-conjectures/blob/e0da6ec78953b17618895a093d4bee90fd3f6f67/FormalConjectures/ErdosProblems/494.lean#L592"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«494»","statement":"∀ k > 2, ¬Erdos494.Erdos494Unique k k","subjects":["5"],"theorem":"Erdos494.erdos_494.variants.k_eq_card"},{"answerKinds":[],"category":"research solved","docstring":"Selfridge and Straus [SeSt58] gave counterexamples to the conjecture\nwhen $k = 2$ and $|A| = 2^l$.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/hjyuh/formal-conjectures/blob/e0da6ec78953b17618895a093d4bee90fd3f6f67/FormalConjectures/ErdosProblems/494.lean#L533"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«494»","statement":"∀ (card : ℕ), (∃ l, card = 2 ^ l) → ¬Erdos494.Erdos494Unique 2 card","subjects":["5"],"theorem":"Erdos494.erdos_494.variants.k_eq_2_card_pow_two"},{"answerKinds":[],"category":"research solved","docstring":"Selfridge and Straus [SeSt58] also showed that the conjecture is true when\n1) $k = 3$ and $|A| > 6$ or\n2) $k = 4$ and $|A| > 12$.\nMore generally, they proved that $A$ is determined by $A_k$ (and $|A|$) if $|A|$ is divisible by\na prime greater than $k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«494»","statement":"∀ card > 6, Erdos494.Erdos494Unique 3 card","subjects":["5"],"theorem":"Erdos494.erdos_494.variants.k_eq_3_card_gt_6"},{"answerKinds":[],"category":"research solved","docstring":"Selfridge and Straus [SeSt58] proved that $A$ is determined by $A_k$\nif $|A|$ is divisible by a prime greater than $k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«494»","statement":"∀ (k card p : ℕ), Nat.Prime p → k ∈ Set.Ioo 0 p → p ∣ card → Erdos494.Erdos494Unique k card","subjects":["5"],"theorem":"Erdos494.erdos_494.variants.card_divisible_by_prime_gt_k"},{"answerKinds":[],"category":"research solved","docstring":"Similarly, Tao noted that the conjecture fails when $|A| = 2k$, by taking $A$ to be a set of\nthe total sum 0 and considering $-A$.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/hjyuh/formal-conjectures/blob/e0da6ec78953b17618895a093d4bee90fd3f6f67/FormalConjectures/ErdosProblems/494.lean#L916"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«494»","statement":"∀ k > 2, ¬Erdos494.Erdos494Unique k (2 * k)","subjects":["5"],"theorem":"Erdos494.erdos_494.variants.card_eq_2k"},{"answerKinds":[],"category":"research solved","docstring":"Selfridge and Straus [SeSt58] showed that the conjecture is true when $k = 2$ and\n$|A| \\ne 2^l$ for $l \\ge 0$.\nThey also gave counterexamples when $k = 2$ and $|A| = 2^l$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«494»","statement":"∀ (card : ℕ), (∀ (l : ℕ), card ≠ 2 ^ l) → Erdos494.Erdos494Unique 2 card","subjects":["5"],"theorem":"Erdos494.erdos_494.variants.k_eq_2_card_not_pow_two"},{"answerKinds":[],"category":"research solved","docstring":"There exists a set `A` with positive density that does not have property `P₁`.\n#TODO: prove this lemma by assuming `erdos_318.contain_single_even`.\n\nThe density sits in an existential, so `HasPosDensity` is the *stronger* reading here and\nweakening it to positive lower density would claim less, which is the opposite of the usual\nsituation for Erdős' \"positive density\". It also costs nothing: by\n`erdos_318.variants.contain_single_even` a witness only needs exactly one even element, and the\nodd numbers together with one even number have density `1 / 2` on the nose. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«318»","statement":"∃ A, A.HasPosDensity ∧ ¬Erdos318.P₁ A","subjects":["11"],"theorem":"Erdos318.erdos_318.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Sattler proved in [Sa75] that the set of odd numbers has property `P₁`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«318»","statement":"Erdos318.P₁ {n | Odd n}","subjects":["11"],"theorem":"Erdos318.erdos_318.variants.odd"},{"answerKinds":[],"category":"test","docstring":"The set of squares does not have property `P₁`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«318»","statement":"¬Erdos318.P₁ {n | IsSquare n}","subjects":["11"],"theorem":"Erdos318.erdos_318.variants.squares"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does the set of squares excluding 1 have property `P₁`?\n\nLarsen [La26] proved that this set does have property `P₁`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«318»","statement":"True ↔ Erdos318.P₁ ({n | IsSquare n} \\ {1})","subjects":["11"],"theorem":"Erdos318.erdos_318.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"`ℕ` has property `P₁`. This is proved in [ErSt75]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«318»","statement":"Erdos318.P₁ Set.univ","subjects":["11"],"theorem":"Erdos318.erdos_318.variants.univ"},{"answerKinds":[],"category":"research solved","docstring":"For any set `A` containing exactly one even number, `A` does not have property `P₁`. Sattler\n[Sa82] credits this observation to Erdős, who presumably found this after [ErGr80]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«318»","statement":"∀ {A : Set ℕ}, {n | n ∈ A ∧ Even n}.ncard = 1 → ¬Erdos318.P₁ A","subjects":["11"],"theorem":"Erdos318.erdos_318.variants.contain_single_even"},{"answerKinds":[],"category":"research solved","docstring":"Every infinite arithmetic progression has property `P₁`. This is proved in [Sa82b]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«318»","statement":"∀ {A : Set ℕ}, A.IsAPOfLength ⊤ → Erdos318.P₁ A","subjects":["11"],"theorem":"Erdos318.erdos_318.variants.infinite_AP"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f:\\mathbb{R}\\to \\mathbb{R}$ be such that\n$$2f(x) \\leq f(x+h)+f(x+2h)$$\nfor every $x\\in \\mathbb{R}$ and $h>0$. Must $f$ be monotonic?\n\nA problem of Kemperman [Ke69], who proved it is true if $f$ is measurable. Erdős [Er81b] wrote 'if it were my problem I would offer \\$500 for it'. This was solved by Laczkovich [La84].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1125.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1125»","statement":"True ↔ ∀ (f : ℝ → ℝ), (∀ (x h : ℝ), h > 0 → 2 * f x ≤ f (x + h) + f (x + 2 * h)) → Monotone f","subjects":["26"],"theorem":"Erdos1125.erdos_1125"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq\\mathbb{N}$ be an infinite set such that $|A\\cap \\{1, ..., N\\}| = o(N)$.\nIs it true that\n$$\n\\limsup_{N\\to\\infty}\\frac{|(A - A)\\cap \\{1, ..., N\\}|}{|A \\cap \\{1, ..., N\\}|} = \\infty?\n$$\n\nThe answer is yes, proved by Ruzsa [Ru78].\n\n[Ru78] Ruzsa, I. Z., _On the cardinality of {$A+A$}\\ and {$A-A$}_. (1978), 933--938.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos899.lean#L793"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«899»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    A.Infinite →\n      Filter.Tendsto (fun N => ↑(A ∩ Set.Icc 1 N).ncard / ↑N) Filter.atTop (nhds 0) →\n        Filter.limsup (fun N => ↑((A - A) ∩ Set.Icc 1 N).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop = ⊤","subjects":["5"],"theorem":"Erdos899.erdos_899"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The anti-Ramsey number $\\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the\nedges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges\nhave different colours).\n\nLet $C_k$ be the cycle on $k$ vertices. Is it true that\n$\\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1)$?\n\nMontellano-Ballesteros and Neumann-Lara [MoNe05] gave an exact formula for $\\mathrm{AR}(n,C_k)$,\nwhich implies in particular that\n$\\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1).$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1105»","statement":"True ↔\n  ∀ (k : ℕ),\n    3 ≤ k →\n      (fun n => ↑((SimpleGraph.cycleGraph k).antiRamseyNum n) - ((↑k - 2) / 2 + 1 / (↑k - 1)) * ↑n) =O[Filter.atTop]\n        fun x => 1","subjects":["5"],"theorem":"Erdos1105.erdos_1105.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $P_k$ be the path on $k$ vertices and $\\ell=\\lfloor\\frac{k-1}{2}\\rfloor$. If $n\\geq k\\geq 5$\nthen is $\\mathrm{AR}(n,P_k)$ equal to $\\max\\left(\\binom{k-2}{2}+1,\n\\binom{\\ell-1}{2}+(\\ell-1)(n-\\ell+1)+\\epsilon\\right)$where $\\epsilon=1$ if $k$ is odd and\n$\\epsilon=2$ otherwise?\n\nA proof of the formula for $\\mathrm{AR}(n,P_k)$ for all $n\\geq k\\geq 5$ has been announced by\nYuan [Yu21].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1105»","statement":"True ↔\n  ∀ (k n : ℕ),\n    5 ≤ k →\n      k ≤ n →\n        have ℓ := (k - 1) / 2;\n        have ε := if Odd k then 1 else 2;\n        (SimpleGraph.pathGraph k).antiRamseyNum n =\n          max ((k - 2).choose 2 + 1) ((ℓ - 1).choose 2 + (ℓ - 1) * (n - ℓ + 1) + ε)","subjects":["5"],"theorem":"Erdos1105.erdos_1105.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"The observation of Zachary Hunter in [that question](https://mathoverflow.net/q/410808)\ncoupled with the bounds of Kelley-Meka [KeMe23](https://arxiv.org/abs/2302.05537) imply that\n$$h(N) \\gg \\exp(c(\\log N)^{\\frac 1 {12}})$$\nfor some $c > 0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«160»","statement":"∃ c > 0, (fun n => Real.exp (c * Real.log ↑n ^ (1 / 12))) =O[Filter.atTop] fun n => ↑(Erdos160.erdos_160.h n)","subjects":["5","51"],"theorem":"Erdos160.erdos_160.variants.known_lower"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Estimate $h(n)$ by finding a better upper bound.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«160»","statement":"have upper_bound := sorry;\n(fun n => ↑(Erdos160.erdos_160.h n)) =O[Filter.atTop] upper_bound ∧ upper_bound =o[Filter.atTop] fun n => ↑n ^ (2 / 3)","subjects":["5","51"],"theorem":"Erdos160.erdos_160.better_upper"},{"answerKinds":[],"category":"research solved","docstring":"On [Mathoverflow](https://mathoverflow.net/a/410815) user\n[leechlattice](https://mathoverflow.net/users/125498/leechlattice) shows that\n$h(n) \\ll n^{\\frac 2 3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«160»","statement":"(fun n => ↑(Erdos160.erdos_160.h n)) =O[Filter.atTop] fun n => ↑n ^ (2 / 3)","subjects":["5","51"],"theorem":"Erdos160.erdos_160.known_upper"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Estimate $h(n)$ by finding a better lower bound.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«160»","statement":"have lower_bound := sorry;\n(lower_bound =O[Filter.atTop] fun n => ↑(Erdos160.erdos_160.h n)) ∧\n  ∀ c > 0,\n    ((fun n => Real.exp (c * Real.log ↑n ^ (1 / 12))) =O[Filter.atTop] fun n => ↑(Erdos160.erdos_160.h n)) →\n      ∀ c > 0, (fun n => Real.exp (c * Real.log ↑n ^ (1 / 12))) =o[Filter.atTop] lower_bound","subjects":["5","51"],"theorem":"Erdos160.erdos_160.better_lower"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\\geq 0$, where\n$\\Omega(m)<\\log\\log m$? (Here $\\Omega(m)$ is the number of prime divisors of $m$ counted with\nmultiplicity.) Or some more slowly growing function?\n\nBarreto and Leeham, using ChatGPT and Aristotle, have proved a negative answer, which was\nquantified by Tao and Alexeev (see the comments).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«205»","statement":"False ↔\n  ∃ f, (f =o[Filter.atTop] fun m => Real.log (Real.log ↑m)) ∧ ∀ᶠ (n : ℕ) in Filter.atTop, Erdos205.IsRepresentable f n","subjects":["11"],"theorem":"Erdos205.erdos_205.parts.iii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\\geq 0$, where\n$\\Omega(m)<\\log\\log m$? (Here $\\Omega(m)$ is the number of prime divisors of $m$ counted with\nmultiplicity.)\n\nBarreto and Leeham, using ChatGPT and Aristotle, have proved a negative answer, which was\nquantified by Tao and Alexeev (see the comments): in fact there are infinitely many $n$ such that,\nfor all $k$ with $2^k<n$, $n-2^k$ has at least\n$$\\gg \\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2}$$\nmany prime factors.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«205»","statement":"False ↔ ∀ᶠ (n : ℕ) in Filter.atTop, Erdos205.IsRepresentable (fun m => Real.log (Real.log ↑m)) n","subjects":["11"],"theorem":"Erdos205.erdos_205.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"In fact there are infinitely many $n$ such that, for all $k$ with $2^k<n$, $n-2^k$ has at least\n$$\\gg \\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2}$$\nmany prime factors.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos205.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«205»","statement":"∃ c > 0,\n  {n |\n      ∀ (k : ℕ),\n        2 ^ k < n →\n          c * √(Real.log ↑n / Real.log (Real.log ↑n)) ≤ ↑(ArithmeticFunction.cardFactors (n - 2 ^ k))}.Infinite","subjects":["11"],"theorem":"Erdos205.erdos_205.variants.many_prime_factors"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\\geq 0$, where\n$\\Omega(m)<\\log\\log m$? (Here $\\Omega(m)$ is the number of prime divisors of $m$ counted with\nmultiplicity.) What about $<\\epsilon \\log\\log m$?\n\nBarreto and Leeham, using ChatGPT and Aristotle, have proved a negative answer, which was\nquantified by Tao and Alexeev (see the comments).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«205»","statement":"False ↔ ∀ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, Erdos205.IsRepresentable (fun m => ε * Real.log (Real.log ↑m)) n","subjects":["11"],"theorem":"Erdos205.erdos_205.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"The $n$ constructed in this way are divisible by a large power of $2$. It remains open whether\nthere exist arbitrarily large odd counterexamples.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«205»","statement":"True ↔ {n | Odd n ∧ ¬Erdos205.IsRepresentable (fun m => Real.log (Real.log ↑m)) n}.Infinite","subjects":["11"],"theorem":"Erdos205.erdos_205.variants.odd_counterexamples"},{"answerKinds":[],"category":"research open","docstring":"If $k>3$ (and $k \\neq 2^l$), and for all primes $p$ there exists $n$ such that $p^{k-2}\\nmid f(n)$,\nthen are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«978»","statement":"True ↔\n  ∀ {f : Polynomial ℤ},\n    Irreducible f →\n      f.natDegree > 3 →\n        (¬∃ l, f.natDegree = 2 ^ l) →\n          0 < f.leadingCoeff →\n            (∀ (p : ℕ), Nat.Prime p → ∃ n, ¬↑p ^ (f.natDegree - 2) ∣ Polynomial.eval (↑n) f) →\n              {n | Powerfree (f.natDegree - 2) (Polynomial.eval (↑n) f)}.Infinite","subjects":["11"],"theorem":"Erdos978.erdos_978.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"If the degree `k` of `f` is larger than or equal to `9`, then the set of `n` such that `f n` is\n`(k - 2)`-th power free has infinitely many elements. This result is proved in [Br11]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«978»","statement":"∀ {f : Polynomial ℤ},\n  Irreducible f →\n    9 ≤ f.natDegree →\n      (∀ (p : ℕ), Nat.Prime p → ∃ n, ¬↑p ^ (f.natDegree - 1) ∣ Polynomial.eval (↑n) f) →\n        {n | Powerfree (f.natDegree - 2) (Polynomial.eval (↑n) f)}.Infinite","subjects":["11"],"theorem":"Erdos978.erdos_978.variants.sub_two"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $k > 3$ (and $k \\neq 2^l$), then are there infinitely many $n$ for which $f(n)$ is\n$(k-2)$-power-free?\n\nThis was disproved by the DeepMind prover agent.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/3b5d6ac2555cd63b83d418c29ff040876be9dee0/FormalConjectures/ErdosProblems/978.lean#L64"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«978»","statement":"False ↔\n  ∀ {f : Polynomial ℤ},\n    Irreducible f →\n      f.natDegree > 3 →\n        (¬∃ l, f.natDegree = 2 ^ l) →\n          0 < f.leadingCoeff →\n            (¬∃ p, Nat.Prime p ∧ ∀ (n : ℕ), ↑p ^ (f.natDegree - 1) ∣ Polynomial.eval (↑n) f) →\n              {n | Powerfree (f.natDegree - 2) (Polynomial.eval (↑n) f)}.Infinite","subjects":["11"],"theorem":"Erdos978.erdos_978.variants.allow_fixed_divisors"},{"answerKinds":[],"category":"research solved","docstring":"Let `f ∈ ℤ[X]` be an irreducible polynomial with positive leading coefficient. Suppose that the\ndegree `k` of `f` is larger than `2` and is not equal to a power of `2`. Then the set of `n` such\nthat `f n` is `(k - 1)`-th power free is infinite, and this is proved in [Er53]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«978»","statement":"∀ {f : Polynomial ℤ},\n  Irreducible f →\n    2 < f.natDegree →\n      (∀ (x : ℕ), f.natDegree ≠ 2 ^ x) →\n        0 < f.leadingCoeff → {n | Powerfree (f.natDegree - 1) (Polynomial.eval (↑n) f)}.Infinite","subjects":["11"],"theorem":"Erdos978.erdos_978.variants.sub_one"},{"answerKinds":[],"category":"research open","docstring":"Does `n ^ 4 + 2` represent infinitely many squarefree numbers? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«978»","statement":"True ↔ {n | Squarefree (n ^ 4 + 2)}.Infinite","subjects":["11"],"theorem":"Erdos978.erdos_978.parts.iii"},{"answerKinds":[],"category":"research solved","docstring":"Let `f ∈ ℤ[X]` be an irreducible polynomial with positive leading coefficient. Suppose that the\ndegree `k` of `f` is larger than `2`, is not equal to a power of `2`, and `f n` has no fixed\n`(k - 1)`-th power divisors other than `1`. Then the set of `n` such that `f n` is `(k - 1)`-th\npower free has positive density, and this is proved in [Ho67]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«978»","statement":"∀ {f : Polynomial ℤ},\n  Irreducible f →\n    2 < f.natDegree →\n      (∀ (x : ℕ), f.natDegree ≠ 2 ^ x) →\n        0 < f.leadingCoeff →\n          (∀ (p : ℕ), Nat.Prime p → ∃ n, ¬↑p ^ (f.natDegree - 1) ∣ Polynomial.eval (↑n) f) →\n            {n | Powerfree (f.natDegree - 1) (Polynomial.eval (↑n) f)}.HasPosDensity","subjects":["11"],"theorem":"Erdos978.erdos_978.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er95] proved that the distribution function of $\\varphi(n)/n$ is purely singular: it is\ncontinuous, but its derivative is zero almost everywhere.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«50»","statement":"∀ (f : ℝ → ℝ), Erdos50.IsDistributionOfPhiRatio f → Erdos50.IsPurelySingular f","subjects":["11"],"theorem":"Erdos50.erdos_50_singular"},{"answerKinds":[],"category":"research open","docstring":"Let $f$ be the asymptotic distribution function of $\\varphi(n)/n$, so that for each $c \\in [0,1]$,\n$f(c)$ is the natural density of $\\{n : \\varphi(n) < cn\\}$. Is it true that there is no $x$ such\nthat the derivative $f'(x)$ exists and is positive?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«50»","statement":"True ↔\n  ∀ (f : ℝ → ℝ), Erdos50.IsDistributionOfPhiRatio f → ¬∃ x ∈ Set.Icc 0 1, ∃ y > 0, HasDerivWithinAt f y (Set.Icc 0 1) x","subjects":["11"],"theorem":"Erdos50.erdos_50"},{"answerKinds":[],"category":"research solved","docstring":"Schoenberg [Sch38] proved that the asymptotic distribution function of $\\varphi(n)/n$ exists.\nThat is, for any $c \\in [0, 1]$, the proportion of integers $n \\le N$ satisfying $\\varphi(n)/n < c$\napproaches a limit as $N \\to \\infty$. This limit function is the cumulative distribution function\nof the values of $\\varphi(n)/n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«50»","statement":"∃ f, Erdos50.IsDistributionOfPhiRatio f","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos50.erdos_50_schoenberg"},{"answerKinds":[],"category":"textbook","docstring":"Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Show that $f$ is undefined at $n=2$, i.e. we get the junk value $0$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1054»","statement":"Erdos1054.f 2 = 0","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos1054.f_undefined_at_2"},{"answerKinds":[],"category":"textbook","docstring":"Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Show that $f$ is undefined at $n=5$, i.e. we get the junk value $0$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1054»","statement":"Erdos1054.f 5 = 0","subjects":["11"],"theorem":"Erdos1054.f_undefined_at_3"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1054»","statement":"True ↔ ∃ A, A.HasDensity 1 ∧ (fun n => ↑(Erdos1054.f ↑n)) =o[Filter.atTop] fun n => ↑↑n","subjects":["11"],"theorem":"Erdos1054.erdos_1054.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Is it true that $\\limsup f(n)/n=\\infty$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1054»","statement":"True ↔ ∃ A, A.HasDensity 1 ∧ Filter.limsup (fun n => ↑(Erdos1054.f n) / ↑n) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos1054.erdos_1054.parts.iii"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors\nof $m$ for some $k\\geq 1$. Is it true that $f(n)=o(n)$?","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1054»","statement":"True ↔ (fun n => ↑(Erdos1054.f n)) =o[Filter.atTop] fun n => ↑n","subjects":["11"],"theorem":"Erdos1054.erdos_1054.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Let $A ⊆ \\mathbb{N}$. Let $B ⊆ \\mathbb{N}$ be the set of integers which are representable\nin exactly one way as the sum of two elements from $A$. Is it true that for all\n$\\epsilon > 0$ and large $N$, $|\\{1,\\ldots,N\\} \\setminus B| \\gg_\\epsilon N^{1/2 - \\epsilon}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«14»","statement":"True ↔ ∀ (A : Set ℕ), ∀ ε > 0, Erdos14.almostSquareRoot ε =O[Filter.atTop] Erdos14.nonUniqueSumCount A","subjects":["11"],"theorem":"Erdos14.erdos_14.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is it possible that $|\\{1,\\ldots,N\\} \\setminus B| = o(N^\\frac{1}{2})$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«14»","statement":"True ↔ ∃ A, Erdos14.nonUniqueSumCount A =o[Filter.atTop] Erdos14.squareRoot","subjects":["11"],"theorem":"Erdos14.erdos_14.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 1050, variant [Er88c] (open).** Erdős conjectured that\n$\\sum_{n=1}^\\infty \\frac{1}{2^n + t}$ is *transcendental* for every integer $t \\ne 0$ — strictly\nstronger than the irrationality established by Borwein [Bo91]. This remains open.\n\nTwo exclusions make the series well-posed and the claim non-vacuous, exactly as in Borwein's theorem:\n$t \\ne -2^n$ for all $n \\ge 1$ (so no denominator vanishes), and $t \\ne 0$ (at $t = 0$ the series is\nthe rational $\\sum_{n \\ge 1} 2^{-n} = 1$, hence not transcendental). ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1050»","statement":"True ↔ ∀ (t : ℤ), t ≠ 0 → (∀ (n : ℕ), 1 ≤ n → t ≠ -2 ^ n) → Transcendental ℚ (∑' (n : ℕ), 1 / (2 ^ (n + 1) + ↑t))","subjects":["11"],"theorem":"Erdos1050.erdos_1050.variants.transcendental"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős Problem 1050, variant [Er48].** Erdős proved that the related series\n$\\sum_{n=1}^\\infty \\frac{1}{2^n - 1}$ (which equals the Lambert series $\\sum_n \\frac{\\tau(n)}{2^n}$,\nwhere $\\tau$ is the divisor function) is irrational. This is the $q = 2$, $r = -1$ case of Borwein's\ngeneral theorem `Erdos1050.erdos_1050.variants.borwein` below. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/gotrevor/lean-gallery/blob/main/LeanGallery/NumberTheory/Erdos1050/Variants.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1050»","statement":"Irrational (∑' (n : ℕ), 1 / (2 ^ (n + 1) - 1))","subjects":["11"],"theorem":"Erdos1050.erdos_1050.variants.two_pow_sub_one"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős Problem 1050.** The series $\\sum_{n=1}^\\infty \\frac{1}{2^n - 3}$ is irrational.\n\nThe source sum runs over $n \\ge 1$; as a `tsum` over `ℕ` (which starts at $0$) it is reindexed\n$n \\mapsto n + 1$, i.e. $\\sum_{n=0}^\\infty 1/(2^{n+1} - 3)$. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/gotrevor/lean-gallery/blob/main/LeanGallery/NumberTheory/Erdos1050/Statement.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1050»","statement":"Irrational (∑' (n : ℕ), 1 / (2 ^ (n + 1) - 3))","subjects":["11"],"theorem":"Erdos1050.erdos_1050"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős Problem 1050, variant [Bo91].** Borwein's general theorem: for every integer $q \\ge 2$\nand rational $r \\ne 0$ with $r \\ne -q^n$ for all $n \\ge 1$, the series $\\sum_{n=1}^\\infty \\frac{1}{q^n + r}$\nis irrational. Problem 1050 is the case $q = 2$, $r = -3$; the [Er48] variant is $q = 2$, $r = -1$. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/gotrevor/lean-gallery/blob/main/LeanGallery/NumberTheory/Erdos1050/Variants.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1050»","statement":"∀ (q : ℤ),\n  2 ≤ q → ∀ (r : ℚ), r ≠ 0 → (∀ (n : ℕ), 1 ≤ n → r ≠ -↑q ^ n) → Irrational (∑' (n : ℕ), 1 / (↑q ^ (n + 1) + ↑r))","subjects":["11"],"theorem":"Erdos1050.erdos_1050.variants.borwein"},{"answerKinds":[],"category":"research solved","docstring":"Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$\nvertices contains either a $K_k$ or an independent set on $l$ vertices.\n\nProve, for fixed $k\\geq 3$, that\n$$\\lim_{l\\to \\infty}\\frac{R(k,l+1)}{R(k,l)}=1.$$\n\nThis has been\n[solved](https://cdn.openai.com/pdf/6dc7175d-d9e7-4b8d-96b8-48fe5798cd5b/Ramsey.pdf)\nby an internal model at OpenAI.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1014.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1014»","statement":"∀ (k : ℕ),\n  3 ≤ k →\n    Filter.Tendsto (fun l => ↑(SimpleGraph.classicalRamsey k (l + 1)) / ↑(SimpleGraph.classicalRamsey k l)) Filter.atTop\n      (nhds 1)","subjects":["5"],"theorem":"Erdos1014.erdos_1014"},{"answerKinds":[],"category":"research solved","docstring":"That proof in fact shows that\n$$R(k,l+1)\\leq (1+O(l^{-c/k^2}))R(k,l)$$\nfor some constant $c>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1014»","statement":"∃ c,\n  0 < c ∧\n    ∀ (k : ℕ),\n      3 ≤ k →\n        ∃ C,\n          ∀ᶠ (l : ℕ) in Filter.atTop,\n            ↑(SimpleGraph.classicalRamsey k (l + 1)) ≤ (1 + C * ↑l ^ (-c / ↑k ^ 2)) * ↑(SimpleGraph.classicalRamsey k l)","subjects":["5"],"theorem":"Erdos1014.erdos_1014.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Erdos, Graham, Ruzsa, and Straus proved that if\n$$\n  f(n) = \\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p}\n$$\nand\n$$\n  \\gamma_0 = \\sum_{k = 2}^{\\infty} \\frac{\\log k}{2^k}\n$$\nthen for almost all integers $f(m) = \\gamma_0 + o(1)$.\n\n[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., _On the prime factors of $\\binom{2n}{n}$_. Math. Comp. (1975), 83-92.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«377»","statement":"∀ (γ₀ : ℝ),\n  γ₀ = ∑' (k : ℕ), Real.log (↑k + 2) / 2 ^ (k + 2) →\n    ∃ o,\n      ∃ (_ : Filter.Tendsto o Filter.atTop (nhds 0)),\n        ∀ᶠ (n : ℕ) in Filter.cofinite, Erdos377.sumInvPrimesNotDvdCentralBinom n = γ₀ + o n","subjects":["11"],"theorem":"Erdos377.erdos_377.variants.ae"},{"answerKinds":[],"category":"research solved","docstring":"Erdos, Graham, Ruzsa, and Straus proved that if\n$$\n  f(n) = \\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p}\n$$\nand\n$$\n  \\gamma_0 = \\sum_{k = 2}^{\\infty} \\frac{\\log k}{2^k}\n$$\nthen\n$$\n  \\lim_{x\\to\\infty} \\frac{1}{x}\\sum_{n\\leq x} f(n)^2 = \\gamma_0^2\n$$\n\n[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., _On the prime factors of $\\binom{2n}{n}$_. Math. Comp. (1975), 83-92.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«377»","statement":"∀ (γ₀ : ℝ),\n  γ₀ = ∑' (k : ℕ), Real.log (↑k + 2) / 2 ^ (k + 2) →\n    Filter.Tendsto (fun x => 1 / ↑x * ∑ n ∈ Finset.Icc 1 x, Erdos377.sumInvPrimesNotDvdCentralBinom n ^ 2) Filter.atTop\n      (nhds (γ₀ ^ 2))","subjects":["11"],"theorem":"Erdos377.erdos_377.variants.limit.ii"},{"answerKinds":[],"category":"research solved","docstring":"Erdos, Graham, Ruzsa, and Straus proved that if\n$$\n  f(n) = \\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p}\n$$\nthen there is some constant $c < 1$ such that for all large $n$\n$$\n  f(n) \\leq c \\log\\log n.\n$$\n\n[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., _On the prime factors of $\\binom{2n}{n}$_. Math. Comp. (1975), 83-92.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«377»","statement":"∃ c < 1, ∀ᶠ (n : ℕ) in Filter.atTop, Erdos377.sumInvPrimesNotDvdCentralBinom n ≤ c * Real.log (Real.log ↑n)","subjects":["11"],"theorem":"Erdos377.erdos_377.variants.ub"},{"answerKinds":[],"category":"research solved","docstring":"Erdos, Graham, Ruzsa, and Straus proved that if\n$$\n  f(n) = \\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p}\n$$\nand\n$$\n  \\gamma_0 = \\sum_{k = 2}^{\\infty} \\frac{\\log k}{2^k}\n$$\nthen\n$$\n  \\lim_{x\\to\\infty} \\frac{1}{x}\\sum_{n\\leq x} f(n) = \\gamma_0\n$$\n\n[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., _On the prime factors of $\\binom{2n}{n}$_. Math. Comp. (1975), 83-92.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«377»","statement":"∀ (γ₀ : ℝ),\n  γ₀ = ∑' (k : ℕ), Real.log (↑k + 2) / 2 ^ (k + 2) →\n    Filter.Tendsto (fun x => 1 / ↑x * ∑ n ∈ Finset.Icc 1 x, Erdos377.sumInvPrimesNotDvdCentralBinom n) Filter.atTop\n      (nhds γ₀)","subjects":["11"],"theorem":"Erdos377.erdos_377.variants.limit.i"},{"answerKinds":[],"category":"research open","docstring":"Is there some absolute constant $C > 0$ such that\n$$\n  \\sum_{p \\leq n} 1_{p\\nmid {2n \\choose n}}\\frac{1}{p} \\leq C\n$$\nfor all $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«377»","statement":"True ↔ ∃ C > 0, ∀ (n : ℕ), Erdos377.sumInvPrimesNotDvdCentralBinom n ≤ C","subjects":["11"],"theorem":"Erdos377.erdos_377"},{"answerKinds":["Prop"],"category":"research solved","docstring":"For any permutation $\\pi\\in S_n$ of $\\{1,\\ldots,n\\}$ let $S(\\pi)$ count the number of distinct consecutive sums, that is, sums of the shape $\\sum_{u\\leq i\\leq v}\\pi(i)$. Is it true that\n$$\nS(\\pi) = o(n^2)\n$$\nfor all $\\pi\\in S_n$?\n\nHegyvári [He86] gave a counterexample.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos34.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«34»","statement":"False ↔ ∀ (c : ℝ), 0 < c → ∃ N, ∀ n ≥ N, ∀ (p : Equiv.Perm (Fin n)), ↑(Erdos34.consecutiveSums n p).card < c * ↑n ^ 2","subjects":["5","11"],"theorem":"Erdos34.erdos_34"},{"answerKinds":[],"category":"research open","docstring":"Is $n! - 1$ powerful for finitely many $n$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«936»","statement":"True ↔ Erdos936.EventuallyNotPowerful fun x => x.factorial - 1","subjects":["11"],"theorem":"Erdos936.erdos_936.variants.factorial_sub_one"},{"answerKinds":[],"category":"research open","docstring":"Is $2^n - 1$ powerful for finitely many $n$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«936»","statement":"True ↔ Erdos936.EventuallyNotPowerful fun x => 2 ^ x - 1","subjects":["11"],"theorem":"Erdos936.erdos_936.variants.two_pow_sub_one"},{"answerKinds":[],"category":"research open","docstring":"Is $2^n + 1$ powerful for finitely many $n$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«936»","statement":"True ↔ Erdos936.EventuallyNotPowerful fun x => 2 ^ x + 1","subjects":["11"],"theorem":"Erdos936.erdos_936.variants.two_pow_add_one"},{"answerKinds":[],"category":"research open","docstring":"Is $n! + 1$ powerful for finitely many $n$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«936»","statement":"True ↔ Erdos936.EventuallyNotPowerful fun x => x.factorial + 1","subjects":["11"],"theorem":"Erdos936.erdos_936.variants.factorial_add_one"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Prove that\n$$R(4,k) \\gg \\frac{k^3}{(\\log k)^{O(1)}}.$$\n\nThis is true, and was proved by Mattheus and Verstraëte [MaVe23], who showed that\n$R(4,k) \\gg \\frac{k^3}{(\\log k)^4}$.\nThis problem is #5 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«166»","statement":"True ↔\n  ∃ c C, 0 < c ∧ 0 < C ∧ ∀ᶠ (k : ℕ) in Filter.atTop, ↑(SimpleGraph.classicalRamsey 4 k) ≥ C * ↑k ^ 3 / Real.log ↑k ^ c","subjects":["5"],"theorem":"Erdos166.erdos_166"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $\\epsilon,\\delta>0$ and $n$ be sufficiently large in terms of $\\epsilon$ and $\\delta$.\nLet $G$ be a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-\\epsilon}$.\nCan $G$ be made into a triangle-free graph with diameter $2$ by adding at most $\\delta n^2$ edges?\n\nAsked by Erdős and Gyárfás, who proved that this is the case when $G$ has maximum degree $\\ll \\log n/\\log\\log n$. A construction of Simonovits shows that this conjecture is false if we just have maximum degree $\\leq Cn^{1/2}$, for some large enough $C$. In this note Alon solves this problem in a strong form, in particular proving that a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-\\epsilon}$ can be made into a triangle-free graph with diameter $2$ by adding at most $O(n^{2-\\epsilon})$ edges.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos134.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«134»","statement":"True ↔\n  ∀ (ε δ : ℝ),\n    0 < ε →\n      0 < δ →\n        ∃ N,\n          ∀ n ≥ N,\n            ∀ (G : SimpleGraph (Fin n)) [inst : DecidableRel G.Adj],\n              G.CliqueFree 3 →\n                (∀ (v : Fin n), ↑(G.degree v) < (↑n).rpow (1 / 2 - ε)) →\n                  ∃ H x,\n                    G ≤ H ∧\n                      H.CliqueFree 3 ∧\n                        (∀ (x y : Fin n), x ≠ y → H.Adj x y ∨ ∃ z, H.Adj x z ∧ H.Adj z y) ∧\n                          ↑(H.edgeFinset \\ G.edgeFinset).card ≤ δ * ↑n ^ 2","subjects":["5"],"theorem":"Erdos134.erdos_134"},{"answerKinds":[],"category":"research solved","docstring":"If we change the condition to $a_t \\leq n$ it can be shown that\n$$\n  A(n) = n - \\frac{n}{\\log n} + o\\left(\\frac{n}{\\log n}\\right)\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«392»","statement":"∀ (A : ℕ → ℕ),\n  (∀ n > 0, IsLeast {x | ∃ t, ∃ (_ : ∃ a, n.factorial = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n), t + 1 = x} (A n)) →\n    (fun n => ↑(A n) - ↑n + ↑n / Real.log ↑n) =o[Filter.atTop] fun n => ↑n / Real.log ↑n","subjects":["11"],"theorem":"Erdos392.erdos_392.variants.lower"},{"answerKinds":[],"category":"research solved","docstring":"Cambie has observed that a positive answer follows from the result above with $a_t \\leq n$, simply\nby pairing variables together, e.g. taking $a'_i = a_{2i-1}a_{2i}$ (and the lower bound follows from\nStirling's approximation).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«392»","statement":"(∀ (A : ℕ → ℕ),\n    (∀ n > 0,\n        IsLeast {x | ∃ t, ∃ (_ : ∃ a, n.factorial = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n ^ 2), t + 1 = x} (A n)) →\n      (fun n => ↑(A n) - ↑n / 2 + ↑n / (2 * Real.log ↑n)) =o[Filter.atTop] fun n => ↑n / Real.log ↑n) →\n  ∀ (A : ℕ → ℕ),\n    (∀ n > 0,\n        IsLeast {x | ∃ t, ∃ (_ : ∃ a, n.factorial = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n), t + 1 = x} (A n)) →\n      (fun n => ↑(A n) - ↑n + ↑n / Real.log ↑n) =o[Filter.atTop] fun n => ↑n / Real.log ↑n","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos392.erdos_392.variants.implication"},{"answerKinds":[],"category":"research solved","docstring":"Let $A(n)$ denote the least value of $t$ such that\n$$\n  n! = a_1 \\cdots a_t\n$$\nwith $a_1 \\leq \\cdots \\leq a_t\\leq n^2$. Then\n$$\n  A(n) = \\frac{n}{2} - \\frac{n}{2\\log n} + o\\left(\\frac{n}{\\log n}\\right).\n$$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/AlexKontorovich/PrimeNumberTheoremAnd/blob/d82a35ecc798ada27f82a88b559f25b753f63a73/PrimeNumberTheoremAnd/IEANTN/Erdos392.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«392»","statement":"∀ (A : ℕ → ℕ),\n  (∀ n > 0,\n      IsLeast {x | ∃ t, ∃ (_ : ∃ a, n.factorial = ∏ i, a i ∧ Monotone a ∧ a (Fin.last t) ≤ n ^ 2), t + 1 = x} (A n)) →\n    (fun n => ↑(A n) - ↑n / 2 + ↑n / (2 * Real.log ↑n)) =o[Filter.atTop] fun n => ↑n / Real.log ↑n","subjects":["11"],"theorem":"Erdos392.erdos_392"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $x>0$ be a real number. For any $n\\geq 1$ let\n$$R_n(x) = \\sum_{i=1}^n\\frac{1}{m_i}<x$$\nbe the maximal sum of $n$ distinct unit fractions which is $<x$.\n\nIs it true that, for almost all $x$, for sufficiently large $n$, we have\n$$R_{n+1}(x)=R_n(x)+\\frac{1}{m},$$\nwhere $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is\n$<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy'\nfashion?)\n\nKovač [Ko24b] has proved that this is false - in fact as false as possible: the set of\n$x\\in (0,\\infty)$ for which the best underapproximations are eventually 'greedy' has Lebesgue\nmeasure zero.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos206.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«206»","statement":"False ↔ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioi 0), Erdos206.EventuallyGreedy x","subjects":["11"],"theorem":"Erdos206.erdos_206"},{"answerKinds":[],"category":"research solved","docstring":"This exponent was improved to $0.3389$ by Lichtman [Li22].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1057»","statement":"∀ᶠ (x : ℝ) in Filter.atTop, Erdos1057.carmichaelCounting x > x ^ 0.3389","subjects":["11"],"theorem":"Erdos1057.erdos_1057.variants.lichtman_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Alford, Granville, and Pomerance [AGP94] proved that $C(x)\\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1057»","statement":"Filter.Tendsto Erdos1057.carmichaelCounting Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos1057.erdos_1057.variants.agp_infinite"},{"answerKinds":[],"category":"research solved","docstring":"Alford, Granville, and Pomerance [AGP94] proved that $C(x)>x^{2/7}$ for large $x$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1057»","statement":"∀ᶠ (x : ℝ) in Filter.atTop, Erdos1057.carmichaelCounting x > x ^ (2 / 7)","subjects":["11"],"theorem":"Erdos1057.erdos_1057.variants.agp_lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $C(x)=x^{1-o(1)}$?\n\nThis is discussed in problem A13 of Guy's collection [Gu04].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1057»","statement":"True ↔ Filter.Tendsto (fun x => Real.log (Erdos1057.carmichaelCounting x) / Real.log x) Filter.atTop (nhds 1)","subjects":["11"],"theorem":"Erdos1057.erdos_1057"},{"answerKinds":[],"category":"research solved","docstring":"The lower bound $C(x)> x^{0.33336704}$ was proved by Harman [Ha08].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1057»","statement":"∀ᶠ (x : ℝ) in Filter.atTop, Erdos1057.carmichaelCounting x > x ^ 0.33336704","subjects":["11"],"theorem":"Erdos1057.erdos_1057.variants.harman_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er56c] proved $C(x) < x \\exp\\left(-c \\frac{\\log x\\log\\log\\log x}{\\log\\log x}\\right)$\nfor some constant $c>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1057»","statement":"∃ c > 0,\n  ∀ᶠ (x : ℝ) in Filter.atTop,\n    Erdos1057.carmichaelCounting x <\n      x * Real.exp (-c * (Real.log x * Real.log (Real.log (Real.log x))) / Real.log (Real.log x))","subjects":["11"],"theorem":"Erdos1057.erdos_1057.variants.upper_bound"},{"answerKinds":[],"category":"research open","docstring":"Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact\n$C(x)= x \\exp\\left(-(1+o(1))\\frac{\\log x\\log\\log\\log x}{\\log\\log x}\\right)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1057»","statement":"True ↔\n  Filter.Tendsto\n    (fun x =>\n      -(Real.log (Erdos1057.carmichaelCounting x / x) * Real.log (Real.log x)) /\n        (Real.log x * Real.log (Real.log (Real.log x))))\n    Filter.atTop (nhds 1)","subjects":["11"],"theorem":"Erdos1057.erdos_1057.variants.pomerance"},{"answerKinds":[],"category":"research solved","docstring":"Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«445»","statement":"∃ c₀ < 1, ∀ c > c₀, ∀ᶠ (p : ℕ) in Filter.atTop, Nat.Prime p → ∀ (n : ℕ), Erdos445.Erdos445Prop c p n","subjects":["11"],"theorem":"Erdos445.erdos_445.variants.heilbronn"},{"answerKinds":[],"category":"test","docstring":"Small example: for $p=5$, $c=1$, $n=0$, the pair $(2,3) \\in (0,5)$ satisfies\n$2 \\cdot 3 = 6 \\equiv 1 \\pmod{5}$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«445»","statement":"Erdos445.Erdos445Prop 1 5 1","subjects":["11"],"theorem":"Erdos445.erdos_445.test.small_example"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any\n$n\\geq 0$, there exist $a,b\\in(n,n+p^c)$ such that $ab\\equiv 1\\pmod{p}$?\n\nThis is discussed in this MathOverflow question [MathOverflow].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«445»","statement":"True ↔ ∀ c > 1 / 2, ∀ᶠ (p : ℕ) in Filter.atTop, Nat.Prime p → ∀ (n : ℕ), Erdos445.Erdos445Prop c p n","subjects":["11"],"theorem":"Erdos445.erdos_445"},{"answerKinds":[],"category":"research solved","docstring":"Heath-Brown [He00] used Kloosterman sums to prove this for all $c>3/4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«445»","statement":"∀ c > 3 / 4, ∀ᶠ (p : ℕ) in Filter.atTop, Nat.Prime p → ∀ (n : ℕ), Erdos445.Erdos445Prop c p n","subjects":["11"],"theorem":"Erdos445.erdos_445.variants.heath_brown"},{"answerKinds":[],"category":"research open","docstring":"Is there a graph with $\\aleph_{\\omega+1}$ vertices and chromatic number $\\aleph_1$ such that\nevery subgraph on $\\aleph_\\omega$ vertices has chromatic number $\\leq\\aleph_0$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«918»","statement":"True ↔\n  ∃ V G,\n    Cardinal.mk V = Cardinal.aleph (Ordinal.omega0 + 1) ∧\n      G.chromaticCardinal = Cardinal.aleph 1 ∧\n        ∀ (W : Set V),\n          Cardinal.mk ↑W = Cardinal.aleph Ordinal.omega0 → (SimpleGraph.induce W G).chromaticCardinal ≤ Cardinal.aleph0","subjects":["5"],"theorem":"Erdos918.erdos_918.parts.ii"},{"answerKinds":[],"category":"textbook","docstring":"In [Er69b] the questions are stated with $= \\aleph_0$ rather than $\\leq\\aleph_0$. This is\na likely typo since it can be shown that no such graph exists in this case.\n\nThis is the first question with all subgraphs. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«918»","statement":"¬∃ V G,\n    Cardinal.mk V = Cardinal.aleph 2 ∧\n      G.chromaticCardinal = Cardinal.aleph 2 ∧\n        ∀ (H : G.Subgraph), Cardinal.mk ↑H.verts = Cardinal.aleph 1 → H.coe.chromaticCardinal = Cardinal.aleph0","subjects":["5"],"theorem":"Erdos918.erdos_918.variants.eq_aleph_0_all_subgraphs.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"A question of Erd\\H{o}s and Hajnal [ErHa68b], who proved that for every finite $k$\nthere is a graph with chromatic number $\\aleph_1$ and $\\aleph_k$ vertices where each subgraph on\nless than $\\aleph_k$ vertices has chromatic number $\\leq \\aleph_0$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«918»","statement":"∀ (k : ℕ),\n  0 < k →\n    ∃ V G,\n      Cardinal.mk V = Cardinal.aleph ↑k ∧\n        G.chromaticCardinal = Cardinal.aleph 1 ∧\n          ∀ (W : Set V),\n            Cardinal.mk ↑W < Cardinal.aleph ↑k → (SimpleGraph.induce W G).chromaticCardinal ≤ Cardinal.aleph0","subjects":["5"],"theorem":"Erdos918.erdos_918.variants.erdos_hajnal"},{"answerKinds":[],"category":"textbook","docstring":"In [Er69b] the questions are stated with $= \\aleph_0$ rather than $\\leq\\aleph_0$. This is\na likely typo since it can be shown that no such graph exists in this case.\n\nThis is the second question with induced subgraphs. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«918»","statement":"¬∃ V G,\n    Cardinal.mk V = Cardinal.aleph (Ordinal.omega0 + 1) ∧\n      G.chromaticCardinal = Cardinal.aleph 1 ∧\n        ∀ (W : Set V),\n          Cardinal.mk ↑W = Cardinal.aleph Ordinal.omega0 → (SimpleGraph.induce W G).chromaticCardinal = Cardinal.aleph0","subjects":["5"],"theorem":"Erdos918.erdos_918.variants.eq_aleph_0.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Is there a graph with $\\aleph_2$ vertices and chromatic number $\\aleph_2$ such that every\nsubgraph on $\\aleph_1$ vertices has chromatic number $\\leq\\aleph_0$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«918»","statement":"True ↔\n  ∃ V G,\n    Cardinal.mk V = Cardinal.aleph 2 ∧\n      G.chromaticCardinal = Cardinal.aleph 2 ∧\n        ∀ (H : G.Subgraph), Cardinal.mk ↑H.verts = Cardinal.aleph 1 → H.coe.chromaticCardinal ≤ Cardinal.aleph0","subjects":["5"],"theorem":"Erdos918.erdos_918.variants.all_subgraphs.parts.i"},{"answerKinds":[],"category":"textbook","docstring":"In [Er69b] the questions are stated with $= \\aleph_0$ rather than $\\leq\\aleph_0$. This is\na likely typo since it can be shown that no such graph exists in this case.\n\nThis is the first question with induced subgraphs. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«918»","statement":"¬∃ V G,\n    Cardinal.mk V = Cardinal.aleph 2 ∧\n      G.chromaticCardinal = Cardinal.aleph 2 ∧\n        ∀ (W : Set V), Cardinal.mk ↑W = Cardinal.aleph 1 → (SimpleGraph.induce W G).chromaticCardinal = Cardinal.aleph0","subjects":["5"],"theorem":"Erdos918.erdos_918.variants.eq_aleph_0.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is there a graph with $\\aleph_2$ vertices and chromatic number $\\aleph_2$ such that every\nsubgraph on $\\aleph_1$ vertices has chromatic number $\\leq\\aleph_0$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«918»","statement":"True ↔\n  ∃ V G,\n    Cardinal.mk V = Cardinal.aleph 2 ∧\n      G.chromaticCardinal = Cardinal.aleph 2 ∧\n        ∀ (W : Set V), Cardinal.mk ↑W = Cardinal.aleph 1 → (SimpleGraph.induce W G).chromaticCardinal ≤ Cardinal.aleph0","subjects":["5"],"theorem":"Erdos918.erdos_918.parts.i"},{"answerKinds":[],"category":"textbook","docstring":"In [Er69b] the questions are stated with $= \\aleph_0$ rather than $\\leq\\aleph_0$. This is\na likely typo since it can be shown that no such graph exists in this case.\n\nThis is the second question with all subgraphs. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«918»","statement":"¬∃ V G,\n    Cardinal.mk V = Cardinal.aleph (Ordinal.omega0 + 1) ∧\n      G.chromaticCardinal = Cardinal.aleph 1 ∧\n        ∀ (H : G.Subgraph),\n          Cardinal.mk ↑H.verts = Cardinal.aleph Ordinal.omega0 → H.coe.chromaticCardinal = Cardinal.aleph0","subjects":["5"],"theorem":"Erdos918.erdos_918.variants.eq_aleph_0_all_subgraphs.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Is there a graph with $\\aleph_{\\omega+1}$ vertices and chromatic number $\\aleph_1$ such that\nevery subgraph on $\\aleph_\\omega$ vertices has chromatic number $\\leq\\aleph_0$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«918»","statement":"True ↔\n  ∃ V G,\n    Cardinal.mk V = Cardinal.aleph (Ordinal.omega0 + 1) ∧\n      G.chromaticCardinal = Cardinal.aleph 1 ∧\n        ∀ (H : G.Subgraph),\n          Cardinal.mk ↑H.verts = Cardinal.aleph Ordinal.omega0 → H.coe.chromaticCardinal ≤ Cardinal.aleph0","subjects":["5"],"theorem":"Erdos918.erdos_918.variants.all_subgraphs.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"If $G$ is a graph which contains odd cycles of $\\leq k$ different lengths then $\\chi(G)\\leq 2k+2$,\nwith equality if and only if $G$ contains $K_{2k+2}$.\n\nConjectured by Bollobás and Erdős. Proved by Gyárfás [Gy92].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«58»","statement":"∀ {V : Type u_1} (G : SimpleGraph V) (k : ℕ),\n  G.oddCycleLengths.Finite →\n    G.oddCycleLengths.ncard ≤ k →\n      G.chromaticNumber ≤ 2 * ↑k + 2 ∧\n        (G.chromaticNumber = 2 * ↑k + 2 ↔ (SimpleGraph.completeGraph (Fin (2 * k + 2))).IsContained G)","subjects":["5"],"theorem":"Erdos58.erdos_58"},{"answerKinds":[],"category":"research open","docstring":"Let $A\\subseteq \\mathbb{R}^2$ be a set of size $n$ and let $\\{d_1,\\ldots,d_k\\}$ be\nthe set of distinct distances determined by $A$. Let $f(d)$ be the number of times\nthe distance $d$ is determined, ordered so that\n$f(d_1)\\geq f(d_2)\\geq \\cdots \\geq f(d_k)$. Estimate\n$$\\max (f(d_1)-f(d_2)),$$\nwhere the maximum is taken over all $A$ of size $n$ (this is `extremalGap n`).\n\nThe asymptotic order of `extremalGap` is not known; a natural formalization of\n\"estimate\" asks whether it has a well-defined polynomial growth exponent.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«959»","statement":"True ↔ ∃ γ, Filter.Tendsto (fun n => Real.log ↑(Erdos959.extremalGap n) / Real.log ↑n) Filter.atTop (nhds γ)","subjects":["52"],"theorem":"Erdos959.erdos_959"},{"answerKinds":[],"category":"research solved","docstring":"A superlinear lower bound: there is a $c>0$ with\n$$\\text{extremalGap}(n)\\geq n^{1 + c/\\log\\log n}$$\nfor all large $n$, so $\\max (f(d_1)-f(d_2))$ grows faster than any linear function\nof $n$. Determining the exact order remains open, which is `erdos_959`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-959/Research/FinalLowerBound.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«959»","statement":"∃ c, 0 < c ∧ ∃ N, ∀ n ≥ N, ↑n ^ (1 + c / Real.log (Real.log ↑n)) ≤ ↑(Erdos959.extremalGap n)","subjects":["52"],"theorem":"Erdos959.erdos_959.lower_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $k\\geq 3$. Does there exist a finite set $A\\subset \\mathbb{R}^2$ such that, in any $2$-colouring of $A$, there exists a line which contains at least $k$ points from $A$, and all the points of $A$ on the line have the same colour?\n\nErdős [Er75f] says Graham and Selfridge proved the answer is yes when $k=3$. Hunter has observed that, for sufficiently large $n$, a generic projection of $[k]^n$ into $\\mathbb{R}^2$ has this property, by the Hales-Jewett theorem.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1090.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1090»","statement":"True ↔\n  ∀ (k : ℕ),\n    3 ≤ k →\n      ∃ A,\n        ∀ (C : ↥A → Fin 2),\n          ∃ S,\n            ∃ (hSA : S ⊆ A),\n              Collinear ℝ ↑S ∧\n                S.card ≥ k ∧ (∀ y ∈ A, y ∈ affineSpan ℝ ↑S → y ∈ S) ∧ ∃ c, ∀ (x : Fin 2 → ℝ) (hx : x ∈ S), C ⟨x, ⋯⟩ = c","subjects":["5"],"theorem":"Erdos1090.erdos_1090"},{"answerKinds":[],"category":"research solved","docstring":"It is easy to prove that $F(n)\\geq 1-o(1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1203»","statement":"∀ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, Erdos1203.F n ≥ 1 - ε","subjects":["11"],"theorem":"Erdos1203.erdos_1203.variants.lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Prove that $F(n)\\to \\infty$ as $n\\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1203»","statement":"True ↔ Filter.Tendsto Erdos1203.F Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos1203.erdos_1203"},{"answerKinds":[],"category":"research solved","docstring":"**Shiu's theorem**: for any $k \\geq 1$ and $(a, q) = 1$ there exist infinitely many $k$-tuples of consecutive primes\n$p_m, \\dots, p_{m + k - 1}$ all of which are congruent to $a$ modulo $q$.\n\nThis is stated ahead of `erdos_427` because the formal proof linked there assumes it, and the\n`assuming` clause must name a declaration that already exists.\n\n[Sh00] Shiu, D. K. L., _Strings of congruent primes_. J. London Math. Soc. (2) (2000), 359-373.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«427»","statement":"Erdos427.ShiuTheorem","subjects":["11"],"theorem":"Erdos427.erdos_427.variants.shiu"},{"answerKinds":[],"category":"research solved","docstring":"Cedric Pilatte has observed that a positive solution to Erdős Problem 427 follows from Shiu's theorem.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«427»","statement":"Erdos427.ShiuTheorem → Erdos427.erdos427","subjects":["11"],"theorem":"Erdos427.erdos_427.variants.of_shiu"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 427**: is it true that, for every $n$ and $d$, there exists $k$ such that\n$$\n  d \\mid p_{n + 1} + \\cdots + p_{n + k},\n$$\nwhere $p_r$ denotes the $r$th prime?\n\nThe linked proof is not complete on its own. It declares Shiu's theorem as an axiom and derives\nthe result from it, so it is marked `conditional` and names `erdos_427.variants.shiu`.\n","formalProofs":[{"conditions":["Erdos427.erdos_427.variants.shiu"],"kind":"lean4","link":"https://gist.githubusercontent.com/JohnEdwardJennings/e2c6ef0daab55857b7cc9d340de7af84/raw/8ff97800e38582c71246a238e7541a9d69488cbd/Erdos427.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«427»","statement":"True ↔ Erdos427.erdos427","subjects":["11"],"theorem":"Erdos427.erdos_427"},{"answerKinds":[],"category":"research solved","docstring":"Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours\nthen there must exist a monochromatic triangle. Determine\n$$\\lim_{k\\to \\infty}R(3;k)^{1/k}.$$\n\nThere is no finite limit: $R(3;k)^{1/k}\\to\\infty$. This was established by OpenAI [OpenAI26]\nalong with the explicit superexponential lower bound in\n`erdos_183.variants.explicit_lower_bound`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/openai/ten-proofs/blob/94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6/MulticolorTriangleRamsey.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«183»","statement":"Filter.Tendsto (fun k => ↑(Erdos183.multicolourTriangleRamsey k) ^ (1 / ↑k)) Filter.atTop Filter.atTop","subjects":["5"],"theorem":"Erdos183.erdos_183"},{"answerKinds":[],"category":"research solved","docstring":"The explicit bound behind `erdos_183`: for every $k\\geq 2$,\n$$R(3;k)\\geq \\left(\\frac{k^{1/3}}{6e^{38}\\log k}\\right)^k.$$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/openai/ten-proofs/blob/94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6/MulticolorTriangleRamsey.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«183»","statement":"∀ (k : ℕ), 2 ≤ k → (1 / (6 * Real.exp 38) * ↑k ^ (1 / 3) / Real.log ↑k) ^ k ≤ ↑(Erdos183.multicolourTriangleRamsey k)","subjects":["5"],"theorem":"Erdos183.erdos_183.variants.explicit_lower_bound"},{"answerKinds":[],"category":"test","docstring":"Sanity check for the encoding of \"chromatic number $\\geq \\aleph_1$\": a countably\ncolourable graph, such as the empty graph on $\\mathbb{N}$, does not satisfy the\nhypothesis `IsEmpty (G.Coloring ℕ)`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«594»","statement":"¬IsEmpty (⊥.Coloring ℕ)","subjects":["5"],"theorem":"Erdos594.erdos_594.variants.bot_countably_colorable"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 594** (Erdős–Hajnal [ErHa66], [Er69b]):\n\nDoes every graph $G$ with chromatic number $\\geq \\aleph_1$ contain all sufficiently\nlarge odd cycles?\n\nThe answer is **Yes**, proved by Erdős, Hajnal, and Shelah [EHS74].\n\nA graph has chromatic number $\\geq \\aleph_1$ (i.e. uncountable chromatic number)\nif and only if it admits no proper colouring with countably many colours; this is\nencoded as `IsEmpty (G.Coloring ℕ)`. The conclusion states that there is some\n$N$ such that for every $k \\geq N$ the graph contains a cycle of odd length $2k + 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«594»","statement":"True ↔\n  ∀ (V : Type) (G : SimpleGraph V),\n    IsEmpty (G.Coloring ℕ) → ∃ N, ∀ (k : ℕ), N ≤ k → ∃ v w, w.IsCycle ∧ w.length = 2 * k + 1","subjects":["3","5"],"theorem":"Erdos594.erdos_594"},{"answerKinds":[],"category":"research solved","docstring":"The earlier result of Erdős and Hajnal [ErHa66]: every graph with chromatic number\n$\\geq \\aleph_2$ contains all sufficiently large odd cycles.\n\nChromatic number $\\geq \\aleph_2$ is encoded as the nonexistence of a proper colouring\nwith any set of at most $\\aleph_1$ colours.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«594»","statement":"∀ (V : Type) (G : SimpleGraph V),\n  (∀ (α : Type), Cardinal.mk α ≤ Cardinal.aleph 1 → IsEmpty (G.Coloring α)) →\n    ∃ N, ∀ (k : ℕ), N ≤ k → ∃ v w, w.IsCycle ∧ w.length = 2 * k + 1","subjects":["3","5"],"theorem":"Erdos594.erdos_594.variants.erdos_hajnal"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$\nif $(a,b)=1$) such that $\\limsup_{p,k} f(p^k) / \\log(p^k) = ∞$ and $f(p^k) = f(p)$\nor $f(p^k) = kf(p)$.\nIs it true that $\\limsup_n f(n+1)/f(n) = ∞$?\n\nThe known counterexample does not satisfy either of these extra hypotheses, so this variant remains\nopen.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«897»","statement":"True ↔\n  ∀ (f : ℕ → ℝ),\n    (∀ a > 0, ∀ b > 0, a.Coprime b → f (a * b) = f a + f b) →\n      Filter.limsup\n            (fun x =>\n              match x with\n              | (p, k) => ↑(f (p ^ k)) / ↑(Real.log (↑p ^ k)))\n            (Filter.atTop ⊓ Filter.principal {(p, k) | Nat.Prime p}) =\n          ⊤ →\n        ((∀ (k p : ℕ), Nat.Prime p → f (p ^ k) = f p) ∨ ∀ (k p : ℕ), Nat.Prime p → f (p ^ k) = ↑k * f p) →\n          Filter.limsup (fun n => ↑(f (n + 1)) / ↑(f n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos897.erdos_897.variants.parts.ii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$\nif $(a,b)=1$ such that $\\limsup_{p,k} f(p^k) / \\log(p^k) = ∞$.\nIs it true that $\\limsup_n (f(n+1)−f(n))/ \\log n = ∞$?\n\nThe answer is no; this follows from a construction of Wirsing [Wi81], rediscovered by\nArchivara [Ar25] and formalised in Lean by Aristotle [ArWu25].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos897.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«897»","statement":"False ↔\n  ∀ (f : ℕ → ℝ),\n    (∀ a > 0, ∀ b > 0, a.Coprime b → f (a * b) = f a + f b) →\n      Filter.limsup\n            (fun x =>\n              match x with\n              | (p, k) => ↑(f (p ^ k)) / ↑(Real.log (↑p ^ k)))\n            (Filter.atTop ⊓ Filter.principal {(p, k) | Nat.Prime p}) =\n          ⊤ →\n        Filter.limsup (fun n => (↑(f (n + 1)) - ↑(f n)) / ↑(Real.log ↑n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos897.erdos_897.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$\nif $(a,b)=1$) such that $\\limsup_{p,k} f(p^k) / \\log(p^k) = ∞$ and $f(p^k) = f(p)$\nor $f(p^k) = kf(p)$.\nIs it true that $\\limsup_n (f(n+1)−f(n))/ \\log n = ∞$?\n\nThe known counterexample does not satisfy either of these extra hypotheses, so this variant remains\nopen.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«897»","statement":"True ↔\n  ∀ (f : ℕ → ℝ),\n    (∀ a > 0, ∀ b > 0, a.Coprime b → f (a * b) = f a + f b) →\n      Filter.limsup\n            (fun x =>\n              match x with\n              | (p, k) => ↑(f (p ^ k)) / ↑(Real.log (↑p ^ k)))\n            (Filter.atTop ⊓ Filter.principal {(p, k) | Nat.Prime p}) =\n          ⊤ →\n        ((∀ (k p : ℕ), Nat.Prime p → f (p ^ k) = f p) ∨ ∀ (k p : ℕ), Nat.Prime p → f (p ^ k) = ↑k * f p) →\n          Filter.limsup (fun n => (↑(f (n + 1)) - ↑(f n)) / ↑(Real.log ↑n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos897.erdos_897.variants.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$\nif $(a,b)=1$) such that $\\limsup_{p,k} f(p^k) / \\log(p^k) = ∞$.\nIs it true that $\\limsup_n f(n+1)/ f(n) = ∞$?\n\nThe answer is no; the same counterexample is formalised in Lean by Aristotle [ArWu25].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos897.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«897»","statement":"False ↔\n  ∀ (f : ℕ → ℝ),\n    (∀ a > 0, ∀ b > 0, a.Coprime b → f (a * b) = f a + f b) →\n      Filter.limsup\n            (fun x =>\n              match x with\n              | (p, k) => ↑(f (p ^ k)) / ↑(Real.log (↑p ^ k)))\n            (Filter.atTop ⊓ Filter.principal {(p, k) | Nat.Prime p}) =\n          ⊤ →\n        Filter.limsup (fun n => ↑(f (n + 1)) / ↑(f n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos897.erdos_897.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Wirsing [Wi70] proved that if $|f(n+1)−f(n)| ≤ C$ then $f(n) = c \\log n + O(1)$ for some constant\n$c$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«897»","statement":"∀ (f : ℕ → ℝ),\n  (∀ a > 0, ∀ b > 0, a.Coprime b → f (a * b) = f a + f b) →\n    ∀ (C : ℝ), (∀ (n : ℕ), |f (n + 1) - f n| ≤ C) → ∃ c O, O =O[Filter.atTop] 1 ∧ ∀ (n : ℕ), f n ≤ c * Real.log ↑n + O n","subjects":["11"],"theorem":"Erdos897.erdos_897.variants.log_growth"},{"answerKinds":[],"category":"research open","docstring":"Every connected graph on $n$ vertices can be partitioned into at most $\\lceil n/2\\rceil$\nedge-disjoint paths.\n\nA problem of Erdős and Gallai.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«583»","statement":"∀ {V : Type u_1} [inst : Fintype V] (G : SimpleGraph V),\n  G.Connected → ∃ D, (∀ H ∈ D, Erdos583.IsPathSubgraph H) ∧ G.IsDecomposition D ∧ D.card ≤ ⌈↑(Fintype.card V) / 2⌉₊","subjects":["5"],"theorem":"Erdos583.erdos_583"},{"answerKinds":[],"category":"research open","docstring":"Show that for $k\\geq 3$\n$$\\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{1}{k}}.$$\n\nThis problem is #46 in Extremal Graph Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«572»","statement":"∀ (k : ℕ),\n  3 ≤ k →\n    ∃ c > 0,\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        c * ↑n ^ (1 + 1 / ↑k) ≤ ↑(SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph (2 * k)))","subjects":["5"],"theorem":"Erdos572.erdos_572"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for every integer $k\\neq 1$, there are infinitely many $n$ such that\n$2^n\\equiv k\\pmod{n}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«479»","statement":"True ↔ ∀ (k k : ℤ), k ≠ 1 → {n | 2 ^ n ≡ k [ZMOD ↑n]}.Infinite","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos479.erdos_479"},{"answerKinds":[],"category":"research solved","docstring":"$v_0(n) > 1$ for all $n$ except $n$ = 0, 1, 2, 3, 4, 7, 8, 16\n\n[ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«889»","statement":"∀ n ∉ {0, 1, 2, 3, 4, 7, 8, 16}, 1 < Erdos889.v₀ n","subjects":["11"],"theorem":"Erdos889.erdos_889.variants.v0_gt_1"},{"answerKinds":[],"category":"research open","docstring":"Does $v_1(n) = 1$ have finite solutions?\n\n[ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«889»","statement":"True ↔ {n | Erdos889.v_l 1 n = 1}.Finite","subjects":["11"],"theorem":"Erdos889.erdos_889.variants.v1_eq_1_finite"},{"answerKinds":[],"category":"research open","docstring":"Let $v(n,k)$ count the prime factors of $n+k$ which\ndo not divide $n+i$ for $0\\leq i < k$. Is it true that\n$v_0(n)=\\max_{k\\geq 0}v(n,k)\\to \\infty$ as $n\\to \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«889»","statement":"Filter.Tendsto Erdos889.v₀ Filter.atTop (nhds ⊤)","subjects":["11"],"theorem":"Erdos889.erdos_889"},{"answerKinds":[],"category":"research open","docstring":"Let $v_l(n) = \\max_{k\\geq l} v(n,k)$. For every fixed $l$,\n$v_l(n) \\to \\infty$ as $n \\to \\infty$\n\n[ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«889»","statement":"∀ (l : ℕ), Filter.Tendsto (Erdos889.v_l l) Filter.atTop (nhds ⊤)","subjects":["11"],"theorem":"Erdos889.erdos_889.variants.general"},{"answerKinds":[],"category":"research open","docstring":"Does $V_1(n) = 1$ have finite solutions?\n\nThis is a modification of `erdos_889.variants.v1_eq_1_finite`,\nwhich might make it more amenable to attack according to [ErSe67].\n\n[ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«889»","statement":"True ↔ {n | Erdos889.V_l 1 n = 1}.Finite","subjects":["11"],"theorem":"Erdos889.erdos_889.variants.V1_eq_1_finite"},{"answerKinds":[],"category":"research solved","docstring":"There are infinitely many $n$ such that $τ(n) = τ(n+1)$. Proved in [He84].\nHere τ is the divisor counting function, which is `σ 0` in mathlib.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«946»","statement":"{n | (ArithmeticFunction.sigma 0) n = (ArithmeticFunction.sigma 0) (n + 1)}.Infinite","subjects":["11"],"theorem":"Erdos946.erdos_946"},{"answerKinds":[],"category":"research solved","docstring":"There are infinitely many $n$ such that $τ(n) = τ(n + 5040)$. Proved in [Sp81].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«946»","statement":"{n | (ArithmeticFunction.sigma 0) n = (ArithmeticFunction.sigma 0) (n + 5040)}.Infinite","subjects":["11"],"theorem":"Erdos946.erdos_946.variants.spiro_5040"},{"answerKinds":[],"category":"research solved","docstring":"Upper bound in [EPS87]: $O(x / \\sqrt{\\log \\log x})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«946»","statement":"Erdos946.erdos946Count =O[Filter.atTop] fun x => x / √(Real.log (Real.log x))","subjects":["11"],"theorem":"Erdos946.erdos_946.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Improved lower bound in [Hi85]: $Ω(x / (\\log \\log x)^3)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«946»","statement":"(fun x => x / Real.log (Real.log x) ^ 3) =O[Filter.atTop] Erdos946.erdos946Count","subjects":["11"],"theorem":"Erdos946.erdos_946.variants.hildebrand_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"The number of $n \\le x$ with $τ(n) = τ(n+1)$ is at least $x / (\\log x)^7$ for all sufficiently\nlarge $x$. Proved in [He84].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«946»","statement":"(fun x => x / Real.log x ^ 7) =O[Filter.atTop] Erdos946.erdos946Count","subjects":["11"],"theorem":"Erdos946.erdos_946.variants.heathbrown_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"The upper bound $\\delta_3(5) \\leq 1/12$ of Erdős, Hajnal, Simonovits, Sós, and Szemerédi\n[EHSSS94]: for every $\\epsilon > 0$ there is a $\\delta > 0$ such that for all sufficiently\nlarge $n$, every $K_5$-free graph $G$ on $n$ vertices in which every triangle-free vertex\nset has at most $\\delta n$ vertices has at most $(1/12 + \\epsilon)n^2$ edges.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«533»","statement":"∀ (ε : ℝ),\n  0 < ε →\n    ∃ δ,\n      0 < δ ∧\n        ∀ᶠ (n : ℕ) in Filter.atTop,\n          ∀ (G : SimpleGraph (Fin n)),\n            G.CliqueFree 5 →\n              (∀ (S : Finset (Fin n)), G.CliqueFreeOn (↑S) 3 → ↑S.card ≤ δ * ↑n) →\n                ↑G.edgeFinset.card ≤ (1 / 12 + ε) * ↑n ^ 2","subjects":["5"],"theorem":"Erdos533.erdos_533.variants.ehsss_upper"},{"answerKinds":[],"category":"research solved","docstring":"The observation $\\delta_3(7) \\geq 1/4$ of Erdős, Hajnal, Simonovits, Sós, and Szemerédi\n[EHSSS94], via a construction of Erdős and Rogers [ErRo62]: for every $\\epsilon, \\delta > 0$\nand all sufficiently large $n$ there is a $K_7$-free graph $G$ on $n$ vertices in which every\ntriangle-free vertex set has at most $\\delta n$ vertices, yet which has at least\n$(1/4 - \\epsilon)n^2$ edges.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«533»","statement":"∀ (ε δ : ℝ),\n  0 < ε →\n    0 < δ →\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∃ G,\n          G.CliqueFree 7 ∧\n            (∀ (S : Finset (Fin n)), G.CliqueFreeOn (↑S) 3 → ↑S.card ≤ δ * ↑n) ∧\n              (1 / 4 - ε) * ↑n ^ 2 ≤ ↑G.edgeFinset.card","subjects":["5"],"theorem":"Erdos533.erdos_533.variants.delta_seven_ge_quarter"},{"answerKinds":[],"category":"research solved","docstring":"The matching lower bound $\\delta_3(5) \\geq 1/12$, from the construction of Liu, Reiher,\nSharifzadeh, and Staden [LRSS21] (improving the earlier $\\delta_3(5) > 0$ of Balogh and Lenz\n[BaLe13]): for every $\\epsilon, \\delta > 0$ and all sufficiently large $n$ there is a\n$K_5$-free graph $G$ on $n$ vertices in which every triangle-free vertex set has at most\n$\\delta n$ vertices, yet which has at least $(1/12 - \\epsilon)n^2$ edges. In particular this\nrefutes `erdos_533`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«533»","statement":"∀ (ε δ : ℝ),\n  0 < ε →\n    0 < δ →\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∃ G,\n          G.CliqueFree 5 ∧\n            (∀ (S : Finset (Fin n)), G.CliqueFreeOn (↑S) 3 → ↑S.card ≤ δ * ↑n) ∧\n              (1 / 12 - ε) * ↑n ^ 2 ≤ ↑G.edgeFinset.card","subjects":["5"],"theorem":"Erdos533.erdos_533.variants.lrss_lower"},{"answerKinds":[],"category":"research solved","docstring":"The contrasting positive result $\\delta_3(4) = 0$ of Erdős, Hajnal, Simonovits, Sós, and\nSzemerédi [EHSSS94]: the $K_4$ analogue of `erdos_533` is **true**. For every $\\delta > 0$\nthere is a $c > 0$ such that for all sufficiently large $n$, every $K_4$-free graph $G$ on\n$n$ vertices with at least $\\delta n^2$ edges contains a triangle-free vertex set of size at\nleast $c n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«533»","statement":"∀ (δ : ℝ),\n  0 < δ →\n    ∃ c,\n      0 < c ∧\n        ∀ᶠ (n : ℕ) in Filter.atTop,\n          ∀ (G : SimpleGraph (Fin n)),\n            G.CliqueFree 4 → δ * ↑n ^ 2 ≤ ↑G.edgeFinset.card → ∃ S, c * ↑n ≤ ↑S.card ∧ G.CliqueFreeOn (↑S) 3","subjects":["5"],"theorem":"Erdos533.erdos_533.variants.delta_four_eq_zero"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $\\delta > 0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no\n$K_5$ and at least $\\delta n^2$ edges, must $G$ contain a set of $\\gg_\\delta n$ vertices\nspanning no triangle?\n\nEquivalently, writing $\\mathrm{RT}_3(n, K_5, m)$ for the maximum number of edges of a\n$K_5$-free graph on $n$ vertices in which every triangle-free vertex set has fewer than $m$\nvertices (the *triangle Ramsey–Turán number*), is\n$$\\delta_3(5) = \\lim_{\\epsilon \\to 0} \\lim_{n \\to \\infty}\n  \\frac{\\mathrm{RT}_3(n, K_5, \\epsilon n)}{n^2} = 0?$$\n\nThis is a problem of Erdős, Hajnal, Simonovits, Sós, and Szemerédi [EHSSS94], who proved\n$\\delta_3(5) \\leq 1/12$ and the analogous $\\delta_3(4) = 0$, and observed $\\delta_3(7) \\geq 1/4$\nvia a construction of Erdős and Rogers [ErRo62].\n\nThe answer is **no**: Balogh and Lenz [BaLe13] disproved it by showing $\\delta_3(5) > 0$, and\nthe exact value $\\delta_3(5) = 1/12$ was determined by the matching lower-bound construction of\nLiu, Reiher, Sharifzadeh, and Staden [LRSS21] (see `erdos_533.variants.lrss_lower`).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«533»","statement":"False ↔\n  ∀ (δ : ℝ),\n    0 < δ →\n      ∃ c,\n        0 < c ∧\n          ∀ᶠ (n : ℕ) in Filter.atTop,\n            ∀ (G : SimpleGraph (Fin n)),\n              G.CliqueFree 5 → δ * ↑n ^ 2 ≤ ↑G.edgeFinset.card → ∃ S, c * ↑n ≤ ↑S.card ∧ G.CliqueFreeOn (↑S) 3","subjects":["5"],"theorem":"Erdos533.erdos_533"},{"answerKinds":[],"category":"test","docstring":"Sanity check for the triangle-free-set condition: in the empty graph every vertex set spans\nno triangle, since the empty graph has no $3$-clique.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«533»","statement":"∀ (n : ℕ) (S : Finset (Fin n)), ⊥.CliqueFreeOn (↑S) 3","subjects":["5"],"theorem":"Erdos533.erdos_533.variants.test_bot"},{"answerKinds":[],"category":"research open","docstring":"Denote by $M(n, k)$ the least common multiple of the finite set $\\{n+1, \\dotsc, n+k\\}$.\nIs it true that for all $m \\geq n + k$, we get $M(m, k) \\neq M(n, k)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«677»","statement":"∀ (m n k : ℕ), k > 0 → m ≥ n + k → Finset.lcmInterval m k ≠ Finset.lcmInterval n k","subjects":["11"],"theorem":"Erdos677.erdos_677"},{"answerKinds":[],"category":"test","docstring":"Erdős expected very few solutions for $M(n, k) = M(m, l)$, where $m \\geq n + k$ and $l > 1$.\nThe only solutions he knew were the following.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«677»","statement":"Finset.lcmInterval 4 3 = Finset.lcmInterval 13 2 ∧ Finset.lcmInterval 3 4 = Finset.lcmInterval 19 2","subjects":["11"],"theorem":"Erdos677.lcmInterval_eq_example1"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"The Hadwiger–Nelson problem asks: How many colors are required to color the plane\nsuch that no two points at distance 1 from each other have the same color?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«508»","statement":"(SimpleGraph.UnitDistancePlaneGraph Set.univ).chromaticNumber = sorry","subjects":["52"],"theorem":"Erdos508.HadwigerNelsonProblem"},{"answerKinds":[],"category":"research solved","docstring":"Aubrey de Grey improved the lower bound for the chromatic number of the plane\nto 5 in 2018 using a graph that has >1000 nodes.\n\n\"The chromatic number of the plane is at least 5\" Aubrey D. N. J. de Grey, 2018\n(https://doi.org/10.48550/arXiv.1804.02385)\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«508»","statement":"5 ≤ (SimpleGraph.UnitDistancePlaneGraph Set.univ).chromaticNumber","subjects":["52"],"theorem":"Erdos508.HadwigerNelsonAtLeastFive"},{"answerKinds":[],"category":"research solved","docstring":"The \"chromatic number of the plane\" is at least 4. This can be\nproven by considering the [Moser-Spindel graph](https://de.wikipedia.org/wiki/Moser-Spindel)\nor the [Golomb graph](https://en.wikipedia.org/wiki/Golomb_graph) graph.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«508»","statement":"4 ≤ (SimpleGraph.UnitDistancePlaneGraph Set.univ).chromaticNumber","subjects":["5"],"theorem":"Erdos508.HadwigerNelsonAtLeast4"},{"answerKinds":[],"category":"textbook","docstring":"The chromatic number of the plane is at least 3.\n\nThis is proven by considering an equilateral triangle in the plane. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«508»","statement":"3 ≤ (SimpleGraph.UnitDistancePlaneGraph Set.univ).chromaticNumber","subjects":["5"],"theorem":"Erdos508.HadwigerNelsonAtLeastThree"},{"answerKinds":[],"category":"textbook","docstring":"This upper bound for the chromatic number of the plane was\nobserved by John R. Isbell. His approach was dividing the\nplane into hexagons of uniform size and coloring them with a repeating\npattern. A proof can probably be found in:\n\nSoifer, Alexander (2008), The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, New York: Springer, ISBN 978-0-387-74640-1\n\nAn alternative approach that uses square tiling was highlighted by László Székely.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«508»","statement":"(SimpleGraph.UnitDistancePlaneGraph Set.univ).chromaticNumber ≤ 7","subjects":["52"],"theorem":"Erdos508.HadwigerNelsonAtMostSeven"},{"answerKinds":[],"category":"research open","docstring":"Carmichael has asked whether there is an integer $n$ for which $\\phi(m) = n$ has\nexactly one solution, that is $\\frac{f_\\max(n)}{f_\\min(n)} = 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«694»","statement":"True ↔ ∃ n > 0, ∃! m, m.totient = n","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos694.erdos_694.variants.carmichael"},{"answerKinds":[],"category":"research solved","docstring":"Let $f_\\max(n)$ be the largest $m$ such that $\\phi(m) = n$, and\n$f_\\min(n)$ be the smallest such $m$, where $\\phi$ is Euler's\ntotient function. Investigate\n$$\n  \\max_{n\\leq x}\\frac{f_\\max(n)}{f_\\min(n)}.\n$$\n\nGPT-5.5 Pro (prompted by Price) has proved (see also the comments for a summary) that\n$$\n\\max_{n\\leq x}\\frac{f_{\\max}(n)}{f_{\\min}(n)}=(e^\\gamma+o(1))\\log\\log x.\n$$\n\nA Lean formalisation of the reduction exists, conditional on Mertens' product theorem and\nLinnik's theorem; see the\n[formal proof](https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P694/Proof.lean).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«694»","statement":"∀ (fmax fmin : ℕ → ℕ),\n  (∀ (n : ℕ), IsGreatest (Nat.totient ⁻¹' {n}) (fmax n)) →\n    (∀ (n : ℕ), IsLeast (Nat.totient ⁻¹' {n}) (fmin n)) →\n      ∃ o,\n        Filter.Tendsto o Filter.atTop (nhds 0) ∧\n          ∀ (x : ℕ),\n            sSup {x_1 | ∃ n, ∃ (_ : n ≤ x) (_ : ∃ m, m.totient = n), ↑(fmax n) / ↑(fmin n) = x_1} =\n              (Real.exp Real.eulerMascheroniConstant + o x) * Real.log (Real.log ↑x)","subjects":["11"],"theorem":"Erdos694.erdos_694"},{"answerKinds":[],"category":"research solved","docstring":"Erdős has proved that if there exists an integer $n$ for which $\\phi(m) = n$ has\nexactly one solution, then there must be infinitely many such $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«694»","statement":"(∃ n > 0, ∃! m, m.totient = n) → {n | ∃! m, m.totient = n}.Infinite","subjects":["11"],"theorem":"Erdos694.erdos_694.variants.inf_unique"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A=\\{n_1<n_2<\\cdots\\}\\subset \\mathbb{N}$ be a lacunary sequence (so there exists some\n$\\epsilon>0$ with $n_{k+1}\\geq (1+\\epsilon)n_k$ for all $k$). Must there exist an irrational\n$\\theta$ such that\n$$\\{ \\|\\theta n_k\\| : k\\geq 1\\}$$\nis not dense in $[0,1]$ (where $\\| x\\|$ is the distance to the nearest integer)?\n\nSolved independently by de Mathan [dM80] and Pollington [Po79b], who showed that, given any\nsuch $A$, there exists such a $\\theta$, with\n$$\\inf_{k\\geq 1}\\| \\theta n_k\\| \\gg \\frac{\\epsilon^4}{\\log(1/\\epsilon)}.$$\nThis bound was improved by Katznelson [Ka01], Akhunzhanov and Moshchevitin [AkMo04], and\nDubickas [Du06], before Peres and Schlag [PeSc10] improved it to\n$$\\inf_{k\\geq 1}\\| \\theta n_k\\| \\gg \\frac{\\epsilon}{\\log(1/\\epsilon)},$$\nand note that the best bound possible here would be $\\gg \\epsilon$.\n\nThis problem has consequences for [894](https://www.erdosproblems.com/894).\n\nThe conclusion \"$\\{\\|\\theta n_k\\|\\}$ is not dense in $[0,1]$\" is formalized as the sequence\n$(\\theta n_k)$ not being dense modulo one; see the formalization notes above.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/110d489ed5c07e5b216453e092e9113127c98c9a/problems/464/Erdos464.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«464»","statement":"True ↔\n  ∀ (n : ℕ → ℕ),\n    StrictMono n → (∀ (k : ℕ), 0 < n k) → IsLacunary n → ∃ θ, Irrational θ ∧ ¬Dense (Set.range fun k => ↑(θ * ↑(n k)))","subjects":["11"],"theorem":"Erdos464.erdos_464"},{"answerKinds":[],"category":"research solved","docstring":"Erdős originally conjectured this (in [Er46b]) with no 3 vertices equidistant,\nbut Danzer found a convex polygon on 9 points such that every vertex has three\nvertices equidistant from it (but this distance depends on the vertex).\nDanzer's construction is explained in [Er87b].\n\n[Er46b] Erdős, P., _On sets of distances of $n$ points_. Amer. Math. Monthly (1946), 248-250.\n[Er87b] Erdős, P., _Some combinatorial and metric problems in geometry_. Intuitive geometry (Siófok, 1985), 167-177.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«97»","statement":"∃ A, A.Nonempty ∧ EuclideanGeometry.ConvexIndep ↑A ∧ Erdos97.HasNEquidistantProperty 3 A","subjects":["52"],"theorem":"Erdos97.erdos_97.variants.three_equidistant"},{"answerKinds":[],"category":"research solved","docstring":"Fishburn and Reeds [FiRe92] have found a convex polygon on 20 points such that\nevery vertex has three vertices equidistant from it (and this distance is the same for all vertices).\n\n[FiRe92] Fishburn, P. C. and Reeds, J. A., _Unit distances between vertices of a convex polygon_. Comput. Geom. (1992), 81-91.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«97»","statement":"∃ A, A.Nonempty ∧ EuclideanGeometry.ConvexIndep ↑A ∧ Erdos97.HasNUnitDistanceProperty 3 A","subjects":["52"],"theorem":"Erdos97.erdos_97.variants.three_unit_distance"},{"answerKinds":[],"category":"research solved","docstring":"Fishburn and Reeds [FiRe92] also proved that the smallest $n$ for which there exists\na convex $n$-gon and a cut $\\{A, B\\}$ of its vertices such that $|\\{b \\in B : d(a, b) = 1\\}| ≥ 3$\nfor all $a \\in A$, and $|\\{a \\in A : d(a, b) = 1\\}| ≥ 3$ for all $b \\in B$, is $n = 20$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«97»","statement":"sInf\n    {n |\n      ∃ V A B,\n        n = V.card ∧\n          EuclideanGeometry.ConvexIndep ↑V ∧\n            A.Nonempty ∧\n              B.Nonempty ∧\n                Erdos97.IsCut V A B ∧ Erdos97.HasNUnitDistancePointsOn 3 B A ∧ Erdos97.HasNUnitDistancePointsOn 3 A B} =\n  20","subjects":["52"],"theorem":"Erdos97.erdos_97.variants.three_unit_distance_cut_min"},{"answerKinds":[],"category":"research open","docstring":"Erdős also conjectured that there is a $k$ for which every convex polygon has a vertex\nwith no other $k$ vertices equidistant from it.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«97»","statement":"True ↔\n  ∃ k,\n    ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))),\n      A.Nonempty → EuclideanGeometry.ConvexIndep ↑A → ¬Erdos97.HasNEquidistantProperty k A","subjects":["52"],"theorem":"Erdos97.erdos_97.variants.k_equidistant"},{"answerKinds":[],"category":"research open","docstring":"Does every convex polygon have a vertex with no other 4 vertices equidistant from it?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«97»","statement":"True ↔\n  ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))),\n    A.Nonempty → EuclideanGeometry.ConvexIndep ↑A → ¬Erdos97.HasNEquidistantProperty 4 A","subjects":["52"],"theorem":"Erdos97.erdos_97"},{"answerKinds":[],"category":"research solved","docstring":"**Mertens' third theorem** [Me1874], in the weakened form the linked proof assumes: the product\nover the primes up to $n$ of $1 - 1/p$ is at least $1/(3\\log n)$.\n\nThe true asymptotic is $e^{-\\gamma}/\\log n$, and $e^{-\\gamma} > 1/3$, so this bound is weaker than\nthe theorem and is what the deduction needs.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1141»","statement":"∀ (n : ℕ), 3 ≤ n → 1 / (3 * Real.log ↑n) ≤ ∏ p ∈ Finset.range (n + 1) with Nat.Prime p, (1 - 1 / ↑p)","subjects":["11"],"theorem":"Erdos1141.erdos_1141.variants.mertens_third"},{"answerKinds":[],"category":"research solved","docstring":"**Theorem 1.3 of [Po17]**: for a quadratic character to a large enough modulus, the primes below\n$m^{1/4+\\varepsilon}$ on which the character is $1$ outnumber any fixed power of $\\log m$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1141»","statement":"∀ (ε A : ℝ),\n  0 < ε →\n    0 < A →\n      ∃ m₀,\n        ∀ (m : ℕ),\n          m₀ < m →\n            ∀ (χ : DirichletCharacter ℂ m),\n              MulChar.IsQuadratic χ → Real.log ↑m ^ A ≤ ↑(Erdos1141.residuePrimesUpTo m χ ε).card","subjects":["11"],"theorem":"Erdos1141.erdos_1141.variants.pollack_1_3"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$?\n\nIn [Va99] it is asked whether $968$ is the largest integer with this property, but this is an\nerror, since for example $968-9=7\\cdot 137$.\n\nThe list of $n$ satisfying the given property is [A214583] in the OEIS. The largest known such $n$\nis $1722$.\n\nThe answer is negative: [APSSV26b] proves a stronger finiteness theorem, deducing it from\nPollack [Po17]. Oriike [Or26] formalised the deduction in Lean.\n\nThe linked proof is the deduction and not the whole result. It declares Theorem 1.3 of [Po17] and\nMertens' third theorem as axioms, so it is marked `conditional` and names both.\n","formalProofs":[{"conditions":["Erdos1141.erdos_1141.variants.pollack_1_3","Erdos1141.erdos_1141.variants.mertens_third"],"kind":"lean4","link":"https://github.com/yuta0x89/ErdosProblems/blob/a1319f732cdee5140faf47d984e2c451c1184803/Erdos1141.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1141»","statement":"False ↔ Infinite ↑{n | Erdos1141.Erdos1141Prop n}","subjects":["11"],"theorem":"Erdos1141.erdos_1141"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er63d] proved\n$$\\frac{n}{4}\\leq H(n) \\ll n^{3/2}.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1028»","statement":"∀ᶠ (n : ℕ) in Filter.atTop, ↑n / 4 ≤ ↑(Erdos1028.H n)","subjects":["5"],"theorem":"Erdos1028.erdos_1028.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Let\n$$H(n)=\\min_f \\max_{X\\subseteq \\{1,\\ldots,n\\}} \\left\\lvert \\sum_{x<y\\in X} f(x,y)\\right\\rvert,$$\nwhere $f$ ranges over all functions $f:\\{1,\\ldots,n\\}^2\\to \\{-1,1\\}$. Estimate $H(n)$.\n\nErdős [Er63d] proved\n$$\\frac{n}{4}\\leq H(n) \\ll n^{3/2}.$$\nErdős and Spencer [ErSp71] proved that $H(n)\\gg n^{3/2}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1028.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1028»","statement":"(fun n => ↑(Erdos1028.H n)) =Θ[Filter.atTop] fun n => ↑n ^ (3 / 2)","subjects":["5"],"theorem":"Erdos1028.erdos_1028"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Spencer [ErSp71] proved that $H(n)\\gg n^{3/2}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1028»","statement":"(fun n => ↑n ^ (3 / 2)) =O[Filter.atTop] fun n => ↑(Erdos1028.H n)","subjects":["5"],"theorem":"Erdos1028.erdos_1028.variants.erdos_spencer"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er63d] proved\n$$\\frac{n}{4}\\leq H(n) \\ll n^{3/2}.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1028»","statement":"(fun n => ↑(Erdos1028.H n)) =O[Filter.atTop] fun n => ↑n ^ (3 / 2)","subjects":["5"],"theorem":"Erdos1028.erdos_1028.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er49c] proved that the statement in `erdos_971` holds for infinitely many values of `d`.\n\n[Er49c] Erdős, P., _On some applications of Brun's method_. Acta Univ. Szeged. Sect. Sci. Math.\n(1949), 57--63.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«971»","statement":"∃ c > 0,\n  ∃ C > 0,\n    {d |\n        C * ↑d.totient ≤\n          ↑{a ∈ Finset.Iio d |\n                a.Coprime d ∧ ↑(Erdos971.leastCongruentPrime a d) > (1 + c) * ↑d.totient * Real.log ↑d}.card}.Infinite","subjects":["11"],"theorem":"Erdos971.erdos_971.variants.infinite_sequence"},{"answerKinds":[],"category":"research open","docstring":"Let `p(a, d)` be the least prime congruent to `a (mod d)`.\nDoes there exist a constant `c > 0` such that for all large `d`,\n`p(a, d) > (1 + c) * φ(d) * log d` for `≫ φ(d)` many values of `a`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«971»","statement":"True ↔\n  ∃ c > 0,\n    ∃ C > 0,\n      ∀ᶠ (d : ℕ) in Filter.atTop,\n        C * ↑d.totient ≤\n          ↑{a ∈ Finset.Iio d |\n                a.Coprime d ∧ ↑(Erdos971.leastCongruentPrime a d) > (1 + c) * ↑d.totient * Real.log ↑d}.card","subjects":["11"],"theorem":"Erdos971.erdos_971"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er49c] proved that for any `ε > 0` we have `p(a, d) < ε * φ(d) * log d` for `≫_ε φ(d)` many\nvalues of `a` (for all large `d`).\n\n[Er49c] Erdős, P., _On some applications of Brun's method_. Acta Univ. Szeged. Sect. Sci. Math.\n(1949), 57--63.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«971»","statement":"∀ ε > 0,\n  ∃ C > 0,\n    ∀ᶠ (d : ℕ) in Filter.atTop,\n      C * ↑d.totient ≤\n        ↑{a ∈ Finset.Iio d | a.Coprime d ∧ ↑(Erdos971.leastCongruentPrime a d) < ε * ↑d.totient * Real.log ↑d}.card","subjects":["11"],"theorem":"Erdos971.erdos_971.variants.many_small"},{"answerKinds":[],"category":"research open","docstring":"Is it true that there are infinitely many $p$ for which $f(p) = p − 1$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1072»","statement":"True ↔ {p | Nat.Prime p ∧ Erdos1072.f p = p - 1}.Infinite","subjects":["11"],"theorem":"Erdos1072.erdos_1072.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Erdős, Hardy, and Subbarao [HaSu02], believed that the number of $p \\le x$ for which $f(p)=p−1$\nis $o(x/\\log x)$.\n\n[HaSu02] Hardy, G. E. and Subbarao, M. V., _A modified problem of Pillai and some related questions._\nAmer. Math. Monthly (2002), 554--559.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1072»","statement":"(fun x => ↑({p | Nat.Prime p ∧ Erdos1072.f p = p - 1} ∩ Set.Icc 0 x).ncard) =o[Filter.atTop] fun x => ↑x / Real.log ↑x","subjects":["11"],"theorem":"Erdos1072.erdos_1072.variants.littleo"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $f(p)/p \\to 0$ for $p \\to \\infty$ in a density 1 subset of the primes? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1072»","statement":"True ↔\n  ∃ P ⊆ {p | Nat.Prime p},\n    P.HasDensity 1 {p | Nat.Prime p} ∧\n      Filter.Tendsto (fun p => ↑(Erdos1072.f p) / ↑p) (Filter.atTop ⊓ Filter.principal P) (nhds 0)","subjects":["11"],"theorem":"Erdos1072.erdos_1072.parts.ii"},{"answerKinds":["non-Prop"],"category":"research solved","docstring":"For $n > 393$ the answer is $\\binom{n-1}{2} + 1 - \\left\\lfloor \\frac{n-1}{2} \\right\\rfloor$\n(Elliott [El67], with the correction of Purdy and Smith [PuSm], also reported in [BaBa94]): this is\nboth a lower bound for every such configuration and is attained, e.g. by a circle with $n - 1$\npoints together with a single point off the circle.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«506»","statement":"∀ (n : ℕ),\n  393 < n →\n    IsLeast {k | ∃ P, P.card = n ∧ ¬Collinear ℝ ↑P ∧ ¬EuclideanGeometry.Cospherical ↑P ∧ Erdos506.numCircles ↑P = k}\n      ((n - 1).choose 2 + 1 - (n - 1) / 2)","subjects":["51","52"],"theorem":"Erdos506.erdos_506.variants.large_n"},{"answerKinds":[],"category":"research solved","docstring":"Segre's observation: the lower bound $\\binom{n-1}{2}$ (without the correction of [PuSm]) is\nalready false for $n = 8$, as witnessed by the projection of a cube onto a plane. Hence the minimum\nnumber of circles determined by $8$ such points is strictly less than $\\binom{7}{2} = 21$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«506»","statement":"∃ P, P.card = 8 ∧ ¬Collinear ℝ ↑P ∧ ¬EuclideanGeometry.Cospherical ↑P ∧ Erdos506.numCircles ↑P < (8 - 1).choose 2","subjects":["51","52"],"theorem":"Erdos506.erdos_506.variants.segre_n_eq_eight"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the minimum number of circles determined by any $n$ points in $\\mathbb{R}^2$, not all on a\ncircle?\n\nThere is clearly some non-degeneracy condition intended here - probably either that not all the\npoints are on a line, or the stronger condition that no three points are on a line.\n\nThe answer is known for $n > 393$ (see `erdos_506.variants.large_n`) but the problem appears to\nremain open for small $n$ (see `erdos_506.variants.small_n`).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«506»","statement":"∀ (n : ℕ),\n  4 ≤ n →\n    IsLeast {k | ∃ P, P.card = n ∧ ¬Collinear ℝ ↑P ∧ ¬EuclideanGeometry.Cospherical ↑P ∧ Erdos506.numCircles ↑P = k}\n      sorry","subjects":["51","52"],"theorem":"Erdos506.erdos_506"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"The problem appears to remain open for small $n$: Elliott's answer is established only for\n$n > 393$, so the minimum number of circles determined by $n$ points, not all on a line and not all\non a circle, is unknown for $n \\le 393$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«506»","statement":"∀ (n : ℕ),\n  4 ≤ n →\n    n ≤ 393 →\n      IsLeast {k | ∃ P, P.card = n ∧ ¬Collinear ℝ ↑P ∧ ¬EuclideanGeometry.Cospherical ↑P ∧ Erdos506.numCircles ↑P = k}\n        sorry","subjects":["51","52"],"theorem":"Erdos506.erdos_506.variants.small_n"},{"answerKinds":[],"category":"research open","docstring":"Is it true that `limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«454»","statement":"True ↔ Filter.limsup (fun n => ↑(Erdos454.f n) - 2 * ↑(Nat.nth Prime n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos454.erdos_454"},{"answerKinds":[],"category":"research solved","docstring":"`limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop ≥ 2`, and this is proved in [Po79]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«454»","statement":"2 ≤ Filter.limsup (fun n => ↑(Erdos454.f n) - 2 * ↑(Nat.nth Prime n)) Filter.atTop","subjects":["11"],"theorem":"Erdos454.erdos_454.variants.two_le_limsup"},{"answerKinds":[],"category":"research open","docstring":"Must every permutation of $\\mathbb{N}$, contain a monotone 4-term arithmetic progression?","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«196»","statement":"True ↔ ∀ (f : ℕ ≃ ℕ), HasMonotoneAP (⇑f) 4","subjects":["5","11"],"theorem":"Erdos196.erdos_196"},{"answerKinds":["Prop"],"category":"research solved","docstring":"For $2 \\le k \\le n$, the interval $A = \\{n, n - 1, \\dots, n - k + 1\\}$ maximises the number\nof integers not representable as the sum of finitely many elements from $A$ (with repetitions\nallowed), as proved by Kiss [Ki02].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://www.erdosproblems.com/forum/thread/434#post-4437"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«434»","statement":"True ↔\n  ∀ n ≥ 1,\n    ∀ k ≥ 2,\n      k ≤ n →\n        IsGreatest\n          {x |\n            ∃ S,\n              ∃ (_ : S ⊆ Finset.Icc 1 n) (_ : S.card = k) (_ : S.gcd id = 1), Erdos434.Nat.NcardUnrepresentable ↑S = x}\n          (Erdos434.Nat.NcardUnrepresentable (Set.Icc (n - k + 1) n))","subjects":["11"],"theorem":"Erdos434.erdos_434.parts.ii"},{"answerKinds":["non-Prop"],"category":"research solved","docstring":"Let $k \\le n$. What choice of $A\\subseteq\\{1, \\dots, n\\}$ (with $\\text{gcd}(A) = 1$) of size $|A| = k$\nmaximises the number of integers not representable as the sum of finitely\nmany elements from $A$ (with repetitions allowed)?\nIs it $\\{n, n - 1, \\dots, n - k + 1\\}$?\n\nThe maximal choice is indeed $\\{n, \\dots, n - k + 1\\}$, as proved by Kiss [Ki02].\n\nThe Lean theorem assumes $2 \\le k$. When $k = 1 < n$, the proposed singleton $\\{n\\}$ does not\nhave gcd $1$; the gcd condition instead forces the unique admissible choice $A = \\{1\\}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://www.erdosproblems.com/forum/thread/434#post-4437"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«434»","statement":"∀ (n k : ℕ),\n  1 ≤ n →\n    2 ≤ k →\n      k ≤ n →\n        IsGreatest\n          {x |\n            ∃ S,\n              ∃ (_ : S ⊆ Finset.Icc 1 n) (_ : S.card = k) (_ : S.gcd id = 1), Erdos434.Nat.NcardUnrepresentable ↑S = x}\n          (Erdos434.Nat.NcardUnrepresentable (Set.Icc (n - k + 1) n))","subjects":["11"],"theorem":"Erdos434.erdos_434.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is there an infinite Sidon set $A\\subset \\mathbb{N}$ such that\n$\\lvert A\\cap \\{1\\ldots,N\\}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}$\nfor all $\\varepsilon > 0$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«39»","statement":"True ↔\n  ∃ A, A.Infinite ∧ IsSidon A ∧ ∀ ε > 0, (fun x => ↑x ^ (1 / 2 - ε)) =O[Filter.atTop] fun N => ↑(Set.Icc 1 N ∩ A).ncard","subjects":["11"],"theorem":"Erdos39.erdos_39"},{"answerKinds":[],"category":"research open","docstring":"Is it true that there are only finitely many pairs of intervals $I_1$, $I_2$ such that\n$$\n\\sum_{n_1 \\in I_1} \\frac{1}{n_1} + \\sum_{n_2 \\in I_2} \\frac{1}{n_2} \\in \\mathbb{N}?\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«288»","statement":"True ↔\n  {I | ∀ (j : Fin 2), (I j).1 ≤ (I j).2 ∧ ∃ n, ∑ j, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑↑nⱼ)⁻¹ = ↑↑n}.Finite","subjects":["11"],"theorem":"Erdos288.erdos_288"},{"answerKinds":[],"category":"research open","docstring":"This is still open even if $|I_2| = 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«288»","statement":"True ↔ {(I, n₂) | I.1 ≤ I.2 ∧ ∃ n, ∑ n₁ ∈ (Set.Icc I.1 I.2).toFinset, (↑↑n₁)⁻¹ + (↑↑n₂)⁻¹ = ↑↑n}.Finite","subjects":["11"],"theorem":"Erdos288.erdos_288.variants.i2_card_eq_1"},{"answerKinds":[],"category":"research open","docstring":"Is it true for any $k > 2$ that only finitely many $k$ intervals satisfy this condition?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«288»","statement":"True ↔\n  ∃ k > 2,\n    {I | ∀ (j : Fin k), (I j).1 ≤ (I j).2 ∧ ∃ n, ∑ j, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑↑nⱼ)⁻¹ = ↑↑n}.Finite","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos288.erdos_288.variants.exists_k_gt_2"},{"answerKinds":[],"category":"research open","docstring":"It is perhaps true with two intervals replaced by any $k$ intervals.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«288»","statement":"True ↔\n  ∀ (k : ℕ),\n    {I | ∀ (j : Fin k), (I j).1 ≤ (I j).2 ∧ ∃ n, ∑ j, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑↑nⱼ)⁻¹ = ↑↑n}.Finite","subjects":["11"],"theorem":"Erdos288.erdos_288.variants.k_intervals"},{"answerKinds":[],"category":"research solved","docstring":"If $C_1,\\ldots,C_n$ are circles in $\\mathbb{R}^2$ with radii $r_1,\\ldots,r_n$ such that no line\ndisjoint from all the circles divides them into two non-empty sets then the circles can be\ncovered by a circle of radius $r=\\sum r_i$.\n\nThis is true, and was proved by Goodman and Goodman [GoGo45] (whose proof also generalises to\nhigher dimensions). A generalisation to convex bodies was proved by Hadwiger [Ha47].\n\nAn alternative proof is given by Bezdek and Litvak [BeLi16].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1121.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1121»","statement":"∀ {n : ℕ} (c : Fin n → EuclideanSpace ℝ (Fin 2)) (r : Fin n → ℝ),\n  (∀ (i : Fin n), 0 < r i) →\n    (∀ (v : EuclideanSpace ℝ (Fin 2)) (t : ℝ),\n        v ≠ 0 →\n          (∀ (i : Fin n), ∀ p ∈ Metric.closedBall (c i) (r i), inner ℝ v p ≠ t) →\n            (∀ (i : Fin n), inner ℝ v (c i) < t) ∨ ∀ (i : Fin n), t < inner ℝ v (c i)) →\n      ∃ z, ⋃ i, Metric.closedBall (c i) (r i) ⊆ Metric.closedBall z (∑ i, r i)","subjects":["52"],"theorem":"Erdos1121.erdos_1121"},{"answerKinds":[],"category":"research solved","docstring":"The proof of Goodman and Goodman [GoGo45] also generalises to higher dimensions: if\n$B_1,\\ldots,B_n$ are balls in $\\mathbb{R}^d$ with radii $r_1,\\ldots,r_n$ such that no hyperplane\ndisjoint from all the balls divides them into two non-empty sets then the balls can be covered by\na ball of radius $r=\\sum r_i$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1121»","statement":"∀ {d n : ℕ} (c : Fin n → EuclideanSpace ℝ (Fin d)) (r : Fin n → ℝ),\n  (∀ (i : Fin n), 0 < r i) →\n    (∀ (v : EuclideanSpace ℝ (Fin d)) (t : ℝ),\n        v ≠ 0 →\n          (∀ (i : Fin n), ∀ p ∈ Metric.closedBall (c i) (r i), inner ℝ v p ≠ t) →\n            (∀ (i : Fin n), inner ℝ v (c i) < t) ∨ ∀ (i : Fin n), t < inner ℝ v (c i)) →\n      ∃ z, ⋃ i, Metric.closedBall (c i) (r i) ⊆ Metric.closedBall z (∑ i, r i)","subjects":["52"],"theorem":"Erdos1121.erdos_1121.variants.higher_dimension"},{"answerKinds":[],"category":"research solved","docstring":"Tang's lower bound [Tang]:\n\n$\\log k(n) \\ge (1/\\sqrt{2} - o(1)) * \\sqrt{\\log n * \\log \\log n}$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«962»","statement":"∃ ε,\n  (∀ δ > 0, ∀ᶠ (n : ℕ) in Filter.atTop, |ε n| < δ) ∧\n    ∀ᶠ (n : ℕ) in Filter.atTop, (1 / √2 - ε n) * √(Real.log ↑n * Real.log (Real.log ↑n)) ≤ Real.log ↑(Erdos962.k n)","subjects":["11"],"theorem":"Erdos962.erdos_962.variants.tang_lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Main conjecture:\n\n$\\log k(n) \\le (\\log n)^{(1/2 + o(1))}$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«962»","statement":"True ↔\n  ∃ ε,\n    (∀ δ > 0, ∀ᶠ (n : ℕ) in Filter.atTop, |ε n| < δ) ∧\n      ∀ᶠ (n : ℕ) in Filter.atTop, Real.log ↑(Erdos962.k n) ≤ (Real.log ↑n).rpow (1 / 2 + ε n)","subjects":["11"],"theorem":"Erdos962.erdos_962"},{"answerKinds":[],"category":"research solved","docstring":"Tao's upper bound [Tao]:\n\n$k(n) \\le (1 + o(1)) * n^{1/2}$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«962»","statement":"∃ ε, (∀ δ > 0, ∀ᶠ (n : ℕ) in Filter.atTop, |ε n| < δ) ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos962.k n) ≤ (1 + ε n) * √↑n","subjects":["11"],"theorem":"Erdos962.erdos_962.variants.tao_upper_bound"},{"answerKinds":[],"category":"research open","docstring":"For $A\\subseteq \\{1,\\ldots,n\\}$ let $G(A)$ be the graph with vertex set $A$, where two\nintegers are joined by an edge if they are coprime.\n\nIs it true that if\n$$|A| > \\lfloor n/2 \\rfloor + \\lfloor n/3 \\rfloor - \\lfloor n/6 \\rfloor$$\nthen $G(A)$ contains all odd cycles of length $\\leq n/3 + 1$?\n\nA problem of Erdős and Sárközy [ErSa97].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«883»","statement":"True ↔\n  ∀ (n : ℕ),\n    ∀ A ⊆ Finset.Icc 1 n,\n      n / 2 + n / 3 - n / 6 < A.card →\n        ∀ (l : ℕ), Odd l → 3 ≤ l → l ≤ n / 3 + 1 → l ∈ (SimpleGraph.induce (↑A) Erdos883.coprimeGraph).oddCycleLengths","subjects":["5","11"],"theorem":"Erdos883.erdos_883.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, for every $\\ell\\geq 1$, if $n$ is sufficiently large and\n$$|A| > \\lfloor n/2\\rfloor + \\lfloor n/3\\rfloor - \\lfloor n/6\\rfloor$$\nthen $G(A)$ must contain a complete $(1,\\ell,\\ell)$ tripartite graph on $2\\ell+1$ vertices?\n\nThe second question was solved by Sárközy [Sa99], who proved this with\n$\\ell \\gg \\log n/\\log\\log n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«883»","statement":"True ↔\n  ∀ (l : ℕ),\n    1 ≤ l →\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ A ⊆ Finset.Icc 1 n,\n          n / 2 + n / 3 - n / 6 < A.card →\n            (SimpleGraph.completeMultipartiteGraph fun i => Fin (![1, l, l] i)).IsContained\n              (SimpleGraph.induce (↑A) Erdos883.coprimeGraph)","subjects":["5","11"],"theorem":"Erdos883.erdos_883.parts.ii"},{"answerKinds":[],"category":"API","docstring":"The hypothesis `¬ Summable (fun n : A ↦ distToNearestInt (θ * n))` used below says exactly\nthat the partial sums of `‖θ n‖` over `n ∈ A` diverge, which is the form the linked proof uses.\n`distToNearestInt` is nonnegative, so this is an instance of\n`not_summable_subtype_iff_tendsto_sum_indicator`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«254»","statement":"∀ (A : Set ℕ) (θ : ℝ),\n  (¬Summable fun n => distToNearestInt (θ * ↑↑n)) ↔\n    Filter.Tendsto (fun N => ∑ n ∈ Finset.range N, A.indicator (fun n => distToNearestInt (θ * ↑n)) n) Filter.atTop\n      Filter.atTop","subjects":["11"],"theorem":"Erdos254.not_summable_iff_tendsto_partial_sums"},{"answerKinds":[],"category":"research solved","docstring":"Cassels [Ca60] proved this under the alternative hypotheses $\\lim \\frac{\\lvert A\\cap [1,2x]\\rvert\n-\\lvert A\\cap [1,x]\\rvert}{\\log\\log x}=\\infty$ and $\\sum_{n\\in A} \\{ \\theta n\\}^2=\\infty$ for every\n$\\theta\\in (0,1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«254»","statement":"∀ (A : Set ℕ),\n  (Filter.Tendsto (fun x => (↑(A ∩ Set.Icc 1 (2 * x)).ncard - ↑(A ∩ Set.Icc 1 x).ncard) / Real.log (Real.log ↑x))\n        Filter.atTop Filter.atTop ∧\n      ∀ (θ : ℝ), 0 < θ → θ < 1 → ¬Summable fun n => distToNearestInt (θ * ↑↑n) ^ 2) →\n    ∀ᶠ (m : ℕ) in Filter.atTop, Erdos254.IsSumOfDistinct A m","subjects":["11"],"theorem":"Erdos254.erdos_254.variants.cassels"},{"answerKinds":[],"category":"research solved","docstring":"Let $A\\subseteq \\mathbb{N}$ be such that $\\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert \\to\n\\infty\\textrm{ as }x\\to \\infty$ and $\\sum_{n\\in A} \\{ \\theta n\\}=\\infty$ for every $\\theta\\in\n(0,1)$, where $\\{x\\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large\ninteger is the sum of distinct elements of $A$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-254/Research/Basic.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«254»","statement":"∀ (A : Set ℕ),\n  (Filter.Tendsto (fun x => (A ∩ Set.Icc 1 (2 * x)).ncard - (A ∩ Set.Icc 1 x).ncard) Filter.atTop Filter.atTop ∧\n      ∀ (θ : ℝ), 0 < θ → θ < 1 → ¬Summable fun n => distToNearestInt (θ * ↑↑n)) →\n    ∀ᶠ (m : ℕ) in Filter.atTop, Erdos254.IsSumOfDistinct A m","subjects":["11"],"theorem":"Erdos254.erdos_254"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the smallest $k$ such that $\\mathbb{R}^2$ can be red/blue coloured with no pair of red\npoints unit distance apart, and no $k$-term arithmetic progression of blue points with distance 1?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«188»","statement":"IsLeast Erdos188.s sorry","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"Erdos188.erdos_188"},{"answerKinds":[],"category":"research solved","docstring":"Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 15):\n\nHow small can $M$ be made? The only estimate currently known is that $M \\le 10000000$ (more or less).\nIn the other direction, it has just been shown by R. Juhász [Ju (79)] that we must have $M \\ge 5$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«188»","statement":"(∀ k ∈ Erdos188.s, 5 ≤ k) ∧ ∃ k ∈ Erdos188.s, k ≤ 10000000","subjects":["5"],"theorem":"Erdos188.erdos_188.variants.estimate"},{"answerKinds":[],"category":"research solved","docstring":"Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 14, 15):\n\nIt has been shown that there is a large $M$ so that it is possible to partition $\\mathbb{E}^2$ into\ntwo sets $A$ and $B$ so that $A$ contains no pair of points with distance 1 and $B$ contains no A.P.\nof length $M$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«188»","statement":"Erdos188.s.Nonempty","subjects":["5"],"theorem":"Erdos188.erdos_188.variants.nonempty"},{"answerKinds":[],"category":"research open","docstring":"If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power\nof $2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«406»","statement":"IsGreatest {n | n.isPowerOfTwo ∧ Nat.digits 3 n ⊆ [1, 2]} (2 ^ 15)","subjects":["11"],"theorem":"Erdos406.erdos_406.variants.one_two"},{"answerKinds":[],"category":"research open","docstring":"Is it true that there are only finitely many powers of $2$ which have only the digits $0$\nand $1$ when written in base $3$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«406»","statement":"True ↔ {n | n.isPowerOfTwo ∧ Nat.digits 3 n ⊆ [0, 1]}.Finite","subjects":["11"],"theorem":"Erdos406.erdos_406"},{"answerKinds":[],"category":"research open","docstring":"Erdős Problem 423 [Er77c, p.71; ErGr80, p.83]:\n\nLet $a(1) = 1$, $a(2) = 2$, and for $k \\ge 3$ let $a(k)$ be the least integer greater\nthan $a(k-1)$ that is a sum of at least two consecutive terms of the sequence.\nWhat is the asymptotic behaviour of this sequence? It seems likely that $a_n = n + o(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«423»","statement":"True ↔ ∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → (fun n => ↑(a n) - ↑n) =o[Filter.atTop] fun n => ↑n","subjects":["5","11"],"theorem":"Erdos423.erdos_423"},{"answerKinds":[],"category":"research solved","docstring":"Bolan and Tang [Ta26] independently proved that $a_n-n\\to\\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«423»","statement":"∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → ∀ (M : ℕ), ∀ᶠ (n : ℕ) in Filter.atTop, M + n ≤ a n","subjects":["5","11"],"theorem":"Erdos423.erdos_423.variants.unbounded"},{"answerKinds":[],"category":"research solved","docstring":"Tang [Ta26] proved the lower bound $a_n=n+\\Omega(\\log\\log n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«423»","statement":"∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → (fun n => Real.log (Real.log ↑n)) =O[Filter.atTop] fun n => ↑(a n) - ↑n","subjects":["5","11"],"theorem":"Erdos423.erdos_423.variants.lower_bound"},{"answerKinds":[],"category":"test","docstring":"The unboundedness of $a_n-n$ is equivalent to the sequence omitting infinitely many positive\nintegers. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«423»","statement":"(∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → ∀ (M : ℕ), ∀ᶠ (n : ℕ) in Filter.atTop, M + n ≤ a n) ↔\n  ∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → (Set.range a)ᶜ.Infinite","subjects":["5","11"],"theorem":"Erdos423.erdos_423.test.unbounded_iff_infinite_complement"},{"answerKinds":[],"category":"research solved","docstring":"Bolan and Tang [Ta26] independently proved that $a_n-n$ is nondecreasing.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«423»","statement":"∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → ∀ (n m : ℕ), 1 ≤ n → n ≤ m → a n - n ≤ a m - m","subjects":["5","11"],"theorem":"Erdos423.erdos_423.variants.nondecreasing"},{"answerKinds":[],"category":"test","docstring":"The fourth term of the Hofstadter sequence is $a(4) = 5 = a(2) + a(3) = 2 + 3$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«423»","statement":"∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → a 4 = 5","subjects":["5","11"],"theorem":"Erdos423.erdos_423.test.a4"},{"answerKinds":[],"category":"research solved","docstring":"Tang [Ta26] proved $a_n \\ll n^{1/(c-1)+o(1)}$ whenever every finite convex set $A$ satisfies\n$|A-A|\\geq |A|^{c-o(1)}$. Using the bound of Cushman [Cu25] gives\n$a_n\\ll n^{688/413+o(1)}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«423»","statement":"∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → ∀ ε > 0, (fun n => ↑(a n)) =O[Filter.atTop] fun n => ↑n ^ (688 / 413 + ε)","subjects":["5","11"],"theorem":"Erdos423.erdos_423.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Bolan and Tang [Ta26] independently proved that infinitely many positive integers do not occur in\nthe Hofstadter sequence.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«423»","statement":"∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → (Set.range a)ᶜ.Infinite","subjects":["5","11"],"theorem":"Erdos423.erdos_423.variants.infinite_complement"},{"answerKinds":[],"category":"test","docstring":"The third term of the Hofstadter sequence is $a(3) = 3 = a(1) + a(2) = 1 + 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«423»","statement":"∀ (a : ℕ → ℕ), Erdos423.IsHofstadterSeq a → a 3 = 3","subjects":["5","11"],"theorem":"Erdos423.erdos_423.test.a3"},{"answerKinds":[],"category":"research solved","docstring":"For sufficiently large $n$ and every $k\\geq n$, $R(C_k,K_n)=(k-1)(n-1)+1$.\n\nKeevash, Long, and Skokan [KLS21] have proved this identity when\n$k\\geq C\\frac{\\log n}{\\log\\log n}$ for some constant $C$, thus establishing the conjecture\nfor sufficiently large $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«551»","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ (k : ℕ), n ≤ k → (SimpleGraph.cycleGraph k).graphRamsey (SimpleGraph.completeGraph (Fin n)) = (k - 1) * (n - 1) + 1","subjects":["5"],"theorem":"Erdos551.erdos_551.variants.sufficiently_large"},{"answerKinds":[],"category":"research open","docstring":"Prove that\n$$R(C_k,K_n)=(k-1)(n-1)+1$$\nfor $k\\geq n\\geq 3$ (except when $n=k=3$).\n\nAsked by Erdős, Faudree, Rousseau, and Schelp.\nThis problem is #18 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«551»","statement":"∀ (k n : ℕ),\n  3 ≤ n →\n    n ≤ k →\n      ¬(n = 3 ∧ k = 3) →\n        (SimpleGraph.cycleGraph k).graphRamsey (SimpleGraph.completeGraph (Fin n)) = (k - 1) * (n - 1) + 1","subjects":["5"],"theorem":"Erdos551.erdos_551"},{"answerKinds":[],"category":"research open","docstring":"Are there $n$ such that $n+2^{2^k}$ is infinitely often squarefree?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1209»","statement":"True ↔ ∃ n, {k | Squarefree (n + 2 ^ 2 ^ k)}.Infinite","subjects":["11"],"theorem":"Erdos1209.erdos_1209.parts.iii.d"},{"answerKinds":[],"category":"research open","docstring":"Are there $n$ such that $n+2^{2^k}$ is always squarefree?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1209»","statement":"True ↔ ∃ n, ∀ (k : ℕ), Squarefree (n + 2 ^ 2 ^ k)","subjects":["11"],"theorem":"Erdos1209.erdos_1209.parts.iii.b"},{"answerKinds":["Prop"],"category":"research solved","docstring":"What if we ask for $n+a_k$ to be squarefree instead of prime?\n\nA similar construction provides a counterexample to the squarefree question.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1209»","statement":"False ↔\n  ∃ f,\n    ∀ (a : ℕ → ℕ),\n      StrictMono a →\n        (∀ (k : ℕ), f k ≤ a k) → (∃ n, ∀ (k : ℕ), Squarefree (n + a k)) → {n | ∀ (k : ℕ), Squarefree (n + a k)}.Infinite","subjects":["11"],"theorem":"Erdos1209.erdos_1209.parts.ii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there $n$ such that $n+2^{2^k}$ is always a prime?\n\nebarschkis and GPT have proved that there are no $n$ such that $n+2^{2^k}$ is always prime: let\n$n\\geq 3$ be any odd integer. If $k$ is chosen sufficiently large, and $p=n+2^{2^{k}}$ is prime,\nthen the multiplicative order of $2^{2^k}\\pmod{p}$, say $m$ is odd, and hence if $l$ is chosen such\nthat $2^l\\equiv 1\\pmod{m}$ then $p\\mid n+2^{2^{k+rl}}$ for all $r\\geq 1$.\n\nThis was formalized in Lean by Barschkis using ChatGPT.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/ebarschkis/ErdosProblem/blob/main/Problem1209/Formalization.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1209»","statement":"False ↔ ∃ n, ∀ (k : ℕ), Nat.Prime (n + 2 ^ 2 ^ k)","subjects":["11"],"theorem":"Erdos1209.erdos_1209.parts.iii.a"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A=\\{a_1<a_2<\\cdots\\}$ be a sequence of integers which tends to infinity sufficiently fast.\nIf there is an $n$ such that all $n+a_k$ are primes then must there exist infinitely many such $n$?\n\nErdős [Er80] wrote 'unless I overlook a trivial way of getting a counterexample these questions\nare quite hopeless'. There is indeed a trivial counterexample (a variant of the construction in\n[erdosproblems.com/429]): define $a_1=2$ and for $k\\geq 2$ let $a_k>a_{k-1}$ be a prime such that\n$a_k+k\\equiv 0\\pmod{q_k}$, where $q_k$ is some prime not dividing $k$. This sequence can be made to\ngrow arbitrarily fast\n\nSee also [erdosproblems.com/429] and [erdosproblems.com/1102].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1209»","statement":"False ↔\n  ∃ f,\n    ∀ (a : ℕ → ℕ),\n      StrictMono a →\n        (∀ (k : ℕ), f k ≤ a k) → (∃ n, ∀ (k : ℕ), Nat.Prime (n + a k)) → {n | ∀ (k : ℕ), Nat.Prime (n + a k)}.Infinite","subjects":["11"],"theorem":"Erdos1209.erdos_1209.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Are there $n$ such that $n+2^{2^k}$ is infinitely often a prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1209»","statement":"True ↔ ∃ n, {k | Nat.Prime (n + 2 ^ 2 ^ k)}.Infinite","subjects":["11"],"theorem":"Erdos1209.erdos_1209.parts.iii.c"},{"answerKinds":[],"category":"research solved","docstring":"If $\\tau(n)$ counts the number of divisors of $n$, then what is the set of limit points of\n$$\n\\frac{\\tau((n+1)!)}{\\tau(n!)}?\n$$\n\nThe limit points are exactly $\\{1\\} \\cup \\{1+1/k : k \\geq 1\\}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos419.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«419»","statement":"{x | MapClusterPt x Filter.atTop Erdos419.factorialDivisorRatio} = Erdos419.limitPointSet","subjects":["11"],"theorem":"Erdos419.erdos_419"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ (k : ℕ), Erdos56.WeaklyDivisible k ∅","subjects":["11"],"theorem":"Erdos56.weaklyDivisible_empty"},{"answerKinds":[],"category":"API","docstring":"A singleton is `k`-weakly divisible if `k ≠ 0`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ {k : ℕ}, k ≠ 0 → ∀ (l : ℕ), Erdos56.WeaklyDivisible k {l}","subjects":["11"],"theorem":"Erdos56.weaklyDivisible_singleton"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ {k : ℕ}, k ≠ 0 → Erdos56.MaxWeaklyDivisible 1 k = 1","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos56.maxWeaklyDivisible_one"},{"answerKinds":[],"category":"API","docstring":"No non-empty set is `1`-weakly divisible. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ {A : Finset ℕ}, A.Nonempty → ¬Erdos56.WeaklyDivisible 0 A","subjects":["11"],"theorem":"Erdos56.not_weaklyDivisible_zero"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ (k : ℕ), Erdos56.MaxWeaklyDivisible 0 k = 0","subjects":["11"],"theorem":"Erdos56.maxWeaklyDivisible_zero"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Suppose $A \\subseteq \\{1,\\dots,N\\}$ is such that there are no $k+1$ elements of $A$ which are\nrelatively prime. An example is the set of all multiples of the first $k$ primes.\nIs this the largest such set?  To avoid trivial counterexamples, we must insist that $N$ be at\nleast the $k$th prime.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos56.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«56»","statement":"False ↔\n  ∀ k > 0, ∀ N ≥ Nat.nth Nat.Prime (k - 1), Erdos56.MaxWeaklyDivisible N k = (Erdos56.FirstPrimesMultiples N k).card","subjects":["11"],"theorem":"Erdos56.erdos_56"},{"answerKinds":[],"category":"API","docstring":"An example of a `k`-weakly divisible set is the subset of `{1, ..., N}`\ncontaining the multiples of the first `k` primes.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ (N k : ℕ), Erdos56.WeaklyDivisible k (Erdos56.FirstPrimesMultiples N k)","subjects":["11"],"theorem":"Erdos56.weaklyDivisible_firstPrimesMultiples"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ {A : Finset ℕ}, Erdos56.WeaklyDivisible 0 A ↔ A = ∅","subjects":["11"],"theorem":"Erdos56.empty_iff_weaklyDivisible_zero"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ (k : ℕ), (Erdos56.FirstPrimesMultiples 1 k).card = 0","subjects":["11"],"theorem":"Erdos56.firstPrimesMultiples_one_card_zero"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ (N : ℕ), (Erdos56.FirstPrimesMultiples N 0).card = 0","subjects":["11"],"theorem":"Erdos56.firstPrimesMultiples_zero_k_card_zero"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«56»","statement":"∀ (N : ℕ), Erdos56.MaxWeaklyDivisible N 0 = 0","subjects":["11"],"theorem":"Erdos56.maxWeaklyDivisible_zero_k"},{"answerKinds":[],"category":"research open","docstring":"Let $σ_1(n) = σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$.\n\nIs it true that $\\lim_{k → ∞} σ_k(n)^{\\frac 1 k} = ∞$?\n\nThis is problem (iii) from\nErdos, Granville, Pomerance, Spiro\n\"On the normal behavior of the iterates of some arithmetical functions\"\n(page 169 of the book \"Analytic Number Theory\", 1990).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«410»","statement":"True ↔ ∀ n > 1, Filter.Tendsto (fun k => ↑((⇑(ArithmeticFunction.sigma 1))^[k] n) ^ (1 / ↑k)) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos410.erdos_410"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Graham, Ruzsa, and Straus observe that the method of Balakran can be further used to prove\nthat there are infinitely many $n$ such that $(n+k)!(n+1)!∣(2n)!$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«727»","statement":"∀ (k : ℕ), 2 ≤ k → {n | (n + k).factorial * (n + 1).factorial ∣ (2 * n).factorial}.Infinite","subjects":["11"],"theorem":"Erdos727.erdos_727.variants.k_1_2"},{"answerKinds":[],"category":"research open","docstring":"It is open even for $k = 2$.\nLet $k = 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many n?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«727»","statement":"True ↔ {n | (n + 2).factorial ^ 2 ∣ (2 * n).factorial}.Infinite","subjects":["11"],"theorem":"Erdos727.erdos_727.variants.k_2"},{"answerKinds":[],"category":"research open","docstring":"Let $k ≥ 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«727»","statement":"True ↔ ∀ k ≥ 2, {n | (n + k).factorial ^ 2 ∣ (2 * n).factorial}.Infinite","subjects":["11"],"theorem":"Erdos727.erdos_727"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Balakran proved this holds for $k = 1$.\n\nLet $k = 1$. Does $((n+k)!)^2∣(2n)!$ for infinitely many $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«727»","statement":"True ↔ {n | (n + 1).factorial ^ 2 ∣ (2 * n).factorial}.Infinite","subjects":["11"],"theorem":"Erdos727.erdos_727.variants.k_1"},{"answerKinds":[],"category":"textbook","docstring":"The greedy construction gives a Sidon set of size approximately `√N`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«44»","statement":"∀ (N : ℕ), 1 ≤ N → ∃ A ⊆ Finset.Icc 1 N, IsSidon ↑A ∧ A.card ≥ N.sqrt","subjects":["5","11"],"theorem":"Erdos44.greedy_sidon_construction"},{"answerKinds":[],"category":"textbook","docstring":"The maximum size of a Sidon set in `{1, ..., N}` is less than or equal to `2 * √N`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«44»","statement":"∀ (N : ℕ), 1 ≤ N → ↑(Finset.Icc 1 N).maxSidonSubsetCard ≤ 2 * √↑N","subjects":["5","11"],"theorem":"Erdos44.maxSidonSubsetCard_icc_bound"},{"answerKinds":[],"category":"research open","docstring":"The case where we start with an empty set (constructing large Sidon sets).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«44»","statement":"True ↔ ∀ ε > 0, ∀ᶠ (M : ℕ) in Filter.atTop, ∃ A ⊆ Finset.Icc 1 M, IsSidon ↑A ∧ (1 - ε) * √↑M ≤ ↑A.card","subjects":["5","11"],"theorem":"Erdos44.erdos_44.variants.empty_start"},{"answerKinds":[],"category":"textbook","docstring":"For any `N`, there exists a Sidon set of size at least `√N/2`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«44»","statement":"∀ (N : ℕ), 1 ≤ N → ∃ A ⊆ Finset.Icc 1 N, IsSidon ↑A ∧ N.sqrt / 2 ≤ A.card","subjects":["5","11"],"theorem":"Erdos44.sidon_set_lower_bound"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 44:** Let N ≥ 1 and `A ⊆ {1,…,N}` be a Sidon set. Is it true that, for any ε > 0,\nthere exist M = M(ε) and `B ⊆ {N+1,…,M}` such that `A ∪ B ⊆ {1,…,M}` is a Sidon set\nof size at least `(1−ε)M^{1/2}`?\n\nThis problem asks whether any Sidon set can be extended to achieve a density\narbitrarily close to the optimal density for Sidon sets.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«44»","statement":"True ↔\n  ∀ N ≥ 1,\n    ∀ A ⊆ Finset.Icc 1 N,\n      IsSidon ↑A → ∀ ε > 0, ∃ M > N, ∃ B ⊆ Finset.Icc (N + 1) M, IsSidon (↑A ∪ ↑B) ∧ (1 - ε) * √↑M ≤ ↑(A ∪ B).card","subjects":["5","11"],"theorem":"Erdos44.erdos_44"},{"answerKinds":[],"category":"textbook","docstring":"The set `{1, 2, 4, 8, 13}` is a Sidon set in `{1, ..., 13}`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«44»","statement":"IsSidon {1, 2, 4, 8, 13}","subjects":["5","11"],"theorem":"Erdos44.example_sidon_set"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $x_1,x_2,\\ldots\\in [0,1]$ be an infinite sequence.\nIs it true that\n$$\\inf_n \\liminf_{m\\to \\infty} n \\lvert x_{m+n}-x_m\\rvert\\leq 5^{-1/2}\\approx 0.447?$$\nA conjecture of Newman.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«480»","statement":"True ↔\n  ∀ (x : ℕ → ℝ),\n    (∀ (n : ℕ), x n ∈ Set.Icc 0 1) → ⨅ n, Filter.liminf (fun m => ↑↑n * |x (m + ↑n) - x m|) Filter.atTop ≤ 1 / √5","subjects":["11"],"theorem":"Erdos480.erdos_480"},{"answerKinds":[],"category":"research solved","docstring":"This was proved by Chung and Graham \\cite{ChGr84}, who in fact prove that\n$$\\inf_n \\liminf_{m\\to \\infty} n \\lvert x_{m+n}-x_m\\rvert\\leq \\frac{1}{c}\\approx 0.3944$$\nwhere\n$$c=1+\\sum_{k\\geq 1}\\frac{1}{F_{2k}}=2.5353705\\cdots$$\nand $F_m$ is the $m$th Fibonacci number.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«480»","statement":"have c := 1 + ∑' (k : ℕ+), 1 / ↑(Nat.fib (2 * ↑k));\n∀ (x : ℕ → ℝ),\n  (∀ (n : ℕ), x n ∈ Set.Icc 0 1) → ⨅ n, Filter.liminf (fun m => ↑↑n * |x (m + ↑n) - x m|) Filter.atTop ≤ 1 / c","subjects":["11"],"theorem":"Erdos480.erdos_480.variants.chung_graham"},{"answerKinds":[],"category":"research solved","docstring":"They also prove that this constant is best possible.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«480»","statement":"have c := 1 + ∑' (k : ℕ+), 1 / ↑(Nat.fib (2 * ↑k));\n∀ ε > 0,\n  ¬∀ (x : ℕ → ℝ),\n      (∀ (n : ℕ), x n ∈ Set.Icc 0 1) → ⨅ n, Filter.liminf (fun m => ↑↑n * |x (m + ↑n) - x m|) Filter.atTop ≤ 1 / c - ε","subjects":["11"],"theorem":"Erdos480.erdos_480.variants.chung_graham_best_possible"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $\\pm1$, such\nthat $$\\sqrt n \\ll |P(z)| \\ll \\sqrt n$$ for all $|z|=1$, with the implied constants independent of\n$z$ and $n$?\n\nThe answer is yes, proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].\n\n[BBMST19] Balister, P. and Bollob\\'{A}s, B. and Morris, R. and Sahasrabudhe, J. and Tiba, M., _Flat Littlewood Polynomials Exist_. arXiv:1907.09464 (2019).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«228»","statement":"True ↔\n  ∃ c₁ c₂,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∃ p,\n        p.degree = ↑n ∧\n          (∀ i ≤ n, p.coeff i = 1 ∨ p.coeff i = -1) ∧\n            ∀ (z : ℂ), ‖z‖ = 1 → √↑n < c₁ * ‖Polynomial.eval z p‖ ∧ ‖Polynomial.eval z p‖ < c₂ * √↑n","subjects":["5","12","41"],"theorem":"Erdos228.erdos_228"},{"answerKinds":[],"category":"research open","docstring":"Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors,\n$x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«850»","statement":"True ↔\n  ∃ x y,\n    x ≠ y ∧\n      x.primeFactors = y.primeFactors ∧\n        (x + 1).primeFactors = (y + 1).primeFactors ∧ (x + 2).primeFactors = (y + 2).primeFactors","subjects":["11"],"theorem":"Erdos850.erdos_850"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$?\n\nKo [Ko40] proved there are none if $(x,y)=1$, but there are in fact infinitely many solutions in\ngeneral - for example, $x=2^{12}3^6$, $y = 2^83^8$, and $z = 2^{11}3^7$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos674.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«674»","statement":"True ↔ Erdos674.solutionSet.Nonempty","subjects":["11"],"theorem":"Erdos674.erdos_674"},{"answerKinds":[],"category":"research solved","docstring":"There are in fact infinitely many integer solutions to $x^xy^y=z^z$ with $x,y,z>1$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos674.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«674»","statement":"Erdos674.solutionSet.Infinite","subjects":["11"],"theorem":"Erdos674.erdos_674.variants.infinite"},{"answerKinds":[],"category":"research open","docstring":"For sufficiently large n, is it the case that any set of n points with minimum distance $1$\nthat minimizes diameter must contain an equilateral triangle of side length 1? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«99»","statement":"True ↔\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))),\n      A.card = n →\n        Erdos99.HasMinDist1 A →\n          IsMinOn (fun B => Metric.diam ↑B) {B | B.card = n ∧ Erdos99.HasMinDist1 B} A →\n            ∃ p ∈ A, ∃ q ∈ A, ∃ r ∈ A, Erdos99.FormsEquilateralTriangle p q r","subjects":["52"],"theorem":"Erdos99.erdos_99"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A$ be the set of all $n$ such that $n = d_1 + ⋯ + d_k$ with $d_i$ distinct\nproper divisors of $n$, but this is not true for any $m ∣ n$ with $m < n$. Does:\n$$\n  \\sum_{n ∈ A} \\frac 1 n\n$$\nconverge?\n\nYes: the sum converges. This was proved by Lewis [Le25], whose proof has been formalized in\nLean.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos469.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«469»","statement":"True ↔ Summable fun n => 1 / ↑↑n","subjects":["11"],"theorem":"Erdos469.erdos_469"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there an absolute constant $c > 0$ such that, for all $1 \\leq k < n$, the binomial coefficient\n$\\binom{n}{k}$ has a divisor in $(cn, n]$?\n\nBui, Naprienko, Pratt, and Zaharescu [BNPZ26] answered this negatively.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«387»","statement":"False ↔ ∃ c, 0 < c ∧ ∀ (n k : ℕ), 1 ≤ k → k < n → ∃ d, ↑d ∈ Set.Ioc (c * ↑n) ↑n ∧ d ∣ n.choose k","subjects":["11"],"theorem":"Erdos387.erdos_387"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true for any $c < 1$ and all $n$ sufficiently large, for all $1 \\leq k < n$,\n$\\binom{n}{k}$ has a divisor in $(cn, n]$?\n\nThis variant appears in [Gu04]. Bui, Naprienko, Pratt, and Zaharescu [BNPZ26] answered it\nnegatively.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«387»","statement":"False ↔ ∀ c < 1, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (k : ℕ), 1 ≤ k → k < n → ∃ d, ↑d ∈ Set.Ioc (c * ↑n) ↑n ∧ d ∣ n.choose k","subjects":["11"],"theorem":"Erdos387.erdos_387.variants.guy"},{"answerKinds":[],"category":"research open","docstring":"The following is Schinzel's conjecture, which appears in [Gu04]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«387»","statement":"True ↔ ∀ᶠ (k : ℕ) in Filter.atTop, ¬IsPrimePow k → ∃ n, ∀ i < k, ¬n - i ∣ n.choose k","subjects":["11"],"theorem":"Erdos387.erdos_387.variants.schinzel"},{"answerKinds":[],"category":"research solved","docstring":"It is easy to see that `n.choose k` has a divisor in `[n / k, n]`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«387»","statement":"∀ {n k : ℕ}, 1 ≤ n → k ≤ n → ∃ d, ↑d ∈ Set.Icc (↑n / ↑k) ↑n ∧ d ∣ n.choose k","subjects":["11"],"theorem":"Erdos387.erdos_387.variants.easy"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there a sequence $A=\\{a_1\\leq a_2\\leq \\cdots\\}$ of integers with\n$$\\lim \\frac{a_{n+1}}{a_n}=2$$\nsuch that\n$$P(A')= \\left\\{\\sum_{n\\in B}n : B\\subseteq A'\\textrm{ finite }\\right\\}$$\nhas density $1$ for every cofinite subsequence $A'$ of $A$?\n\nThis has been solved in the affirmative by ebarschkis in the comments (based on idea of Tao and\nvan Doorn, also in the comments).\n\nThis was formalized in Lean by Barschkis using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/ebarschkis/ErdosProblem/blob/main/Problem347/Formalization.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«347»","statement":"True ↔\n  ∃ a,\n    Monotone a ∧\n      Filter.Tendsto (fun n => ↑(a (n + 1)) / ↑(a n)) Filter.atTop (nhds 2) ∧\n        ∀ (ι : ℕ → ℕ), (Set.range ι)ᶜ.Finite → (subsetSums (Set.range (a ∘ ι))).HasDensity 1","subjects":["11"],"theorem":"Erdos347.erdos_347"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A_1(N)$ be the number of maximal Sidon subsets of $\\{1,\\ldots,N\\}$. Is it true that\n$$A_1(N) < 2^{o(N^{1/2})}?$$\n\nA problem of Cameron and Erdős. This is resolved as a consequence of results of Saxton and\nThomason [SaTh15] - they prove that the number of Sidon sets in $\\{1,\\ldots,N\\}$ is at least\n$2^{(1.16+o(1))N^{1/2}}$. Since each Sidon set is contained in a maximal Sidon set, and each\nmaximal Sidon set contains at most $2^{(1+o(1))N^{1/2}}$ Sidon sets, it follows that\n$$A_1(N) \\geq 2^{(0.16+o(1))N^{1/2}}.$$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos862.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«862»","statement":"False ↔ (fun N => Real.logb 2 ↑(Erdos862.numMaximalSidonSets N)) =o[Filter.atTop] fun N => ↑N ^ (1 / 2)","subjects":["5","11"],"theorem":"Erdos862.erdos_862.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"This is resolved as a consequence of results of Saxton and Thomason [SaTh15] - they prove that\nthe number of Sidon sets in $\\{1,\\ldots,N\\}$ is at least $2^{(1.16+o(1))N^{1/2}}$. Since each\nSidon set is contained in a maximal Sidon set, and each maximal Sidon set contains at most\n$2^{(1+o(1))N^{1/2}}$ Sidon sets, it follows that\n$$A_1(N) \\geq 2^{(0.16+o(1))N^{1/2}}.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«862»","statement":"∀ (ε : ℝ), 0 < ε → ∀ᶠ (N : ℕ) in Filter.atTop, 2 ^ ((0.16 - ε) * ↑N ^ (1 / 2)) ≤ ↑(Erdos862.numMaximalSidonSets N)","subjects":["5","11"],"theorem":"Erdos862.erdos_862.variants.lower_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A_1(N)$ be the number of maximal Sidon subsets of $\\{1,\\ldots,N\\}$. Is it true that\n$$A_1(N) > 2^{N^c}$$\nfor some constant $c>0$?\n\nA problem of Cameron and Erdős. This is resolved as a consequence of results of Saxton and\nThomason [SaTh15] - they prove that the number of Sidon sets in $\\{1,\\ldots,N\\}$ is at least\n$2^{(1.16+o(1))N^{1/2}}$. Since each Sidon set is contained in a maximal Sidon set, and each\nmaximal Sidon set contains at most $2^{(1+o(1))N^{1/2}}$ Sidon sets, it follows that\n$$A_1(N) \\geq 2^{(0.16+o(1))N^{1/2}}.$$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos862.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«862»","statement":"True ↔ ∃ c, 0 < c ∧ ∀ᶠ (N : ℕ) in Filter.atTop, 2 ^ ↑N ^ c < ↑(Erdos862.numMaximalSidonSets N)","subjects":["5","11"],"theorem":"Erdos862.erdos_862.parts.ii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does every graph with infinite chromatic number contain a cycle of length $2^n$ for infinitely\nmany $n$?\n\nConjectured by Mihók and Erdős. Solved affirmatively following the work of Liu and Montgomery\n[LiMo20].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«63»","statement":"True ↔ ∀ {V : Type u_1} (G : SimpleGraph V), G.chromaticNumber = ⊤ → ∀ (N : ℕ), ∃ n ≥ N, 2 ^ n ∈ G.cycleLengths","subjects":["5"],"theorem":"Erdos63.erdos_63"},{"answerKinds":[],"category":"research open","docstring":"Is it true that for every $\\epsilon,\\eta>0$ there exists a $k$ such that the density of $n$\nfor which $P(n(n+1)\\cdots(n+k))>n^{1-\\epsilon}$ is at least $1-\\eta$ (where $P(m)$ is the greatest\nprime divisor of $m$)?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1201»","statement":"True ↔\n  ∀ ε > 0,\n    ∀ η > 0,\n      ∃ k, Filter.liminf (fun x => ↑↑(Nat.count (fun x => x ∈ Erdos1201.Erdos1201Set ε k) x) / ↑↑x) Filter.atTop ≥ 1 - η","subjects":["11"],"theorem":"Erdos1201.erdos_1201"},{"answerKinds":[],"category":"research solved","docstring":"Erdős wrote he could prove this for $\\epsilon=1/2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1201»","statement":"∀ η > 0,\n  ∃ k,\n    Filter.liminf (fun x => ↑↑(Nat.count (fun x => x ∈ Erdos1201.Erdos1201Set (1 / 2) k) x) / ↑↑x) Filter.atTop ≥ 1 - η","subjects":["11"],"theorem":"Erdos1201.erdos_1201.variants.epsilon_half"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq \\mathbb{F}_p$. Let\n$$\nA\\hat{+}A = \\{ a+b : a\\neq b \\in A\\}.\n$$\nIs it true that\n$$\n\\lvert A\\hat{+}A\\rvert \\geq \\min(2\\lvert A\\rvert-3,p)?\n$$\n\nThis is the Erdős-Heilbronn inequality, proved by Dias da Silva and Hamidoune.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos476.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«476»","statement":"True ↔ ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (A : Finset (ZMod p)), A.restrictedSumset.card ≥ min (2 * A.card - 3) p","subjects":["5","11"],"theorem":"Erdos476.erdos_476"},{"answerKinds":[],"category":"research solved","docstring":"Let $k\\geq 2$. Erdős and Selfridge [ES75] proved that the product of any $k$ consecutive\nintegers $N$ cannot be a perfect power.\n\n[ES75] P. Erdös, J. L. Selfridge, \"The product of consecutive integers is never a power\",\n  Illinois J. Math. 19(2): 292-301, 1975\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«137»","statement":"∀ k ≥ 2, ∀ (n x l : ℕ), 2 ≤ l → ∏ x ∈ Finset.Ioc n (n + k), x ≠ x ^ l","subjects":["11"],"theorem":"Erdos137.erdos_137.variants.perfect_power"},{"answerKinds":[],"category":"research open","docstring":"Let $k\\geq 3$. Can the product of any $k$ consecutive integers $N$ ever be powerful? That is,\nmust there always exist a prime $p\\mid N$ such that $p^2\\nmid N$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«137»","statement":"True ↔ ∀ k ≥ 3, ∀ (n : ℕ), ¬(∏ x ∈ Finset.Ioc n (n + k), x).Powerful","subjects":["11"],"theorem":"Erdos137.erdos_137"},{"answerKinds":[],"category":"research open","docstring":"Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all\npositive integers $m$, there must be at least $k$ distinct primes $p$ such that\n$p\\mid m(m+1)\\cdots (m+n)$ and yet $p^2$ does not divide the right hand side.\n\n[Er82c] Erdős, Paul, \"Miscellaneous problems in number theory\". Congr. Numer. (1982), 25-45.,\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«137»","statement":"∀ (k : ℕ),\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∀ (m : ℕ),\n      0 < m →\n        ∃ P,\n          P.card = k ∧ ∀ p ∈ P, Nat.Prime p ∧ p ∣ ∏ x ∈ Finset.Ioc m (m + n), x ∧ ¬p ^ 2 ∣ ∏ x ∈ Finset.Ioc m (m + n), x","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos137.erdos_137.variants.multiple_powerful_factors"},{"answerKinds":[],"category":"API","docstring":"Sanity Check: the trivial ruler is actually a perfect ruler if $K \\geq N$ ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«170»","statement":"∀ (N : ℕ), Erdos170.TrivialRuler N ∈ Erdos170.PerfectRulersLengthN N","subjects":["5"],"theorem":"Erdos170.trivial_ruler_is_perfect"},{"answerKinds":[],"category":"research solved","docstring":"The existence of the limit has been proved by Erdős and Gál [ErGa48].\nThe lower bound has been proven by Leech [Le56], who refined an argument of Rédei and Rényi.\nThe upper bound is due to Wichmann [Wi63]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«170»","statement":"∃ x ∈ Set.Icc Erdos170.lower_bound Erdos170.upper_bound,\n  Filter.Tendsto (fun N => ↑(Erdos170.F N) / √↑N) Filter.atTop (nhds x)","subjects":["5"],"theorem":"Erdos170.erdos170.existing_bounds"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"The problem is to determine the limit of the sequence $\\frac{F(N)}{\\sqrt{N}}$ as $N \\to \\infty$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«170»","statement":"Filter.Tendsto (fun N => ↑(Erdos170.F N) / √↑N) Filter.atTop (nhds sorry)","subjects":["5"],"theorem":"Erdos170.erdos170"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $C>0$ be a constant. Are there infinitely many integers $a,b,n$ with $a+b> n+C\\log n$ such\nthat the denominator of $$\\frac{n!}{a!b!}$$contains only primes $\\ll_C 1$?\n\nErdős [Er68c] proved that if $a!b!\\mid n!$ then $a+b\\leq n+O(\\log n)$. This has been proved in the affirmative by Barreto and Leeham, using ChatGPT and Aristotle, with a modification of the argument used for [728].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos729.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«729»","statement":"True ↔\n  ∀ C > 0,\n    ∃ K ≥ 3,\n      {(a, b, n) |\n          a > 0 ∧\n            b > 0 ∧\n              n > 0 ∧\n                ↑a + ↑b > ↑n + C * Real.log ↑n ∧\n                  ∀ (p : ℕ),\n                    Nat.Prime p → p > K → padicValNat p (↑n.factorial / (↑a.factorial * ↑b.factorial)).den = 0}.Infinite","subjects":["11"],"theorem":"Erdos729.erdos_729"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1098»","statement":"∀ {G : Type u_1} [inst : Group G] (g h : G), (Erdos1098.nonCommutingGraph G).Adj g h ↔ g * h ≠ h * g","subjects":["5","20"],"theorem":"Erdos1098.nonCommutingGraph_adj"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a group and $\\Gamma=\\Gamma(G)$ be the non-commuting graph, with vertices the elements\nof $G$ and an edge between $g$ and $h$ if and only if $g$ and $h$ do not commute, $gh\\neq hg$.\n\nIf $\\Gamma$ contains no infinite complete subgraph, then is there a finite bound on the size of\ncomplete subgraphs of $\\Gamma$?\n\nThis was solved by Neumann [Ne76], who proved that $\\Gamma$ contains no infinite complete subgraph\nif and only if the centre of the group has finite index, and noted that if the centre has index $n$\nthen $\\Gamma$ contains no complete subgraph on $>n$ vertices.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1098.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1098»","statement":"True ↔\n  ∀ (G : Type u_1) [inst : Group G],\n    (∀ (s : Set G), (Erdos1098.nonCommutingGraph G).IsClique s → s.Finite) →\n      ∃ n, ∀ (s : Finset G), (Erdos1098.nonCommutingGraph G).IsClique ↑s → s.card ≤ n","subjects":["5","20"],"theorem":"Erdos1098.erdos_1098"},{"answerKinds":[],"category":"research solved","docstring":"Neumann [Ne76] proved that $\\Gamma$ contains no infinite complete subgraph if and only if the\ncentre of the group has finite index.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1098»","statement":"∀ (G : Type u_1) [inst : Group G],\n  (∀ (s : Set G), (Erdos1098.nonCommutingGraph G).IsClique s → s.Finite) ↔ (Subgroup.center G).FiniteIndex","subjects":["5","20"],"theorem":"Erdos1098.erdos_1098.variants.center_finiteIndex"},{"answerKinds":[],"category":"research solved","docstring":"Neumann [Ne76] noted that if the centre has index $n$ then $\\Gamma$ contains no complete subgraph\non $>n$ vertices.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1098»","statement":"∀ (G : Type u_1) [inst : Group G] [(Subgroup.center G).FiniteIndex],\n  (Erdos1098.nonCommutingGraph G).CliqueFree ((Subgroup.center G).index + 1)","subjects":["5","20"],"theorem":"Erdos1098.erdos_1098.variants.upper_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A$ be a finite collection of $d\\geq 4$ non-parallel lines in $\\mathbb{R}^2$ such that\nthere are no points where at least four lines from $A$ meet. Must there exist a 'Gallai\ntriangle' (or 'ordinary triangle'): three lines from $A$ which intersect in three points, and\neach of these intersection points only intersects two lines from $A$?\n\nFüredi and Palásti [FuPa84] showed this is false when $d\\geq 4$ is not divisible by $9$.\nEscudero [Es16] showed this is false for all $d\\geq 4$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/110d489ed5c07e5b216453e092e9113127c98c9a/problems/209/Erdos209.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«209»","statement":"False ↔\n  ∀ (d : ℕ),\n    4 ≤ d →\n      ∀ (A : Finset (AffineSubspace ℝ (EuclideanSpace ℝ (Fin 2)))),\n        A.card = d →\n          (∀ L ∈ A, Erdos209.IsLine L) →\n            ((↑A).Pairwise fun L₁ L₂ => ¬L₁.Parallel L₂) →\n              (∀ (p : EuclideanSpace ℝ (Fin 2)), Erdos209.pointMultiplicity A p ≤ 3) → Erdos209.HasGallaiTriangle A","subjects":["52"],"theorem":"Erdos209.erdos_209"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«219»","statement":"∀ {p : ℕ}, Nat.Prime p → {p} ∈ Erdos219.primeArithmeticProgressions","subjects":["5","11"],"theorem":"Erdos219.singleton_mem_primeArithmeticProgressions"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«219»","statement":"{3, 5, 7} ∈ Erdos219.primeArithmeticProgressions","subjects":["5","11"],"theorem":"Erdos219.primeArithmeticProgression_3_5_7"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«219»","statement":"∀ {p q : ℕ}, Nat.Prime p → Nat.Prime q → p < q → {p, q} ∈ Erdos219.primeArithmeticProgressions","subjects":["5","11"],"theorem":"Erdos219.pair_mem_primeArithmeticProgressions"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«219»","statement":"∅ ∉ Erdos219.primeArithmeticProgressions","subjects":["5","11"],"theorem":"Erdos219.empty_not_primeArithmeticProgression"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«219»","statement":"{1, 2} ∉ Erdos219.primeArithmeticProgressions","subjects":["5","11"],"theorem":"Erdos219.not_primeArithmeticProgression_1_2"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there arbitrarily long arithmetic progressions of primes?\nSolution: yes.\nRef: Green, Ben and Tao, Terence, _The primes contain arbitrarily long arithmetic progressions_\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«219»","statement":"True ↔ ∀ (N : ℕ), ∃ l ∈ Erdos219.primeArithmeticProgressions, ↑N ≤ ENat.card ↑l","subjects":["5","11"],"theorem":"Erdos219.erdos_219"},{"answerKinds":[],"category":"research open","docstring":"If $A ⊆ \\mathbb{N}$ is such that $A + A$ contains all but finitely many integers then\n$\\limsup 1_A ∗ 1_A(n) = \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«28»","statement":"∀ (A : Set ℕ), (A + A)ᶜ.Finite → Filter.limsup (fun n => ↑(AdditiveCombinatorics.sumRep A n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos28.erdos_28"},{"answerKinds":[],"category":"research open","docstring":"Let $a_1 < a_2 < \\cdots$ be an increasing sequence such that $\\frac{a_n}{n} \\to \\infty$.\nIs the sum $\\sum_{n}^{\\infty} \\frac{a_n}{2^{a_n}}$ irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«260»","statement":"True ↔\n  ∀ (a : ℕ → ℤ) (s : ℝ),\n    StrictMono a →\n      Filter.Tendsto (fun n => ↑(a n) / ↑n) Filter.atTop Filter.atTop →\n        HasSum (fun n => ↑(a n) / 2 ^ a n) s → Irrational s","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos260.erdos_260"},{"answerKinds":["Prop"],"category":"research solved","docstring":"How large can a union-free collection $\\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean\nthere are no solutions to $A\\cup B=C$ with distinct $A,B,C\\in \\mathcal{F}$. Perhaps even\n$$\\lvert \\mathcal{F}\\rvert <(1+o(1))\\binom{n}{\\lfloor n/2\\rfloor}?$$\n\nSolved by Kleitman [Kl71], who proved\n$$\\lvert \\mathcal{F}\\rvert <(1+o(1))\\binom{n}{\\lfloor n/2\\rfloor}.$$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos447.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«447»","statement":"True ↔\n  ∃ c, c =o[Filter.atTop] 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos447.maxUnionFree n) < (1 + c n) * ↑(n.choose (n / 2))","subjects":["5"],"theorem":"Erdos447.erdos_447.parts.ii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"How large can a union-free collection $\\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean\nthere are no solutions to $A\\cup B=C$ with distinct $A,B,C\\in \\mathcal{F}$. Must\n$\\lvert \\mathcal{F}\\rvert =o(2^n)$?\n\nIn [Er65b] Erdős reported that the estimate $\\lvert \\mathcal{F}\\rvert=o(2^n)$ was proved in\nunpublished work by Sárközy and Szemerédi.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«447»","statement":"True ↔ (fun n => ↑(Erdos447.maxUnionFree n)) =o[Filter.atTop] fun n => 2 ^ n","subjects":["5"],"theorem":"Erdos447.erdos_447.parts.i"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges\nof $K_n$ contains a monochromatic copy of $K_k$, then find the value of\n$$\\lim_{k\\to \\infty}R(k)^{1/k}.$$\n\nThis problem is #3 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«77»","statement":"Filter.Tendsto (fun k => ↑(SimpleGraph.diagonalRamsey k) ^ (1 / ↑k)) Filter.atTop (nhds sorry)","subjects":["5"],"theorem":"Erdos77.erdos_77"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős Problem 505** (disproved). Borsuk's conjecture is false for\nsufficiently large $n$: there exists a dimension $n$ and a bounded set\n$S \\subseteq \\mathbb{R}^n$ with positive diameter such that $S$ cannot be\ncovered by $n + 1$ subsets each of diameter strictly less than $\\operatorname{diam}(S)$.\n\nErdős [Er44] suspected this. Disproved by Kahn–Kalai [KK93] for\n$n \\geq 2015$. Currently known to be false for $n \\geq 64$.\nA formal proof was formalised by Boris Alexeev using Aristotle. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/96cd54930d844e3655e6bb89b96b65516397dae9/src/v4.24.0/ErdosProblems/Erdos505.lean#L1153"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«505»","statement":"∃ n S,\n  Bornology.IsBounded S ∧\n    0 < Metric.diam S ∧\n      ∀ (F : Fin (n + 1) → Set (EuclideanSpace ℝ (Fin n))), S ⊆ ⋃ i, F i → ∃ i, Metric.diam S ≤ Metric.diam (F i)","subjects":["52"],"theorem":"Erdos505.erdos_505"},{"answerKinds":[],"category":"research solved","docstring":"**Borsuk's conjecture, small dimensions** (open / true for $n \\leq 3$).\nEvery bounded set $S \\subseteq \\mathbb{R}^n$ with $n \\leq 3$ can be\ncovered by $n + 1$ subsets each of strictly smaller diameter.\n\nTrivial for $n \\leq 2$; proved for $n = 3$ by Eggleston [Eg55]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«505»","statement":"∀ n ≤ 3,\n  ∀ (S : Set (EuclideanSpace ℝ (Fin n))),\n    Bornology.IsBounded S →\n      0 < Metric.diam S → ∃ F, S ⊆ ⋃ i, F i ∧ ∀ (i : Fin (n + 1)), Metric.diam (F i) < Metric.diam S","subjects":["52"],"theorem":"Erdos505.erdos_505.small_dim"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«505»","statement":"∀ (S : Set (EuclideanSpace ℝ (Fin 1))),\n  Bornology.IsBounded S → 0 < Metric.diam S → ∃ F, S ⊆ ⋃ i, F i ∧ ∀ (i : Fin 2), Metric.diam (F i) < Metric.diam S","subjects":["52"],"theorem":"Erdos505.erdos_505.test_dim_one"},{"answerKinds":[],"category":"research open","docstring":"Let $n\\geq k+1$. Every graph on $n$ vertices with at least $\\frac{k-1}{2}n+1$ edges contains every tree on $k+1$ vertices.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«548»","statement":"∀ (n k : ℕ),\n  k + 1 ≤ n →\n    ∀ (G : SimpleGraph (Fin n)),\n      (↑k - 1) / 2 * ↑n + 1 ≤ ↑G.edgeSet.ncard → ∀ (T : SimpleGraph (Fin (k + 1))), T.IsTree → T.IsContained G","subjects":["5"],"theorem":"Erdos548.erdos_548"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"∀ (k : ℕ), 0 < k → ((SimpleGraph.johnson (2 * k) k).chromaticNumber = ↑k + 1 ↔ Erdos835.Property k)","subjects":["5"],"theorem":"Erdos835.property_iff_chromaticNumber"},{"answerKinds":[],"category":"research solved","docstring":"Johnson's bound for the independence number of the Johnson graph. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"∀ {n k : ℕ}, (SimpleGraph.johnson n k).indepNum ≤ Erdos835.johnsonBound n 4 k","subjects":["5"],"theorem":"Erdos835.indepNum_johnson_le_johnsonBound"},{"answerKinds":[],"category":"research open","docstring":"Is the chromatic number of `J(2 * k, k)` always at least `k + 2`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"True ↔ ∀ k ≥ 3, ↑k + 2 ≤ (SimpleGraph.johnson (2 * k) k).chromaticNumber","subjects":["5"],"theorem":"Erdos835.johnson_chromaticNumber"},{"answerKinds":[],"category":"research solved","docstring":"It can be seen that the chromatic number of $J(2k,k)$ is $>k+1$ for all odd $k>2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"∀ (k : ℕ), 2 < k → Odd k → ↑k + 1 < (SimpleGraph.johnson (2 * k) k).chromaticNumber","subjects":["5"],"theorem":"Erdos835.johnson_chromaticNumber_odd"},{"answerKinds":[],"category":"test","docstring":"Ma and Tang's result implies the cases for odd $k$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«835»","statement":"(∀ (k : ℕ), 2 < k → (k + 1).Composite → ↑k + 1 < (SimpleGraph.johnson (2 * k) k).chromaticNumber) →\n  ∀ (k : ℕ), 2 < k → Odd k → ↑k + 1 < (SimpleGraph.johnson (2 * k) k).chromaticNumber","subjects":["5"],"theorem":"Erdos835.johnsonGraph_chromaticNumber_odd_of_johnson_chromaticNumber_composite"},{"answerKinds":[],"category":"research solved","docstring":"Johnson's bound for the chromatic number of the Johnson graph. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«835»","statement":"∀ {n k : ℕ}, ↑⌈↑(n.choose k) / ↑(Erdos835.johnsonBound n 4 k)⌉₊ ≤ (SimpleGraph.johnson n k).chromaticNumber","subjects":["5"],"theorem":"Erdos835.div_johnsonBound_le_chromaticNum_johnson"},{"answerKinds":[],"category":"research solved","docstring":"It is known that for $3 \\leq k \\leq 8$, the chromatic number of $J(2k, k)$ is greater than\n$k+1$, see [Johnson graphs](https://aeb.win.tue.nl/graphs/Johnson.html). ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"∀ {k : ℕ}, 3 ≤ k → k ≤ 8 → ↑k + 1 < (SimpleGraph.johnson (2 * k) k).chromaticNumber","subjects":["5"],"theorem":"Erdos835.chromaticNumber_johnson_2k_k_lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Ma and Tang have proved that the chromatic number of $J(2k,k)$ is $>k+1$ for all $k>2$ not of the\nform $p-1$ for prime $p$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"∀ (k : ℕ), 2 < k → (k + 1).Composite → ↑k + 1 < (SimpleGraph.johnson (2 * k) k).chromaticNumber","subjects":["5"],"theorem":"Erdos835.johnson_chromaticNumber_composite"},{"answerKinds":[],"category":"research solved","docstring":"It is known that for $3 \\leq k \\leq 8$, the chromatic number of $J(2k, k)$ is greater than $k+1$,\nsee [Johnson graphs](https://aeb.win.tue.nl/graphs/Johnson.html).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"∀ (k : ℕ), 3 ≤ k → k ≤ 8 → (SimpleGraph.johnson (2 * k) k).chromaticNumber > ↑k + 1","subjects":["5"],"theorem":"Erdos835.johnsonGraph_2k_k_chromaticNumber_known_cases"},{"answerKinds":[],"category":"research solved","docstring":"The smallest case not on this page is $k=9$:\nBut that one can be solved as well:\nThe chromatic number of $J(18, 9)$ is at least $11$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"(SimpleGraph.johnson 18 9).chromaticNumber > 9 + 1","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos835.johnsonGraph_18_9_chromaticNumber"},{"answerKinds":[],"category":"research solved","docstring":"It is also known that for $3 \\leq k \\leq 203$ odd, the chromatic number of $J(2k, k)$ is\ngreater than $k+1$, see [Johnson graphs](https://aeb.win.tue.nl/graphs/Johnson.html). ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«835»","statement":"∀ {k : ℕ}, 3 ≤ k → k ≤ 300 → Odd k → ↑k + 1 < (SimpleGraph.johnson (2 * k) k).chromaticNumber","subjects":["5"],"theorem":"Erdos835.chromaticNumber_johnson_2k_k_lower_bound_odd"},{"answerKinds":[],"category":"research open","docstring":"Alternative statement of Erdős Problem 835 using the chromatic number of the Johnson graph.\nThis is equivalent to asking whether there exists $k > 2$ such that the chromatic number of the\nJohnson graph $J(2k, k)$ is $k+1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"(∃ l, (SimpleGraph.johnson (2 * (l + 3)) (l + 3)).chromaticNumber = ↑(l + 3) + 1) ↔ True","subjects":["5"],"theorem":"Erdos835.erdos_835.variants.johnson"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a $k>2$ such that the $k$-sized subsets of {1,...,2k} can be coloured with\n$k+1$ colours such that for every $A\\subset \\{1,\\ldots,2k\\}$ with $\\lvert A\\rvert=k+1$ all $k+1$\ncolours appear among the $k$-sized subsets of $A$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«835»","statement":"(∃ k > 2, Erdos835.Property k) ↔ True","subjects":["5"],"theorem":"Erdos835.erdos_835"},{"answerKinds":[],"category":"research open","docstring":"Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$ with $$\\geq \\left(\\frac{1}{2}+o(1)\\right)n2^{n-1}$$ many edges contains a $C_4$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«86»","statement":"True ↔\n  ∀ (ε : ℝ),\n    0 < ε →\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ H ≤ SimpleGraph.hypercube n,\n          (1 / 2 + ε) * ↑n * 2 ^ (n - 1) ≤ ↑H.edgeSet.ncard → (SimpleGraph.cycleGraph 4).IsContained H","subjects":["5"],"theorem":"Erdos86.erdos_86"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a finite unit distance graph in $\\mamthbb{R}^2$.\nIs there some $k$ such that if $G$ has girth $≥ k$, then $\\chi(G) ≤ 3$?\n\nThe general case was solved by O'Donnell [OD99], who constructed finite unit distance graphs with\nchromatic number $4$ and arbitrarily large girth.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«705»","statement":"False ↔\n  ∃ k,\n    ∀ (V : Set (EuclideanSpace ℝ (Fin 2))),\n      V.Finite →\n        (SimpleGraph.UnitDistancePlaneGraph V).girth ≥ k → (SimpleGraph.UnitDistancePlaneGraph V).chromaticNumber ≤ 3","subjects":["5"],"theorem":"Erdos705.erdos_705"},{"answerKinds":[],"category":"research open","docstring":"Let $q_1 < q_2 < \\cdots$ be a sequence of primes such that $q_{i + 1} \\equiv 1 \\pmod{q_i}$. Is it\ntrue that\n$$\n\\lim_{k \\to \\infty} q_k^{1/k} = \\infty?\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«695»","statement":"True ↔\n  ∀ {q : ℕ → ℕ},\n    StrictMono q →\n      (∀ (i : ℕ), Nat.Prime (q i)) →\n        (∀ (i : ℕ), q (i + 1) % q i = 1) → Filter.Tendsto (fun k => ↑(q k) ^ (1 / ↑k)) Filter.atTop Filter.atTop","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos695.erdos_695"},{"answerKinds":[],"category":"research open","docstring":"Is there a sequence of primes $q_1 < q_2 < \\cdots$ such that $q_{i + 1} \\equiv 1 \\pmod{q_i}$ and\n$$\nq(k) \\leq \\exp(k (\\log k)^{1 + o(1)})?\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«695»","statement":"True ↔\n  ∃ q,\n    StrictMono q ∧\n      (∀ (i : ℕ), Nat.Prime (q i)) ∧\n        (∀ (i : ℕ), q (i + 1) % q i = 1) ∧\n          ∃ o, o =o[Filter.atTop] 1 ∧ ∀ (k : ℕ), ↑(q k) ≤ Real.exp ((↑k + 1) * Real.log (↑k + 1) ^ (1 + o k))","subjects":["11"],"theorem":"Erdos695.erdos_695.variants.upperBound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$\nedges). Is it true that, for every $\\epsilon>0$, if $n$ is sufficiently large, every subgraph of\n$Q_n$ with\n$$\\geq \\epsilon  n2^{n-1}$$\nmany edges contains a $C_6$?\n\nThe answer to this problem is no: Chung [Ch92] and Brouwer, Dejter, and Thomassen [BDT93]\nconstructed an edge-partition of $Q_n$ into four subgraphs, each containing no $C_6$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos666.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«666»","statement":"False ↔\n  ∀ (ε : ℝ),\n    0 < ε →\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ H ≤ SimpleGraph.hypercube n,\n          ε * ↑n * 2 ^ (n - 1) ≤ ↑H.edgeSet.ncard → (SimpleGraph.cycleGraph 6).IsContained H","subjects":["5"],"theorem":"Erdos666.erdos_666"},{"answerKinds":["Prop"],"category":"research solved","docstring":"We say $G$ is Ramsey size linear if $R(G,H)\\ll m$ for all graphs $H$ with $m$ edges and\nno isolated vertices.\n\nAre there infinitely many graphs $G$ which are not Ramsey size linear but such that all of\nits proper subgraphs are?\n\nAsked by Erdős, Faudree, Rousseau, and Schelp [EFRS93]. $K_4$ was long the only known example.\nWigderson [Wi24] proved that there are infinitely many such graphs.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«79»","statement":"True ↔ ∀ (N : ℕ), ∃ n, ∃ (_ : N ≤ n), ∃ G, ¬G.IsRamseySizeLinear ∧ ∀ H < ⊤, H.coe.IsRamseySizeLinear","subjects":["5"],"theorem":"Erdos79.erdos_79"},{"answerKinds":[],"category":"research open","docstring":"Is $B=\\{2^m3^n : m,n\\geq 0\\}$ an essential component?\n\nIn [Ru99] Ruzsa states \"The simplest set with a chance to be an essential component is the\ncollection of numbers in the form $2^m3^n$ and Erdős often asked whether it is an essential\ncomponent or not; I do not even have a plausible guess.\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1146»","statement":"True ↔ Erdos1146.IsEssentialComponent {k | ∃ m n, k = 2 ^ m * 3 ^ n}","subjects":["11"],"theorem":"Erdos1146.erdos_1146"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $T$ is a tree which is a bipartite graph with $k$ vertices in one class and $2k$ vertices\nin the other class then\n$$R(T)=4k-1.$$\n\nThis is false: Norin, Sun, and Zhao [NSZ16] have proved that if $T$ is the union of two stars on $k$\nand $2k$ vertices, with an edge joining the centre of the two stars, then $R(T)\\geq (4.2-o(1))k$,\nand conjectured that $R(T)=(4.2+o(1))k$.\n\nThis problem is #15 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«549»","statement":"False ↔\n  ∀ (k : ℕ),\n    2 ≤ k →\n      ∀ (T : SimpleGraph (Fin k ⊕ Fin (2 * k))),\n        T.IsTree →\n          (∀ (x₁ x₂ : Fin k), ¬T.Adj (Sum.inl x₁) (Sum.inl x₂)) →\n            (∀ (y₁ y₂ : Fin (2 * k)), ¬T.Adj (Sum.inr y₁) (Sum.inr y₂)) → T.diagonalGraphRamsey = 4 * k - 1","subjects":["5"],"theorem":"Erdos549.erdos_549"},{"answerKinds":[],"category":"research solved","docstring":"Let $3\\leq d_1 < d_2 < \\cdots < d_r$ be integers such that all sufficiently large integers can be\nwritten as a sum of the shape $\\sum_i c_ia_i$ where $c_i \\in \\{0, 1\\}$ and $a_i$ has only the digits\n$0, 1$ when written in base $d_i$. Then\n$$\\sum_{1 \\le i \\le r}\\frac 1{d_i - 1} \\ge 1.$$\n\nReported by Burr, Erdős, Graham, and Li [BEGL96] as an observation of Pomerance\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/arex1337/formal-conjectures-proofs/blob/3b7d581c75fd00e482deabfb20675cf3ccfaf49f/Erdos124/Converse.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«124»","statement":"∀ {D : Finset ℕ},\n  (∀ d ∈ D, 3 ≤ d) →\n    (∀ᶠ (n : ℕ) in Filter.atTop, n ∈ ∑ d ∈ D, Erdos124.sumsOfDistinctPowers d 0) → 1 ≤ ∑ d ∈ D, (↑d - 1)⁻¹","subjects":["11"],"theorem":"Erdos124.erdos124.converse"},{"answerKinds":[],"category":"research open","docstring":"Let $k \\ne 0$ and $3\\leq d_1 < d_2 < \\cdots < d_r$ be integers of gcd equal to $1$ such that\n$$\\sum_{1 \\le i \\le r}\\frac 1{d_i - 1} \\ge 1.$$\nCan all sufficiently large integers be written as a sum of the shape $\\sum_i c_ia_i$\nwhere $c_i \\in \\{0, 1\\}$ and $a_i$ is divisible by $d_i ^ k$ and has only the digits $0, 1$ when\nwritten in base $d_i$?\n\nConjectured by Burr, Erdős, Graham, and Li [BEGL96]\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«124»","statement":"True ↔\n  ∀ (k : ℕ),\n    k ≠ 0 →\n      ∀ (D : Finset ℕ),\n        (∀ d ∈ D, 3 ≤ d) →\n          1 ≤ ∑ d ∈ D, (↑d - 1)⁻¹ →\n            D.gcd id = 1 → ∀ᶠ (n : ℕ) in Filter.atTop, n ∈ ∑ d ∈ D, Erdos124.sumsOfDistinctPowers d k","subjects":["11"],"theorem":"Erdos124.erdos124.ne_zero"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let  $3 \\le d_1 < d_2 < \\dots < d_r$ be integers such that\n$$\\sum_{1 \\le i \\le r}\\frac 1{d_i - 1} \\ge 1.$$\nCan all sufficiently large integers be written as a sum of the shape $\\sum_i c_ia_i$\nwhere $c_i \\in \\{0, 1\\}$ and $a_i$ has only the digits $0, 1$ when written in base $d_i$?\n\nConjectured by Erdős [Er97], solved by Boris Alexeev using Aristotle.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«124»","statement":"True ↔\n  ∀ (D : Finset ℕ),\n    (∀ d ∈ D, 3 ≤ d) →\n      1 ≤ ∑ d ∈ D, (↑d - 1)⁻¹ → ∀ᶠ (n : ℕ) in Filter.atTop, n ∈ ∑ d ∈ D, Erdos124.sumsOfDistinctPowers d 0","subjects":["11"],"theorem":"Erdos124.erdos124.zero"},{"answerKinds":[],"category":"research solved","docstring":"For any $\\varepsilon > 0$, there exists an infinite sequence $2 \\le d_0 < d_1 < \\dots$ such\nthat all sufficiently large integer can be written as $\\sum_{i \\in I} a_i$ where $a_i$ has only\nthe digits $0, 1$ when written in base $d_i$,\nbut $\\sum_{i \\in I} \\frac 1{d_i - 1} \\le \\varepsilon$.\n\nProved by Melfi [Me04]\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«124»","statement":"∀ {ε : ℝ},\n  0 < ε →\n    ∃ d,\n      StrictMono d ∧\n        ∑' (i : ℕ), (↑(d i) - 1)⁻¹ ≤ ε ∧\n          ∀ᶠ (n : ℕ) in Filter.atTop, ∃ I a, (∀ i ∈ I, a i ∈ Erdos124.sumsOfDistinctPowers (d i) 0) ∧ ∑ i ∈ I, a i = n","subjects":["11"],"theorem":"Erdos124.erdos124.melfi_construction"},{"answerKinds":[],"category":"research solved","docstring":"All sufficiently large integers can be written as $a + b + c$, where $a$, $b$, and $c$ are\ndivisible by $3$, $4$, and $7$ respectively, and after division by that base have only the\ndigits $0, 1$ in that base.\n\nProved by Burr, Erdős, Graham, and Li [BEGL96]\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«124»","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  n ∈ Erdos124.sumsOfDistinctPowers 3 1 + Erdos124.sumsOfDistinctPowers 4 1 + Erdos124.sumsOfDistinctPowers 7 1","subjects":["11"],"theorem":"Erdos124.erdos124.ne_zero_three_four_seven"},{"answerKinds":[],"category":"research open","docstring":"Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and\n$n2^{n-1}$ edges). Prove that $$R(Q_n) \\ll 2^n.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«181»","statement":"∃ C > 0, ∀ (n : ℕ), ↑(SimpleGraph.hypercube n).diagonalGraphRamsey ≤ C * 2 ^ n","subjects":["5"],"theorem":"Erdos181.erdos_181"},{"answerKinds":[],"category":"research solved","docstring":"A result independently proved by Beck [Be83] and Szemerédi and Trotter [SzTr83] (see [211])\nimplies it is true with $n-3$ replaced by $cn$ for some constant $c>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«105»","statement":"∃ c > 0,\n  ∀ (A B : Finset (EuclideanSpace ℝ (Fin 2))),\n    Disjoint A B → ↑B.card ≤ c * ↑A.card → ¬Collinear ℝ ↑A → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q]","subjects":["5","52"],"theorem":"Erdos105.erdos_105.variants.beck_szemeredi_trotter"},{"answerKinds":[],"category":"research open","docstring":"It remains possible that this holds with $n-4$ (or in general with $n-O(1)$ or $(1-o(1))n$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«105»","statement":"True ↔\n  ∀ (A B : Finset (EuclideanSpace ℝ (Fin 2))),\n    Disjoint A B → A.card = B.card + 4 → ¬Collinear ℝ ↑A → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q]","subjects":["5","52"],"theorem":"Erdos105.erdos_105.variants.sub_four"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A,B\\subset \\mathbb{R}^2$ be disjoint sets of size $n$ and $n-3$ respectively, with not all\nof $A$ contained on a single line. Is there a line which contains at least two points from $A$\nand no points from $B$?\n\nThis has been disproved by Xichuan in the comments, who has found three explicit\ncounterexamples.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos105.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«105»","statement":"False ↔\n  ∀ (A B : Finset (EuclideanSpace ℝ (Fin 2))),\n    Disjoint A B → A.card = B.card + 3 → ¬Collinear ℝ ↑A → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q]","subjects":["5","52"],"theorem":"Erdos105.erdos_105"},{"answerKinds":[],"category":"research solved","docstring":"A construction of Hickerson shows that this fails with $n-2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«105»","statement":"¬∀ (A B : Finset (EuclideanSpace ℝ (Fin 2))),\n    Disjoint A B → A.card = B.card + 2 → ¬Collinear ℝ ↑A → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q]","subjects":["5","52"],"theorem":"Erdos105.erdos_105.variants.hickerson"},{"answerKinds":[],"category":"research open","docstring":"Let $p$ be a prime and $$A_p = \\{ k! \\pmod{p} : 1\\leq k<p\\}.$$ Is it true that $$\\lvert A_p\\rvert \\sim (1-\\tfrac{1}{e})p?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«478»","statement":"True ↔\n  Filter.Tendsto (fun p => ↑(Finset.image (fun k => k.factorial % p) (Finset.Ico 1 p)).card / ↑p)\n    (Filter.atTop ⊓ Filter.principal {p | Nat.Prime p}) (nhds (1 - 1 / Real.exp 1))","subjects":["11"],"theorem":"Erdos478.erdos_478"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $k\\geq 3$. Is it true that, if $m$ is sufficiently large, for any graph $H$ on $m$ edges\nwithout isolated vertices,\n$$R(C_k,H) \\leq 2m+\\left\\lfloor\\frac{k-1}{2}\\right\\rfloor?$$\n\nThis was proved for even $k$ by Erdős, Faudree, Rousseau, and Schelp [EFRS93]. This was proved\nfor $k=3$ independently by Goddard and Kleitman [GoKl94] and Sidorenko [Si91]. This was proved\nfor $k=5$ by Jayawardene [Ja99]. Finally it was proved for all odd $k\\geq 7$ by Cambie, Freschi,\nMorawski, Petrova, and Pokrovskiy [CFMPP26].\n\nThis problem is #35 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«570»","statement":"True ↔\n  ∀ (k : ℕ),\n    3 ≤ k →\n      ∀ᶠ (m : ℕ) in Filter.atTop,\n        ∀ (W : Type) [inst : Fintype W] (H : SimpleGraph W) [inst_1 : DecidableRel H.Adj],\n          (∀ (v : W), 0 < H.degree v) →\n            H.edgeSet.ncard = m → (SimpleGraph.cycleGraph k).graphRamsey H ≤ 2 * m + (k - 1) / 2","subjects":["5"],"theorem":"Erdos570.erdos_570"},{"answerKinds":[],"category":"research open","docstring":"Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ be a function from the finite subsets of\n$X$ to $X$ such that $f(A)\\not\\in A$ for all $A$. Must there exist an infinite $Y\\subseteq X$\nthat is independent - that is, for all finite $B\\subset Y$ we have $f(B)\\not\\in Y$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«623»","statement":"True ↔\n  ∀ (X : Type u),\n    Cardinal.mk X = Cardinal.aleph Ordinal.omega0 →\n      ∀ (f : Finset X → X), (∀ (A : Finset X), f A ∉ A) → ∃ Y, Y.Infinite ∧ ∀ (B : Finset X), ↑B ⊆ Y → f B ∉ Y","subjects":["3"],"theorem":"Erdos623.erdos_623"},{"answerKinds":[],"category":"research open","docstring":"Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$\nvertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform\nhypergraph on $n$ vertices.\n\nIs there some constant $c>0$ such that\n$$ R_3(n) \\geq 2^{2^{cn}}? $$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«564»","statement":"True ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, 2 ^ 2 ^ (c * n) ≤ ↑(Combinatorics.hypergraphRamsey 3 n)","subjects":["5"],"theorem":"Erdos564.erdos_564"},{"answerKinds":[],"category":"research solved","docstring":"Spencer, Szemerédi, and Trotter showed $f_2(n) = O(n^{4/3})$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1085»","statement":"(fun n => ↑(Erdos1085.f 2 n)) =O[Filter.atTop] fun n => ↑n ^ (4 / 3)","subjects":["52"],"theorem":"Erdos1085.erdos_1085.variants.upper_d2"},{"answerKinds":[],"category":"research solved","docstring":"Erdős showed $f_3(n) = Ω(n^{4/3}\\log\\log n)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1085»","statement":"(fun n => ↑n ^ (4 / 3) * Real.log (Real.log ↑n)) =O[Filter.atTop] fun n => ↑(Erdos1085.f 3 n)","subjects":["52"],"theorem":"Erdos1085.erdos_1085.variants.lower_d3"},{"answerKinds":[],"category":"research solved","docstring":"Lenz showed that, for $d \\ge 4$, $f_d(n) \\ge \\frac{p - 1}{2p} n^2 - O(1)$ where\n$p = \\lfloor\\frac d2\\rfloor$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1085»","statement":"∀ {d : ℕ}, 4 ≤ d → ∃ C, ∀ (n : ℕ), ↑(d / 2 - 1) / (2 * ↑(d / 2)) * ↑n ^ 2 - C ≤ ↑(Erdos1085.f d n)","subjects":["52"],"theorem":"Erdos1085.erdos_1085.variants.lower_d4_lenz"},{"answerKinds":[],"category":"research open","docstring":"Is the $n^{4/3}\\log\\log n$ lower bound in 3D also an upper bound?. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1085»","statement":"True ↔ (fun n => ↑(Erdos1085.f 3 n)) =O[Filter.atTop] fun n => ↑n ^ (4 / 3) * Real.log (Real.log ↑n)","subjects":["52"],"theorem":"Erdos1085.erdos_1085.variants.upper_d3"},{"answerKinds":[],"category":"research solved","docstring":"Erdős showed that, for $d \\ge 4$, $f_d(n) \\le \\left(\\frac{p - 1}{2p} + o(1)\\right) n^2$ where\n$p = \\lfloor\\frac d2\\rfloor$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1085»","statement":"∀ {d : ℕ},\n  4 ≤ d →\n    ∃ g,\n      Filter.Tendsto g Filter.atTop (nhds 0) ∧\n        ∀ (n : ℕ), ↑(Erdos1085.f d n) ≤ (↑(d / 2 - 1) / (2 * ↑(d / 2)) + g n) * ↑n ^ 2","subjects":["52"],"theorem":"Erdos1085.erdos_1085.variants.upper_d4_erdos"},{"answerKinds":[],"category":"research solved","docstring":"Erdős showed $f_2(n) > n^{1+c/\\log\\log n}$ for some $c > 0$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1085»","statement":"∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑n ^ (1 + c / Real.log (Real.log ↑n)) < ↑(Erdos1085.f 2 n)","subjects":["52"],"theorem":"Erdos1085.erdos_1085.variants.lower_d2"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Pach showed that, for $d \\ge 5$ odd, there exist constants $c_1(d), c_2(d) > 0$\nsuch that $\\frac{p - 1}{2p} n^2 - c_1 n^{4/3} ≤ f_d(n) \\le \\frac{p - 1}{2p} n^2 + c_2 n^{4/3}$ where\n$p = \\lfloor\\frac d2\\rfloor$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1085»","statement":"∀ {d : ℕ},\n  5 ≤ d →\n    Odd d →\n      ∃ c₁ > 0,\n        ∃ c₂,\n          ∀ᶠ (n : ℕ) in Filter.atTop,\n            ↑(d / 2 - 1) / (2 * ↑(d / 2)) * ↑n ^ 2 + c₁ * ↑n ^ (4 / 3) ≤ ↑(Erdos1085.f d n) ∧\n              ↑(Erdos1085.f d n) ≤ ↑(d / 2 - 1) / ↑(d / 2) * ↑n ^ 2 + c₂ * ↑n ^ (4 / 3)","subjects":["52"],"theorem":"Erdos1085.erdos_1085.variants.upper_lower_d5_odd"},{"answerKinds":[],"category":"research solved","docstring":"Harris has provided the following simple counterexample to the problem as stated: the\n$3$-uniform graph on $\\{1,\\ldots,9\\}$ with $28$ edges, formed by taking $27$ edges by choosing\none element each from $\\{1,2,3\\},\\{4,5,6\\},\\{7,8,9\\}$, and then adding the edge $\\{1,2,3\\}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«794»","statement":"Erdos794.harrisHypergraph.IsThreeUniform ∧\n  Erdos794.harrisHypergraph.card = 28 ∧\n    ¬Erdos794.harrisHypergraph.ContainsSubgraph 4 3 ∧ ¬Erdos794.harrisHypergraph.ContainsSubgraph 5 7","subjects":["5"],"theorem":"Erdos794.erdos_794.variants.harris"},{"answerKinds":[],"category":"research solved","docstring":"Balogh has observed that this problem is probably misstated by Erdős - indeed, every graph with\n$5$ vertices spanning $7$ edges contains a graph on $4$ vertices spanning $3$ edges, so the\nsecond condition can be dropped.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«794»","statement":"∀ {V : Type u_1} [inst : DecidableEq V] (H : Finset (Finset V)),\n  H.IsThreeUniform → H.ContainsSubgraph 5 7 → H.ContainsSubgraph 4 3","subjects":["5"],"theorem":"Erdos794.erdos_794.variants.balogh"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that every $3$-uniform hypergraph on $3n$ vertices with at least $n^3+1$ edges must\ncontain either a subgraph on $4$ vertices with $3$ edges or a subgraph on $5$ vertices with $7$\nedges?\n\nHarris has provided the following simple counterexample to the problem as stated: the\n$3$-uniform graph on $\\{1,\\ldots,9\\}$ with $28$ edges, formed by taking $27$ edges by choosing\none element each from $\\{1,2,3\\},\\{4,5,6\\},\\{7,8,9\\}$, and then adding the edge $\\{1,2,3\\}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos794.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«794»","statement":"False ↔\n  ∀ (n : ℕ) (H : Finset (Finset (Fin (3 * n)))),\n    H.IsThreeUniform → n ^ 3 + 1 ≤ H.card → H.ContainsSubgraph 4 3 ∨ H.ContainsSubgraph 5 7","subjects":["5"],"theorem":"Erdos794.erdos_794"},{"answerKinds":[],"category":"research solved","docstring":"This problem is then now asking how many edges a $3$-uniform hypergraph can have before it\ncontains $K_4$ minus an edge, and whether the critical edge density is $2/9$. In fact there is a\nconstruction of Frankl and Füredi [FrFu84] showing it must be at least $2/7$, which is the\nconjectured truth (although Turán conjectured before [Er69] that the edge density was $1/4$, and\nso likely there is simply a typo in this problem's statement).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«794»","statement":"∀ (ε : ℝ),\n  0 < ε →\n    ∀ᶠ (n : ℕ) in Filter.atTop, ∃ H, H.IsThreeUniform ∧ ¬H.ContainsSubgraph 4 3 ∧ (2 / 7 - ε) * ↑(n.choose 3) ≤ ↑H.card","subjects":["5"],"theorem":"Erdos794.erdos_794.variants.frankl_furedi"},{"answerKinds":[],"category":"research open","docstring":"Is there and $A \\subset \\mathbb{N}$ is such that\n$$\\lim_{n\\to \\infty}\\frac{1_A\\ast 1_A(n)}{\\log n}$$\nexists and is $\\ne 0$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«66»","statement":"True ↔ ∃ A c, c ≠ 0 ∧ Filter.Tendsto (fun n => ↑(AdditiveCombinatorics.sumRep A n) / Real.log ↑n) Filter.atTop (nhds c)","subjects":["11"],"theorem":"Erdos66.erdos_66"},{"answerKinds":[],"category":"research solved","docstring":"If $n$ distinct points in $\\mathbb{R}^2$ form a convex polygon then they determine at least\n$\\lfloor \\frac{n}{2}\\rfloor$ distinct distances.\n\nSolved by Altman [Al63].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos93.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«93»","statement":"∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), EuclideanGeometry.ConvexIndep ↑A → A.card / 2 ≤ distinctDistances A","subjects":["52"],"theorem":"Erdos93.erdos_93"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there infinitely many four-term arithmetic progressions of coprime powerful numbers?\n(A number $n$ is *powerful* if $p \\mid n \\to p^2 \\mid n$; `Nat.Powerful`.)\n\nErdős [Er76d] asked this; the answer is **yes**: Bajpai, Bennett and Chan [BBC24] proved that\nthere are infinitely many four-term arithmetic progressions of pairwise coprime powerful numbers.\n(Without coprimality this is easy, and by a theorem of Fermat there are no four *squares* in\narithmetic progression.)\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos937.lean#L1031"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«937»","statement":"True ↔ {p | Erdos937.IsCoprimePowerfulAP4 p.1 p.2}.Infinite","subjects":["11"],"theorem":"Erdos937.erdos_937"},{"answerKinds":[],"category":"test","docstring":"Sanity check for `IsCoprimePowerfulAP4`: the progression $0, 1, 2, 3$ is not a valid\nexample, since $2$ is not powerful. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«937»","statement":"¬Erdos937.IsCoprimePowerfulAP4 0 1","subjects":["11"],"theorem":"Erdos937.not_isCoprimePowerfulAP4_zero_one"},{"answerKinds":[],"category":"research solved","docstring":"If a finite system of $r$ congruences $\\{ a_i\\pmod{n_i} : 1\\leq i\\leq r\\}$ (the $n_i$ are not\nnecessarily distinct) covers $2^r$ consecutive integers then it covers all integers.\n\nThis is best possible as the system $2^{i-1}\\pmod{2^i}$ shows. This was proved independently by\nSelfridge and Crittenden and Vanden Eynden [CrVE70].\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos275.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«275»","statement":"∀ (r : ℕ) (a : Fin r → ℤ) (n : Fin r → ℕ),\n  (∃ k, ∀ x ∈ Set.Ico k (k + 2 ^ r), ∃ i, x ≡ a i [ZMOD ↑(n i)]) → ∀ (x : ℤ), ∃ i, x ≡ a i [ZMOD ↑(n i)]","subjects":["11"],"theorem":"Erdos275.erdos_275"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $p_1,\\ldots,p_k$ be distinct primes. Are there infinitely many $n$ such that $n!$ is\ndivisible by an even power of each of the $p_i$?\n\nThe answer is yes, proved by Berend [Be97], who further proved that the sequence of such $n$\nhas bounded gaps (where the bound depends on the initial set of primes).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos646.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«646»","statement":"True ↔ ∀ (S : Finset ℕ), (∀ p ∈ S, Nat.Prime p) → {n | ∀ p ∈ S, Even (padicValNat p n.factorial)}.Infinite","subjects":["11"],"theorem":"Erdos646.erdos_646"},{"answerKinds":[],"category":"research open","docstring":"Given $n$ points in $\\mathbb{R}^2$, no five of which are on a line, the number of\nlines containing four points is $o(n^2)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«101»","statement":"(fun n => ↑(Erdos101.numLinesWithFourPointMax n)) =o[Filter.atTop] fun n => ↑n ^ 2","subjects":["52"],"theorem":"Erdos101.erdos_101"},{"answerKinds":[],"category":"research solved","docstring":"Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence of\n$$s_k=\\sum_{1\\leq i\\leq n}z_i^k$$\ncontains infinitely many $(n-1)$-tuples of consecutive values of $s_k$ which are all $0$. Then\n(essentially)\n$$z_j=e(j/n),$$\nwhere $e(x)=e^{2\\pi ix}$.\n\nA conjecture of Turán.\n\nThis is true (in the stronger form with only two such tuples) - in fact if $n$ is odd then the\n$z_i$ must be exactly the $n$th roots of unity, and if $n$ is even they must be the vertices of\ntwo regular $(n/2)$-gons with the same circumscribed circle centred at the origin. This was first\nproved by Tijdeman [Ti66]. An independent proof of this was given in the comments section by Hu,\nTang, and Zhang.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos974.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«974»","statement":"∀ {n : ℕ} [inst : NeZero n] (z : Fin n → ℂ),\n  z 0 = 1 → {k | ∀ j < n - 1, ∑ i, z i ^ (k + j) = 0}.Infinite → Erdos974.IsTuranConfiguration z","subjects":["11","30"],"theorem":"Erdos974.erdos_974"},{"answerKinds":[],"category":"research solved","docstring":"Erdős speculates that this may be true if there are two distinct $(n-1)$-tuples of consecutive\nvalues of $s_k$ which are $0$. He does not elaborate on what the 'essentially' may mean precisely.\n\nThis is true (in the stronger form with only two such tuples) - in fact if $n$ is odd then the\n$z_i$ must be exactly the $n$th roots of unity, and if $n$ is even they must be the vertices of\ntwo regular $(n/2)$-gons with the same circumscribed circle centred at the origin. This was first\nproved by Tijdeman [Ti66]. An independent proof of this was given in the comments section by Hu,\nTang, and Zhang.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«974»","statement":"∀ {n : ℕ} [inst : NeZero n] (z : Fin n → ℂ),\n  z 0 = 1 →\n    ∀ {a b : ℕ},\n      a ≠ b →\n        (∀ j < n - 1, ∑ i, z i ^ (a + j) = 0) → (∀ j < n - 1, ∑ i, z i ^ (b + j) = 0) → Erdos974.IsTuranConfiguration z","subjects":["11","30"],"theorem":"Erdos974.erdos_974.variants.two_tuples"},{"answerKinds":[],"category":"research open","docstring":"Seems to be open, as of January 2025.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«189»","statement":"¬Erdos189.Erdos189For (fun a b c d => line[ℝ, a, b].Parallel line[ℝ, c, d] ∧ line[ℝ, a, d].Parallel line[ℝ, b, c])\n    fun a b c d => dist a b * dist b c * (EuclideanGeometry.oangle a b c).sin","subjects":["5","51"],"theorem":"Erdos189.erdos_189.variants.parallelogram"},{"answerKinds":[],"category":"research solved","docstring":"Graham claims this is \"easy to see\". ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«189»","statement":"¬Erdos189.Erdos189For\n    (fun a b c d =>\n      line[ℝ, a, b].direction ⟂ line[ℝ, b, c].direction ∧\n        line[ℝ, b, c].direction ⟂ line[ℝ, c, d].direction ∧\n          line[ℝ, c, d].direction ⟂ line[ℝ, d, a].direction ∧ dist a b = dist b c)\n    fun a b c d => dist a b * dist b c","subjects":["5","51"],"theorem":"Erdos189.erdos_189.variants.square"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $\\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the\nvertices of a rectangle of every area?\n\nGraham, \"On Partitions of 𝔼ⁿ\", Journal of Combinatorial Theory, Series A 28, 89-91 (1980).\n(See \"Concluding Remarks\" on page 96.)\n\nSolved (with answer `False`, as formalised below) in:\nVjekoslav Kovač, \"Coloring and density theorems for configurations of a given volume\", 2023\nhttps://arxiv.org/abs/2309.09973\nIn fact, Kovač's colouring is even Jordan measurable (the topological boundary of each\nmonochromatic region is Lebesgue measurable and has measure zero).\n\nThis was formalized in Lean by Alexeev and Kovac using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos189.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«189»","statement":"False ↔\n  Erdos189.Erdos189For\n    (fun a b c d =>\n      line[ℝ, a, b].direction ⟂ line[ℝ, b, c].direction ∧\n        line[ℝ, b, c].direction ⟂ line[ℝ, c, d].direction ∧ line[ℝ, c, d].direction ⟂ line[ℝ, d, a].direction)\n    fun a b c d => dist a b * dist b c","subjects":["5","51"],"theorem":"Erdos189.erdos_189"},{"answerKinds":[],"category":"research solved","docstring":"A positive answer to this question for all $k\\geq 5$ follows from the lower bound in\n[erdosproblems.com/986] given by Bradač [Br26].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«920»","statement":"∀ k ≥ 3, ∃ c > 0, (fun n => ↑n ^ (1 - 2 / (↑k + 1)) / Real.log ↑n ^ c) =O[Filter.atTop] fun n => ↑(Erdos920.f k n)","subjects":["5"],"theorem":"Erdos920.erdos_920.variants.lower_bound_k_ge_5"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $f_3(n)\\asymp (n/\\log n)^{1/2}$ (see [erdosproblems.com/1104]).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«920»","statement":"(fun n => ↑(Erdos920.f 3 n)) =Θ[Filter.atTop] fun n => (↑n / Real.log ↑n) ^ (1 / 2)","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos920.erdos_920.variants.k_eq_3"},{"answerKinds":[],"category":"research solved","docstring":"Graver and Yackel [GrYa68] proved that\n$f_k(n) \\ll \\left(n\\frac{\\log\\log n}{\\log n}\\right)^{1-\\frac{1}{k-1}}.$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«920»","statement":"∀ k ≥ 3,\n  (fun n => ↑(Erdos920.f k n)) =O[Filter.atTop] fun n =>\n    (↑n * Real.log (Real.log ↑n) / Real.log ↑n) ^ (1 - 1 / (↑k - 1))","subjects":["5"],"theorem":"Erdos920.erdos_920.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"The lower bound $R(4,m) \\gg m^3/(\\log m)^4$ of Mattheus and Verstraete [MaVe23]\n(see [erdosproblems.com/166]) implies $f_4(n) \\gg \\frac{n^{2/3}}{(\\log n)^{4/3}}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«920»","statement":"(fun n => ↑n ^ (2 / 3) / Real.log ↑n ^ (4 / 3)) =O[Filter.atTop] fun n => ↑(Erdos920.f 4 n)","subjects":["5"],"theorem":"Erdos920.erdos_920.variants.lower_bound_k_eq_4"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, for $k\\geq 4$, $f_k(n) \\gg \\frac{n^{1-\\frac{1}{k-1}}}{(\\log n)^{c_k}}$ for some\nconstant $c_k>0$?\n\nThis problem follows immediately from Mattheus and Verstraete's lower bound [MaVe23] for k = 4 and\nBradač's lower bound [Br26] for k ≥ 5.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«920»","statement":"True ↔\n  ∀ k ≥ 4, ∃ c > 0, (fun n => ↑n ^ (1 - 1 / (↑k - 1)) / Real.log ↑n ^ c) =O[Filter.atTop] fun n => ↑(Erdos920.f k n)","subjects":["5"],"theorem":"Erdos920.erdos_920"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that if $A\\subseteq \\mathbb{Z}/N\\mathbb{Z}$ has size $\\gg N^{1/2}$ then there exists\nsome non-empty $S\\subseteq A$ such that $\\sum_{n\\in S}n\\equiv 0\\pmod{N}$?\n\nSzemerédi proved the answer is yes, in fact for arbitrary finite abelian groups.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos540.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«540»","statement":"True ↔ ∃ C, 0 < C ∧ ∀ (N : ℕ), 0 < N → ∀ (A : Finset (ZMod N)), C * √↑N ≤ ↑A.card → Erdos540.HasZeroSubsetSum A","subjects":["5","11"],"theorem":"Erdos540.erdos_540"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subset\\mathbb{R}^2$ be an infinite set which contains no three points on a\nline and no four points on a circle. Consider the graph with vertices the points\nin $A$, where two vertices are joined by an edge if and only if they are an\ninteger distance apart. How large can the chromatic number and clique number of\nthis graph be? In particular, can the chromatic number be infinite?\n\nThe chromatic number can be infinite: there is an infinite general-position set\nwhose integer-distance graph admits no finite proper colouring. How large the\n*clique* number can be is not addressed here.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-130/Research/Basic.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«130»","statement":"True ↔\n  ∃ A,\n    A.Infinite ∧ EuclideanGeometry.InGeneralPosition A ∧ (SimpleGraph.IntegerDistancePlaneGraph A).chromaticNumber = ⊤","subjects":["5"],"theorem":"Erdos130.erdos_130"},{"answerKinds":[],"category":"research open","docstring":"Writing $f_{k, 3}(x)$ for the number of integers $\\leq x$ which are the sum of three $k$th powers,\nis it even true that $f_{k, 3}(x) \\gg_{\\epsilon} x ^ (3 / k - \\epsilon)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«325»","statement":"True ↔\n  ∀ ε > 0,\n    ∀ (k : ℕ), 3 ≤ k → (fun x => ↑x ^ (3 / ↑k - ε)) =O[Filter.atTop] fun x => ↑(Erdos325.cardIsSumThreePowerBelow k x)","subjects":["11"],"theorem":"Erdos325.erdos_325.variants.weaker"},{"answerKinds":[],"category":"research open","docstring":"Writing $f_{k, 3}(x)$ for the number of integers $\\leq x$ which are the sum of three $k$th powers,\nis it true that $f_{k, 3}(x) \\gg x ^ (3 / k)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«325»","statement":"True ↔ ∀ (k : ℕ), 3 ≤ k → (fun x => ↑x ^ (3 / ↑k)) =O[Filter.atTop] fun x => ↑(Erdos325.cardIsSumThreePowerBelow k x)","subjects":["11"],"theorem":"Erdos325.erdos_325"},{"answerKinds":[],"category":"research solved","docstring":"For $k = 3$, the best known is due to Wooley [Wo15]\n[Wo15] Wooley, Trevor D., Sums of three cubes, II. Acta Arith. (2015), 73-100.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«325»","statement":"(fun x => ↑x ^ 0.917) =O[Filter.atTop] fun x => ↑(Erdos325.cardIsSumThreePowerBelow 3 x)","subjects":["11"],"theorem":"Erdos325.erdos_325.variants.wooley"},{"answerKinds":[],"category":"research solved","docstring":"Equivalently, every graph with $rm$ vertices and maximum degree at most $m-1$ has a proper vertex\ncolouring with $m$ colours in which every colour class has exactly $r$ vertices (an equitable\ncolouring).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«914»","statement":"∀ {r m : ℕ},\n  2 ≤ r →\n    1 ≤ m →\n      ∀ {V : Type u_1} [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n        Fintype.card V = r * m → G.maxDegree ≤ m - 1 → ∃ C, ∀ (i : Fin m), (C.colorClass i).ncard = r","subjects":["5"],"theorem":"Erdos914.erdos_914.variants.equitable_colouring"},{"answerKinds":[],"category":"research solved","docstring":"Let $r\\geq 2$ and $m\\geq 1$. Every graph with $rm$ vertices and minimum degree at least $m(r-1)$\ncontains $m$ vertex disjoint copies of $K_r$.\n\nWhen $r=2$ this follows from Dirac's theorem. Corrádi and Hajnal [CoHa63] proved this when $r=3$.\nHajnal and Szemerédi [HaSz70] proved this for all $r\\geq 4$.\n\nA shorter proof was given by Kierstead and Kostochka [KiKo08].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos914.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«914»","statement":"∀ {r m : ℕ},\n  2 ≤ r →\n    1 ≤ m →\n      ∀ {V : Type u_1} [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n        Fintype.card V = r * m →\n          m * (r - 1) ≤ G.minDegree →\n            ∃ K, (∀ (i : Fin m), G.IsNClique r (K i)) ∧ Pairwise fun i j => Disjoint (K i) (K j)","subjects":["5"],"theorem":"Erdos914.erdos_914"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?\n\nGyőri proved this with $1.03n^5$, which has been improved by Füredi. The answer is yes, as proved\nindependently by Grzesik [Gr12] and Hatami, Hladky, Král, Norine, and Razborov [HHKNR13].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos24.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«24»","statement":"True ↔ ∀ (n : ℕ) (G : SimpleGraph (Fin (5 * n))), G.CliqueFree 3 → G.copyCount (SimpleGraph.cycleGraph 5) ≤ n ^ 5","subjects":["5"],"theorem":"Erdos24.erdos_24"},{"answerKinds":[],"category":"research solved","docstring":"In September 2025, Terence Tao gave a conditional _negative_ answer to Erdos conjecture 884,\ndisproving it under the assumption of the *Qualitative Hardy-Littlewood Conjecture*.\nSee [here](https://terrytao.wordpress.com/wp-content/uploads/2025/09/erdos-884.pdf).\nThe *qualitative* version of the conjecture only states that there are infinitely many tuples\nof primes and does not require any asymptotical bounds and as such is a corollary of the general\nform of the Hardy-Littlewood Conjecture.\nWe state the 'weaker' implication using general Hardy-Littlewood here, since this conjecture is\nalready formalized.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«884»","statement":"∀ (k : ℕ) (m : Fin k.succ → ℕ), HardyLittlewood.FirstHardyLittlewoodConjectureFor m → ¬Erdos884.Erdos884Prop","subjects":["11"],"theorem":"Erdos884.erdos_884_false_of_hardy_littlewood"},{"answerKinds":["Prop"],"category":"research solved","docstring":"For a natural number n, let $1 = d_1 < \\dotsc < d_{\\tau(n)} = n$ denote the divisors of $n$\nin increasing order.\nDoes it hold that\n$\\sum_{1 \\le i < j \\le \\tau(n)} \\frac{1}{d_j - d_i} \\ll 1 + \\sum_{1 \\le i < \\tau(n)}\n \\frac{1}{d_{i + 1} - d_i}$\nfor $n \\to \\infty$, i.e.\n$\\sum_{1 \\le i < j \\le \\tau(n)} \\frac{1}{d_j - d_i} \\in O \\left( 1 + \\sum_{1 \\le i < \\tau(n)}\n \\frac{1}{d_{i + 1} - d_i} \\right)$?\n\nThis conjecture has been **disproved**:\n- In September 2025, Terence Tao gave a conditional _negative_ answer assuming the prime tuples\n  conjecture, see `erdos_884_false_of_hardy_littlewood` for this implication.\n- Daniel Larsen subsequently gave an\n  [unconditional disproof](https://github.com/Larsen-Daniel/Erdos-884/blob/main/884.pdf).\n\n*Reference:* [erdosproblems.com/884](https://www.erdosproblems.com/884)\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/f8a51976fd2e66a52b4928c109fb9ae877a1a507/problems/884/Erdos884.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«884»","statement":"False ↔ Erdos884.Erdos884Prop","subjects":["11"],"theorem":"Erdos884.erdos_884"},{"answerKinds":[],"category":"research open","docstring":"Is it true that in any 2-colouring of $\\mathbb{N}$ there exists an infinite set $A$\nsuch that all elements of $A+A$ are the same colour?\n\nA conjecture of Owings [Ow74].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1199»","statement":"True ↔ ∀ (color : ℕ → Fin 2), ∃ A, A.Infinite ∧ ∀ n ∈ A + A, ∀ m ∈ A + A, color n = color m","subjects":["5"],"theorem":"Erdos1199.erdos_1199"},{"answerKinds":[],"category":"research solved","docstring":"Hindman [Hi79] has shown that this is false for 3-colourings.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/HowieHwong/lean-erdos-proofs/blob/b8b641ba2d00dc4d1fe205a078a4159372672459/Erdos/P1199.lean#L85"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1199»","statement":"∃ color, ∀ (A : Set ℕ), A.Infinite → ∃ n ∈ A + A, ∃ m ∈ A + A, color n ≠ color m","subjects":["5"],"theorem":"Erdos1199.erdos_1199.variants.three"},{"answerKinds":[],"category":"research solved","docstring":"The answer is yes, proved by Juhász [Ju79], who proved more generally that the complement of $S$\nmust contain a congruent copy of any set of four points.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«214»","statement":"Erdos214.HasRedCopies 4","subjects":["5","52"],"theorem":"Erdos214.erdos_214.variants.juhasz"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $S\\subset \\mathbb{R}^2$ be such that no two points in $S$ are distance $1$ apart. Must the\ncomplement of $S$ contain four points which form a unit square?\n\nThe answer is yes, proved by Juhász [Ju79], who proved more generally that the complement of $S$\nmust contain a congruent copy of any set of four points.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos214.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«214»","statement":"True ↔\n  ∀ (S : Set (EuclideanSpace ℝ (Fin 2))),\n    Erdos214.UnitDistanceAvoiding S → ∃ p, (∀ (i : Fin 4), p i ∈ Sᶜ) ∧ Congruent p Erdos214.unitSquare","subjects":["5","52"],"theorem":"Erdos214.erdos_214"},{"answerKinds":[],"category":"research solved","docstring":"The upper bound was improved further by Csizmadia and Tóth [CsTo94] to $k\\leq 7$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«214»","statement":"¬Erdos214.HasRedCopies 8","subjects":["5","52"],"theorem":"Erdos214.erdos_214.variants.csizmadia_toth"},{"answerKinds":[],"category":"research solved","docstring":"The best known bounds currently are\n$$4\\leq k\\leq 7.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«214»","statement":"4 ≤ Erdos214.k ∧ Erdos214.k ≤ 7","subjects":["5","52"],"theorem":"Erdos214.erdos_214.variants.bounds"},{"answerKinds":[],"category":"research solved","docstring":"Juhász [Ju79] also improved the upper bound to $k\\leq 11$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«214»","statement":"¬Erdos214.HasRedCopies 12","subjects":["5","52"],"theorem":"Erdos214.erdos_214.variants.juhasz_twelve_points"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus [EGMRSS75] had earlier showed that\n$3\\leq k\\leq 10^{12}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«214»","statement":"3 ≤ Erdos214.k ∧ Erdos214.k ≤ 10 ^ 12","subjects":["5","52"],"theorem":"Erdos214.erdos_214.variants.egmrss"},{"answerKinds":[],"category":"research open","docstring":"Let $k(N)$ denote the smallest $k$ such that there exists\n$N ≤ n_1 < ⋯ < n_k$ with $\\frac 1 {n_1} + ... + \\frac 1 {n_k} = 1$\n\nIs it true that $\\lim_{N \\to \\infty} k(N) - (e - 1)N = \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«295»","statement":"True ↔ Filter.Tendsto (fun N => ↑(Erdos295.k N) - (Real.exp 1 - 1) * ↑N) Filter.atTop Filter.atTop","subjects":["5","11"],"theorem":"Erdos295.erdos_295"},{"answerKinds":[],"category":"textbook","docstring":"Helper lemma: for each $N$, there exists $k$ and $n_1 < ... < n_k$ such that\n$N ≤ n_1 < ⋯ < n_k$ with $\\frac 1 {n_1} + ... + \\frac 1 {n_k} = 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«295»","statement":"∀ (N : ℕ), ∃ k n, (∀ (i : Fin k.succ), N ≤ n i) ∧ StrictMono n ∧ ∑ i, 1 / ↑(n i) = 1","subjects":["5","11"],"theorem":"Erdos295.exists_k"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Straus have proved the existence of some constant $c>0$\nsuch that $-c < k(N)-(e-1)N \\ll \\frac N {\\log N}$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«295»","statement":"∃ C > 0,\n  ∃ O > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos295.k N) - (Real.exp 1 - 1) * ↑N ∈ Set.Ioc (-C) (O * ↑N / Real.log ↑N)","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos295.erdos_295.variants.erdos_straus"},{"answerKinds":[],"category":"research open","docstring":"Let `A ⊂ ℕ` be an additive basis of order `k` which is minimal in the sense that\nif `B ⊂ A` is any infinite set, then `A \\ B` is not a basis of order `k`.\n\nMust there exist an infinite `B ⊂ A` such that `A \\ B`\nis an additive basis of order `k + 1`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«881»","statement":"True ↔\n  ∀ (k : ℕ) (A : Set ℕ),\n    Erdos881.IsMinimalAsymptoticAddBasisOfOrder k A → ∃ B ⊆ A, B.Infinite ∧ (A \\ B).IsAsymptoticAddBasisOfOrder (k + 1)","subjects":["5","11"],"theorem":"Erdos881.erdos_881"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Simonovits conjectured that at least 2 copies of $C_4$ are guaranteed.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«60»","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ (G : SimpleGraph (Fin n)) [inst : DecidableRel G.Adj],\n    SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph 4) < G.edgeFinset.card →\n      2 ≤ {H' | Nonempty (H'.coe ≃g SimpleGraph.cycleGraph 4)}.ncard","subjects":["5"],"theorem":"Erdos60.erdos_60.variants.two_copies"},{"answerKinds":[],"category":"research open","docstring":"Does every graph on $n$ vertices with $>\\mathrm{ex}(n;C_4)$ edges contain $\\gg n^{1/2}$ many\ncopies of $C_4$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«60»","statement":"∃ c > 0,\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∀ (G : SimpleGraph (Fin n)) [inst : DecidableRel G.Adj],\n      SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph 4) < G.edgeFinset.card →\n        c * √↑n ≤ ↑{H' | Nonempty (H'.coe ≃g SimpleGraph.cycleGraph 4)}.ncard","subjects":["5"],"theorem":"Erdos60.erdos_60"},{"answerKinds":[],"category":"research solved","docstring":"He, Ma, and Yang [HeMaYa21] proved the conjecture when $n = q^2 + q + 1$ for some even integer $q$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«60»","statement":"∃ c > 0,\n  ∀ (q : ℕ),\n    Even q →\n      ∀ (G : SimpleGraph (Fin (q ^ 2 + q + 1))) [inst : DecidableRel G.Adj],\n        SimpleGraph.extremalNumber (q ^ 2 + q + 1) (SimpleGraph.cycleGraph 4) < G.edgeFinset.card →\n          c * √(↑q ^ 2 + ↑q + 1) ≤ ↑{H' | Nonempty (H'.coe ≃g SimpleGraph.cycleGraph 4)}.ncard","subjects":["5"],"theorem":"Erdos60.erdos_60.variants.he_ma_yang"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $k\\geq 2$. Is there an integer $n_k$ such that, if $D=\\{ 1<d<n_k : d\\mid n_k\\}$, then for any\n$k$-colouring of $D$ there is a monochromatic subset $D'\\subseteq D$ such that\n$\\sum_{d\\in D'}\\frac{1}{d}=1$?\n\nThis follows from the colouring result of Croot [Cr03]. Croot's result allows for\n$n_k \\leq e^{C^k}$ for some constant $C>1$ (simply taking $n_k$ to be the lowest common multiple of\nsome interval $[1,C^k]$). Sawhney has observed that there is also a doubly exponential lower bound,\nand hence this bound is essentially sharp.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos45.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«45»","statement":"True ↔\n  ∀ (k : ℕ),\n    2 ≤ k →\n      ∃ n,\n        ∀ (colouring : ℕ → Fin k),\n          ∃ colour, ∃ D' ⊆ {d ∈ n.divisors | 1 < d ∧ d < n}, (∀ d ∈ D', colouring d = colour) ∧ D'.reciprocalSum = 1","subjects":["5","11"],"theorem":"Erdos45.erdos_45"},{"answerKinds":[],"category":"textbook","docstring":"Is\n$$\n\\sum_{n\\geq 2}\\frac{\\omega(n)}{2^n}\n$$\nirrational? (Here $\\omega(n)$ counts the number of distinct prime divisors of $n$.)\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«69»","statement":"Irrational (∑' (n : ℕ), ↑(ArithmeticFunction.cardDistinctFactors (n + 2)) / 2 ^ (n + 2))","subjects":["11"],"theorem":"Erdos69.erdos_69"},{"answerKinds":[],"category":"research solved","docstring":"Tao observed that `erdos_69` is a special case of `erdos_257`, since\n$$\n\\sum_{n\\geq 2}\\frac{\\omega(n)}{2^n} = \\sum_p \\frac{1}{2^p - 1}.\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«69»","statement":"have A := {n | Nat.Prime n};\n∑' (n : ℕ), ↑(ArithmeticFunction.cardDistinctFactors (n + 2)) / 2 ^ (n + 2) = ∑' (p : ↑A), 1 / (2 ^ ↑p - 1)","subjects":["11"],"theorem":"Erdos69.erdos_69.variants.specialisation_of_erdos_257"},{"answerKinds":[],"category":"research open","docstring":"Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function\n$f:\\{A : A\\subseteq X\\}\\to X$ so that for every $Y\\subseteq X$ with $\\lvert Y\\rvert \\geq H(n)$\nwe have $\\left\\{ f(A) : A\\subseteq Y\\right\\}=X$.\nProve that $H(n)-\\log_2 n \\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«624»","statement":"Filter.Tendsto (fun n => ↑(Erdos624.H n) - Real.logb 2 ↑n) Filter.atTop Filter.atTop","subjects":["5"],"theorem":"Erdos624.erdos_624"},{"answerKinds":[],"category":"research solved","docstring":"Every graph with $n$ vertices and $>n^2/4$ edges contains an edge which is in at least $n/6$\ntriangles.\n\nA conjecture of Bollobás and Erdős. This was proved independently by Edwards (unpublished) and\nKhadzhiivanov and Nikiforov [KhNi79].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos905.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«905»","statement":"∀ (n : ℕ) (G : SimpleGraph (Fin n)) [inst : DecidableRel G.Adj],\n  ↑n ^ 2 / 4 < ↑G.edgeFinset.card → ∃ e ∈ G.edgeFinset, ↑n / 6 ≤ ↑(G.trianglesContaining e).card","subjects":["5"],"theorem":"Erdos905.erdos_905"},{"answerKinds":[],"category":"research open","docstring":"Let $A=\\{1\\leq a_1 < a_2 < \\cdots\\}$ and $B=\\{1\\leq b_1 < b_2 < \\cdots\\}$ be sets of integers with\n$a_n/b_n\\to 1$.\n\nIf $A+B$ contains all sufficiently large positive integers then is it true that\n$\\limsup 1_A\\ast 1_B(n)=\\infty$?\n\nA conjecture of Erdős and Sárközy.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1145»","statement":"True ↔ Erdos1145.Erdos1145Prop","subjects":["5"],"theorem":"Erdos1145.erdos_1145"},{"answerKinds":[],"category":"test","docstring":"A stronger form of [erdosproblems.com/28].\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1145»","statement":"Erdos1145.Erdos1145Prop →\n  ∀ (A : Set ℕ), (A + A)ᶜ.Finite → Filter.limsup (fun n => ↑(AdditiveCombinatorics.sumRep A n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos1145.erdos_1145.test_implies_erdos_28"},{"answerKinds":[],"category":"API","docstring":"`f d n` is the least cardinality with the required distinct-distance property. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1088»","statement":"∀ (d n : ℕ), IsLeast (Erdos1088.cardSet d n) (Erdos1088.f d n)","subjects":["51"],"theorem":"Erdos1088.f_isLeast"},{"answerKinds":[],"category":"research open","docstring":"Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\\mathbb{R}^d$ contains a set of\n$n$ points for which any two determined distances are distinct. Erdős Problem 1088 asks to\nestimate $f_d(n)$. In particular, is it true that, for every fixed $n \\geq 3$,\n$$\nf_d(n) = 2^{o(d)}\n$$\nas $d \\to \\infty$?\n\nThe little-$o$ condition is stated after taking the base-$2$ logarithm.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1088»","statement":"True ↔ ∀ n ≥ 3, (fun d => Real.logb 2 ↑(Erdos1088.f d n)) =o[Filter.atTop] fun d => ↑d","subjects":["51"],"theorem":"Erdos1088.erdos_1088"},{"answerKinds":[],"category":"research solved","docstring":"How many antichains in $[n]$ are there? That is, how many families of subsets of $[n]$ are\nthere such that, if $\\mathcal{F}$ is such a family and $A,B\\in \\mathcal{F}$, then\n$A\\not\\subseteq B$?\n\nSperner's theorem states that $\\lvert \\mathcal{F}\\rvert \\leq \\binom{n}{\\lfloor n/2\\rfloor}$.\nThis is also known as Dedekind's problem. Resolved by Kleitman [Kl69], who proved that the\nnumber of such families is\n$$2^{(1+o(1))\\binom{n}{\\lfloor n/2\\rfloor}}.$$\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos497.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«497»","statement":"∃ o, ∃ (_ : o =o[Filter.atTop] 1), ∀ (n : ℕ), ↑(DedekindNumber.M' n) = 2 ^ ((1 + o n) * ↑(n.choose (n / 2)))","subjects":["5","6"],"theorem":"Erdos497.erdos_497"},{"answerKinds":[],"category":"research open","docstring":"There is a $k$, such that $2 \\le k \\le n - 2$ and\n$\\binom{n}{k}$ can be the product of consecutive primes infinitely often?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«386»","statement":"True ↔ ∃ k ≥ 2, ∃ᶠ (n : ℕ) in Filter.atTop, k ≤ n - 2 ∧ ∃ p q, n.choose k = ∏ i ∈ Finset.Ico p q, Nat.nth Nat.Prime i","subjects":["11"],"theorem":"Erdos386.erdos_386"},{"answerKinds":[],"category":"research open","docstring":"Can $\\binom{n}{2}$ be the product of consecutive primes infinitely often?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«386»","statement":"True ↔ ∃ᶠ (n : ℕ) in Filter.atTop, 2 ≤ n - 2 ∧ ∃ p q, n.choose 2 = ∏ i ∈ Finset.Ico p q, Nat.nth Nat.Prime i","subjects":["11"],"theorem":"Erdos386.erdos_386.variants.two"},{"answerKinds":[],"category":"research open","docstring":"For all $2 \\le k \\le n - 2$,\ncan $\\binom{n}{k}$ be the product of consecutive primes infinitely often?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«386»","statement":"True ↔ ∀ k ≥ 2, ∃ᶠ (n : ℕ) in Filter.atTop, k ≤ n - 2 ∧ ∃ p q, n.choose k = ∏ i ∈ Finset.Ico p q, Nat.nth Nat.Prime i","subjects":["11"],"theorem":"Erdos386.erdos_386.variants.forall"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for every integer $t\\geq1$, there is some integer $a$ such that ${n \\choose k} = a$\nwith $1\\leq k \\le \\frac{n}{2}$ has exactly $t$ solutions?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«849»","statement":"True ↔ ∀ t ≥ 1, ∃ a, {n | ∃ k ≥ 1, 2 * k ≤ n ∧ n.choose k = a}.ncard = t","subjects":["11"],"theorem":"Erdos849.erdos_849"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive upper density.\nCan one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$\nboth have positive upper density?\n\nThis was proved by the DeepMind prover agent.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/9d492049e42167b0d2fd58a9e91da3bf160172b5/FormalConjectures/ErdosProblems/741.lean#L228"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«741»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    0 < (A + A).upperDensity →\n      ∃ A₁ A₂, A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ 0 < (A₁ + A₁).upperDensity ∧ 0 < (A₂ + A₂).upperDensity","subjects":["5"],"theorem":"Erdos741.erdos_741.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive lower density.\nCan one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$\nboth have positive lower density?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«741»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    0 < (A + A).lowerDensity →\n      ∃ A₁ A₂, A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ 0 < (A₁ + A₁).lowerDensity ∧ 0 < (A₂ + A₂).lowerDensity","subjects":["5"],"theorem":"Erdos741.erdos_741.variants.lower"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive density in the literal sense that its\nnatural density exists and is positive.\nCan one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$\nboth have positive density in this sense?\n\nThis was disproved by the DeepMind prover agent.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/486bc8afae062b6711cd16d3466d651ee2880a52/FormalConjectures/ErdosProblems/741.lean#L1449"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«741»","statement":"False ↔\n  ∀ (A : Set ℕ),\n    (A + A).HasPosDensity → ∃ A₁ A₂, A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ (A₁ + A₁).HasPosDensity ∧ (A₂ + A₂).HasPosDensity","subjects":["5"],"theorem":"Erdos741.erdos_741.variants.exact_density"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$\ncannot both have bounded gaps?\n\nThis was proved by DeepMind prover agent.\n ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/486bc8afae062b6711cd16d3466d651ee2880a52/FormalConjectures/ErdosProblems/741.lean#L1629"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«741»","statement":"True ↔\n  ∃ A,\n    (A ∪ {0}).IsAddBasisOfOrder 2 ∧\n      ∀ (A₁ A₂ : Set ℕ), A = A₁ ∪ A₂ → Disjoint A₁ A₂ → ¬(IsSyndetic (A₁ + A₁) ∧ IsSyndetic (A₂ + A₂))","subjects":["5"],"theorem":"Erdos741.erdos_741.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $d\\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that $$f_d(n)=n^{\\frac{2}{d}-o(1)}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1083»","statement":"True ↔\n  ∀ (d : ℕ),\n    3 ≤ d →\n      ∃ o,\n        o =o[Filter.atTop] 1 ∧\n          ∀ᶠ (n : ℕ) in Filter.atTop, ↑(minimalDistinctDistances (EuclideanSpace ℝ (Fin d)) n) = ↑n ^ (2 / ↑d - o n)","subjects":["52"],"theorem":"Erdos1083.erdos_1083"},{"answerKinds":["Prop"],"category":"research solved","docstring":"A weaker version of Erdős' problem 499, which asks whether for every doubly stochastic matrix, there\nexists a permutation $σ \\in S_n$ with $M_{i, σ(i)} ≠ 0$ and such that\n$$\n\\sum_{1 \\leq i \\leq n} M_{i, σ(i)} \\geq 1\n$$\nProved by Marcus and Ree [MaRe59].\n\n[MaRe59] Marcus, M. and Ree, R., Diagonals of doubly stochastic matrices. Quart. J. Math. Oxford Ser. (2) (1959), 296-302.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«499»","statement":"True ↔ ∀ n > 0, ∀ M ∈ doublyStochastic ℝ (Fin n), ∃ σ, (∀ (i : Fin n), M i (σ i) ≠ 0) ∧ 1 ≤ ∑ i, M i (σ i)","subjects":["15"],"theorem":"Erdos499.erdos_499.variants.one_le"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $M$ be a real $n \\times n$ doubly stochastic matrix. Does there exist some $σ \\in S_n$ such that\n$$\n\\prod_{1 \\leq i \\leq n} M_{i, σ(i)} \\geq n^{-n}?\n$$\nThis is true, and was proved by Marcus and Minc [MaMi62]\n\n[MaMi62] Marcus, Marvin and Minc, Henryk, Some results on doubly stochastic matrices. Proc. Amer. Math. Soc. (1962), 571-579.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos499.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«499»","statement":"True ↔ ∀ (n : ℕ), ∀ M ∈ doublyStochastic ℝ (Fin n), ∃ σ, ↑n ^ (-↑n) ≤ ∏ i, M i (σ i)","subjects":["15"],"theorem":"Erdos499.erdos_499"},{"answerKinds":[],"category":"research solved","docstring":"The conjecture of van der Waerden, which states that the permanent of a doubly stochastic matrix is\nat least $n^{-n} n!$.\n\nProved by Gyires [Gy80], Egorychev [Eg81], and Falikman [Fa81].\n\n[Gy80] Gyires, B., The common source of several inequalities concerning doubly stochastic matrices. Publ. Math. Debrecen (1980), 291-304.\n[Eg81] Egorychev, G. P., The solution of the van der Waerden problem for permanents. Dokl. Akad. Nauk SSSR (1981), 1041-1044.\n[Fa81] Falikman, D. I., Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix. Mat. Zametki (1981), 931-938, 957.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«499»","statement":"∀ (n : ℕ), ∀ M ∈ doublyStochastic ℝ (Fin n), ↑n ^ (-↑n) * ↑n.factorial ≤ M.permanent","subjects":["15"],"theorem":"Erdos499.vanDerWaerden"},{"answerKinds":[],"category":"research open","docstring":"Let $\\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in\n$(n^{1/2},n^{1/2}+n^{1/2-\\epsilon})$ is $O_\\epsilon(1)$?\n\nErdős attributes this conjecture to Ruzsa.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«886»","statement":"True ↔ ∀ ε > 0, ∃ K, ∀ᶠ (n : ℕ) in Filter.atTop, (Erdos886.Erdos886Divisors n ε 1).card ≤ K","subjects":["11"],"theorem":"Erdos886.erdos_886"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Rosenfeld [ErRo97] proved that, for any constant $C>0$, all large $n$ have at most\n$1+C^2$ many divisors in $[n^{1/2}, n^{1/2}+Cn^{1/4}]$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«886»","statement":"∀ C > 0,\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ↑{d ∈ n.divisors | ↑n ^ (1 / 2) ≤ ↑d ∧ ↑d ≤ ↑n ^ (1 / 2) + C * ↑n ^ (1 / 4)}.card ≤ 1 + C ^ 2","subjects":["11"],"theorem":"Erdos886.erdos_886.variants.rosenfeld_bound"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Rosenfeld [ErRo97] proved that there are infinitely many $n$ such that there are\nfour divisors of $n$ in $(n^{1/2},n^{1/2}+16n^{1/4})$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«886»","statement":"{n | 4 ≤ (Erdos886.Erdos886Divisors n (1 / 4) 16).card}.Infinite","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos886.erdos_886.variants.rosenfeld_infinite"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq \\mathbb{N}$ be a set such that $\\lvert A\\cap [1,x]\\rvert=o(x^{1/2})$. Let\n$B=\\{ n\\geq 1 : a\\nmid n\\textrm{ for all }a\\in A\\}$.\nIf $B=\\{b_1 < b_2 < \\cdots\\}$ then is it true that\n$$\\lim_{x \\to \\infty} \\frac{1}{x}\\sum_{b_i < x}(b_{i+1}-b_i)^2$$\nexists (and is finite)?\n\nFor example, when $A=\\{p^2: p\\textrm{ prime}\\}$ then $B$ is the set of squarefree numbers,\nand the existence of this limit was proved by Erdős.\n\nSee also [208].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-489/F061/Erdos489.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«489»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    ((fun x => ↑{x ∈ Finset.Icc 1 x | x ∈ A}.card) =o[Filter.atTop] fun x => √↑x) →\n      (Erdos489.sievedSet A).Infinite → ∃ L, Filter.Tendsto (fun x => Erdos489.GapSumSq A x / ↑x) Filter.atTop (nhds L)","subjects":["11"],"theorem":"Erdos489.erdos_489"},{"answerKinds":[],"category":"research solved","docstring":"When $A = \\{p^2 : p \\textrm{ prime}\\}$, $B$ is the set of squarefree numbers, and the\nexistence of this limit was proved by Erdős. This is the $\\alpha = 2$ case of Erdős Problem 145. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«489»","statement":"∃ L, Filter.Tendsto (fun x => Erdos489.GapSumSq {n | ∃ p, Nat.Prime p ∧ n = p ^ 2} x / ↑x) Filter.atTop (nhds L)","subjects":["11"],"theorem":"Erdos489.erdos_489.variants.squarefree"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subset \\{1,\\ldots,N\\}$ be a Sidon set with $\\lvert A\\rvert\\sim N^{1/2}$. Must $A+A$ be\nwell-distributed over all small moduli? In particular, must about half the elements of $A+A$ be\neven and half odd?\n\nThe answer is yes. Lindström [Li98] proved the analogous statement for $A$ itself (see\n`erdos_154.variants.lindstrom`), later strengthened by Kolountzakis [Ko99]; well-distribution of\n$A+A$ then follows using the Sidon property.\n\nWe state the question for the sumset: for any sequence of Sidon sets\n$A_k\\subseteq\\{0,\\ldots,N_k\\}$ with $N_k\\to\\infty$ and $\\lvert A_k\\rvert\\sim N_k^{1/2}$, and any\nmodulus $m\\geq 2$, the proportion of elements of $A_k+A_k$ congruent to $i\\pmod m$ (i.e. the count\ndivided by $\\lvert A_k+A_k\\rvert$) tends to $1/m$ for every residue $i<m$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/willblair0708/lean-proofs/blob/main/ErdosProblems/Erdos154Sumset.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«154»","statement":"True ↔\n  ∀ (m : ℕ),\n    2 ≤ m →\n      ∀ (N : ℕ → ℕ) (A : ℕ → Finset ℕ),\n        Filter.Tendsto (fun k => ↑(N k)) Filter.atTop Filter.atTop →\n          (∀ (k x : ℕ), x ∈ A k → x ≤ N k) →\n            (∀ (k : ℕ), IsSidon ↑(A k)) →\n              Filter.Tendsto (fun k => ↑(A k).card / √↑(N k)) Filter.atTop (nhds 1) →\n                ∀ i < m,\n                  Filter.Tendsto (fun k => ↑{s ∈ A k + A k | s % m = i}.card / ↑(A k + A k).card) Filter.atTop\n                    (nhds (1 / ↑m))","subjects":["5","11"],"theorem":"Erdos154.erdos_154"},{"answerKinds":[],"category":"research solved","docstring":"Lindström's result for $A$ itself [Li98], later strengthened by Kolountzakis [Ko99]: for any\nsequence of Sidon sets $A_k\\subseteq\\{0,\\ldots,N_k\\}$ with $N_k\\to\\infty$ and\n$\\lvert A_k\\rvert\\sim N_k^{1/2}$, and any modulus $m\\geq 2$, the number of elements of $A_k$\ncongruent to $i\\pmod m$, divided by $N_k^{1/2}$, tends to $1/m$ for every residue $i<m$.\n\nWell-distribution of $A+A$ (the actual question, `erdos_154`) follows from this using the Sidon\nproperty.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem154.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«154»","statement":"∀ (m : ℕ),\n  2 ≤ m →\n    ∀ (N : ℕ → ℕ) (A : ℕ → Finset ℕ),\n      Filter.Tendsto (fun k => ↑(N k)) Filter.atTop Filter.atTop →\n        (∀ (k x : ℕ), x ∈ A k → x ≤ N k) →\n          (∀ (k : ℕ), IsSidon ↑(A k)) →\n            Filter.Tendsto (fun k => ↑(A k).card / √↑(N k)) Filter.atTop (nhds 1) →\n              ∀ i < m, Filter.Tendsto (fun k => ↑{a ∈ A k | a % m = i}.card / √↑(N k)) Filter.atTop (nhds (1 / ↑m))","subjects":["5","11"],"theorem":"Erdos154.erdos_154.variants.lindstrom"},{"answerKinds":[],"category":"research solved","docstring":"The best construction to date, due to Kreisel and Kurz, has $n = 7$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«213»","statement":"Erdos213.Erdos213For 7","subjects":["52"],"theorem":"Erdos213.erdos_213.variants.KK08"},{"answerKinds":[],"category":"research open","docstring":"Let $n \\geq 4$. Are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle,\nsuch that all pairwise distances are integers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«213»","statement":"True ↔ ∀ n ≥ 4, Erdos213.Erdos213For n","subjects":["52"],"theorem":"Erdos213.erdos_213"},{"answerKinds":[],"category":"research solved","docstring":"Let $n\\in\\mathbb{N}$ with $n\\neq p^k$ for any prime $p$ and $k\\geq 0$. What is the largest\ninteger not of the form\n$$\\sum_{1\\leq i<n}c_i\\binom{n}{i}$$\nwhere the $c_i\\geq 0$ are integers?\n\nIf $n=\\prod p_k^{a_k}$ then the largest integer not of this form is\n$$\\sum_k \\left( \\sum_{1\\leq d\\leq a_k}\\binom{n}{p_k^d}\\right)(p_k-1)-n.$$\nThis was first proved by Hwang and Song [HwSo24]. Independently this was found in the comment\nsection by Peake and Cambie.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos435.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«435»","statement":"∀ (n : ℕ),\n  n ≠ 0 →\n    (∀ (p k : ℕ), Nat.Prime p → n ≠ p ^ k) →\n      IsGreatest {m | ¬∃ c, m = ∑ i ∈ Finset.Ico 1 n, ↑(c i) * ↑(n.choose i)}\n        (∑ p ∈ n.primeFactors, (∑ d ∈ Finset.Icc 1 (n.factorization p), ↑(n.choose (p ^ d))) * (↑p - 1) - ↑n)","subjects":["5","11"],"theorem":"Erdos435.erdos_435"},{"answerKinds":[],"category":"research solved","docstring":"**Erdős–Herzog–Piranian Component Lemma** (Metric Properties of Polynomials, 1958):\nIf $f$ is a monic degree $n$ polynomial with all roots in the unit disk,\nthen some connected component\nof $\\{z \\mid |f(z)| < 1\\}$ contains at least two roots with multiplicity.\n\nSee p. 139, above Problem 5:\n[EHP58] Erdős, P. and Herzog, F. and Piranian, G., _Metric properties of polynomials_.\n  J. Analyse Math. (1958), 125-148.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1041»","statement":"∀ (n : ℕ) (f : Polynomial ℂ),\n  n ≥ 2 →\n    f.natDegree = n →\n      f.Monic →\n        f.rootSet ℂ ⊆ Metric.ball 0 1 →\n          ∃ C ⊆ {z | ‖Polynomial.eval z f‖ < 1}, IsConnected C ∧ 2 ≤ (Multiset.filter (fun x => x ∈ C) f.roots).card","subjects":["32"],"theorem":"Erdos1041.exists_connected_component_contains_two_roots"},{"answerKinds":[],"category":"research open","docstring":"Let\n$$ f(z) = \\prod_{i=1}^{n} (z - z_i) \\in \\mathbb{C}[x] $$\nwith $|z_i| < 1$ for all $i$.\n\nConjecture: Must there always exist a path of length less than 2 in\n$$ \\{ z \\in \\mathbb{C} \\mid |f(z)| < 1 \\} $$\nwhich connects two of the roots of $f$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1041»","statement":"∀ (n : ℕ) (f : Polynomial ℂ),\n  n ≥ 2 →\n    f.natDegree = n →\n      f.Monic →\n        f.rootSet ℂ ⊆ Metric.ball 0 1 →\n          ∃ z₁ z₂,\n            ∃ (_ : {z₁, z₂} ≤ f.roots),\n              ∃ γ, Set.range ⇑γ ⊆ {z | ‖Polynomial.eval z f‖ < 1} ∧ Erdos1041.length (Set.range ⇑γ) < 2","subjects":["32"],"theorem":"Erdos1041.erdos_1041"},{"answerKinds":["Prop"],"category":"research open","docstring":"**Brocard's Problem**\nDoes $n! + 1 = m^2$ have integer solutions other than $n = 4, 5, 7$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«398»","statement":"sorry ↔ {n | ∃ m, n.factorial + 1 = m ^ 2} = {4, 5, 7}","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos398.erdos_398"},{"answerKinds":[],"category":"research solved","docstring":"A problem of Erdős and Purdy, who proved $t(n) \\gg n^{2/3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«798»","statement":"(fun n => ↑n ^ (2 / 3)) =O[Filter.atTop] fun n => ↑(Erdos798.t n)","subjects":["5","52"],"theorem":"Erdos798.erdos_798.variants.lower_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $t(n)$ be the minimum number of points in $\\{1,\\ldots,n\\}^2$ such that the $\\binom{t}{2}$\nlines determined by these points cover all points in $\\{1,\\ldots,n\\}^2$.\n\nEstimate $t(n)$. In particular, is it true that $t(n)=o(n)$?\n\nA problem of Erdős and Purdy, who proved $t(n) \\gg n^{2/3}$.\n\nResolved by Alon [Al91] who proved $t(n) \\ll n^{2/3}\\log n$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos798.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«798»","statement":"True ↔ (fun n => ↑(Erdos798.t n)) =o[Filter.atTop] fun n => ↑n","subjects":["5","52"],"theorem":"Erdos798.erdos_798"},{"answerKinds":[],"category":"research solved","docstring":"Resolved by Alon [Al91] who proved $t(n) \\ll n^{2/3}\\log n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«798»","statement":"(fun n => ↑(Erdos798.t n)) =O[Filter.atTop] fun n => ↑n ^ (2 / 3) * Real.log ↑n","subjects":["5","52"],"theorem":"Erdos798.erdos_798.variants.upper_bound"},{"answerKinds":[],"category":"research open","docstring":"Are there only finitely many unitary perfect numbers? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1052»","statement":"True ↔ {n | Erdos1052.IsUnitaryPerfect n}.Finite","subjects":["11"],"theorem":"Erdos1052.erdos_1052"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1052»","statement":"Erdos1052.IsUnitaryPerfect 87360","subjects":["11"],"theorem":"Erdos1052.isUnitaryPerfect_87360"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1052»","statement":"Erdos1052.IsUnitaryPerfect 60","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos1052.isUnitaryPerfect_60"},{"answerKinds":[],"category":"test","docstring":null,"formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/Sanexxxx777/formal-conjectures/blob/be8aed15e8888a08bbe723170698e26c046412a4/FormalConjectures/ErdosProblems/1052.lean#L346"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1052»","statement":"Erdos1052.IsUnitaryPerfect 146361946186458562560000","subjects":["11"],"theorem":"Erdos1052.isUnitaryPerfect_146361946186458562560000"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1052»","statement":"Erdos1052.IsUnitaryPerfect 90","subjects":["11"],"theorem":"Erdos1052.isUnitaryPerfect_90"},{"answerKinds":[],"category":"research solved","docstring":"All unitary perfect numbers are even.\n\nFormal proof linked here provided by AlphaProof.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mzhorvath1/formal-conjectures/blob/b70a2ddf5e55f743aac9d4f4a907786b39bc9807/FormalConjectures/ErdosProblems/1052.lean#L46"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1052»","statement":"∀ (n : ℕ), Erdos1052.IsUnitaryPerfect n → Even n","subjects":["11"],"theorem":"Erdos1052.even_of_isUnitaryPerfect"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1052»","statement":"Erdos1052.IsUnitaryPerfect 6","subjects":["11"],"theorem":"Erdos1052.isUnitaryPerfect_6"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq\\mathbb{N}$ be an infinite set such that $|A\\cap \\{1, ..., N\\}| = o(N)$.\nIs it true that\n$$\n\\limsup_{N\\to\\infty}\\frac{|(A + A)\\cap \\{1, ..., N\\}|}{|A \\cap \\{1, ..., N\\}|} \\geq 3?\n$$\n\nThe answer is yes, proved by Freiman [Fr73].\n\n[Fr73] Fre\\u{\\i}man, G. A., _Foundations of a structural theory of set addition_. (1973), vii+108.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«245»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    A.Infinite →\n      Filter.Tendsto (fun N => ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard / N) Filter.atTop (nhds 0) →\n        3 ≤ Filter.limsup (fun N => ↑((A + A) ∩ Set.Icc 1 ⌊N⌋₊).ncard / ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard) Filter.atTop","subjects":["5","11"],"theorem":"Erdos245.erdos_245"},{"answerKinds":[],"category":"research solved","docstring":"Let $A\\subseteq\\mathbb{N}$ be an infinite set such that $|A\\cap \\{1, ..., N\\}| = o(N)$.\nThen\n$$\n\\limsup_{N\\to\\infty}\\frac{|(A + A)\\cap \\{1, ..., N\\}|}{|A \\cap \\{1, ..., N\\}|} \\geq 2.\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«245»","statement":"∀ (A : Set ℕ),\n  A.Infinite →\n    Filter.Tendsto (fun N => ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard / N) Filter.atTop (nhds 0) →\n      2 ≤ Filter.limsup (fun N => ↑((A + A) ∩ Set.Icc 1 ⌊N⌋₊).ncard / ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard) Filter.atTop","subjects":["5","11"],"theorem":"Erdos245.erdos_245.variants.two"},{"answerKinds":[],"category":"research solved","docstring":"For the similar problem in $\\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by\nAltman [Al63].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«660»","statement":"∀ (n : ℕ) (P : Finset (EuclideanSpace ℝ (Fin 2))),\n  P.card = n → ConvexIndependent ℝ Subtype.val → affineSpan ℝ ↑P = ⊤ → n / 2 ≤ distinctDistances P","subjects":["51","52"],"theorem":"Erdos660.erdos_660.variants.altman_planar"},{"answerKinds":[],"category":"research open","docstring":"Let $x_1, \\ldots, x_n \\in \\mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least\n$$(1 - o(1)) \\frac{n}{2}$$\nmany distinct distances between the $x_i$?\n\nThe $(1 - o(1)) \\frac{n}{2}$ lower bound is formalised as: for every $\\varepsilon > 0$, every set\nof $n$ vertices of a convex polyhedron with $n$ sufficiently large determines at least\n$(1 - \\varepsilon) \\frac{n}{2}$ distinct distances.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«660»","statement":"True ↔\n  ∀ (ε : ℝ),\n    0 < ε →\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ (P : Finset (EuclideanSpace ℝ (Fin 3))),\n          P.card = n → Erdos660.IsPolyhedronVertices P → (1 - ε) * (↑n / 2) ≤ ↑(distinctDistances P)","subjects":["51","52"],"theorem":"Erdos660.erdos_660"},{"answerKinds":[],"category":"research open","docstring":"In [Er75f] Erdős claims that Altman proved that the vertices determine $\\gg n$ many distinct\ndistances, but gives no reference.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«660»","statement":"True ↔\n  ∃ c > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (P : Finset (EuclideanSpace ℝ (Fin 3))),\n        P.card = n → Erdos660.IsPolyhedronVertices P → c * ↑n ≤ ↑(distinctDistances P)","subjects":["51","52"],"theorem":"Erdos660.erdos_660.variants.Er75f"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $g = o(h)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"sorry =o[Filter.atTop] fun n => ↑(Erdos357.h n)","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.monotone.parts.ii.littleO_version"},{"answerKinds":[],"category":"research solved","docstring":"Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nIt is known that $f(n) \\geq (2+o(1))\\sqrt{n}$.\nSource: See comment by Desmond Weisenberg here: https://www.erdosproblems.com/forum/thread/357.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"∃ o, o =o[Filter.atTop] 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, (2 + o n) * √↑n ≤ ↑(Erdos357.f n)","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.weisenberg"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $h = o(g)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"(fun n => ↑(Erdos357.h n)) =o[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.monotone.parts.ii.littleO_version_symm"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $f = \\Theta(g)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"(fun n => ↑(Erdos357.f n)) =Θ[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos357.erdos_357.parts.ii.bigTheta_version"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. Is $f(n)=o(n)$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"(fun n => ↑(Erdos357.f n)) =o[Filter.atTop] fun n => ↑n","subjects":["11"],"theorem":"Erdos357.erdos_357.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct.\nThen $A$ has lower density 0. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"∀ (A : ℕ → ℕ), StrictMono A → Erdos357.HasDistinctSums A → (Set.range A).lowerDensity = 0","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.infinite_set_lower_density"},{"answerKinds":[],"category":"research open","docstring":"Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct.\nThen it is conjectured that the sum $\\sum_k \\frac{1}{a_k}$ converges. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"∀ (A : ℕ → ℕ), StrictMono A → Erdos357.HasDistinctSums A → Summable fun i => 1 / ↑(A i)","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.infinite_set_sum"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $h = \\Theta(g)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"(fun n => ↑(Erdos357.h n)) =Θ[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.monotone.parts.ii.bigTheta_version"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $g = O(f)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"sorry =O[Filter.atTop] fun n => ↑(Erdos357.f n)","subjects":["11"],"theorem":"Erdos357.erdos_357.parts.ii.bigO_version"},{"answerKinds":[],"category":"research open","docstring":"Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct.\nThen it is conjectured that $A$ has density 0. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"∀ (A : ℕ → ℕ), StrictMono A → Erdos357.HasDistinctSums A → (Set.range A).HasDensity 0","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.infinite_set_density"},{"answerKinds":[],"category":"research open","docstring":"Let $g(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1, \\dotsc, a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. It is known that\n$$\\left(\\frac 1 3 + o(1) \\right)n \\leq g(n) \\leq \\left(\\frac 2 3 + o(1) \\right)n.$$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"∃ o o',\n  o =o[Filter.atTop] 1 ∧\n    o' =o[Filter.atTop] 1 ∧\n      ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos357.g n) ∈ Set.Icc ((1 / 3 + o n) * ↑n) ((2 / 3 + o' n) * ↑n)","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.hegyvari"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $f = o(g)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"(fun n => ↑(Erdos357.f n)) =o[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos357.erdos_357.parts.ii.littleO_version_symm"},{"answerKinds":[],"category":"research open","docstring":"Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. Is $h(n)=o(n)$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"(fun n => ↑(Erdos357.h n)) =o[Filter.atTop] fun n => ↑n","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.monotone.parts.i"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $g = O(h)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"sorry =O[Filter.atTop] fun n => ↑(Erdos357.h n)","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.monotone.parts.ii.bigO_version"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $h(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 \\leq \\dotsc \\leq a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct. How does $h(n)$ grow?\nCan we find a (good) explicit function $g$ such that $h = O(g)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"(fun n => ↑(Erdos357.h n)) =O[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos357.erdos_357.variants.monotone.parts.ii.bigO_version_symm"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $g = o(f)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"sorry =o[Filter.atTop] fun n => ↑(Erdos357.f n)","subjects":["11"],"theorem":"Erdos357.erdos_357.parts.ii.littleO_version"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $f(n)$ be the maximal $k$ such that there exist integers $1 \\le a_1 < \\dotsc < a_k \\le n$\nsuch that all sums of the shape $\\sum_{u \\le i \\le v} a_i$ are distinct.\nHow does $f(n)$ grow? Can we find a (good) explicit function $g$ such that $f = O(g)$ ? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«357»","statement":"(fun n => ↑(Erdos357.f n)) =O[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos357.erdos_357.parts.ii.bigO_version_symm"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If\n$$f(x+y)=f(x)+f(y)$$\nfor almost all $x,y\\in \\mathbb{R}$ then there exists a function $g$ such that\n$$g(x+y)=g(x)+g(y)$$\nfor all $x,y\\in\\mathbb{R}$ such that $f(x)=g(x)$ for almost all $x$.\n\nProved independently by de Bruijn [dB66] and Jurkat [Ju65].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1126.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1126»","statement":"True ↔\n  ∀ (f : ℝ → ℝ),\n    (∀ᵐ (p : ℝ × ℝ) ∂MeasureTheory.volume.prod MeasureTheory.volume, f (p.1 + p.2) = f p.1 + f p.2) →\n      ∃ h, (∀ (x y : ℝ), h (x + y) = h x + h y) ∧ ∀ᵐ (x : ℝ), f x = h x","subjects":["26","28"],"theorem":"Erdos1126.erdos_1126"},{"answerKinds":[],"category":"research solved","docstring":"The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$\nwith sets of positive density, as follows from [Bl21].\n\nThe statement is as follows:\nIf $A \\subset \\mathbb{N}$ has positive upper density (and hence certainly if $A$ has positive\ndensity) then there is a finite $S \\subset A$ such that $\\sum_{n \\in S} \\frac{1}{n} = 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«299»","statement":"∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity → ∃ S, ↑S ⊆ A ∧ ∑ n ∈ S, 1 / ↑n = 1","subjects":["11","40"],"theorem":"Erdos299.erdos_299.variants.density"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there an infinite sequence $a_1 < a_2 < \\dots$ such that $a_{i+1} - a_i = O(1)$ and no finite\nsum of $\\frac{1}{a_i}$ is equal to 1?\n\nThere does not exist such a sequence, which follows from the positive solution to\n[erdosproblems.com/298] by Bloom [Bl21].\n\nThis was formalized in Lean 3 by Bloom and Mehta.\n","formalProofs":[{"conditions":[],"kind":"other_system","link":"https://github.com/b-mehta/unit-fractions/blob/master/src/final_results.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«299»","statement":"False ↔\n  ∃ a,\n    StrictMono a ∧\n      (∀ (n : ℕ), 0 < a n) ∧\n        (fun n => ↑(a (n + 1)) - ↑(a n)) =O[Filter.atTop] 1 ∧ ∀ (S : Finset ℕ), ∑ i ∈ S, 1 / ↑(a i) ≠ 1","subjects":["11","40"],"theorem":"Erdos299.erdos_299"},{"answerKinds":[],"category":"research open","docstring":"Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime\ndivisor of\n$$\n  \\prod_{i = 0}^k (p ^ 2 + i)\n$$\nis $p$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«383»","statement":"True ↔ ∀ (k : ℕ), {p | Nat.Prime p ∧ (∏ i ∈ Finset.Icc 0 k, (p ^ 2 + i)).maxPrimeFac = p}.Infinite","subjects":["11"],"theorem":"Erdos383.erdos_383"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there only finitely many solutions to\n$$\n  \\prod_i \\binom{2m_i}{m_i}=\\prod_j \\binom{2n_j}{n_j}\n$$\nwith the $m_i,n_j$ distinct?\n\nSomani, using ChatGPT, has given a negative answer. In fact, for any $a\\geq 2$, if $c=8a^2+8a+1$,\n$\\binom{2a}{a}\\binom{4a+4}{2a+2}\\binom{2c}{c}= \\binom{2a+2}{a+1}\\binom{4a}{2a}\\binom{2c+2}{c+1}.$\nFurther families of solutions are given in the comments by SharkyKesa.\n\nThis was earlier asked about in a [MathOverflow] question, in response to which Elkies also gave an\nalternative construction which produces solutions - at the moment it is not clear whether Elkies'\nargument gives infinitely many solutions (although Bloom believes that it can).\n\nThis was formalized in Lean by Wu using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/3c356a50a21bcbf3543f960b0c92d7fb26228cb6/FormalConjectures/ErdosProblems/397.lean#L147"},{"conditions":[],"kind":"lean4","link":"https://gist.github.com/llllvvuu/40d68cfa9de9f43eece07ff4fdc3b0ef"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«397»","statement":"False ↔ {(M, N) | Disjoint M N ∧ ∏ i ∈ M, i.centralBinom = ∏ j ∈ N, j.centralBinom}.Finite","subjects":["11"],"theorem":"Erdos397.erdos_397"},{"answerKinds":[],"category":"research solved","docstring":"These always exist if $n$ is a prime power.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«723»","statement":"∀ (n : ℕ), IsPrimePow n → ∃ P L x x_1 x_2 pp, Configuration.ProjectivePlane.order P L = n","subjects":["5"],"theorem":"Erdos723.erdos_723.variants.prime_power_is_projplane_order"},{"answerKinds":[],"category":"research solved","docstring":"Bruck and Ryser have proved that if $n \\equiv 1 (\\mod 4)$ or $n \\equiv 2 (\\mod 4)$ then $n$ must be\nthe sum of two squares.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«723»","statement":"∀ {P L : Type} [inst : Membership P L] [Fintype P] [Fintype L] (n : ℕ) (pp : Configuration.ProjectivePlane P L),\n  Configuration.ProjectivePlane.order P L = n → n ≡ 1 [MOD 4] ∨ n ≡ 2 [MOD 4] → ∃ a b, n = a ^ 2 + b ^ 2","subjects":["5"],"theorem":"Erdos723.erdos_723.variants.bruck_ryser"},{"answerKinds":[],"category":"research open","docstring":"It is open whether there exists a projective plane of order 12.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«723»","statement":"True ↔ ∃ P L x x_1 x_2 pp, Configuration.ProjectivePlane.order P L = 12","subjects":["5"],"theorem":"Erdos723.erdos_723.variants.eq_12"},{"answerKinds":[],"category":"research open","docstring":"If there is a finite projective plane of order $n$ then must $n$ be a prime power?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«723»","statement":"True ↔\n  ∀ {P L : Type} (x : Membership P L) (x_1 : Fintype P) (x_2 : Fintype L) (pp : Configuration.ProjectivePlane P L),\n    IsPrimePow (Configuration.ProjectivePlane.order P L)","subjects":["5"],"theorem":"Erdos723.erdos_723"},{"answerKinds":[],"category":"research solved","docstring":"This conjecture has been proved for $n \\leq 11$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«723»","statement":"∀ {P L : Type} [inst : Membership P L] [Fintype P] [Fintype L] (pp : Configuration.ProjectivePlane P L),\n  Configuration.ProjectivePlane.order P L ≤ 11 → IsPrimePow (Configuration.ProjectivePlane.order P L)","subjects":["5"],"theorem":"Erdos723.erdos_723.variants.leq_11"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $p(z)$ is a polynomial of degree $n$ such that $\\{z : \\lvert p(z)\\rvert\\leq 1\\}$ is connected\nthen is it true that\n$$\\max_{\\substack{z\\in\\mathbb{C}\\\\ \\lvert p(z)\\rvert\\leq 1}} \\lvert p'(z)\\rvert\n\\leq (\\tfrac{1}{2}+o(1))n^2?$$\n\nEremenko and Lempert [ErLe94] have shown this is true, and in fact Chebyshev polynomials are the\nextreme examples.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos115.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«115»","statement":"True ↔\n  ∀ ε > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (p : Polynomial ℂ),\n        p.Monic →\n          p.natDegree = n →\n            IsConnected {z | ‖Polynomial.eval z p‖ ≤ 1} →\n              ∀ (z : ℂ),\n                ‖Polynomial.eval z p‖ ≤ 1 → ‖Polynomial.eval z (Polynomial.derivative p)‖ ≤ (1 / 2 + ε) * ↑n ^ 2","subjects":["30"],"theorem":"Erdos115.erdos_115"},{"answerKinds":[],"category":"research open","docstring":"**Erdős problem 972.**\nLet $\\alpha > 1$ be irrational. Are there infinitely many primes $p$\nsuch that $\\lfloor p\\alpha \\rfloor$ is also prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«972»","statement":"True ↔ ∀ α > 1, Irrational α → (Erdos972.primeSet α).Infinite","subjects":["11"],"theorem":"Erdos972.erdos_972"},{"answerKinds":[],"category":"research open","docstring":"Let $r\\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at\nleast one colour missing on the edges of the induced $K_{r+1}$.\n\nIn other words, there is no balanced colouring.\n\nA conjecture of Erdős and Gyárfás [ErGy99].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«617»","statement":"∀ r ≥ 3,\n  ∀ {V : Type} [inst : Fintype V] [DecidableEq V],\n    Fintype.card V = r ^ 2 + 1 →\n      ∀ (coloring : Sym2 V → Fin r), ∃ S k, S.card = r + 1 ∧ ∀ u ∈ S, ∀ v ∈ S, u ≠ v → coloring s(u, v) ≠ k","subjects":["5"],"theorem":"Erdos617.erdos_617"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Gyárfás [ErGy99] proved the conjecture for $r=3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«617»","statement":"∀ r ≥ 3,\n  ∀ {V : Type} [inst : Fintype V] [DecidableEq V],\n    Fintype.card V = 3 ^ 2 + 1 →\n      ∀ (coloring : Sym2 V → Fin 3), ∃ S k, S.card = 3 + 1 ∧ ∀ u ∈ S, ∀ v ∈ S, u ≠ v → coloring s(u, v) ≠ k","subjects":["5"],"theorem":"Erdos617.erdos_617.variants.r_eq_3"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Gyárfás [ErGy99] proved the conjecture for $r=4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«617»","statement":"∀ r ≥ 3,\n  ∀ {V : Type} [inst : Fintype V] [DecidableEq V],\n    Fintype.card V = 4 ^ 2 + 1 →\n      ∀ (coloring : Sym2 V → Fin 4), ∃ S k, S.card = 4 + 1 ∧ ∀ u ∈ S, ∀ v ∈ S, u ≠ v → coloring s(u, v) ≠ k","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos617.erdos_617.variants.r_eq_4"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Gyárfás [ErGy99] showed this property fails for infinitely many $r$ if we replace $r^2+1$\nby $r^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«617»","statement":"{r |\n    ∃ V x x_1,\n      Fintype.card V = r ^ 2 ∧\n        ∃ coloring,\n          ∀ (S : Finset V), S.card = r + 1 → ∀ (k : Fin r), ∃ u ∈ S, ∃ v ∈ S, u ≠ v ∧ coloring s(u, v) = k}.Infinite","subjects":["5"],"theorem":"Erdos617.erdos_617.variants.r2"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $\\delta>0$ and $N$ is sufficiently large in terms of $\\delta$, and $A\\subseteq\\{1,\\ldots,N\\}$ is such that $\\sum_{a\\in A}\\frac{1}{a}>\\delta \\log N$ then must there exist $S\\subseteq A$ such that $\\sum_{n\\in S}\\frac{1}{n}=1$?\n\nBloom [Bl21] proved this in the affirmative.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos47.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«47»","statement":"True ↔\n  ∀ (δ : ℝ),\n    0 < δ →\n      ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, δ * Real.log ↑N < A.reciprocalSum → ∃ S ⊆ A, S.reciprocalSum = 1","subjects":["11"],"theorem":"Erdos47.erdos_47"},{"answerKinds":[],"category":"research open","docstring":"Is there some constant $c > 0$ such that $h(n) < (\\log n)^{c + o(1)}$ and, for infinitely many $n$,\n$h(n) > (\\log n)^{c - o(1)}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«942»","statement":"True ↔\n  ∃ c > 0,\n    ∃ o,\n      o =o[Filter.atTop] 1 ∧\n        (∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos942.erdos_942.h n) < Real.log ↑n ^ (c + o n)) ∧\n          {n | ↑(Erdos942.erdos_942.h n) > Real.log ↑n ^ (c - o n)}.Infinite","subjects":["11"],"theorem":"Erdos942.erdos_942"},{"answerKinds":[],"category":"textbook","docstring":"It is not hard to prove that $\\limsup h(n) = \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«942»","statement":"Filter.limsup ((fun n => ↑n) ∘ Erdos942.erdos_942.h) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos942.erdos_942.variants.limsup"},{"answerKinds":[],"category":"textbook","docstring":"It is not hard to prove that the density $\\delta_l$ of integers for which $h(n) = l$ exists\nand satisfies $$\\sum_l \\delta_l = 1$$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«942»","statement":"∃ δ, ∀ (l : ℕ), {n | Erdos942.erdos_942.h n = l}.HasDensity (δ l) ∧ ∑' (l : ℕ), δ l = 1","subjects":["11"],"theorem":"Erdos942.erdos_942.variants.density"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $\\epsilon > 0$ and let $n$ be sufficiently large depending on $\\epsilon$. Is there a graph\non $n$ vertices with at least $n^2/8$ many edges which contains no $K_4$, such that the largest\nindependent set has size at most $\\epsilon n$?\n\nThis is true, as proved by Fox, Loh, and Zhao [FLZ15].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«22»","statement":"True ↔\n  ∀ (ε : ℝ),\n    0 < ε → ∀ᶠ (n : ℕ) in Filter.atTop, ∃ G, G.CliqueFree 4 ∧ ↑G.indepNum ≤ ε * ↑n ∧ ↑n ^ 2 / 8 ≤ ↑G.edgeFinset.card","subjects":["5"],"theorem":"Erdos22.erdos_22"},{"answerKinds":[],"category":"test","docstring":"A sanity check for `erdos_22.variants.szemeredi_upper`: the empty graph is $K_4$-free and\ntrivially satisfies the upper bound $(1/8 + \\epsilon)n^2$ on the number of edges.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«22»","statement":"∀ (n : ℕ) (ε : ℝ), 0 < ε → ⊥.CliqueFree 4 ∧ ↑⊥.edgeFinset.card ≤ (1 / 8 + ε) * ↑n ^ 2","subjects":["5"],"theorem":"Erdos22.erdos_22.variants.test_bot"},{"answerKinds":[],"category":"research solved","docstring":"The quantitative strengthening proved by Fox, Loh, and Zhao [FLZ15]: there is a constant\n$C > 0$ such that for all sufficiently large $n$ there exists a $K_4$-free graph on $n$\nvertices with at least $n^2/8$ edges whose largest independent set has size at most\n$$C \\cdot \\frac{(\\log\\log n)^{3/2}}{(\\log n)^{1/2}} \\cdot n.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«22»","statement":"∃ C,\n  0 < C ∧\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∃ G,\n        G.CliqueFree 4 ∧\n          ↑G.indepNum ≤ C * Real.log (Real.log ↑n) ^ (3 / 2) / Real.log ↑n ^ (1 / 2) * ↑n ∧\n            ↑n ^ 2 / 8 ≤ ↑G.edgeFinset.card","subjects":["5"],"theorem":"Erdos22.erdos_22.variants.fox_loh_zhao"},{"answerKinds":[],"category":"research solved","docstring":"The matching upper bound, due to Szemerédi [Sz72]: a $K_4$-free graph on $n$ vertices whose\nindependence number is sublinear in $n$ has at most $(1/8 + o(1))n^2$ edges. That is, for every\n$\\epsilon > 0$ there is a $\\delta > 0$ such that for all sufficiently large $n$, every\n$K_4$-free graph $G$ on $n$ vertices with $\\alpha(G) \\leq \\delta n$ has at most\n$(1/8 + \\epsilon)n^2$ edges.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«22»","statement":"∀ (ε : ℝ),\n  0 < ε →\n    ∃ δ,\n      0 < δ ∧\n        ∀ᶠ (n : ℕ) in Filter.atTop,\n          ∀ (G : SimpleGraph (Fin n)), G.CliqueFree 4 → ↑G.indepNum ≤ δ * ↑n → ↑G.edgeFinset.card ≤ (1 / 8 + ε) * ↑n ^ 2","subjects":["5"],"theorem":"Erdos22.erdos_22.variants.szemeredi_upper"},{"answerKinds":[],"category":"research solved","docstring":"The construction of Bollobás and Erdős [BoEr76]: for every $\\epsilon > 0$ and $\\delta > 0$,\nfor all sufficiently large $n$ there is a $K_4$-free graph on $n$ vertices with independence\nnumber at most $\\delta n$ and at least $(1/8 - \\epsilon)n^2$ edges. Together with\n`erdos_22.variants.szemeredi_upper` this shows that the Ramsey–Turán density of $K_4$ is $1/8$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«22»","statement":"∀ (ε δ : ℝ),\n  0 < ε →\n    0 < δ →\n      ∀ᶠ (n : ℕ) in Filter.atTop, ∃ G, G.CliqueFree 4 ∧ ↑G.indepNum ≤ δ * ↑n ∧ (1 / 8 - ε) * ↑n ^ 2 ≤ ↑G.edgeFinset.card","subjects":["5"],"theorem":"Erdos22.erdos_22.variants.bollobas_erdos_lower"},{"answerKinds":[],"category":"research open","docstring":"In [Er80] he claims he \"did not state this quite correctly\" in [Er77c]. The problem in [Er77c] which\nErdős is presumably referring to states that if $n < q_1 < \\cdots < q_k\\leq m$ is the set of primes\nin $(n,m]$ then $\\sum \\frac{1}{q_i-n} < \\sum_{p < m-n}\\frac{1}{p}+O(1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1210»","statement":"True ↔\n  ∃ C,\n    ∀ (n m : ℕ),\n      n < m → ∑ q ∈ Finset.Ioc n m with Prime q, 1 / (↑q - ↑n) < ∑ p ∈ Finset.range (m - n) with Prime p, 1 / ↑p + C","subjects":["11"],"theorem":"Erdos1210.erdos_1210.variants.er80_correction"},{"answerKinds":[],"category":"research open","docstring":"Let $A\\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\\in A$.\nIs it true that $\\sum_{a\\in A}\\frac{1}{n-a}\\leq \\sum_{p < n}\\frac{1}{p}+O(1)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1210»","statement":"True ↔\n  ∃ C,\n    ∀ (n : ℕ) (A : Finset ℕ),\n      (∀ a ∈ A, 1 ≤ a ∧ a < n) →\n        (∀ a ∈ A, ∀ b ∈ A, a ≠ b → a.Coprime b) → ∑ a ∈ A, 1 / (↑n - ↑a) ≤ ∑ p ∈ Finset.range n with Prime p, 1 / ↑p + C","subjects":["11"],"theorem":"Erdos1210.erdos_1210"},{"answerKinds":[],"category":"research open","docstring":"Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that $$h(k) \\ll k^2?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«970»","statement":"True ↔ ∃ C > 0, ∀ (k : ℕ), 0 < k → ↑(Erdos970.jacobsthalFunction k) ≤ C * ↑k ^ 2","subjects":["11"],"theorem":"Erdos970.erdos_970"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does the equation\n$$2^m=a_1!+\\cdots+a_k!$$\nwith $a_1<a_2<\\cdots <a_k$ have only finitely many solutions?\n\nAsked by Burr and Erdős. Frankl and Lin [Li76] independently showed that the answer is yes, and\nthe largest solution is\n$$2^7=2!+3!+5!.$$\nIn fact Lin showed that the largest power of $2$ which can divide a sum of distinct factorials\ncontaining $2$ is $2^{254}$, and that there are only 5 solutions to $3^m=a_1!+\\cdots+a_k!$\n(when $m=0,1,2,3,6$).\n\nSee also [404].\n\nA solution is encoded below as a pair $(m, s)$ where $s$ is the finite set\n$\\{a_1 < a_2 < \\cdots < a_k\\}$ of positive integers, so the distinctness of the $a_i$ is\ngiven by set membership. The empty set contributes no solutions since $2^m \\geq 1 > 0$.\n\nThe linked proof gives more than finiteness: it classifies the solutions outright, as\n$(0,\\{1\\})$, $(1,\\{2\\})$, $(3,\\{2,3\\})$, $(5,\\{2,3,4\\})$ and $(7,\\{2,3,5\\})$, so the set below\nhas exactly five elements.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/f8a51976fd2e66a52b4928c109fb9ae877a1a507/problems/403/Erdos403.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«403»","statement":"True ↔ {p | (∀ a ∈ p.2, 0 < a) ∧ 2 ^ p.1 = ∑ a ∈ p.2, a.factorial}.Finite","subjects":["11"],"theorem":"Erdos403.erdos_403"},{"answerKinds":[],"category":"research open","docstring":"Determine whether there exists a constant $C>1$ such that the following holds.\n\nLet $P$ be a finite [projective plane](https://en.wikipedia.org/wiki/Projective_plane). Must there exist a set of points $S$ such that $1\\leq \\lvert S\\cap \\ell\\rvert \\leq C$ for all lines $\\ell$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1159»","statement":"True ↔\n  ∃ C,\n    1 < C ∧\n      ∀ (P L : Type) (x : Membership P L) (x_1 : Fintype P) (x_2 : Fintype L) (x_3 : Configuration.ProjectivePlane P L),\n        ∃ S, ∀ (l : L), 1 ≤ (S ∩ {p | p ∈ l}).ncard ∧ (S ∩ {p | p ∈ l}).ncard ≤ C","subjects":["5","51"],"theorem":"Erdos1159.erdos_1159"},{"answerKinds":[],"category":"research open","docstring":"Does there exist some `ε > 0` such that there are infinitely many `ε`-barriers for `ω`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«413»","statement":"True ↔ ∃ ε > 0, {n | Erdos413.IsBarrier (fun n => ε * ↑(ArithmeticFunction.cardDistinctFactors n)) n}.Infinite","subjects":["11"],"theorem":"Erdos413.erdos_413.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many barriers for `ω`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«413»","statement":"True ↔ {n | Erdos413.IsBarrier (fun m => ↑(ArithmeticFunction.cardDistinctFactors m)) n}.Infinite","subjects":["11"],"theorem":"Erdos413.erdos_413.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Erdős proved that the barrier set for `expProd` is infinite and even has positive density.\n\n`HasPosDensity` is the right reading rather than positive lower density. In [Er79d] this is\nTheorem 1, \"the density of integers satisfying (2) is positive\", where `d₀(n) = ∏ αᵢ` is\n`expProd`. The averaging argument there bounds the density below, but Erdős states the existence\nseparately on the last page: \"With a little more trouble, I can prove that the density of\nintegers `n` for which `n` is a barrier for `d₀(n)` exists.\" He goes further, that if `αᵢ` is the\ndensity of `n` with `max_{m<n} (m + d₀(m)) = n + i`, then every `αᵢ` exists and they sum to `1`.\n\n[Er79d] Erdős, P., *Some unconventional problems in number theory*.\nActa Math. Acad. Sci. Hungar. (1979), 71-80. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«413»","statement":"{n | Erdos413.IsBarrier (fun m => ↑(Erdos413.expProd m)) n}.HasPosDensity","subjects":["11"],"theorem":"Erdos413.erdos_413.variants.hasPosDensity_barrier_expProd"},{"answerKinds":[],"category":"research open","docstring":"Erdős believed there should be infinitely many barriers for `Ω`, the total prime multiplicity. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«413»","statement":"True ↔ {n | Erdos413.IsBarrier (fun m => ↑(ArithmeticFunction.cardFactors m)) n}.Infinite","subjects":["11"],"theorem":"Erdos413.erdos_413.variants.bigOmega"},{"answerKinds":[],"category":"research solved","docstring":"Selfridge computed that the largest `Ω`-barrier below `10^5` is `99840`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«413»","statement":"IsGreatest {n | n < 10 ^ 5 ∧ Erdos413.IsBarrier (fun m => ↑(ArithmeticFunction.cardFactors m)) n} 99840","subjects":["11"],"theorem":"Erdos413.erdos_413.variants.bigOmega_largest_barrier_lt_100k"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many $n$ such that if\n$$\n  n(n + 1) = \\prod_i p_i^{k_i}\n$$\nis the factorisation into distinct primes then all exponents $k_i$ are distinct?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«913»","statement":"True ↔ {n | Set.InjOn ⇑(n * (n + 1)).factorization ↑(n * (n + 1)).primeFactors}.Infinite","subjects":["11"],"theorem":"Erdos913.erdos_913"},{"answerKinds":[],"category":"research solved","docstring":"If there are infinitely many primes $p$ such that $8p^2 - 1$ is prime, then this is true. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«913»","statement":"{p | Nat.Prime p ∧ Nat.Prime (8 * p ^ 2 - 1)}.Infinite →\n  {n | Set.InjOn ⇑(n * (n + 1)).factorization ↑(n * (n + 1)).primeFactors}.Infinite","subjects":["11"],"theorem":"Erdos913.erdos_913.variants.conditional"},{"answerKinds":[],"category":"research open","docstring":"It is likely that there are infinitely many primes $p$ such that $8p^2 - 1$ is also prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«913»","statement":"{p | Nat.Prime p ∧ Nat.Prime (8 * p ^ 2 - 1)}.Infinite","subjects":["11"],"theorem":"Erdos913.erdos_913.variants.infinite_many_8p_sq_add_one_primes"},{"answerKinds":[],"category":"research open","docstring":"Let $A$ be a finite Sidon set and $A+A=\\{s_1<\\cdots<s_t\\}$. Is it true that\n$$\\frac{1}{t}\\sum_{1\\leq i<t}(s_{i+1}-s_i)^2 \\to \\infty$$\nas $\\lvert A\\rvert\\to \\infty$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«153»","statement":"True ↔ Filter.Tendsto Erdos153.f Filter.atTop Filter.atTop","subjects":["5"],"theorem":"Erdos153.erdos_153"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, if $A\\subset \\mathbb{Z}$ is a finite set of size $N$, then\n$$\\int_0^1 \\left\\lvert \\sum_{n\\in A}e(n\\theta)\\right\\rvert \\mathrm{d}\\theta \\gg \\log N,$$\nwhere $e(x)=e^{2\\pi ix }$?\n\nLittlewood's conjecture, proved independently by Konyagin [Ko81] and McGehee, Pigno, and\nSmith [MPS81].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/f8a51976fd2e66a52b4928c109fb9ae877a1a507/problems/512/Erdos512.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«512»","statement":"True ↔\n  ∃ c > 0, ∀ (N : ℕ) (A : Finset ℤ), A.card = N → c * Real.log ↑N ≤ ∫ (θ : ℝ) in 0..1, ‖∑ n ∈ A, additiveChar (↑n * θ)‖","subjects":["11","42"],"theorem":"Erdos512.erdos_512"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges.\nIs it true that, if $H$ has $m$ edges and no isolated vertices, then $R(G,H) \\ll m$?\n\nIn other words: if $G$ is sparse (every induced subgraph on $k$ vertices has $≤ 2k-3$ edges),\nis $G$ Ramsey size linear?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«566»","statement":"True ↔\n  ∀ (p : ℕ) (G : SimpleGraph (Fin p)),\n    (∀ (S : Finset (Fin p)), 2 ≤ S.card → (SimpleGraph.induce (↑S) G).edgeSet.ncard ≤ 2 * S.card - 3) →\n      G.IsRamseySizeLinear","subjects":["5"],"theorem":"Erdos566.erdos_566"},{"answerKinds":[],"category":"research open","docstring":"Let $m_1\\leq\\cdots\\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices\nand $G$ is the complete multipartite graph with vertex class sizes $m_1,\\ldots,m_k$ then prove that\n$$R(T,G)\\leq (\\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.$$\n\nThis problem is #16 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«550»","statement":"∀ (k : ℕ) (hk : 2 ≤ k) (m : Fin k → ℕ),\n  Monotone m →\n    (∀ (i : Fin k), 0 < m i) →\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ (T : SimpleGraph (Fin n)),\n          T.IsTree →\n            T.graphRamsey (SimpleGraph.completeMultipartiteGraph fun i => Fin (m i)) ≤\n              (k - 1) * (T.graphRamsey (completeBipartiteGraph (Fin (m ⟨0, ⋯⟩)) (Fin (m ⟨1, ⋯⟩))) - 1) + m ⟨0, ⋯⟩","subjects":["5"],"theorem":"Erdos550.erdos_550"},{"answerKinds":[],"category":"research open","docstring":"Let $P$ be a finite set of primes with $|P| \\ge 2$ and let\n$\\{a_1 < a_2 < \\dots\\}$ be the set of positive integers whose prime factors\nare all in $P$. Is the sum\n$$ \\sum_{n=1}^\\infty \\frac{1}{[a_1,\\ldots,a_n]} $$\nirrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«269»","statement":"True ↔ ∀ (P : Finset ℕ), (∀ p ∈ P, Nat.Prime p) → P.card ≥ 2 → Irrational (Erdos269.series ↑P)","subjects":["11"],"theorem":"Erdos269.erdos_269.variants.irrational"},{"answerKinds":[],"category":"research solved","docstring":"This theorem addresses the case where the set of primes $P$ is infinite. In this case the sum is\nirrational.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«269»","statement":"∀ (P : Set ℕ), (∀ p ∈ P, Nat.Prime p) → P.Infinite → Irrational (Erdos269.series P)","subjects":["11"],"theorem":"Erdos269.erdos_269.variants.infinite"},{"answerKinds":[],"category":"research open","docstring":"Let $P$ be a finite set of primes with $|P| \\ge 2$ and let\n$\\{a_1 < a_2 < \\dots\\}$ be the set of positive integers whose prime factors\nare all in $P$. Is the sum\n$$ \\sum_{n=1}^\\infty \\frac{1}{[a_1,\\ldots,a_n]} $$\nrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«269»","statement":"True ↔ ∀ (P : Finset ℕ), (∀ p ∈ P, Nat.Prime p) → P.card ≥ 2 → ∃ q, ↑q = Erdos269.series ↑P","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos269.erdos_269.variants.rational"},{"answerKinds":[],"category":"research open","docstring":"Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that no circle whose centre is one\nof the $x_i$ contains three other points. Are there at least$$(1+c)\\frac{n}{2}$$\ndistinct distances determined between the $x_i$, for some constant $c>0$ and\nall $n$ sufficiently large?\n\nIn the spirit of related conjectures of Erdős and others, presumably\nsome kind of assumption that the points are in general position\n(e.g. no three on a line and no four on a circle) was intended.","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«655»","statement":"True ↔\n  ∃ c > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (X : Finset (EuclideanSpace ℝ (Fin 2))),\n        X.card = n →\n          Erdos655.IsValid X → EuclideanGeometry.InGeneralPosition ↑X → (1 + c) * ↑n / 2 ≤ ↑(distinctDistances X)","subjects":["5","52"],"theorem":"Erdos655.erdos_655.variants.general_position"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that no circle whose centre is one\nof the $x_i$ contains three other points. Are there at least\n$$(1+c)\\frac{n}{2}$$\ndistinct distances determined between the $x_i$, for some constant $c>0$ and\nall $n$ sufficiently large?\n\nThe answer is **no**: as Zach Hunter observed, the regular `n`-gon (`n` points equally spaced on a\ncircle) is valid and determines only `⌊n/2⌋ < (1+c)n/2` distinct distances, for every `c > 0`.\n(In the spirit of related conjectures of Erdős and others, presumably some kind of assumption that\nthe points are in general position was intended; see `erdos_655.variants.general_position`.)\n\nThe disproof — the regular `n`-gon construction together with its supporting lemmas — is formalised\nat the linked commit. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/AlperTheKing/formal-conjectures/blob/4aaaf544b6ed0ef22580787a8d8a19e85dc49556/FormalConjectures/ErdosProblems/655.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«655»","statement":"False ↔\n  ∃ c > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (X : Finset (EuclideanSpace ℝ (Fin 2))),\n        X.card = n → Erdos655.IsValid X → (1 + c) * ↑n / 2 ≤ ↑(distinctDistances X)","subjects":["5","52"],"theorem":"Erdos655.erdos_655"},{"answerKinds":[],"category":"research solved","docstring":"Gao, Hamidoune, and Wang [GHW10] solved this for all moduli `p` (not necessarily prime). ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«541»","statement":"∀ (p : ℕ) (a : Fin p → ZMod p),\n  (∃ r, ∀ (S : Finset (Fin p)), S ≠ ∅ → ∑ i ∈ S, a i = 0 → S.card = r) → (Set.range a).ncard ≤ 2","subjects":["11"],"theorem":"Erdos541.erdos_541.variants.general_moduli"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $a_1, \\dots, a_p$ be (not necessarily distinct) residues modulo a prime $p$, such that there\nexists some $r$ so that if $S \\subseteq [p]$ is non-empty and\n$$\\sum_{i \\in S} a_i \\equiv 0 \\pmod{p}$$\nthen $|S| = r$.\n\nMust there be at most two distinct residues amongst the $a_i$?\n\nThis was formalized in Lean by Alexeev using Aristotle and ChatGPT.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos541.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«541»","statement":"True ↔\n  ∀ (p : ℕ),\n    Fact (Nat.Prime p) →\n      ∀ (a : Fin p → ZMod p),\n        (∃ r, ∀ (S : Finset (Fin p)), S ≠ ∅ → ∑ i ∈ S, a i = 0 → S.card = r) → (Set.range a).ncard ≤ 2","subjects":["11"],"theorem":"Erdos541.erdos_541"},{"answerKinds":[],"category":"research solved","docstring":"This was proved by Erdős and Szemerédi [ErSz76] for p sufficiently large. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«541»","statement":"∀ᶠ (p : ℕ) in Filter.atTop,\n  Nat.Prime p →\n    ∀ (a : Fin p → ZMod p),\n      (∃ r, ∀ (S : Finset (Fin p)), S ≠ ∅ → ∑ i ∈ S, a i = 0 → S.card = r) → (Set.range a).ncard ≤ 2","subjects":["11"],"theorem":"Erdos541.erdos_541.variants.large_primes"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, for all sufficiently large $n$, there exists some $i<n$ such that\n$$\np_n^2 < p_{n+i}p_{n-i},\n$$\nwhere $p_k$ is the $k$th prime?\n\nPomerance proved that the answer is no.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos453.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«453»","statement":"False ↔ Erdos453.EventuallyHasPrimeWitness","subjects":["11"],"theorem":"Erdos453.erdos_453"},{"answerKinds":[],"category":"research solved","docstring":"There are infinitely many $n$ such that $d_n < d_{n+1} < d_{n+2}$, where $d$\ndenotes the prime gap function.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«6»","statement":"{n | primeGap n < primeGap (n + 1) ∧ primeGap (n + 1) < primeGap (n + 2)}.Infinite","subjects":["11"],"theorem":"Erdos6.erdos_6"},{"answerKinds":[],"category":"research solved","docstring":"For all $m$, there are infinitely many $n$ such that $d_n < d_{n+1} < \\dots < d_{n+m}$,\nwhere $d$ denotes the prime gap function.\n\nProved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the\nMaynard-Tao machinery concerning bounded gaps between primes [Ma15]\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«6»","statement":"∀ (m : ℕ), {n | ∀ i ∈ Finset.range m, primeGap (n + i) < primeGap (n + i + 1)}.Infinite","subjects":["11"],"theorem":"Erdos6.erdos_6.variants.increasing"},{"answerKinds":[],"category":"research solved","docstring":"For all $m$, there are infinitely many $n$ such that $d_n > d_{n+1} \\dots > d_{n+m}$,\nwhere $d$ denotes the prime gap function.\n\nProved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the\nMaynard-Tao machinery concerning bounded gaps between primes [Ma15]\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«6»","statement":"∀ (m : ℕ), {n | ∀ i ∈ Finset.range m, primeGap (n + i) > primeGap (n + i + 1)}.Infinite","subjects":["11"],"theorem":"Erdos6.erdos_6.variants.decreasing"},{"answerKinds":[],"category":"research open","docstring":"Let $A$ be a finite set of integers. Is it true that for every $\\epsilon>0$\n$\\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg_\\epsilon \\lvert A\\rvert^{2-\\epsilon}?$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«52»","statement":"True ↔ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∃ C, 0 < C ∧ ∀ (A : Finset ℤ), max ↑(A + A).card ↑(A * A).card ≥ C * ↑A.card ^ (2 - ε)","subjects":["11"],"theorem":"Erdos52.erdos_52"},{"answerKinds":[],"category":"research open","docstring":"Let $A$ be the set of powerful numbers. Is is true that $1_A\\ast 1_A(n)=n^{o(1)}$ for every $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«943»","statement":"True ↔ ∃ o, o =o[Filter.atTop] 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑(AdditiveCombinatorics.sumRep Nat.Powerful n) ≤ ↑n ^ o n","subjects":["11"],"theorem":"Erdos943.erdos_943"},{"answerKinds":[],"category":"test","docstring":"Note that trivially $F(n) \\leq n + \\sqrt{n}$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«385»","statement":"∀ (n : ℕ), ↑(Erdos385.F n) ≤ ↑n + √↑n","subjects":["11"],"theorem":"Erdos385.trivial_ub"},{"answerKinds":[],"category":"research open","docstring":"A question of Erdős, Eggleton, and Selfridge, who write that in fact it is possible that\nthis quantity is always at least $n+(1-o(1))\\sqrt{n}$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«385»","statement":"True ↔ ∃ e, ∃ (_ : e =o[Filter.atTop] 1), ∀ (n : ℕ), ↑n + (1 - e n) * √↑n ≤ ↑(Erdos385.F n)","subjects":["11"],"theorem":"Erdos385.erdos_385.variants.lb"},{"answerKinds":[],"category":"research open","docstring":"Let $F(n) := \\max\\{m + p(m) \\mid  \\textrm{$m < n$ composite}\\}\\}$ where $p(m)$ is the least\nprime divisor of $m$. Does $F(n) - n \\to \\infty$ as $n\\to\\infty$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«385»","statement":"True ↔ Filter.Tendsto (fun n => Erdos385.F n - n) Filter.atTop Filter.atTop","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos385.erdos_385.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $F(n) := \\max\\{m + p(m) \\mid  \\textrm{$m < n$ composite}\\}\\}$ where $p(m)$ is the least\nprime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«385»","statement":"True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, n < Erdos385.F n","subjects":["11"],"theorem":"Erdos385.erdos_385.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Can $\\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone\n3-term arithmetic progressions?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«197»","statement":"True ↔ ∃ A B, IsCompl A B ∧ (∃ f, ¬HasMonotoneAP (Subtype.val ∘ ⇑f) 3) ∧ ∃ g, ¬HasMonotoneAP (Subtype.val ∘ ⇑g) 3","subjects":["5"],"theorem":"Erdos197.erdos_197"},{"answerKinds":[],"category":"research solved","docstring":"Bogdan Grechuk has observed that `1117175146` is not the sum of a prime\nand at most $3$ powers of $2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«10»","statement":"1117175146 ∉ Erdos10.sumPrimeAndTwoPows 3","subjects":["5","11"],"theorem":"Erdos10.erdos_10.variants.grechuk_example"},{"answerKinds":[],"category":"research open","docstring":"Is there some $k$ such that every integer is the sum of a prime and at most $k$\npowers of $2$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«10»","statement":"True ↔ ∃ k, Erdos10.sumPrimeAndTwoPows k = Set.univ \\ {0, 1}","subjects":["5","11"],"theorem":"Erdos10.erdos_10"},{"answerKinds":[],"category":"research solved","docstring":"There are infinitely many even integers not the sum of a prime and $2$ powers of $2$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«10»","statement":"({n | Even n} \\ Erdos10.sumPrimeAndTwoPows 2).Infinite","subjects":["5","11"],"theorem":"Erdos10.erdos_10.variants.two_pows"},{"answerKinds":[],"category":"research solved","docstring":"Gallagher [Ga75] has shown that for any $ϵ > 0$ there exists $k(ϵ)$\nsuch that the set of integers which are the sum of a prime and at most $k(ϵ)$\nmany powers of $2$ has lower density at least $1 - ϵ$.\n\nRef: Gallagher, P. X., _Primes and powers of 2_.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«10»","statement":"∀ (ε : ℝ), 0 < ε → ∃ k, 1 - ε ≤ (Erdos10.sumPrimeAndTwoPows k).lowerDensity","subjects":["5","11"],"theorem":"Erdos10.erdos_10.variants.gallagher"},{"answerKinds":[],"category":"research open","docstring":"Granville and Soundararajan [GrSo98] have conjectured that at most $3$\npowers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$\nsuffice for all even integers.\n\nRef: Granville, A. and Soundararajan, K., _A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$_\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«10»","statement":"{n | Odd n ∧ 1 < n} ⊆ Erdos10.sumPrimeAndTwoPows 3 ∧ {n | Even n ∧ n ≠ 0} ⊆ Erdos10.sumPrimeAndTwoPows 4","subjects":["5","11"],"theorem":"Erdos10.erdos_10.variants.granville_soundararajan_odd"},{"answerKinds":[],"category":"research open","docstring":"Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$\npowers of $2$, and pointed out that parity considerations, coupled with the fact that there\nare many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist\ninfinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«10»","statement":"({n | Even n} \\ Erdos10.sumPrimeAndTwoPows 3).Infinite","subjects":["5","11"],"theorem":"Erdos10.erdos_10.variants.grechuk"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $\\epsilon>0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a\nsequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?\n\nThe problem is trivially true as written (simply taking $\\{1,\\ldots,k\\}$ and $n>k^{1/\\epsilon}$).\nThere are (at least) two possible variants which are non-trivial, and it is not clear which\nErdős and Graham meant. We formalize the second: each $m\\in P$ (where $P$ is the sequence of $k$\nconsecutive integers sought for) must be in $[n/2,n]$. In this case a positive answer also\nfollows directly from the result of Balog and Wooley [BaWo98] for infinitely many $n$. Proving\nthis is true for all large $n$ does not follow immediately from [BaWo98], but can be deduced\nusing a similar construction, as shown by SkyYang.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos369.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«369»","statement":"True ↔\n  ∀ (ε : ℝ),\n    0 < ε →\n      ∀ (k : ℕ),\n        2 ≤ k →\n          ∀ᶠ (n : ℕ) in Filter.atTop,\n            ∃ a, n / 2 ≤ a + 1 ∧ a + k ≤ n ∧ ∀ j < k, ∀ p ∈ (a + 1 + j).primeFactors, ↑p ≤ ↑n ^ ε","subjects":["11"],"theorem":"Erdos369.erdos_369"},{"answerKinds":[],"category":"textbook","docstring":"The trivial lower bound from Parseval's identity: for any polynomial $P$ of degree $n$ with\ncoefficients in $\\{-1, 1\\}$, we have $\\max_{|z|=1} |P(z)| \\geq \\sqrt{n+1}$.\n\nThis follows from Parseval's identity:\n$$\\frac{1}{2\\pi} \\int_0^{2\\pi} |P(e^{i\\theta})|^2 d\\theta = \\sum_{k=0}^{n} |a_k|^2 = n+1$$\nsince each $|a_k|^2 = 1$. The circle average is bounded by the pointwise supremum squared, so\n$\\max_{|z|=1} |P(z)|^2 \\ge n+1$, whence $\\max_{|z|=1} |P(z)| \\ge \\sqrt{n+1}$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1150»","statement":"∀ (P : Polynomial ℂ) (n : ℕ),\n  (∀ i ≤ P.natDegree, P.coeff i = -1 ∨ P.coeff i = 1) → P.natDegree = n → ⨆ z, ‖Polynomial.eval (↑z) P‖ ≥ √(↑n + 1)","subjects":["12","30"],"theorem":"Erdos1150.erdos_1150.variants.parseval_lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of\ndegree $n$ with coefficients in $\\{-1, 1\\}$,\n$$\\max_{|z|=1} |P(z)| > (1 + c) \\sqrt{n}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1150»","statement":"True ↔\n  ∃ c > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (P : Polynomial ℂ),\n        (∀ i ≤ P.natDegree, P.coeff i = -1 ∨ P.coeff i = 1) →\n          P.natDegree = n → ⨆ z, ‖Polynomial.eval (↑z) P‖ > (1 + c) * √↑n","subjects":["12","30"],"theorem":"Erdos1150.erdos_1150"},{"answerKinds":[],"category":"test","docstring":"Every graph has a triangle-free subgraph: the bottom subgraph (with no edges)\nwitnesses triangle-freeness, so the existential\n`∃ H : G.Subgraph, H.coe.CliqueFree 3` in `erdos_1175` is non-vacuous. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1175»","statement":"∀ (V : Type u_1) (G : SimpleGraph V), ∃ H, H.coe.CliqueFree 3","subjects":["5"],"theorem":"Erdos1175.erdos_1175.test.exists_triangle_free_subgraph"},{"answerKinds":[],"category":"research open","docstring":"Let $\\kappa$ be an uncountable cardinal. Must there exist a cardinal $\\lambda$ such that every\ngraph with chromatic number $\\lambda$ contains a triangle-free subgraph with chromatic number\n$\\kappa$?\n\nShelah proved that a negative answer is consistent when\n$\\kappa = \\lambda = \\aleph_1$ (see `erdos_1175.variants.shelah_consistency`).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1175»","statement":"True ↔\n  ∀ (κ : Cardinal.{u_1}),\n    Cardinal.aleph0 < κ →\n      ∃ μ,\n        ∀ (V : Type u_1) (G : SimpleGraph V),\n          G.chromaticCardinal = μ → ∃ H, H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ","subjects":["5"],"theorem":"Erdos1175.erdos_1175"},{"answerKinds":[],"category":"research open","docstring":"**Threshold reformulation variant.** Replaces `chromaticCardinal = λ` in the hypothesis\nof `erdos_1175` with `λ ≤ chromaticCardinal` (a graph of chromatic number ≥ λ has a\ntriangle-free subgraph of chromatic number κ). This is a strengthening of `erdos_1175`\n(see `erdos_1175.test.threshold_implies_exact`).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1175»","statement":"True ↔\n  ∀ (κ : Cardinal.{u_1}),\n    Cardinal.aleph0 < κ →\n      ∃ μ,\n        ∀ (V : Type u_1) (G : SimpleGraph V),\n          μ ≤ G.chromaticCardinal → ∃ H, H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ","subjects":["5"],"theorem":"Erdos1175.erdos_1175.variants.threshold_formulation"},{"answerKinds":[],"category":"test","docstring":"The threshold variant `threshold_formulation` is stronger than the exact-equality\nform `erdos_1175`: if every graph with `chromaticCardinal ≥ μ` has the desired\ntriangle-free subgraph, then in particular every graph with `chromaticCardinal = μ`\ndoes too. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1175»","statement":"(∀ (κ : Cardinal.{0}),\n    Cardinal.aleph0 < κ →\n      ∃ μ,\n        ∀ (V : Type) (G : SimpleGraph V),\n          μ ≤ G.chromaticCardinal → ∃ H, H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ) →\n  ∀ (κ : Cardinal.{0}),\n    Cardinal.aleph0 < κ →\n      ∃ μ,\n        ∀ (V : Type) (G : SimpleGraph V),\n          G.chromaticCardinal = μ → ∃ H, H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ","subjects":["5"],"theorem":"Erdos1175.erdos_1175.test.threshold_implies_exact"},{"answerKinds":[],"category":"research solved","docstring":"**Shelah's consistency result**: it is consistent with ZFC that there exists a graph $G$ with\nchromatic number $\\aleph_1$ such that every triangle-free subgraph of $G$ has chromatic number\nstrictly less than $\\aleph_1$.\n\nThis shows that a negative answer to Problem 1175 (with $\\kappa = \\lambda = \\aleph_1$) is\nconsistent, so the main statement `erdos_1175` is not provable in ZFC.\n\n**Formalization caveat (consistency placeholder).** Shelah's result is a *consistency*\nstatement — it asserts the existence of a model of ZFC, not a ZFC theorem. Lean operates\ninside a single (fixed) model of its set theory, so we cannot directly express \"consistent\nwith ZFC\" without leaving ZFC. Rather than pretend that Shelah's theorem is a bare ZFC\nnegation, we record it here as an explicit `answer(sorry)` consistency placeholder: the\nintended conjecture is the model-theoretic statement, and any concrete formalisation must\neither appeal to an explicit extra axiom (such as Shelah's specific forcing extension)\nor to a meta-theoretic consistency proof. Until such a wrapper exists in `FormalConjectures`,\nwe leave the body as `sorry`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1175»","statement":"True ↔\n  ¬∀ (V : Type u_1) (G : SimpleGraph V),\n      G.chromaticCardinal = Cardinal.aleph 1 → ∃ H, H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = Cardinal.aleph 1","subjects":["5"],"theorem":"Erdos1175.erdos_1175.variants.shelah_consistency"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«107»","statement":"Erdos107.f 3 = 3","subjects":["52"],"theorem":"Erdos107.f_three_eq"},{"answerKinds":[],"category":"test","docstring":"Depending on details of definitions,\nthe statement is false or trivial for $n < 3$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«107»","statement":"Erdos107.f 0 = 0","subjects":["52"],"theorem":"Erdos107.f_zero_eq"},{"answerKinds":[],"category":"research solved","docstring":"The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20],\nwho prove\n$$\n  f(n) ≤ 2^{n+O(\\sqrt{n\\log n})}.\n$$\n\n[HMPT20] Holmsen, Andreas F. and Mojarrad, Hossein Nassajian and Pach, János and Tardos, Gábor,\n  _Two extensions of the Erdős-Szekeres problem_. J. Eur. Math. Soc. (JEMS) (2020), 3981-3995.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«107»","statement":"∃ r, (r =O[Filter.atTop] fun n => √(↑n * Real.log ↑n)) ∧ ∀ n ≥ 3, ↑(Erdos107.f n) ≤ 2 ^ (↑n + r n)","subjects":["52"],"theorem":"Erdos107.variants.hmpt_bound"},{"answerKinds":[],"category":"research solved","docstring":"Suk [Su17] proved\n$$\n  f(n) ≤ 2^{(1+o(1))n}.\n$$\n\n[Su17] Suk, Andrew, _On the Erdős-Szekeres convex polygon problem_.\n  J. Amer. Math. Soc. (2017), 1047-1053.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«107»","statement":"∃ r, (r =o[Filter.atTop] fun n => ↑n) ∧ ∀ n ≥ 3, ↑(Erdos107.f n) ≤ 2 ^ (↑n + r n)","subjects":["52"],"theorem":"Erdos107.variants.su_bound"},{"answerKinds":["Prop"],"category":"research open","docstring":"Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line,\ncontain $n$ points which form the vertices of a convex $n$-gon.\nProve that $f(n) = 2^{n-2} + 1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«107»","statement":"sorry ↔ ∀ n ≥ 3, Erdos107.f n = 2 ^ (n - 2) + 1","subjects":["52"],"theorem":"Erdos107.erdos_107"},{"answerKinds":[],"category":"research solved","docstring":"For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line,\ncontain a convex $n$-gon. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«107»","statement":"∀ n ≥ 3, (Erdos107.cardSet n).Nonempty","subjects":["52"],"theorem":"Erdos107.nonempty_cardSet"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Szekeres proved the bounds\n$$\n  2^{n-2} + 1 ≤ f(n) ≤ \\binom{2n-4}{n-2} + 1\n$$\n([ErSz60] and [ErSz35] respectively).\n\n[ErSz60] Erdős, P. and Szekeres, G., _On some extremum problems in elementary geometry_.\n  Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (1960/61), 53-62.\n\n[ErSz35] Erdős, P. and Szekeres, G., _A combinatorial problem in geometry_.\n  Compos. Math. (1935), 463-470.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«107»","statement":"∀ n ≥ 3, 2 ^ (n - 2) + 1 ≤ Erdos107.f n ∧ Erdos107.f n ≤ (2 * n - 4).choose (n - 2) + 1","subjects":["52"],"theorem":"Erdos107.variants.ersz_bounds"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $\\sum_{k\\geq 1}\\frac{1}{u_k+1}$ and\n$u_k=\\lfloor c_0^{2^k}+1\\rfloor$ for $k\\geq 1$, where\n$$c_0=\\lim u_n^{1/2^n}=1.264085\\cdots.$$\nLet $a_1<a_2<\\cdots $ be any other sequence with $\\sum \\frac{1}{a_k}=1$. Is it true that\n$$\\liminf a_n^{1/2^n}<c_0=1.264085\\cdots?$$\n\nThis is true, and was proved independently by Kamio [Ka25] and Li and Tang [LiTa25].\n\nAn earlier interpretation of this question on this site defined $u_1=2$ and\n$u_{n+1}=u_n^2-u_n+1$ (Sylvester's sequence), which is the same sequence shifted by $1$; we use\nthe phrasing above as more faithful to [ErGr80]. The constant $c_0$ is called the Vardi constant.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos315.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«315»","statement":"True ↔\n  ∀ (a : ℕ → ℕ),\n    (∀ (i : ℕ), 0 < a i) →\n      StrictMono a →\n        (∃ i, a i ≠ Erdos315.u i + 1) →\n          ∑' (i : ℕ), 1 / ↑(a i) = 1 → Filter.liminf (fun i => ↑(a i) ^ (1 / 2) ^ (i + 1)) Filter.atTop < Erdos315.c₀","subjects":["11"],"theorem":"Erdos315.erdos_315"},{"answerKinds":[],"category":"research solved","docstring":"Cambie, Chan, and Hunter have in the comment section given a simple construction of a graph on\n$n$ vertices with at least $\\frac{3}{4}n$ distinct degrees, every degree appears at most twice,\nand the largest trivial subgraph has size $O(\\log n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1037»","statement":"∃ C,\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∃ G,\n      (∀ (d : ℕ), {v | (G.neighborSet v).ncard = d}.ncard ≤ 2) ∧\n        3 / 4 * ↑n ≤ ↑(Set.range fun v => (G.neighborSet v).ncard).ncard ∧\n          ∀ (s : Set (Fin n)), Erdos1037.IsTrivialSet G s → ↑s.ncard ≤ C * Real.log ↑n","subjects":["5"],"theorem":"Erdos1037.erdos_1037.variants.cambie_chan_hunter"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a graph on $n$ vertices in which every degree occurs at most twice, and the number of\ndistinct degrees is $>(\\frac{1}{2}+\\epsilon)n$. Must $G$ contain a trivial (empty or complete)\nsubgraph of size 'much larger' than $\\log n$?\n\nA question of Chen and Erdős.\n\nThe answer is no - Cambie, Chan, and Hunter have in the comment section given a simple\nconstruction of a graph on $n$ vertices with at least $\\frac{3}{4}n$ distinct degrees, every\ndegree appears at most twice, and the largest trivial subgraph has size $O(\\log n)$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1037.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1037»","statement":"False ↔\n  ∀ (ε : ℝ),\n    0 < ε →\n      ∀ (C : ℝ),\n        ∀ᶠ (n : ℕ) in Filter.atTop,\n          ∀ (G : SimpleGraph (Fin n)),\n            (∀ (d : ℕ), {v | (G.neighborSet v).ncard = d}.ncard ≤ 2) →\n              (1 / 2 + ε) * ↑n < ↑(Set.range fun v => (G.neighborSet v).ncard).ncard →\n                ∃ s, Erdos1037.IsTrivialSet G s ∧ C * Real.log ↑n < ↑s.ncard","subjects":["5"],"theorem":"Erdos1037.erdos_1037"},{"answerKinds":[],"category":"research open","docstring":"For what functions $g(N) → \\infty$ is it true that\n$$\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg \\frac{N^{1/2}}{g(N)}$$\nimplies $\\limsup 1_A\\ast 1_A(n)=\\infty$?\n\nAsked here in decision form: is there any such $g$ at all? Establishing the\nimplication for even one $g(N) → \\infty$ already answers Erdős Problem 28\npositively, because a basis of order $2$ satisfies\n$\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg N^{1/2}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«40»","statement":"True ↔ ∃ g, Filter.Tendsto g Filter.atTop Filter.atTop ∧ Erdos40.Erdos40For g","subjects":["11"],"theorem":"Erdos40.erdos_40"},{"answerKinds":[],"category":"textbook","docstring":"If we don't pose additional conditions on the functions, then this is a stronger form of the\nErdős-Turán conjecture, see Erdõs Problem 28,\n(since establishing this for any function $g(N) → \\infty$ would imply a positive solution to Erdős\nProblem 28).\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«40»","statement":"Erdos40.Erdos40ForSet Set.univ →\n  ∀ (A : Set ℕ), (A + A)ᶜ.Finite → Filter.limsup (fun n => ↑(AdditiveCombinatorics.sumRep A n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos40.erdos_40.variants.implies_erdos_28"},{"answerKinds":[],"category":"research open","docstring":"Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no three on a line.\nDoes $A$ determine at least $\\lfloor n/2\\rfloor$ distinct distances?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1082»","statement":"True ↔ ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), EuclideanGeometry.NonTrilinear ↑A → A.card / 2 ≤ distinctDistances A","subjects":["51"],"theorem":"Erdos1082.erdos_1082.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no three on a line.\nMust there exist a single point from which there are at least $\\lfloor n/2\\rfloor$ distinct\ndistances?\n\nThis question has been answered negatively by Xichuan in the\n[comments](https://www.erdosproblems.com/forum/thread/1082), who gave a set of $42$ points in\n$\\mathbb{R}^2$, with no three on a line, such that each point determines only $20$ distinct distances.\n\nA smaller counterexample has been formalised here: it comprised of $8$ points, where each point only\ndetermines $3$ distances.\n\nThis counterexample has originally been found by Heiko Harborth.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/0aca4d71095301c0fd2dca32611b7addb2ea735c/FormalConjectures/ErdosProblems/1082.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1082»","statement":"False ↔\n  ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))),\n    A.Nonempty → EuclideanGeometry.NonTrilinear ↑A → ∃ a, ∃ (_ : a ∈ A), A.card / 2 ≤ distinctDistancesFrom A a","subjects":["51"],"theorem":"Erdos1082.erdos_1082.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Erdős proved that, if $n_k$ is a lacunary sequence, then the sequence $\\{ \\alpha n_k\\}$ is not\nwell-distributed for almost all $\\alpha$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«997»","statement":"∀ (n : ℕ → ℕ), IsLacunary n → ∀ᵐ (α : ℝ), ¬Erdos997.IsWellDistributed fun k => Int.fract (α * ↑(n k))","subjects":["11"],"theorem":"Erdos997.erdos_997.variants.lacunary"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that, for every $\\alpha$, the sequence $\\{ \\alpha p_n\\}$ is not well-distributed,\nif $p_n$ is the sequence of primes?\n\nThe answer is yes, by [APSSV26, Section 4]; a Lean formalisation is available in [Mo26].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://live.lean-lang.org/#project=mathlib-v4.28.0&url=https://gist.githubusercontent.com/pitmonticone/016f2ed66b4cd1c4c4b9998095170e60/raw/b7dfc05c525ae385b5835f89f1ada721443e4305/Erdos997.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«997»","statement":"True ↔ ∀ (α : ℝ), ¬Erdos997.IsWellDistributed fun n => Int.fract (α * ↑(Nat.nth Nat.Prime n))","subjects":["11"],"theorem":"Erdos997.erdos_997"},{"answerKinds":[],"category":"research solved","docstring":"He also claimed in [Er64b] to have proved that there exists an irrational $\\alpha$ for which\n$\\{\\alpha p_n\\}$ is not well-distributed. He later retracted this claim in [Er85e], saying \"The\ntheorem is no doubt correct and perhaps will not be difficult to prove but I never was able to\nreconstruct my 'proof' which perhaps never existed.\"\n\nThe existence of such an $\\alpha$ was established by Champagne, Le, Liu, and Wooley [CLLW24].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«997»","statement":"∃ α, Irrational α ∧ ¬Erdos997.IsWellDistributed fun n => Int.fract (α * ↑(Nat.nth Nat.Prime n))","subjects":["11"],"theorem":"Erdos997.erdos_997.variants.irrational"},{"answerKinds":["Prop"],"category":"research open","docstring":"Let `n` be sufficiently large. Is there some choice of congruence class `a_p` for all primes\n`2 ≤ p ≤ n` such that every integer in `[1,n]` satisfies at least two of the congruences\n`≡ a_p (mod p)`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«689»","statement":"sorry ↔\n  ∀ᶠ (n : ℕ) in Filter.atTop, ∃ a, ∀ m ∈ Finset.Icc 1 n, 2 ≤ {p ∈ Finset.Icc 1 n | Nat.Prime p ∧ a p ≡ m [MOD p]}.card","subjects":["11"],"theorem":"Erdos689.erdos_689"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $C > 0$. Is it true that the set of integers of the form $n = b_1 + \\cdots + b_t$,\nwith $b_1 < \\cdots < b_t$, where $b_i = 2^{k_i}3^{l_i}$ for $1 \\leq i\\leq t$ and\n$b_t \\leq Cb_1$ has density $0$?\n\nvan Doorn and Everts \\cite{vDEv25} have disproved this with $C=6$ - in fact, they prove that all\nintegers can be written as such a sum in which $b_t<6b_1$.\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos845.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«845»","statement":"False ↔\n  ∀ (C : ℝ),\n    0 < C →\n      have f := fun x =>\n        match x with\n        | (k, l) => 2 ^ k * 3 ^ l;\n      {x | ∃ B, ∃ (h : B.Nonempty) (_ : ↑(B.sup f) ≤ C * ↑(B.inf' h f)), ∑ x ∈ B, f x = x}.HasDensity 0","subjects":["11"],"theorem":"Erdos845.erdos_845"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does every set $A \\subseteq \\mathbb{N}$ of positive density contain some finite $S \\subset A$ such that\n$\\sum_{n \\in S} \\frac{1}{n} = 1$?\n\nThe answer is yes, proved by Bloom [Bl21] (even if 'positive density' is interpreted as 'positive\nupper density', which is likely what Erdős intended).\n\nThe theorem below uses the positive-upper-density interpretation; the literal natural-density\ninterpretation is recorded separately.\n\nThis was formalized in Lean 3 by Bloom and Mehta.\n","formalProofs":[{"conditions":[],"kind":"other_system","link":"https://github.com/b-mehta/unit-fractions/blob/master/src/final_results.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«298»","statement":"True ↔ ∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity → ∃ S, ↑S ⊆ A ∧ ∑ n ∈ S, 1 / ↑n = 1","subjects":["11"],"theorem":"Erdos298.erdos_298"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The literal natural-density interpretation of Erdős Problem 298 follows from [Bl21].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«298»","statement":"True ↔ ∀ (A : Set ℕ), 0 ∉ A → A.HasPosDensity → ∃ S, ↑S ⊆ A ∧ ∑ n ∈ S, 1 / ↑n = 1","subjects":["11"],"theorem":"Erdos298.erdos_298.variants.natural_density"},{"answerKinds":[],"category":"research open","docstring":"There exists some constant $c>0$ such that\n$$R(C_4,K_n) \\ll n^{2-c}.$$\n\nThe prize of $100 is offered in [Er78] for a proof or disproof.\n\nThis problem is #17 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«159»","statement":"∃ c,\n  ∃ (_ : 0 < c),\n    ∃ C,\n      ∀ (n : ℕ),\n        1 ≤ n → ↑((SimpleGraph.cycleGraph 4).graphRamsey (SimpleGraph.completeGraph (Fin n))) ≤ C * ↑n ^ (2 - c)","subjects":["5"],"theorem":"Erdos159.erdos_159"},{"answerKinds":["Prop"],"category":"research solved","docstring":"More generally, is $f_r(n)\\gg_r n$? Disproved by Rödl, who showed $f_r(n) = o(n)$ for all\nfixed $r \\geq 2$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1092»","statement":"False ↔ ∀ (r : ℕ), (fun n => ↑r * ↑n) =o[Filter.atTop] fun n => ↑(Erdos1092.f r n)","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"Erdos1092.f_asymptotic_general"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that $f_2(n) \\gg n$? Disproved by Rödl, who showed $f_r(n) = o(n)$ for all fixed\n$r \\geq 2$. A conjecture of Erdős, Hajnal, and Szemerédi.\n\nThis seems to be closely related to, but distinct from, [744](https://www.erdosproblems.com/744).\n\nTang notes in the comments that Rödl [Ro82] constructed, for any $\\epsilon>0$ and $k$, a graph\nwith chromatic number $\\geq k$ such that every graph on $m$ vertices is bipartite after deleting at\nmost $\\epsilon m$ edges. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1092»","statement":"False ↔ (fun n => ↑n) =o[Filter.atTop] fun n => ↑(Erdos1092.f 2 n)","subjects":["5"],"theorem":"Erdos1092.f_asymptotic_2"},{"answerKinds":["Prop"],"category":"research solved","docstring":"A set $A\\subset \\mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum\n$$\\sum_{n\\in A}\\frac{1}{n\\log n}$$\nmaximised over all primitive sets when $A$ is the set of primes?\n\nErdős [Er35] proved that this sum always converges for a primitive set. Lichtman [Li23] proved\nthat the answer is yes. An alternative, simpler, proof is given by Alexeev, Barreto, Li, Lichtman,\nPrice, Shah, Tang, and Tao [ABLLPSTT26].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos164.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«164»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    (∀ a ∈ A, 2 ≤ a) →\n      Erdos1196.IsPrimitive A →\n        ∑' (a : ↑A), 1 / (↑↑a * Real.log ↑↑a) ≤ ∑' (p : ↑{p | Nat.Prime p}), 1 / (↑↑p * Real.log ↑↑p)","subjects":["11"],"theorem":"Erdos164.erdos_164"},{"answerKinds":[],"category":"research solved","docstring":"Clemen, Dumitrescu, and Liu [CDL25b] proved the stronger version where\nequilateral triangles of all positive side lengths are counted.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«755»","statement":"Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Erdos755.TAnySize 6 n)) fun n => 1 / 27 * ↑n ^ 3","subjects":["52"],"theorem":"Erdos755.erdos_755.variants.any_size_cdl"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«755»","statement":"Erdos755.unitEquilateralTriangleCount 1 ∅ = 0","subjects":["52"],"theorem":"Erdos755.erdos_755.test_dim_one"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Erdős asked whether every $n$-point set in\n$\\mathbb{R}^6$ spans at most $(1/27 + o(1)) n^3$ unit equilateral triangles.\n\nClemen, Dumitrescu, and Liu [CDL25b] proved the stronger any-size statement\n$T_6(n) = (1/27 + o(1)) n^3$. The unit-triangle upper bound follows as a\ncorollary, since unit equilateral triangles are a subset of equilateral\ntriangles of any positive side length: $T_\\mathrm{unit} \\leq T_\\mathrm{anysize}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos755.lean#L1344"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«755»","statement":"True ↔\n  ∃ o, (o =o[Filter.atTop] fun x => 1) ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos755.TUnit 6 n) ≤ (1 / 27 + o n) * ↑n ^ 3","subjects":["52"],"theorem":"Erdos755.erdos_755"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours\nthen there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«609»","statement":"(fun n => ↑(Erdos609.f n)) =Θ[Filter.atTop] sorry","subjects":["5"],"theorem":"Erdos609.erdos_609"},{"answerKinds":[],"category":"research solved","docstring":"Sylvester and Schur [Er34] proved that $P(n, k) > k$ for $k \\le n/2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«683»","statement":"∀ (n k : ℕ), 0 < k ∧ k ≤ n / 2 → Erdos683.P n k > k","subjects":["11"],"theorem":"Erdos683.erdos_683.variant.sylvester_schur"},{"answerKinds":[],"category":"research open","docstring":"Let $P(n, k)$ be the largest prime factor of $\\binom{n}{k}$.\nThere exists $c > 0$ such that $P(n, k) \\ge k^{1 + c}$ for all $0 < k \\le n/2$.\n\nErdős originally stated this for $1 \\le k \\le n$ with a $\\min(n-k+1, k^{1+c})$ bound [Er79d],\nbut as discussed on [erdosproblems.com](https://www.erdosproblems.com/forum/discuss/683),\nthe $\\min$ term was introduced only to handle $k > n/2$; the problem is naturally stated\nfor $k \\le n/2$ (cf. [#961](https://www.erdosproblems.com/961)).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«683»","statement":"True ↔ ∃ c > 0, ∀ (n k : ℕ), 0 < k ∧ k ≤ n / 2 → ↑(Erdos683.P n k) ≥ ↑k ^ (1 + c)","subjects":["11"],"theorem":"Erdos683.erdos_683"},{"answerKinds":[],"category":"research solved","docstring":"Erdos [Er55d] improved this to $P(n, k) \\gg k \\log k $ for $k \\le n/2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«683»","statement":"∃ c > 0, ∀ (n k : ℕ), 0 < k ∧ k ≤ n / 2 → ↑(Erdos683.P n k) > c * ↑k * Real.log ↑k","subjects":["11"],"theorem":"Erdos683.erdos_683.variant.erdos_log"},{"answerKinds":[],"category":"research open","docstring":"Standard heuristics suggest that $P(n, k) > e^{c\\sqrt{k}}$ for some constant $c > 0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«683»","statement":"∃ c > 0, ∀ (n k : ℕ), 0 < k ∧ k ≤ n / 2 → ↑(Erdos683.P n k) > Real.exp (c * √↑k)","subjects":["11"],"theorem":"Erdos683.erdos_683.variant.exp_sqrt"},{"answerKinds":[],"category":"research solved","docstring":"Erdos observed that this value is finite and > 1. This records the lower bound `> 1`,\ni.e. that `1` is a strict lower bound for every admissible `A`; the finiteness is\n`erdos_33.variants.vanDoorn`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«33»","statement":"1 < ⨅ A, Filter.limsup (fun N => ↑(↑A ∩ Set.Icc 1 N).ncard / ↑√↑N) Filter.atTop","subjects":["11"],"theorem":"Erdos33.erdos_33.variants.one_mem_lowerBounds"},{"answerKinds":[],"category":"research solved","docstring":"The smallest possible value of `lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)`\nis at most `2φ^(5/2) ≈ 6.66`, with `φ` equal to the golden ratio. Proven by\nWouter van Doorn.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«33»","statement":"⨅ A, Filter.limsup (fun N => ↑(↑A ∩ Set.Icc 1 N).ncard / ↑√↑N) Filter.atTop ≤ ↑(2 * Real.goldenRatio ^ (5 / 2))","subjects":["11"],"theorem":"Erdos33.erdos_33.variants.vanDoorn"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let `A ⊆ ℕ` be a set such that every integer can be written as `n^2 + a`\nfor some `a` in `A` and `n ≥ 0`. What is the smallest possible value of\n`lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«33»","statement":"⨅ A, Filter.limsup (fun N => ↑(↑A ∩ Set.Icc 1 N).ncard / ↑√↑N) Filter.atTop = sorry","subjects":["11"],"theorem":"Erdos33.erdos_33"},{"answerKinds":[],"category":"research open","docstring":"Schinzel conjectured (see [Si56]) the generalisation that, for any fixed $a$, if $n$ is sufficiently\nlarge in terms of $a$ then there exist distinct integers $1\\leq x < y < z$ such that\n$\\frac{a}{n} = \\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}.$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«242»","statement":"∀ (a : ℕ), 0 < a → ∀ᶠ (n : ℕ) in Filter.atTop, ∃ x y z, 1 ≤ x ∧ x < y ∧ y < z ∧ ↑a / ↑n = 1 / ↑x + 1 / ↑y + 1 / ↑z","subjects":["11"],"theorem":"Erdos242.erdos_242.variants.schinzel_generalization"},{"answerKinds":[],"category":"research open","docstring":"For every $n>2$ there exist distinct integers $1 ≤ x < y < z$\nsuch that $\\frac 4 n = \\frac 1 x + \\frac 1 y + \\frac 1 z$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«242»","statement":"∀ (n : ℕ), 2 < n → ∃ x y z, 1 ≤ x ∧ x < y ∧ y < z ∧ 4 / ↑n = 1 / ↑x + 1 / ↑y + 1 / ↑z","subjects":["11"],"theorem":"Erdos242.erdos_242"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that, for all $a,b\\in A$, the product $ab$ is not\nsquarefree.\n\nIs the maximum size of such an $A$ achieved by taking $A$ to be the set of even numbers and\nodd non-squarefree numbers?\n\nA problem of Erdős and Sárközy.\n\nWeisenberg has provided the following positive proof. It is clear that such a maximal $A$ must\ncontain all non-squarefree numbers. It therefore suffices to find the largest size of a subset\nof all squarefree numbers in $\\{1,\\ldots,N\\}$ such that any two have at least one prime factor\nin common. By the result of Chvátal [Ch74] discussed in [701] this is maximised by the set of\nall even squarefree numbers.\n\nAn alternative proof was independently found by Alexeev, Mixon, and Sawin [AMS25].\n\nThis was formalized in Lean by Jennings using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://gist.githubusercontent.com/JohnEdwardJennings/e32f2c412b0225091e7519d60741bd2d/raw/7d811ea413e2f7c0c0442749958aaac421eb6807/Erdos844.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«844»","statement":"True ↔\n  ∀ (N : ℕ),\n    IsGreatest {k | ∃ A ⊆ Finset.Icc 1 N, (∀ a ∈ A, ∀ b ∈ A, ¬Squarefree (a * b)) ∧ A.card = k}\n      (Erdos844.evenOrOddNonSquarefree N).card","subjects":["5","11"],"theorem":"Erdos844.erdos_844"},{"answerKinds":[],"category":"research solved","docstring":"**Explicit form of Prikry–Mills**:\n\nThere exists a 2-colouring of $\\omega_1 \\times \\omega_1 \\times \\omega_1$ such that\nfor every countably infinite $A_1, B_1, C_1 \\subseteq \\omega_1$, the box\n$A_1 \\times B_1 \\times C_1$ is not monochromatic.\n\nThis is the content of the Prikry–Mills theorem (1978, unpublished), stated\nusing Lean's ordinal type `{o : Ordinal // o < ω_ 1}` as the\nrepresentation of $\\omega_1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1128»","statement":"∃ f,\n  ∀ (A₁ : Set { o // o < Ordinal.omega 1 }) (B₁ : Set { o // o < Ordinal.omega 1 })\n    (C₁ : Set { o // o < Ordinal.omega 1 }),\n    Cardinal.mk ↑A₁ = Cardinal.aleph 0 →\n      Cardinal.mk ↑B₁ = Cardinal.aleph 0 → Cardinal.mk ↑C₁ = Cardinal.aleph 0 → ¬Erdos1128.IsMonochromaticBox f A₁ B₁ C₁","subjects":["3","5"],"theorem":"Erdos1128.erdos_1128.variants.prikryMills_explicit"},{"answerKinds":[],"category":"research solved","docstring":"The claim that every 2-colouring of $\\omega_1 \\times \\omega_1$ has an uncountable\nmonochromatic product rectangle is **false** in ZFC.\n\n**Counterexample:** The ordering colouring $f(\\alpha, \\beta) = 0$ iff $\\alpha < \\beta$\nhas no uncountable monochromatic product rectangle $A_1 \\times B_1$.\n\n**Proof:** If $A_1 \\times B_1$ were monochromatic with colour 0, then every element of\n$A_1$ would be strictly less than every element of $B_1$, making $A_1$ bounded above in\n$\\omega_1$; but any bounded subset of $\\omega_1$ is countable (since initial segments are\ncountable), contradicting $A_1$ being uncountable. The colour-1 case is symmetric with\nthe roles of $A_1$ and $B_1$ swapped.\n\nNote: The correct classical result for 2-colourings of pairs (not products) is the\nErdős–Rado theorem $\\omega_1 \\to (\\omega_1)^2_2$, which concerns unordered pairs.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1128»","statement":"¬∀ (f : Erdos1128.Omega1✝ → Erdos1128.Omega1✝ → Fin 2),\n    ∃ A₁ B₁, ¬A₁.Countable ∧ ¬B₁.Countable ∧ ∃ c, ∀ a ∈ A₁, ∀ b ∈ B₁, f a b = c","subjects":["3","5"],"theorem":"Erdos1128.erdos_1128.variants.two_dimensional_false"},{"answerKinds":[],"category":"research solved","docstring":"**Prikry–Mills counterexample** (key lemma):\n\nThere exists a 2-colouring $f$ of a set of cardinality $\\aleph_1$ cubed such that\nno countable box $A_1 \\times B_1 \\times C_1$ is monochromatic.\n\nThis is the unpublished result of Prikry and Mills (1978). The proof proceeds by\ntransfinite induction along $\\omega_1$, which has uncountable cofinality, ensuring\nevery countable box is non-monochromatic.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1128»","statement":"∃ X,\n  ∃ (_ : Cardinal.mk X = Cardinal.aleph 1),\n    ∃ f,\n      ∀ (A₁ B₁ C₁ : Set X),\n        Cardinal.mk ↑A₁ = Cardinal.aleph 0 →\n          Cardinal.mk ↑B₁ = Cardinal.aleph 0 →\n            Cardinal.mk ↑C₁ = Cardinal.aleph 0 → ¬Erdos1128.IsMonochromaticBox f A₁ B₁ C₁","subjects":["3","5"],"theorem":"Erdos1128.erdos_1128.prikryMills"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 1128** (disproved by Prikry–Mills, 1978):\n\nErdős asked whether every 2-colouring of $A \\times B \\times C$, where\n$|A| = |B| = |C| = \\aleph_1$, must contain a monochromatic countable box\n$A_1 \\times B_1 \\times C_1$ with $|A_1| = |B_1| = |C_1| = \\aleph_0$.\n\nThe answer is **No**: Prikry and Mills constructed a 2-colouring of $\\omega_1^3$\nwith no monochromatic countable box.\n\nNote: The positive statement asserts that every 2-colouring of every $\\aleph_1^3$\ncontains a monochromatic countably infinite box. Since the answer is False, this\npositive statement fails.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1128»","statement":"False ↔\n  ∀ (A B C : Type),\n    Cardinal.mk A = Cardinal.aleph 1 →\n      Cardinal.mk B = Cardinal.aleph 1 →\n        Cardinal.mk C = Cardinal.aleph 1 →\n          ∀ (f : A → B → C → Fin 2),\n            ∃ A₁ B₁ C₁,\n              Cardinal.mk ↑A₁ = Cardinal.aleph 0 ∧\n                Cardinal.mk ↑B₁ = Cardinal.aleph 0 ∧\n                  Cardinal.mk ↑C₁ = Cardinal.aleph 0 ∧ Erdos1128.IsMonochromaticBox f A₁ B₁ C₁","subjects":["3","5"],"theorem":"Erdos1128.erdos_1128"},{"answerKinds":[],"category":"API","docstring":"Every triangle is cuttable into any non-zero square number of congruent triangles. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«633»","statement":"∀ {n : ℕ} {T : Affine.Triangle ℝ (EuclideanSpace ℝ (Fin 2))}, n ≠ 0 → IsSquare n → Erdos633.IsCuttable n T","subjects":["5","51"],"theorem":"Erdos633.IsCuttable.of_isSquare"},{"answerKinds":[],"category":"API","docstring":"A triangle isn't simili-cuttable into zero triangles. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«633»","statement":"∀ {n : ℕ} {T : Affine.Triangle ℝ (EuclideanSpace ℝ (Fin 2))}, Erdos633.IsSimiliCuttable n T → n ≠ 0","subjects":["5","51"],"theorem":"Erdos633.IsSimiliCuttable.ne_zero"},{"answerKinds":[],"category":"research solved","docstring":"A triangle whose side lengths and angles are integrally independent is cuttable only into\na non-zero square number of congruent triangles. This is proved in [So09c]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«633»","statement":"∀ {n : ℕ} {T : Affine.Triangle ℝ (EuclideanSpace ℝ (Fin 2))},\n  LinearIndependent ℤ\n      ![dist (T.points 0) (T.points 1), dist (T.points 1) (T.points 2), dist (T.points 2) (T.points 0)] →\n    LinearIndependent ℤ\n        ![EuclideanGeometry.angle (T.points 0) (T.points 1) (T.points 2),\n          EuclideanGeometry.angle (T.points 1) (T.points 2) (T.points 0),\n          EuclideanGeometry.angle (T.points 2) (T.points 0) (T.points 1)] →\n      (Erdos633.IsCuttable n T ↔ n ≠ 0 ∧ IsSquare n)","subjects":["5","51"],"theorem":"Erdos633.isCuttable_iff_isSquare_of_linearIndependent"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Which triangles can only be decomposed into a square number of congruent triangles? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«633»","statement":"∀ {T : Affine.Triangle ℝ (EuclideanSpace ℝ (Fin 2))}, T ∈ sorry ↔ ∀ (n : ℕ), Erdos633.IsCuttable n T → IsSquare n","subjects":["5","51"],"theorem":"Erdos633.erdos_633"},{"answerKinds":[],"category":"research solved","docstring":"There exists a triangle which isn't simili-cuttable into 0, 2, 3, 5 parts.\nThis is proved in [So09]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«633»","statement":"∃ T, ∀ (n : ℕ), Erdos633.IsSimiliCuttable n T ↔ n ≠ 0 ∧ n ≠ 2 ∧ n ≠ 3 ∧ n ≠ 5","subjects":["5","51"],"theorem":"Erdos633.exists_isSimiliCuttable_iff_ne_zero_two_three_five"},{"answerKinds":[],"category":"research solved","docstring":"Every triangle is simili-cuttable into any number of similar triangles, except 0, 2, 3, 5.\nThis is proved in [So09]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«633»","statement":"∀ {n : ℕ} {T : Affine.Triangle ℝ (EuclideanSpace ℝ (Fin 2))},\n  n ≠ 0 → n ≠ 2 → n ≠ 3 → n ≠ 5 → Erdos633.IsSimiliCuttable n T","subjects":["5","51"],"theorem":"Erdos633.IsSimiliCuttable.of_ne_zero_two_three_five"},{"answerKinds":[],"category":"API","docstring":"Every triangle is cuttable into any non-zero square number of congruent triangles. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«633»","statement":"∀ {n : ℕ} {T : Affine.Triangle ℝ (EuclideanSpace ℝ (Fin 2))}, n ≠ 0 → Erdos633.IsCuttable (n ^ 2) T","subjects":["5","51"],"theorem":"Erdos633.IsCuttable.sq"},{"answerKinds":[],"category":"API","docstring":"A triangle isn't cuttable into zero triangles. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«633»","statement":"∀ {n : ℕ} {T : Affine.Triangle ℝ (EuclideanSpace ℝ (Fin 2))}, Erdos633.IsCuttable n T → n ≠ 0","subjects":["5","51"],"theorem":"Erdos633.IsCuttable.ne_zero"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $\\varepsilon$ be sufficiently small and $C, C' > 0$. Are there integers $a, b, n$ such that\n$$a, b > \\varepsilon n\\quad a!\\, b! \\mid n!\\, (a + b - n)!, $$\nand\n$$C \\log n < a + b - n < C' \\log n ?$$\n\nNote that the website currently displays a simpler (trivial) version of this problem because\n$a + b$ isn't assumed to be in the $n + O(\\log n)$ regime.\n\nBarreto and ChatGPT-5.2 have proved that, for any $0 < C_1 < C_2$, there are infinitely many\n$a, b, n$ with $b = n/2$, $a = n/2 + O(\\log n)$, and $C_1 \\log n < a + b - n < C_2 \\log n$ such\nthat $a! b! \\mid n! (a + b - n)!$\n\nThis appears to answer the question in the spirit it was intended.\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos728p.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«728»","statement":"True ↔\n  ∀ᶠ (ε : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n    ∀ C > 0,\n      ∀ C' > C,\n        ∃ a b n,\n          0 < n ∧\n            ε * ↑n < ↑a ∧\n              ε * ↑n < ↑b ∧\n                a.factorial * b.factorial ∣ n.factorial * (a + b - n).factorial ∧\n                  ↑a + ↑b > ↑n + C * Real.log ↑n ∧ ↑a + ↑b < ↑n + C' * Real.log ↑n","subjects":["11"],"theorem":"Erdos728.erdos_728"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Can the smallest modulus of a covering system be arbitrarily large?\n\nThis problem has a negative answer: there is a universal bound on the least modulus of any\ndistinct covering system.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«2»","statement":"False ↔ ∀ (B : ℕ), ∃ c, ∀ (i : c.ι), ∃ m, c.moduli i = Ideal.span {↑m} ∧ B < m","subjects":["11"],"theorem":"Erdos2.erdos_2"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there a set $A\\subset\\mathbb{N}$ such that, for all large $N$,\n$$\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll N/\\log N$$\nand such that every large integer can be written as $2^k+a$ for some\n$k\\geq 0$ and $a\\in A$?\n\nLorentz [Lo54] proved there is such a set with, for all large $N$,\n$$\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll \\frac{\\log\\log N}{\\log N}N$$\nThe answer is yes, proved by Ruzsa [Ru72].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem221.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«221»","statement":"True ↔\n  ∃ A,\n    ((fun N => ↑{a | a ∈ A ∧ a ≤ N}.ncard) =O[Filter.atTop] fun N => ↑N / Real.log ↑N) ∧\n      ∀ᶠ (N : ℕ) in Filter.atTop, ∃ k a, 0 ≤ k ∧ a ∈ A ∧ N = 2 ^ k + a","subjects":["11"],"theorem":"Erdos221.erdos_221"},{"answerKinds":[],"category":"test","docstring":null,"formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/6c7a16e8998d1c597fa2a5c6329bc9301fcc56e2/FormalConjectures/ErdosProblems/138.lean#L79"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«138»","statement":"Erdos138.W 1 = 1","subjects":["11"],"theorem":"Erdos138.monoAPNumber_two_one"},{"answerKinds":["Prop"],"category":"research solved","docstring":"In [Er81] Erdős asks whether $W(k+1) - W(k) \\to \\infty$.\n\nThe DeepMind prover agent has found a formal proof of this statement.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/6ac8d0cbe1a85e71747c62c1391a84788015ebc1/FormalConjectures/ErdosProblems/138.lean#L844"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«138»","statement":"True ↔ Filter.Tendsto (fun k => Erdos138.W (k + 1) - Erdos138.W k) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos138.erdos_138.variants.difference"},{"answerKinds":[],"category":"research open","docstring":"In [Er80] Erdős asks whether $W(k)/2^k\\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«138»","statement":"True ↔ Filter.Tendsto (fun k => ↑(Erdos138.W k) / 2 ^ k) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos138.erdos_138.variants.dvd_two_pow"},{"answerKinds":[],"category":"research open","docstring":"In [Er81] Erdős asks whether $\\frac{W(k+1)}{W(k)} \\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«138»","statement":"True ↔ Filter.Tendsto (fun k => ↑(Erdos138.W (k + 1)) / ↑(Erdos138.W k)) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos138.erdos_138.variants.quotient"},{"answerKinds":[],"category":"test","docstring":null,"formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/6c7a16e8998d1c597fa2a5c6329bc9301fcc56e2/FormalConjectures/ErdosProblems/138.lean#L142"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«138»","statement":"Erdos138.W 2 = 3","subjects":["11"],"theorem":"Erdos138.monoAPNumber_two_two"},{"answerKinds":[],"category":"research solved","docstring":"Gowers [Go01] has proved $$W(k) \\leq 2^{2^{2^{2^{2^{k+9}}}}.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«138»","statement":"∀ (k : ℕ), Erdos138.W k ≤ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ (k + 9)","subjects":["11"],"theorem":"Erdos138.erdos_138.variants.upper"},{"answerKinds":[],"category":"research solved","docstring":"When $p$ is prime Berlekamp [Be68] has proved $W(p+1) ≥ p^{2^p}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«138»","statement":"∀ (p : ℕ), Nat.Prime p → p * 2 ^ p ≤ Erdos138.W (p + 1)","subjects":["11"],"theorem":"Erdos138.erdos_138.variants.prime"},{"answerKinds":[],"category":"research solved","docstring":"Asserts that for any number of colors `r` and any progression length `k`, there\nalways exists some number `N` large enough to guarantee a monochromatic arithmetic progression.\nIn other words, the set `monoAP_guarantee_set` is non-empty. This is the fundamental existence\nresult that allows for the definition of the van der Waerden numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«138»","statement":"∀ (r k : ℕ), (Erdos138.monoAP_guarantee_set r k).Nonempty","subjects":["11"],"theorem":"Erdos138.monoAP_guarantee_set_nonempty"},{"answerKinds":[],"category":"research open","docstring":"In [Er80] Erdős asks whether\n$$ \\lim_{k \\to \\infty} (W(k))^{1/k} = \\infty $$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«138»","statement":"True ↔ Filter.Tendsto (fun k => ↑(Erdos138.W k) ^ (1 / ↑k)) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos138.erdos_138"},{"answerKinds":[],"category":"research solved","docstring":"Cambie has also found solutions when $r=7$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«939»","statement":"(Erdos939.Erdos939Sums 7).Nonempty","subjects":["11"],"theorem":"Erdos939.erdos_939.variants.seven"},{"answerKinds":[],"category":"research solved","docstring":"Euler had conjectured that the sum of $k - 1$ many $k$-th powers is never a\n$k$-th power, but this is false for $k=5$, as Lander and Parkin [LaPa67] found\n$$27^5+84^5+110^5+133^5=144^5$$.\n\nThe summands must be positive. Without that condition a set containing `0` would count, so the\nnegation would be satisfied by a sum of fewer than $k-1$ powers and would claim less than the\nrefutation of Euler's conjecture that this theorem records.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«939»","statement":"¬∀ k ≥ 4, ∀ (S : Finset ℕ), S.card = k - 1 → (∀ s ∈ S, 0 < s) → ¬∃ q, ∑ s ∈ S, s ^ k = q ^ k","subjects":["11"],"theorem":"Erdos939.erdos_939.variants.euler"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Are there infinitely many triples of coprime $3$-powerful numbers $a, b, c$ such that $a + b = c$?\n\nThe answer is yes. Nitaj [Ni95] proved it, with $2^3\\cdot 3^5\\cdot 73^3 + 271^3 = 919^3$ as an\nexample. In Nitaj's construction at least two of $a, b, c$ are perfect cubes. Cohn [Co98]\nconstructed infinitely many triples of which none is a perfect cube, and Walsh [Wa24] gave a\nfurther construction.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«939»","statement":"True ↔ {(a, b, c) | {a, b, c}.Coprime ∧ 0 < a ∧ 0 < b ∧ Nat.Full 3 a ∧ Nat.Full 3 b ∧ Nat.Full 3 c ∧ a + b = c}.Infinite","subjects":["11"],"theorem":"Erdos939.erdos_939.variants.triples"},{"answerKinds":[],"category":"research solved","docstring":"Cambie has also found solutions when $r=8$.\n\nThe source adds that the $r=8$ solution works \"even with the sum of $5$ $8$-powerful numbers\".\nThat is a stronger result than this statement, which asks for the $r - 2 = 6$ summands of\n`Erdos939Sums`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«939»","statement":"(Erdos939.Erdos939Sums 8).Nonempty","subjects":["11"],"theorem":"Erdos939.erdos_939.variants.eight"},{"answerKinds":[],"category":"research open","docstring":"If $r≥4$ then can the sum of $r-2$ coprime $r$-powerful numbers ever be itself $r$-powerful?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«939»","statement":"True ↔ ∀ r ≥ 4, (Erdos939.Erdos939Sums r).Nonempty","subjects":["11"],"theorem":"Erdos939.erdos_939"},{"answerKinds":[],"category":"research solved","docstring":"Cambie has found several examples of the sum of $r - 2$ coprime $r$-powerful numbers being itself\n$r$-powerful. For example when $r=5$ we have\n$$3^7\\cdot 61^5 = 2^8\\cdot3^{10}\\cdot 5^7 + 2^{12}\\cdot 23^6 + 11^5\\cdot 13^5$$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«939»","statement":"∃ r ≥ 4, (Erdos939.Erdos939Sums r).Nonempty","subjects":["11"],"theorem":"Erdos939.erdos_939.variants.examples"},{"answerKinds":[],"category":"research open","docstring":"If $r≥4$ are there infinitely many sums of $r-2$ coprime $r$-powerful numbers\nthat are themselves $r$-powerful?\n\nA construction in the site's comments, from GPT-5.5 Pro prompted by Price, gives infinitely\nmany for every $r \\ge 6$. This statement quantifies over every $r \\ge 4$, so it stays open at\n$r = 4$ and $r = 5$. The category is unchanged because the construction is recorded in the\ncomments and not in the literature.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«939»","statement":"True ↔ ∀ r ≥ 4, (Erdos939.Erdos939Sums r).Infinite","subjects":["11"],"theorem":"Erdos939.erdos_939.variants.infinite"},{"answerKinds":[],"category":"research open","docstring":"Is it true that for all `c ≥ 0`, the density `f c` of integers for which\n`(p (n + 1) - p n) / log n < c` exists and is a continuous function of `c`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«234»","statement":"True ↔ ∃ f, Continuous f ∧ ∀ (c : NNReal), {n | ↑(primeGap n) / Real.log ↑n < ↑c}.HasDensity (f c)","subjects":["11"],"theorem":"Erdos234.erdos_234"},{"answerKinds":[],"category":"research open","docstring":"Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime. Is it true that\n$$\\frac{\\max_{n < x}d_{n}d_{n-1}}{(\\max_{n < x}d_n)^2}\\to 0$$ as $x\\to \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1137»","statement":"True ↔\n  Filter.Tendsto\n    (fun x => ↑((Finset.range x).sup fun n => primeGap n * primeGap (n - 1)) / ↑((Finset.range x).sup primeGap) ^ 2)\n    Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Erdos1137.erdos_1137"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $z_1,\\ldots,z_n\\in\\mathbb{C}$ with $1\\leq \\lvert z_i\\rvert$ for $1\\leq i\\leq n$. Let $D$ be an\narbitrary disc of radius $1$. Is it true that the number of sums of the shape\n$$\\sum_{i=1}^n\\epsilon_iz_i \\textrm{ for }\\epsilon_i\\in \\{-1,1\\}$$\nwhich lie in $D$ is at most $\\binom{n}{\\lfloor n/2\\rfloor}$?\n\nA strong form of the Littlewood-Offord problem. Erdős [Er45] proved this is true if\n$z_i\\in\\mathbb{R}$, and for general $z_i\\in\\mathbb{C}$ proved a weaker upper bound of\n$$\\ll \\frac{2^n}{\\sqrt{n}}.$$\nThis was solved in the affirmative by Kleitman [Kl65], who also later generalised this to\narbitrary Hilbert spaces [Kl70].\n\nSee also [395].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«498»","statement":"True ↔\n  ∀ (n : ℕ) (z : Fin n → ℂ),\n    (∀ (i : Fin n), 1 ≤ ‖z i‖) →\n      ∀ (c : ℂ),\n        {ε | (∀ (i : Fin n), ε i = -1 ∨ ε i = 1) ∧ ∑ i, ↑(ε i) * z i ∈ Metric.ball c 1}.ncard ≤ n.choose (n / 2)","subjects":["5"],"theorem":"Erdos498.erdos_498"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«340»","statement":"Finset.greedySidon 1 = 2","subjects":["5"],"theorem":"Erdos340.greedySidon_one"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«340»","statement":"Finset.greedySidon 2 = 4","subjects":["5"],"theorem":"Erdos340.greedySidon_two"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«340»","statement":"↑(Finset.greedySidon.go {1, 2} ⋯ 3) = 4","subjects":["5"],"theorem":"Erdos340.greedySidon_go_pair_three"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«340»","statement":"Finset.greedySidon 4 = 13","subjects":["5"],"theorem":"Erdos340.greedySidon_four"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«340»","statement":"↑(Finset.greedySidon.go {1} ⋯ 2) = 2","subjects":["5"],"theorem":"Erdos340.greedySidon_go_singleton_two"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«340»","statement":"Finset.greedySidon 3 = 8","subjects":["5"],"theorem":"Erdos340.greedySidon_three"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«340»","statement":"Finset.greedySidon 5 = 21","subjects":["5"],"theorem":"Erdos340.greedySidon_five"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $A = \\{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \\ldots\\}$ be the greedy Sidon sequence:\nwe begin with $1$ and iteratively include the next smallest integer that preserves the\nSidon property (i.e. there are no non-trivial solutions to $a + b = c + d$). What is the\norder of growth of $A$? Is it true that $|A \\cap \\{1, \\ldots, N\\}| \\gg N^{1/2 - \\varepsilon}$\nfor all $\\varepsilon > 0$ and large $N$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«340»","statement":"∀ ε > 0, (fun n => ↑(Set.range Finset.greedySidon ∩ Set.Icc 1 n).ncard) =Θ[Filter.atTop] sorry","subjects":["5"],"theorem":"Erdos340.erdos_340.variants.isTheta"},{"answerKinds":[],"category":"research open","docstring":"Let $A = \\{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \\ldots\\}$ be the greedy Sidon sequence:\nwe begin with $1$ and iteratively include the next smallest integer that preserves the\nSidon property (i.e. there are no non-trivial solutions to $a + b = c + d$). What is the\norder of growth of $A$? Is it true that $|A \\cap \\{1, \\ldots, N\\}| \\gg N^{1/2 - \\varepsilon}$\nfor all $\\varepsilon > 0$ and large $N$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«340»","statement":"∀ ε > 0, (fun n => √↑n / ↑n ^ ε) =O[Filter.atTop] fun n => ↑(Set.range Finset.greedySidon ∩ Set.Icc 1 n).ncard","subjects":["5"],"theorem":"Erdos340.erdos_340"},{"answerKinds":[],"category":"textbook","docstring":"It is trivial that this sequence grows at least like $\\gg N^{1/3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«340»","statement":"∀ ε > 0, (fun n => ↑n ^ (1 / 3)) =O[Filter.atTop] fun n => ↑(Set.range Finset.greedySidon ∩ Set.Icc 1 n).ncard","subjects":["5"],"theorem":"Erdos340.erdos_340.variants.third"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«340»","statement":"Finset.greedySidon 0 = 1","subjects":["5"],"theorem":"Erdos340.greedySidon_zero"},{"answerKinds":[],"category":"research open","docstring":"Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this has\npositive density.\n\n[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number\ntheory. Monographies de L'Enseignement Mathematique (1980).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«340»","statement":"(Set.range Finset.greedySidon - Set.range Finset.greedySidon).HasPosDensity","subjects":["5"],"theorem":"Erdos340.erdos_340.variants.sub_hasPosDensity"},{"answerKinds":[],"category":"research open","docstring":"It may be true that all or almost all integers are in $A - A$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«340»","statement":"True ↔ ∀ᶠ (n : ℕ) in Filter.cofinite, n ∈ Set.range Finset.greedySidon - Set.range Finset.greedySidon","subjects":["5"],"theorem":"Erdos340.erdos_340.variants.cofinite_sub"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«340»","statement":"Finset.greedySidon 10 = 97","subjects":["5"],"theorem":"Erdos340.greedySidon_ten"},{"answerKinds":[],"category":"research open","docstring":"It may be true that all or almost all integers are in $A - A$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«340»","statement":"True ↔ ∃ S, S.HasDensity 0 ∧ ∀ n ∈ Sᶜ, n ∈ Set.range Finset.greedySidon - Set.range Finset.greedySidon","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"Erdos340.erdos_340.variants.co_density_zero_sub"},{"answerKinds":["Prop"],"category":"research solved","docstring":"For a triangle-free graph $G$ let $h_2(G)$ be the smallest number of edges that need to be\nadded to $G$ so that it has diameter $2$ and is still triangle-free. Is it true that if $G$\nhas maximum degree $o(n^{1/2})$ then $h(G)=o(n^2)$?\n\nA problem of Erdős, Gyárfás, and Ruszinkó [EGR98]. Simonovits showed that there exist graphs\n$G$ with maximum degree $\\gg n^{1/2}$ and $h_2(G)\\gg n^2$. Alon has observed this problem is\nessentially identical to [134], and his solution in\n[this note](https://web.math.princeton.edu/~nalon/PDFS/remark1901.pdf) also solves this\nproblem in the affirmative.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos618.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«618»","statement":"True ↔\n  ∀ (G : (n : ℕ) → SimpleGraph (Fin n)),\n    (∀ (n : ℕ), (G n).CliqueFree 3) →\n      ((fun n => ↑(G n).maxDegree) =o[Filter.atTop] fun n => ↑n ^ (1 / 2)) →\n        (fun n => ↑(Erdos618.h2 (G n))) =o[Filter.atTop] fun n => ↑n ^ 2","subjects":["5"],"theorem":"Erdos618.erdos_618"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $R(n+1)-R(n) \\gg n^2$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«812»","statement":"True ↔\n  (fun n => ↑n ^ 2) =O[Filter.atTop] fun n =>\n    ↑(Combinatorics.hypergraphRamsey 2 (n + 1)) - ↑(Combinatorics.hypergraphRamsey 2 n)","subjects":["5"],"theorem":"Erdos812.erdos_812.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $\\frac{R(n+1)}{R(n)}\\geq 1+c$ for some constant $c>0$, for all large $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«812»","statement":"True ↔\n  ∃ c > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ↑(Combinatorics.hypergraphRamsey 2 (n + 1)) / ↑(Combinatorics.hypergraphRamsey 2 n) ≥ 1 + c","subjects":["5"],"theorem":"Erdos812.erdos_812.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Burr, Erdős, Faudree, and Schelp [BEFS89] proved that $R(n+1)-R(n) \\geq 4n-8$ for all $n\\geq 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«812»","statement":"∀ n ≥ 2, ↑(Combinatorics.hypergraphRamsey 2 (n + 1)) - ↑(Combinatorics.hypergraphRamsey 2 n) ≥ 4 * ↑n - 8","subjects":["5"],"theorem":"Erdos812.erdos_812.variants.lower_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does there exist a graph $G$ which contains no $K_4$, and yet any $2$-colouring of the edges\nproduces a monochromatic $K_3$?\n\nErdős and Hajnal [ErHa67] first asked for the existence of any such graph. Existence was proved\nby Folkman [Fo70], but with very poor quantitative bounds. (As a result these quantities are\noften called Folkman numbers.) The current best bounds on the minimal number of vertices $N$ of\nsuch a graph are $21 \\leq N \\leq 786$, where the lower bound is due to Bikov and Nenov [BiNe20]\nand the upper bound is due to Lange, Radziszowski, and Xu [LRX14].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos582.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«582»","statement":"True ↔ ∃ V x G, G.CliqueFree 4 ∧ Erdos582.EdgeRamseyTriangle G","subjects":["5"],"theorem":"Erdos582.erdos_582"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $1<a_1<\\cdots$ be a sequence of integers such that $\\sum\\frac{1}{a_i}<\\infty$. Is it true\nthat, for every $t\\in \\mathbb{R}$,\n$$1+\\sum_{k}\\frac{1}{a_k^{1+it}}\\neq 0?$$\n\nYip [Yi25] has proved that this is not always true - in fact, for any real $t\\neq 0$, there\nexists a sequence of integers $1<a_1<\\cdots$ such that $\\sum \\frac{1}{a_i}<\\infty$ and\n$1+\\sum_{k}\\frac{1}{a_k^{1+it}}=0$.\n\nThis was formalized in Lean by Wu using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://gist.github.com/llllvvuu/d25f037d1f1000bdabd6ca928c74c9bb"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«967»","statement":"False ↔\n  ∀ (a : ℕ → ℕ),\n    StrictMono a → 1 < a 0 → (Summable fun k => 1 / ↑(a k)) → ∀ (t : ℝ), 1 + ∑' (k : ℕ), Erdos967.summand t (a k) ≠ 0","subjects":["11","30"],"theorem":"Erdos967.erdos_967"},{"answerKinds":[],"category":"research solved","docstring":"Their interest in this problem arose from their proof that the statement that there are no such\nzeros is equivalent to the fact that, for any non-decreasing $f:\\mathbb{R}\\to \\mathbb{R}_{\\geq 0}$\nwhich is bounded on every bounded interval and is $=0$ for $x<1$, the relationship\n$$f(x)+\\sum_k f(x/a_k)=\\left(1+\\sum_k \\frac{1}{a_k}+o(1)\\right)x$$\nimplies $f(x)=(1+o(1))x$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«967»","statement":"∀ (a : ℕ → ℕ),\n  StrictMono a →\n    1 < a 0 →\n      (Summable fun k => 1 / ↑(a k)) →\n        ((∀ (t : ℝ), 1 + ∑' (k : ℕ), Erdos967.summand t (a k) ≠ 0) ↔\n          ∀ (f : ℝ → ℝ),\n            Monotone f →\n              (∀ (x : ℝ), 0 ≤ f x) →\n                (∀ x < 1, f x = 0) →\n                  (∀ (x y : ℝ), BddAbove (f '' Set.Icc x y)) →\n                    Filter.Tendsto (fun x => (f x + ∑' (k : ℕ), f (x / ↑(a k))) / x) Filter.atTop\n                        (nhds (1 + ∑' (k : ℕ), 1 / ↑(a k))) →\n                      Filter.Tendsto (fun x => f x / x) Filter.atTop (nhds 1))","subjects":["11","40"],"theorem":"Erdos967.erdos_967.variants.tauberian"},{"answerKinds":[],"category":"research open","docstring":"It remains open whether this is true for every finite sequence of integers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«967»","statement":"True ↔ ∀ (A : Finset ℕ), (∀ n ∈ A, 1 < n) → ∀ (t : ℝ), 1 + ∑ n ∈ A, Erdos967.summand t n ≠ 0","subjects":["11","30"],"theorem":"Erdos967.erdos_967.variants.finite"},{"answerKinds":[],"category":"research open","docstring":"A question of Erdős and Ingham [ErIn64]. The simplest case they could not decide this question\nfor was the finite sequence $\\{2,3,5\\}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«967»","statement":"True ↔ ∀ (t : ℝ), 1 + ∑ n ∈ {2, 3, 5}, Erdos967.summand t n ≠ 0","subjects":["11","30"],"theorem":"Erdos967.erdos_967.variants.two_three_five"},{"answerKinds":[],"category":"research solved","docstring":"Yip [Yi25] has proved that this is not always true - in fact, for any real $t\\neq 0$, there\nexists a sequence of integers $1<a_1<\\cdots$ such that $\\sum \\frac{1}{a_i}<\\infty$ and\n$1+\\sum_{k}\\frac{1}{a_k^{1+it}}=0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«967»","statement":"∀ (t : ℝ),\n  t ≠ 0 → ∃ a, StrictMono a ∧ 1 < a 0 ∧ (Summable fun k => 1 / ↑(a k)) ∧ 1 + ∑' (k : ℕ), Erdos967.summand t (a k) = 0","subjects":["11","30"],"theorem":"Erdos967.erdos_967.variants.yip"},{"answerKinds":[],"category":"research open","docstring":"Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the primes up to finitely\nmany exceptions? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«431»","statement":"True ↔ ∃ A B, A.Infinite ∧ B.Infinite ∧ A + B =ᶠ[Filter.cofinite] {n | Nat.Prime n}","subjects":["11"],"theorem":"Erdos431.erdos_431"},{"answerKinds":[],"category":"research solved","docstring":"For sufficiently large $n$, every tree $T$ on $n$ vertices satisfies $R(T)\\leq 2n-2$.\nProved by Zhao [Zh11].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«547»","statement":"∀ᶠ (n : ℕ) in Filter.atTop, ∀ (T : SimpleGraph (Fin n)), T.IsTree → T.diagonalGraphRamsey ≤ 2 * n - 2","subjects":["5"],"theorem":"Erdos547.erdos_547.variants.sufficiently_large"},{"answerKinds":[],"category":"research open","docstring":"If $T$ is a tree on $n$ vertices then\n$$R(T) \\leq 2n-2.$$\n\nThis problem is #14 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«547»","statement":"∀ (n : ℕ), 2 ≤ n → ∀ (T : SimpleGraph (Fin n)), T.IsTree → T.diagonalGraphRamsey ≤ 2 * n - 2","subjects":["5"],"theorem":"Erdos547.erdos_547"},{"answerKinds":[],"category":"research open","docstring":"Let `c₁, c₂ > 0`. Is it true that for any sufficiently large `x`, there exists more than\n`c₁ * log x` many consecutive primes `≤ x` such that the difference between any two is `> c₂`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«238»","statement":"True ↔\n  ∀ c₁ > 0,\n    ∀ c₂ > 0,\n      ∀ᶠ (x : ℝ) in Filter.atTop,\n        ∃ k,\n          c₁ * Real.log x < ↑k ∧\n            ∃ f m,\n              (∀ (i : Fin k), ↑(f i) ≤ x ∧ f i = Nat.nth Nat.Prime (m + ↑i)) ∧\n                ∀ (i : Fin (k - 1)), c₂ < primeGap (m + ↑i)","subjects":["11"],"theorem":"Erdos238.erdos_238"},{"answerKinds":[],"category":"research solved","docstring":"It is well-known that the conjecture above is true when `c₁` is sufficiently small.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«238»","statement":"∀ c₂ > 0,\n  ∀ᶠ (c₁ : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n    ∀ᶠ (x : ℝ) in Filter.atTop,\n      ∃ k,\n        c₁ * Real.log x < ↑k ∧\n          ∃ f m,\n            (∀ (i : Fin k), ↑(f i) ≤ x ∧ f i = Nat.nth Nat.Prime (m + ↑i)) ∧ ∀ (i : Fin (k - 1)), c₂ < primeGap (m + ↑i)","subjects":["11"],"theorem":"Erdos238.erdos_238.variants.small_c1"},{"answerKinds":[],"category":"research solved","docstring":"What is the size of the largest $A\\subseteq \\mathbb{R}^n$ such that there are only two distinct\ndistances between elements of $A$? That is,\n$$\\# \\{ \\lvert x-y\\rvert : x\\neq y\\in A\\} = 2.$$\n\nAsked to Erdős by Coxeter. Bannai, Bannai, and Stanton [BBS83] have proved that\n$$\\lvert A\\rvert \\leq \\binom{n+2}{2}.$$\nA simple proof of this upper bound was given by Petrov and Pohoata [PePo21].\n\nThe exact maximum is not known in general: a lower bound of $\\binom{n+1}{2}$ follows from the\nconstruction of Alweiss (see [503]). ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos502.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«502»","statement":"∀ (n : ℕ) (A : Set (EuclideanSpace ℝ (Fin n))),\n  A.Finite → {d | ∃ x ∈ A, ∃ y ∈ A, x ≠ y ∧ dist x y = d}.ncard = 2 → A.ncard ≤ (n + 2).choose 2","subjects":["51","52"],"theorem":"Erdos502.erdos_502"},{"answerKinds":[],"category":"research solved","docstring":"Cambie [Ca25] has shown that, for fixed $n$, the sequence $\\delta_1(n,m)$ has superpolynomially\nmany local maxima $m$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«692»","statement":"∀ (k : ℕ) (δ : ℕ → ℕ → ℝ),\n  (∀ (a b : ℕ), Erdos692.IsDelta₁ a b (δ a b)) →\n    ∀ᶠ (n : ℕ) in Filter.atTop, ↑n ^ k ≤ ↑{m | n + 1 < m ∧ δ n (m - 1) ≤ δ n m ∧ δ n (m + 1) ≤ δ n m}.ncard","subjects":["11"],"theorem":"Erdos692.erdos_692.variants.cambie_local_maxima"},{"answerKinds":[],"category":"research solved","docstring":"Erdős proved that\n$$\\delta_1(n,m) \\ll \\frac{1}{(\\log n)^c}$$\nfor all $m$, for some constant $c>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«692»","statement":"∀ (δ : ℕ → ℕ → ℝ),\n  (∀ (a b : ℕ), Erdos692.IsDelta₁ a b (δ a b)) →\n    ∃ c > 0, ∃ C > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (m : ℕ), δ n m ≤ C / Real.log ↑n ^ c","subjects":["11"],"theorem":"Erdos692.erdos_692.variants.erdos_upper_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $\\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$.\nIs $\\delta_1(n,m)$ unimodular for $m>n+1$ (i.e. increases until some $m$ then decreases\nthereafter)?\n\nCambie has calculated that unimodularity fails even for $n=2$ and $n=3$.\n\nThis was formalized in Lean by Monticone using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://gist.githubusercontent.com/pitmonticone/96516af9100a37a1da81908dc0b0410c/raw/a1d6ca7f3835c58b257e5e715c8fdf3a224e1bd0/Erdos692.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«692»","statement":"False ↔ ∀ (δ : ℕ → ℕ → ℝ), (∀ (a b : ℕ), Erdos692.IsDelta₁ a b (δ a b)) → ∀ (n : ℕ), UnimodularOn (δ n) (n + 1)","subjects":["11"],"theorem":"Erdos692.erdos_692.parts.i"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $\\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$.\nFor fixed $n$, where does $\\delta_1(n,m)$ achieve its maximum?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«692»","statement":"∀ (n : ℕ) (δ : ℕ → ℕ → ℝ),\n  (∀ (a b : ℕ), Erdos692.IsDelta₁ a b (δ a b)) → IsGreatest (δ n '' Set.Ioi (n + 1)) (δ n sorry)","subjects":["11"],"theorem":"Erdos692.erdos_692.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Cambie has calculated that unimodularity fails even for $n=2$ and $n=3$. For example,\n$$\\delta_1(3,6)= 0.35\\quad \\delta_1(3,7)\\approx 0.33\\quad \\delta_1(3,8)\\approx 0.3619.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«692»","statement":"∀ (δ : ℕ → ℕ → ℝ), (∀ (a b : ℕ), Erdos692.IsDelta₁ a b (δ a b)) → δ 3 7 < δ 3 6 ∧ δ 3 7 < δ 3 8","subjects":["11"],"theorem":"Erdos692.erdos_692.variants.cambie_three"},{"answerKinds":[],"category":"research open","docstring":"Case 1:\nDoes $A + B$ have zero upper and lower density?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«125»","statement":"True ↔\n  ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).upperDensity = 0 ∧\n    ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).lowerDensity = 0","subjects":["11"],"theorem":"Erdos125.erdos_125.variants.zero_density"},{"answerKinds":[],"category":"research open","docstring":"Literature question:\nDoes $A + B$ have positive upper density?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«125»","statement":"True ↔ 0 < ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).upperDensity","subjects":["11"],"theorem":"Erdos125.erdos_125.variants.positive_upper_density"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Case 3:\nDoes $A + B$ have positive upper and lower density that are equal?\nThis is the literal interpretation of \"positive density\" which was falsified.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/300bf771bdbef43d7b9aa2521e633a50fd54dd28/FormalConjectures/ErdosProblems/125.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«125»","statement":"False ↔ ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).HasPosDensity","subjects":["11"],"theorem":"Erdos125.erdos_125"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Case 4:\nDoes $A + B$ have positive upper and lower density that are unequal?\n\nThis follows from the disproof `erdos_125.variants.positive_lower_density` above.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/0bc740d2351c53713e66d9340e83f7d2c1ddecab/FormalConjectures/ErdosProblems/125.lean#L860"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«125»","statement":"False ↔\n  0 < ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).lowerDensity ∧\n    ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).lowerDensity <\n      ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).upperDensity","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos125.erdos_125.variants.positive_unequal_density"},{"answerKinds":[],"category":"research open","docstring":"Case 2:\nDoes $A + B$ have zero lower density, but positive upper density?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«125»","statement":"True ↔\n  ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).lowerDensity = 0 ∧\n    0 < ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).upperDensity","subjects":["11"],"theorem":"Erdos125.erdos_125.variants.zero_lower_positive_upper_density"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Literature question:\nDoes $A + B$ have positive lower density?\n\nThis has been falsified.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/c27415379b5dbe34105d1fdd707994540c4c6fc7/FormalConjectures/ErdosProblems/125.lean#L468"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«125»","statement":"False ↔ 0 < ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).lowerDensity","subjects":["11"],"theorem":"Erdos125.erdos_125.variants.positive_lower_density"},{"answerKinds":[],"category":"research open","docstring":"**$r = 2$ case.** The stepping-down from 3-uniform to 2-uniform partition\nrelations: $2^\\lambda \\to (\\kappa_\\alpha + 1)_{\\alpha<\\gamma}^3$ implies\n$\\lambda \\to (\\kappa_\\alpha)_{\\alpha<\\gamma}^2$. Generalises the classical\nErdős–Rado stepping-up/down theorem for pairs.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1167»","statement":"∀ (lam : Cardinal.{u}),\n  Cardinal.aleph0 ≤ lam →\n    ∀ (γ : Ordinal.{u}),\n      2 ≤ γ →\n        ∀ (κ : γ.ToType → Cardinal.{u}),\n          (Combinatorics.cardinalPartitionRel (2 ^ lam) 3 γ fun α => κ α + 1) →\n            Combinatorics.cardinalPartitionRel lam 2 γ κ","subjects":["5"],"theorem":"Erdos1167.erdos_1167.variants.r_eq_two"},{"answerKinds":[],"category":"research open","docstring":"**Finite-target case.** When all $\\kappa_\\alpha$ are finite, $\\kappa_\\alpha + 1$\nis the ordinary natural-number successor. Special case of `erdos_1167`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1167»","statement":"∀ (r : ℕ),\n  2 ≤ r →\n    ∀ (lam : Cardinal.{u}),\n      Cardinal.aleph0 ≤ lam →\n        ∀ (γ : Ordinal.{u}),\n          2 ≤ γ →\n            ∀ (n : γ.ToType → ℕ),\n              (Combinatorics.cardinalPartitionRel (2 ^ lam) (r + 1) γ fun α => ↑(n α) + 1) →\n                Combinatorics.cardinalPartitionRel lam r γ fun α => ↑(n α)","subjects":["5"],"theorem":"Erdos1167.erdos_1167.variants.finite_targets"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 1167.** Let $r \\geq 2$ be finite, $\\gamma \\geq 2$, and $\\lambda$ be an infinite\ncardinal. Let $\\kappa_\\alpha$ be cardinals for all $\\alpha < \\gamma$. Is it true\nthat\n$$2^\\lambda \\to (\\kappa_\\alpha + 1)_{\\alpha < \\gamma}^{r+1}$$\nimplies\n$$\\lambda \\to (\\kappa_\\alpha)_{\\alpha < \\gamma}^r?$$\nHere $+$ means cardinal addition, so that $\\kappa_\\alpha + 1 = \\kappa_\\alpha$\nif $\\kappa_\\alpha$ is infinite.\n\nA problem of Erdős, Hajnal, and Rado.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1167»","statement":"True ↔\n  ∀ (r : ℕ),\n    2 ≤ r →\n      ∀ (lam : Cardinal.{u}),\n        Cardinal.aleph0 ≤ lam →\n          ∀ (γ : Ordinal.{u}),\n            2 ≤ γ →\n              ∀ (κ : γ.ToType → Cardinal.{u}),\n                (Combinatorics.cardinalPartitionRel (2 ^ lam) (r + 1) γ fun α => κ α + 1) →\n                  Combinatorics.cardinalPartitionRel lam r γ κ","subjects":["5"],"theorem":"Erdos1167.erdos_1167"},{"answerKinds":[],"category":"research open","docstring":"**Binary-color case.** The $\\gamma = 2$ specialization (two color classes).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1167»","statement":"∀ (r : ℕ),\n  2 ≤ r →\n    ∀ (lam : Cardinal.{u}),\n      Cardinal.aleph0 ≤ lam →\n        ∀ (κ : Ordinal.ToType 2 → Cardinal.{u}),\n          (Combinatorics.cardinalPartitionRel (2 ^ lam) (r + 1) 2 fun α => κ α + 1) →\n            Combinatorics.cardinalPartitionRel lam r 2 κ","subjects":["5"],"theorem":"Erdos1167.erdos_1167.variants.binary_colors"},{"answerKinds":[],"category":"test","docstring":"The unrestricted version of Erdős Problem 1167 (without the condition $\\gamma \\geq 2$ from the\noriginal Erdős–Hajnal list) is false. Taking $\\gamma = 1$, $\\kappa_0 = \\aleph_1$, $\\lambda =\n\\aleph_0$: the premise $2^{\\aleph_0} \\to (\\aleph_1 + 1)^3_1$ holds since $2^{\\aleph_0} \\geq\n\\aleph_1$, but the conclusion $\\aleph_0 \\to (\\aleph_1)^2_1$ fails since $\\aleph_0 < \\aleph_1$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1167»","statement":"¬∀ (r : ℕ),\n    2 ≤ r →\n      ∀ (lam : Cardinal.{u}),\n        Cardinal.aleph0 ≤ lam →\n          ∀ (γ : Ordinal.{u}) (κ : γ.ToType → Cardinal.{u}),\n            (Combinatorics.cardinalPartitionRel (2 ^ lam) (r + 1) γ fun α => κ α + 1) →\n              Combinatorics.cardinalPartitionRel lam r γ κ","subjects":["5"],"theorem":"Erdos1167.erdos_1167.unrestricted_is_false"},{"answerKinds":[],"category":"API","docstring":"The partition relation $\\mu \\to (\\nu)^r_1$ with a single color is equivalent to $\\nu \\le \\mu$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1167»","statement":"∀ (μ : Cardinal.{u}) (r : ℕ) (ν : Ordinal.ToType 1 → Cardinal.{u}),\n  Combinatorics.cardinalPartitionRel μ r 1 ν ↔ μ ≥ ν Erdos1167.i0","subjects":["5"],"theorem":"Erdos1167.cardinalPartitionRel_one"},{"answerKinds":[],"category":"test","docstring":"The `infinite_targets` variant without the bound $\\kappa_\\alpha \\leq \\lambda$ is false.\nTaking $\\gamma = 1$, $\\kappa_0 = 2^{\\aleph_0}$, $\\lambda = \\aleph_0$: the premise\n$2^{\\aleph_0} \\to (2^{\\aleph_0})^3_1$ holds since $2^{\\aleph_0} \\leq 2^{\\aleph_0}$, but the\nconclusion $\\aleph_0 \\to (2^{\\aleph_0})^2_1$ fails by Cantor's theorem.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1167»","statement":"¬∀ (r : ℕ),\n    2 ≤ r →\n      ∀ (lam : Cardinal.{u}),\n        Cardinal.aleph0 ≤ lam →\n          ∀ (γ : Ordinal.{u}) (κ : γ.ToType → Cardinal.{u}),\n            (∀ (i : γ.ToType), Cardinal.aleph0 ≤ κ i) →\n              Combinatorics.cardinalPartitionRel (2 ^ lam) (r + 1) γ κ → Combinatorics.cardinalPartitionRel lam r γ κ","subjects":["5"],"theorem":"Erdos1167.erdos_1167.variants.infinite_targets_needs_bound"},{"answerKinds":[],"category":"research open","docstring":"**Infinite-target case.** When all $\\kappa_\\alpha \\geq \\aleph_0$ are infinite and bounded by\n$\\lambda$, $\\kappa_\\alpha + 1 = \\kappa_\\alpha$, so the hypothesis simplifies to a \"pure\"\nstepping-down lemma:\n$$2^\\lambda \\to (\\kappa_\\alpha)_{\\alpha<\\gamma}^{r+1} \\implies\n   \\lambda \\to (\\kappa_\\alpha)_{\\alpha<\\gamma}^r.$$\nThe condition $\\kappa_\\alpha \\leq \\lambda$ is needed to avoid a size obstruction: without it, the\nconclusion would require a subset of $\\lambda$ of size $\\kappa_\\alpha > \\lambda$, which is impossible\n(see `infinite_targets_needs_bound`).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1167»","statement":"∀ (r : ℕ),\n  2 ≤ r →\n    ∀ (lam : Cardinal.{u}),\n      Cardinal.aleph0 ≤ lam →\n        ∀ (γ : Ordinal.{u}),\n          2 ≤ γ →\n            ∀ (κ : γ.ToType → Cardinal.{u}),\n              (∀ (i : γ.ToType), Cardinal.aleph0 ≤ κ i) →\n                (∀ (i : γ.ToType), κ i ≤ lam) →\n                  Combinatorics.cardinalPartitionRel (2 ^ lam) (r + 1) γ κ →\n                    Combinatorics.cardinalPartitionRel lam r γ κ","subjects":["5"],"theorem":"Erdos1167.erdos_1167.variants.infinite_targets"},{"answerKinds":[],"category":"research solved","docstring":"A machine-checked **barrier** for Erdős 307 (Bonfioli, 2026): any solution with `Q` nonempty uses at\nleast 59 primes in total, and `(∏_{p ∈ P} p)² ≥ 4·10¹¹²` — i.e. `∏_{p ∈ P} p ≥ 2·10⁵⁶` (and, by\nsymmetry, the same for `∏ Q`); so no solution lies below a prime-product of `2.09·10⁵⁶`. The full\n`sorry`-free proof is in the linked repository (`Closed.lean`): the left conjunct is `card_ge_59`,\nthe right is `erdos307_barrier_closed`. Both depend on `propext, Classical.choice, Quot.sound` and\nnothing further: the evaluation of the first 59 primes is kernel-checked by `decide`, so no\n`native_decide` enters this statement.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/ElVec1o/erdos307/blob/76d242b024102f32d8411c714be4ad140a8b7c4b/lean/Erdos307/Closed.lean#L121"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«307»","statement":"∀ {P Q : Finset ℕ},\n  (∀ p ∈ P, Nat.Prime p) →\n    (∀ q ∈ Q, Nat.Prime q) →\n      Q.Nonempty → 1 = (∑ p ∈ P, (↑p)⁻¹) * ∑ q ∈ Q, (↑q)⁻¹ → 59 ≤ (P ∪ Q).card ∧ 4 * 10 ^ 112 ≤ (∏ p ∈ P, ↑p) ^ 2","subjects":["11"],"theorem":"Erdos307.erdos_307.barrier"},{"answerKinds":["Prop"],"category":"textbook","docstring":"Instead of asking for sets of primes, ask only that all elements in the sets be relatively coprime.\n\nCambie has found several examples when this weakened version is true. For example,\n$$\n1=\\left(1+\\frac{1}{5}\\right)\\left(\\frac{1}{2}+\\frac{1}{3}\\right)\n$$\nand\n$$\n1=\\left(1+\\frac{1}{41}\\right)\\left(\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{7}\\right).\n$$\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«307»","statement":"True ↔\n  ∃ P Q,\n    0 ∉ P ∩ Q ∧\n      1 < P.card ∧\n        1 < Q.card ∧ (↑P).Pairwise Nat.Coprime ∧ (↑Q).Pairwise Nat.Coprime ∧ 1 = (∑ p ∈ P, (↑p)⁻¹) * ∑ q ∈ Q, (↑q)⁻¹","subjects":["5","11"],"theorem":"Erdos307.erdos_307.variants.coprime"},{"answerKinds":[],"category":"research open","docstring":"Are there two finite set of primes $P$ and $Q$ such that\n\n$$\n1 = \\left( \\sum_{p \\in P} \\frac{1}{p} \\right) \\left( \\sum_{q \\in Q} \\frac{1}{q} \\right)\n$$\n?\n\nAsked by Barbeau [Ba76].\n\n[Ba76] Barbeau, E. J., _Computer challenge corner: Problem 477: A brute force program._\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«307»","statement":"True ↔ ∃ P Q, (∀ p ∈ P, Nat.Prime p) ∧ (∀ q ∈ Q, Nat.Prime q) ∧ 1 = (∑ p ∈ P, (↑p)⁻¹) * ∑ q ∈ Q, (↑q)⁻¹","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos307.erdos_307"},{"answerKinds":[],"category":"research open","docstring":"There are no examples known of the weakened coprime version if we insist that $1\\not\\in P\\cup Q$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«307»","statement":"True ↔\n  ∃ P Q,\n    0 ∉ P ∩ Q ∧\n      1 ∉ P ∪ Q ∧\n        1 < P.card ∧\n          1 < Q.card ∧ (↑P).Pairwise Nat.Coprime ∧ (↑Q).Pairwise Nat.Coprime ∧ 1 = (∑ p ∈ P, (↑p)⁻¹) * ∑ q ∈ Q, (↑q)⁻¹","subjects":["5","11"],"theorem":"Erdos307.erdos_307.variants.coprime_one_notMem"},{"answerKinds":[],"category":"research open","docstring":"**Erdős problem 681.**\nIs it true that for all large $n$ there exists $k$\nsuch that $n + k$ is composite and $p(n+k) > k^2$,\nwhere $p(m)$ is the least prime factor of $m$ ?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«681»","statement":"True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∃ k > 0, (n + k).Composite ∧ ∀ (p : ℕ), Erdos681.IsLPF p (n + k) → p > k ^ 2","subjects":["11"],"theorem":"Erdos681.erdos_681"},{"answerKinds":[],"category":"textbook","docstring":"A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nShow that for each $r$ there exists a prime $p$ of class $r$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1055»","statement":"∀ (r : ℕ+), ∃ p, Nat.Prime p ∧ Erdos1055.IsOfClass r p","subjects":["11"],"theorem":"Erdos1055.exists_p"},{"answerKinds":[],"category":"research open","docstring":"A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nIf $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?\nSelfridge conjectured that this is bounded. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1055»","statement":"∃ M, ∀ (r : ℕ+), ↑(Erdos1055.p r) ^ (1 / ↑↑r) ≤ M","subjects":["11"],"theorem":"Erdos1055.erdos_1055.variants.selfridge_limit"},{"answerKinds":[],"category":"research open","docstring":"A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nAre there infinitely many primes in each class?","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1055»","statement":"∀ (r : ℕ+), {p | Nat.Prime p ∧ Erdos1055.IsOfClass r p}.Infinite","subjects":["11"],"theorem":"Erdos1055.erdos_1055"},{"answerKinds":[],"category":"research open","docstring":"A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are\n$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor\nof $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nIf $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?\nErdos conjectured that this tends to infinity. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1055»","statement":"Filter.Tendsto (fun r => ↑(Erdos1055.p r) ^ (1 / ↑↑r)) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos1055.erdos_1055.variants.erdos_limit"},{"answerKinds":[],"category":"research solved","docstring":"If $A,B,C\\in \\mathbb{R}^2$ form a triangle and $P$ is a point in the interior then, if $N$ is\nwhere the perpendicular from $P$ to $AB$ meets the triangle, and similarly for $M$ and $L$,\n$$\n\\overline{PA}+\\overline{PB}+\\overline{PC}\\geq 2(\\overline{PM}+\\overline{PN}+\\overline{PL}).\n$$\n\nConjectured by Erdős in 1932 (according to [Er82e]) and proved by Mordell soon afterwards, now\nknown as the Erdős-Mordell inequality.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos898.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«898»","statement":"∀ (A B C P L M N : EuclideanSpace ℝ (Fin 2)),\n  AffineIndependent ℝ ![A, B, C] →\n    P ∈ interior ((convexHull ℝ) {A, B, C}) →\n      N ∈ line[ℝ, A, B] →\n        line[ℝ, P, N].direction ⟂ line[ℝ, A, B].direction →\n          M ∈ line[ℝ, B, C] →\n            line[ℝ, P, M].direction ⟂ line[ℝ, B, C].direction →\n              L ∈ line[ℝ, C, A] →\n                line[ℝ, P, L].direction ⟂ line[ℝ, C, A].direction →\n                  dist P A + dist P B + dist P C ≥ 2 * (dist P M + dist P N + dist P L)","subjects":["51"],"theorem":"Erdos898.erdos_898"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a graph on $n$ vertices which does not contain a trivial (empty or complete) graph on\nmore than $c\\log n$ vertices. Must $G$ contain at least $2^{\\Omega_c(n)}$ many induced subgraphs\nwhich are not pairwise isomorphic?\n\nThis is true, and was proved by Shelah [Sh98].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1036.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1036»","statement":"True ↔\n  ∀ (c : ℝ),\n    0 < c →\n      ∃ δ,\n        0 < δ ∧\n          ∀ᶠ (n : ℕ) in Filter.atTop,\n            ∀ (G : SimpleGraph (Fin n)),\n              ↑G.cliqueNum ≤ c * Real.log ↑n →\n                ↑G.indepNum ≤ c * Real.log ↑n → Erdos1036.HasManyNonIsomorphicInducedSubgraphs G (2 ^ (δ * ↑n))","subjects":["5"],"theorem":"Erdos1036.erdos_1036"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Hajnal [ErHa89b] proved that if $G$ does not contain a complete bipartite graph or its\ncomplement on more than $c\\log n$ vertices then $G$ contains at least $2^{\\Omega_c(n)}$ many\nnon-isomorphic induced subgraphs.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1036»","statement":"∀ (c : ℝ),\n  0 < c →\n    ∃ δ,\n      0 < δ ∧\n        ∀ᶠ (n : ℕ) in Filter.atTop,\n          ∀ (G : SimpleGraph (Fin n)),\n            Erdos1036.NoLargeInducedBipartite G (c * Real.log ↑n) →\n              Erdos1036.HasManyNonIsomorphicInducedSubgraphs G (2 ^ (δ * ↑n))","subjects":["5"],"theorem":"Erdos1036.erdos_1036.variants.erdos_hajnal"},{"answerKinds":[],"category":"research solved","docstring":"Alon and Hajnal [AlHa91] proved that $G$ must contain at least\n$$\\exp\\left(n(\\log n)^{-O(\\log\\log n)}\\right)$$\nmany non-isomorphic induced subgraphs.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1036»","statement":"∀ (c : ℝ),\n  0 < c →\n    ∃ C,\n      0 < C ∧\n        ∀ᶠ (n : ℕ) in Filter.atTop,\n          ∀ (G : SimpleGraph (Fin n)),\n            ↑G.cliqueNum ≤ c * Real.log ↑n →\n              ↑G.indepNum ≤ c * Real.log ↑n →\n                Erdos1036.HasManyNonIsomorphicInducedSubgraphs G\n                  (Real.exp (↑n * Real.log ↑n ^ (-(C * Real.log (Real.log ↑n)))))","subjects":["5"],"theorem":"Erdos1036.erdos_1036.variants.alon_hajnal"},{"answerKinds":[],"category":"research open","docstring":"If $A \\subset \\mathbb{N}$ has $\\sum_{n \\in A}\\frac 1 n = \\infty$, then must $A$ contain arbitrarily\nlong arithmetic progressions?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«3»","statement":"True ↔ ∀ (A : Set ℕ), (¬Summable fun a => 1 / ↑↑a) → ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength ↑k","subjects":["11"],"theorem":"Erdos3.erdos_3"},{"answerKinds":[],"category":"research open","docstring":"Fix a $k \\geq 3$. Is it true that there are infinitely many arithmetic prime progressions of length $k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«141»","statement":"True ↔ ∀ k ≥ 3, (Erdos141.consecutivePrimeArithmeticProgressions k).Infinite","subjects":["5","11"],"theorem":"Erdos141.erdos_141.variants.infinite_general_case"},{"answerKinds":[],"category":"test","docstring":"The first three odd primes are an example of three consecutive primes.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«141»","statement":"Erdos141.Set.IsPrimeProgressionOfLength {3, 5, 7} 3","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos141.first_three_odd_primes"},{"answerKinds":[],"category":"research solved","docstring":"The existence of such progressions has been verified for $k≤10$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«141»","statement":"∀ k ≥ 3, k ≤ 10 → ∃ s, Erdos141.Set.IsAPAndPrimeProgressionOfLength s k","subjects":["5","11"],"theorem":"Erdos141.erdos_141.variants.first_cases"},{"answerKinds":[],"category":"research open","docstring":"It is open, even for $k=3$, whether there are infinitely many such progressions.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«141»","statement":"True ↔ (Erdos141.consecutivePrimeArithmeticProgressions 3).Infinite","subjects":["5","11"],"theorem":"Erdos141.erdos_141.variants.infinite_three"},{"answerKinds":[],"category":"research open","docstring":"Are there $11$ consecutive primes in arithmetic progression?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«141»","statement":"True ↔ ∃ s, Erdos141.Set.IsAPAndPrimeProgressionOfLength s 11","subjects":["5","11"],"theorem":"Erdos141.erdos_141.variants.eleven"},{"answerKinds":[],"category":"test","docstring":"There are 3 consecutive primes in arithmetic progression.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«141»","statement":"∃ s, Erdos141.Set.IsAPAndPrimeProgressionOfLength s 3","subjects":["5","11"],"theorem":"Erdos141.exists_three_consecutive_primes_in_ap"},{"answerKinds":[],"category":"research open","docstring":"Let $k≥3$. Are there $k$ consecutive primes in arithmetic progression?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«141»","statement":"True ↔ ∀ k ≥ 3, ∃ s, Erdos141.Set.IsAPAndPrimeProgressionOfLength s k","subjects":["5","11"],"theorem":"Erdos141.erdos_141"},{"answerKinds":[],"category":"research solved","docstring":"**Stiebitz's theorem** [St85]: every graph in $M_r$ has chromatic number at least $r$. The\nmatching upper bound follows from the construction, so the chromatic number is exactly $r$.\n\nThis is stated ahead of `erdos_750` because the formal proof linked there assumes it, and the\n`assuming` clause must name a declaration that already exists. See also [SaSt89] and [MuSt19].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«750»","statement":"∀ {V : Type u} (G : SimpleGraph V) (r : ℕ), Erdos750.IsRecursivelyBuiltMr r G → ↑r ≤ G.chromaticNumber","subjects":["5"],"theorem":"Erdos750.erdos_750.variants.stiebitz"},{"answerKinds":["Prop"],"category":"research solved","docstring":"In [ErHa67b] Erdős and Hajnal prove this for $f(m)\\geq cm$ for all $c>1/4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«750»","statement":"True ↔\n  ∀ c > 1 / 4,\n    ∃ V G,\n      G.chromaticNumber = ⊤ ∧\n        ∀ (m : ℕ) (S : Set V), 0 < m → S.ncard = m → ∃ I ⊆ S, G.IsIndepSet I ∧ ↑m / 2 - c * ↑m ≤ ↑I.ncard","subjects":["5"],"theorem":"Erdos750.erdos_750.variants.c_gt_quarter"},{"answerKinds":["Prop"],"category":"research solved","docstring":"In [Er69b] Erdős conjectures this for $f(m)=\\epsilon m$ for any fixed $\\epsilon>0$. This follows\nfrom a result of Erdős, Hajnal, and Szemerédi [EHS82], as described by Sellke in the comments.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«750»","statement":"True ↔\n  ∀ ε > 0,\n    ∃ V G,\n      G.chromaticNumber = ⊤ ∧\n        ∀ (m : ℕ) (S : Set V), 0 < m → S.ncard = m → ∃ I ⊆ S, G.IsIndepSet I ∧ ↑m / 2 - ε * ↑m ≤ ↑I.ncard","subjects":["5"],"theorem":"Erdos750.erdos_750.variants.epsilon"},{"answerKinds":[],"category":"test","docstring":"Two level-`0` vertices of $M_s(G)$ are adjacent exactly when the underlying vertices are. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«750»","statement":"∀ {V : Type u} (G : SimpleGraph V) {s : ℕ} (hs : 0 < s) {u v : V},\n  G.Adj u v → (Erdos750.genMyc s G).Adj (Sum.inl (⟨0, hs⟩, u)) (Sum.inl (⟨0, hs⟩, v))","subjects":["5"],"theorem":"Erdos750.genMyc_adj_level_zero"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f(m)$ be some function such that $f(m)\\to \\infty$ as $m\\to \\infty$. Does there exist a\ngraph $G$ of infinite chromatic number such that every subgraph on $m$ vertices contains\nan independent set of size at least $\\frac{m}{2}-f(m)$?\n\nNote that in [Er94b] the function $f$ generalises a (proven) result for $f(m) = \\epsilon m$,\nwhere $\\epsilon > 0$. Hence we should assume it is non-negative valued.\n\nThe existence of such a graph was proved [UlamErdos750] by GPT 5.5 Pro (prompted by Chojecki).\nIndeed, this constructs a graph with infinite chromatic number such that every subgraph on $m$\nvertices can be made bipartite after deleting at most $f(m)$ many vertices.\n\nThis was formalized in Lean by Ammanamanchi using Claude Code 4.7 and GPT-5.5 Pro.\n\nThe linked proof is not complete on its own. It declares Stiebitz's theorem as an axiom and\nderives the result from it, so it is marked `conditional` and names\n`erdos_750.variants.stiebitz`.\n","formalProofs":[{"conditions":["Erdos750.erdos_750.variants.stiebitz"],"kind":"lean4","link":"https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P750/Proof.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«750»","statement":"True ↔\n  ∀ (f : ℕ → NNReal),\n    Filter.Tendsto f Filter.atTop Filter.atTop →\n      ∃ V G,\n        G.chromaticNumber = ⊤ ∧\n          ∀ (m : ℕ) (S : Set V), 0 < m → S.ncard = m → ∃ I ⊆ S, G.IsIndepSet I ∧ ↑m / 2 - f m ≤ ↑I.ncard","subjects":["5"],"theorem":"Erdos750.erdos_750"},{"answerKinds":[],"category":"test","docstring":"The apex of $M_s(G)$ is adjacent to every vertex on the top level. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«750»","statement":"∀ {V : Type u} (G : SimpleGraph V) (s : ℕ) (hs : 0 < s) (v : V),\n  (Erdos750.genMyc s G).Adj (Sum.inr ()) (Sum.inl (⟨s - 1, ⋯⟩, v))","subjects":["5"],"theorem":"Erdos750.genMyc_apex_adj_top"},{"answerKinds":[],"category":"research open","docstring":"Let $A=\\{a_1 < \\cdots < a_k\\}$ be a finite set of integers and extend it to an infinite\nsequence $\\overline{A}=\\{a_1 < a_2 < \\cdots \\}$ by defining $a_{n+1}$ for $n \\geq k$ to be\nthe least integer exceeding $a_n$ which is not of the form $a_i + a_j$ with $i,j \\leq n$.\nIs it true that the sequence of differences $a_{m+1}-a_m$ is eventually periodic?\n\nThis problem is discussed under Problem 7 on Green's open problems list.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«341»","statement":"True ↔\n  ∀ (a : ℕ → ℤ),\n    (∀ᶠ (n : ℕ) in Filter.atTop, IsLeast {x | a n < x ∧ x ∉ {x | ∃ i ≤ n, ∃ j ≤ n, a i + a j = x}} (a (n + 1))) →\n      have d := fun i => a (i + 1) - a i;\n      ∃ p > 0, ∀ᶠ (m : ℕ) in Filter.atTop, d (m + p) = d m","subjects":["11"],"theorem":"Erdos341.erdos_341"},{"answerKinds":[],"category":"research solved","docstring":"He could not even decide whether $\\epsilon_m\\to 0$ as $m\\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1190»","statement":"Filter.Tendsto Erdos1190.eps Filter.atTop (nhds 0)","subjects":["5","11"],"theorem":"Erdos1190.erdos_1190.variants.tendsto_zero"},{"answerKinds":[],"category":"research solved","docstring":"The work of de la Bretèche, Ford, and Vandehey [BFV13] implies\n$$L(m)^{-1+o(1)}< \\epsilon_m < L(m)^{-\\sqrt{3}/2+o(1)},$$\nwhere $L(m)=\\exp(\\sqrt{\\log m\\log\\log m})$. The lower bound is implicit in their construction.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1190»","statement":"∀ (ε : ℝ), 0 < ε → ∀ᶠ (m : ℕ) in Filter.atTop, scaleL m ^ (-1 - ε) < Erdos1190.eps m","subjects":["5","11"],"theorem":"Erdos1190.erdos_1190.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Let\n$$\\epsilon_m=\\max \\sum \\frac{1}{n_i}$$\nwhere the maximum is taken over all finite sequences $m<n_1<\\cdots<n_k$ for which there exist\ncongruences $a_i\\pmod{n_i}$ such that no integer satisfies two such congruences.\n\nEstimate $\\epsilon_m$.\n\nThe resolution of [202] by GPT-5.4 Pro implies via the same reduction that\n$$\\epsilon_m=L(m)^{-1+o(1)},$$\nwhere $L(m)=\\exp(\\sqrt{\\log m\\log\\log m})$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1190.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1190»","statement":"∀ (ε : ℝ),\n  0 < ε → ∀ᶠ (m : ℕ) in Filter.atTop, scaleL m ^ (-1 - ε) < Erdos1190.eps m ∧ Erdos1190.eps m < scaleL m ^ (-1 + ε)","subjects":["5","11"],"theorem":"Erdos1190.erdos_1190"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er80] seems to credit Mirsky and Newman with the result that $\\epsilon_m<1$, but gives no\nreference.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1190»","statement":"∀ (m : ℕ),\n  1 ≤ m →\n    ∀ (S : Finset ℕ) (a : ℕ → ℤ),\n      (∀ n ∈ S, m < n) →\n        (∀ n ∈ S, ∀ n' ∈ S, n ≠ n' → ¬∃ x, x ≡ a n [ZMOD ↑n] ∧ x ≡ a n' [ZMOD ↑n']) → ∑ n ∈ S, (↑n)⁻¹ < 1","subjects":["5","11"],"theorem":"Erdos1190.erdos_1190.variants.lt_one"},{"answerKinds":[],"category":"research solved","docstring":"The work of de la Bretèche, Ford, and Vandehey [BFV13] implies\n$$L(m)^{-1+o(1)}< \\epsilon_m < L(m)^{-\\sqrt{3}/2+o(1)},$$\nwhere $L(m)=\\exp(\\sqrt{\\log m\\log\\log m})$. The upper bound follows immediately from their upper\nbound as reported in [202] and partial summation.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1190»","statement":"∀ (ε : ℝ), 0 < ε → ∀ᶠ (m : ℕ) in Filter.atTop, Erdos1190.eps m < scaleL m ^ (-(√3 / 2) + ε)","subjects":["5","11"],"theorem":"Erdos1190.erdos_1190.variants.upper_bound"},{"answerKinds":[],"category":"research open","docstring":"Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two\nintegers $p_r < n < p_{r+1}$ all of whose prime factors are $< p_{r + 1} - p_r$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«932»","statement":"{r |\n    2 ≤\n      {m ∈ Finset.Ioo (Nat.nth Nat.Prime r) (Nat.nth Nat.Prime r.succ) |\n          m.maxPrimeFac < Nat.nth Nat.Prime r.succ - Nat.nth Nat.Prime r}.card}.Infinite","subjects":["11"],"theorem":"Erdos932.erdos_932"},{"answerKinds":[],"category":"research solved","docstring":"Erdős could show that the density of $r$ such that at least one such $n$ exists is $0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«932»","statement":"{r |\n      1 ≤\n        {m ∈ Finset.Ioo (Nat.nth Nat.Prime r) (Nat.nth Nat.Prime r.succ) |\n            m.maxPrimeFac < Nat.nth Nat.Prime r.succ - Nat.nth Nat.Prime r}.card}.HasDensity\n  0","subjects":["11"],"theorem":"Erdos932.erdos_932.variants.one_le"},{"answerKinds":[],"category":"research open","docstring":"Is there an infinite Lucas sequence $a_0, a_1, \\ldots$ where $a_{n+2} = a_{n+1} + a_n$ for\n$n \\ge 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every\nterm of the sequence?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«276»","statement":"True ↔ ∃ a, Erdos276.IsLucasSequence a ∧ (∀ (k : ℕ), (a k).Composite) ∧ ∀ n > 1, ∃ k, n.gcd (a k) = 1","subjects":["11"],"theorem":"Erdos276.erdos_276"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $a\\geq 1$. Must there exist some $b>a$ such that\n$$\\sum_{a\\leq n\\leq b}\\frac{1}{n}=\\frac{r_1}{s_1}\\textrm{ and }\n\\sum_{a\\leq n\\leq b+1}\\frac{1}{n}=\\frac{r_2}{s_2},$$\nwith $(r_i,s_i)=1$ and $s_2<s_1$? If so, how does this $b(a)$ grow with $a$?\n\nThis was resolved in the affirmative by van Doorn [vD24], who proved $b=b(a)$ always exists, and in\nfact $b(a) \\ll a$. Indeed, if $a\\in (3^k,3^{k+1}]$ then one can take $b=2\\cdot 3^{k+1}-1$. van Doorn\nalso proves that $b(a)>a+(1/2-o(1))\\log a$, and considers various generalisations of the original\nproblem.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem290.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«290»","statement":"True ↔ ∀ (a : ℕ), 1 ≤ a → ∃ b, a < b ∧ Erdos290.harmonicDen a (b + 1) < Erdos290.harmonicDen a b","subjects":["11"],"theorem":"Erdos290.erdos_290"},{"answerKinds":[],"category":"research open","docstring":"Suppose monotone sequence $A$ satisfies the following: `A 0 = 1` and for all `j`, `A (j + 1)` is the\nsmallest natural number that cannot be written as a sum of consecutive terms of `A 0, ..., A j`.\nThen it is conjectured that $$a_k ~ \\frac{k \\log k}{\\log \\log k}$$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«359»","statement":"∀ (A : ℕ → ℕ),\n  Erdos359.IsGoodFor A 1 →\n    Asymptotics.IsEquivalent Filter.atTop (fun k => ↑(A k)) fun k => ↑k * Real.log ↑k / Real.log (Real.log ↑k)","subjects":["11"],"theorem":"Erdos359.erdos_359.variants.isGoodFor_1_asymptotic"},{"answerKinds":[],"category":"research open","docstring":"Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the\nleast integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k ^ {1 + c} \\to 0$\nfor any $c > 0$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«359»","statement":"∀ (A : ℕ → ℕ),\n  Erdos359.IsGoodFor A 1 → ∀ (c : ℝ), 0 < c → Filter.Tendsto (fun k => ↑(A k) / ↑k ^ (1 + c)) Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Erdos359.erdos_359.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the\nleast integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k \\to \\infty$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«359»","statement":"∀ (A : ℕ → ℕ), Erdos359.IsGoodFor A 1 → Filter.Tendsto (fun k => ↑(A k) / ↑k) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos359.erdos_359.parts.i"},{"answerKinds":[],"category":"test","docstring":"Suppose monotone sequence $A$ satisfies the following: `A 0 = 1` and for all `j`, `A (j + 1)` is the\nsmallest natural number that cannot be written as a sum of consecutive terms of `A 0, ..., A j`.\nThen the first few terms of $A$ are $1,2,4,5,8,10,14,15,...$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«359»","statement":"∀ (A : ℕ → ℕ), Erdos359.IsGoodFor A 1 → A '' Set.Iic 7 = {1, 2, 4, 5, 8, 10, 14, 15}","subjects":["11"],"theorem":"Erdos359.erdos_359.variants.isGoodFor_1_low_values"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there a set of $n$ points in $\\mathbb{R}^2$ such that every subset of $4$ points determines at\nleast $3$ distances, yet the total number of distinct distances is $\\ll \\frac{n}{\\sqrt{\\log n}}$?\n\nThere does exist such a set: a suitable truncation of the lattice\n$\\{(a,b\\sqrt{2}): a,b\\in\\mathbb{Z}\\}$ suffices. This construction appears to have been first\nconsidered by Moree and Osburn \\cite{MoOs06}, who proved that it has\n $\\ll \\frac{n}{\\sqrt{\\log n}}$ many distinct distances. This construction was independently found by\n [Lund and Sheffer](https://adamsheffer.wordpress.com/2014/07/16/point-sets-with-few-distinct-distances/),\n who further noted that this configuration contains no squares or equilateral triangles.\n\nThere are only six possible configurations of $4$ points which determine only $2$ distances\n(first noted by Erdős and Fishburn [ErFi96]), and five of them contain either a square or an\nequilateral triangle. The remaining configuration contains four points from a regular pentagon,\nand Grayzel [Gr26] (using Gemini) has noted in the comments that this configuration can also be\nruled out, thus giving a complete solution to this problem.\n\nBoris Alexeev provides a formalisation of the reduction, which is conditional on Bernays' theorem\n(assumed as an axiom in the proof to obtain the $O(n/\\sqrt{\\log n})$ bound).\nSee the [formal proof](https://github.com/plby/lean-proofs/blob/226d5fad7143dcebea2bbb5ec87f18a3a1dcea69/src/v4.24.0/ErdosProblems/Erdos659.lean).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«659»","statement":"True ↔\n  ∃ A,\n    (∀ (n : ℕ), (A n).card = n ∧ ∀ S ⊆ A n, S.card = 4 → 3 ≤ distinctDistances S) ∧\n      (fun n => ↑(distinctDistances (A n))) =O[Filter.atTop] fun n => ↑n / √(Real.log ↑n)","subjects":["52"],"theorem":"Erdos659.erdos_659"},{"answerKinds":[],"category":"research open","docstring":"If $n$ distinct points in $\\mathbb{R}^2$ form a convex polygon then some vertex has at least\n$\\lfloor\\frac{n}{2}\\rfloor$ different distances to other vertices.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«982»","statement":"∀ (n : ℕ),\n  3 ≤ n →\n    ∀ (p : Fin n → EuclideanSpace ℝ (Fin 2)),\n      Function.Injective p →\n        EuclideanGeometry.IsConvexPolygon p → ∃ i, {d | ∃ j, j ≠ i ∧ d = dist (p i) (p j)}.ncard ≥ n / 2","subjects":["52"],"theorem":"Erdos982.erdos_982"},{"answerKinds":[],"category":"research solved","docstring":"Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for quasi-line graphs.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«628»","statement":"∀ (V : Type u_1) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj],\n  G.IsQuasiLineGraph →\n    ∀ (k : ℕ),\n      G.chromaticNumber = ↑k →\n        G.CliqueFree k →\n          ∀ (a b : ℕ),\n            a ≥ 2 →\n              b ≥ 2 →\n                a + b = k + 1 →\n                  ∃ s, (SimpleGraph.induce s G).chromaticNumber ≥ ↑a ∧ (SimpleGraph.induce sᶜ G).chromaticNumber ≥ ↑b","subjects":["5"],"theorem":"Erdos628.erdos_628.variants.quasi_line"},{"answerKinds":[],"category":"research solved","docstring":"Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for graphs with independence number 2.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«628»","statement":"∀ (V : Type u_1) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj],\n  G.indepNum = 2 →\n    ∀ (k : ℕ),\n      G.chromaticNumber = ↑k →\n        G.CliqueFree k →\n          ∀ (a b : ℕ),\n            a ≥ 2 →\n              b ≥ 2 →\n                a + b = k + 1 →\n                  ∃ s, (SimpleGraph.induce s G).chromaticNumber ≥ ↑a ∧ (SimpleGraph.induce sᶜ G).chromaticNumber ≥ ↑b","subjects":["5"],"theorem":"Erdos628.erdos_628.variants.independence_number_2"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ be a graph with chromatic number $k$ containing no $K_k$. If $a,b\\geq 2$ and $a+b=k+1$\nthen must there exist two disjoint subgraphs of $G$ with chromatic numbers $\\geq a$ and $\\geq b$\nrespectively?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«628»","statement":"∀ (V : Type u_1) [Fintype V] (G : SimpleGraph V) (k : ℕ),\n  G.chromaticNumber = ↑k →\n    G.CliqueFree k →\n      ∀ (a b : ℕ),\n        a ≥ 2 →\n          b ≥ 2 →\n            a + b = k + 1 →\n              ∃ s, (SimpleGraph.induce s G).chromaticNumber ≥ ↑a ∧ (SimpleGraph.induce sᶜ G).chromaticNumber ≥ ↑b","subjects":["5"],"theorem":"Erdos628.erdos_628"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er68b] originally asked about $a=b=3$ which was proved by Brown and Jung [BrJu69] (who in fact prove that $G$ must contain two vertex disjoint odd cycles)..\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«628»","statement":"∀ (V : Type u_1) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj],\n  G.chromaticNumber = 5 →\n    G.CliqueFree 5 → ∃ s, (SimpleGraph.induce s G).chromaticNumber ≥ 3 ∧ (SimpleGraph.induce sᶜ G).chromaticNumber ≥ 3","subjects":["5"],"theorem":"Erdos628.erdos_628.variants.k_5_a_3_b_3"},{"answerKinds":[],"category":"research solved","docstring":"Let $g(n)$ denote the largest $t$ such that there exist integers $2\\leq a_1<a_2<\\cdots <a_t <n$\nsuch that $$P(a_1)>P(a_2)>\\cdots >P(a_t)$$ where $P(m)$ is the greatest prime factor of $m$.\nEstimate $g(n)$.\n\nStijn Cambie has proved [Ca25b] $$g(n) \\asymp \\left(\\frac{n}{\\log n}\\right)^{1/2}.$$\nCambie further asks whether there exists a constant $c$ such that\n$$g(n) \\sim c \\left(\\frac{n}{\\log n}\\right)^{1/2}.$$\nCambie's proof shows that such a $c$ must satisfy $2\\leq c\\leq 2\\sqrt{2}$.\n\nThe sequence $a_1<a_2<\\cdots<a_t$ is packaged as a strictly monotone map `a : Fin t → ℕ`\nwith $2\\leq a_i<n$, the greatest prime factor $P$ is `Nat.maxPrimeFac`, and $g(n)$ is the\nsupremum in `ℕ` of the achievable lengths $t$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos648.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«648»","statement":"(fun n => ↑(Erdos648.g n)) =Θ[Filter.atTop] fun n => √(↑n / Real.log ↑n)","subjects":["11"],"theorem":"Erdos648.erdos_648"},{"answerKinds":[],"category":"research open","docstring":"Let $k \\geq 3$. Define $g_k(n)$ to be the minimal $N$ such that\n$\\{1, ..., N\\}$ contains some $A$ of size $|A| = n$ such that\n$$\n  \\langle A\\rangle = \\left\\{\\sum_{a \\in A} \\epsilon_a a : \\epsilon_a \\in\\{0, 1\\}\\right\\}\n$$\ncontains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In\nparticular, is it true that\n$$\n  g_3(n) \\gg 3^n\n$$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«817»","statement":"True ↔ (fun n => 3 ^ n) =O[Filter.atTop] fun n => ↑(Erdos817.g 3 n)","subjects":["5","11"],"theorem":"Erdos817.erdos_817"},{"answerKinds":[],"category":"research solved","docstring":"A problem of Erdős and Sárközy who proved\n$$\n  g_3(n) \\gg \\frac{3^n}{n^{O(1)}}.\n$$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«817»","statement":"∃ O > 0, (fun n => 3 ^ n / ↑n ^ O) =O[Filter.atTop] fun n => ↑(Erdos817.g 3 n)","subjects":["5","11"],"theorem":"Erdos817.erdos_817.variants.bdd_power"},{"answerKinds":[],"category":"API","docstring":"Sanity check: if `S` is a maximal non ternary subset of `{1,..., N}` then `F N` is given by the\ncardinality of `S`\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«168»","statement":"∀ (N : ℕ),\n  ∀ S ⊆ Finset.Icc 1 N,\n    Erdos168.NonTernary S →\n      (∀ T ⊆ Finset.Icc 1 N, Erdos168.NonTernary T → S.card ≤ T.card → T.card = S.card) → Erdos168.F N = S.card","subjects":["5","11"],"theorem":"Erdos168.F_eq_card"},{"answerKinds":[],"category":"API","docstring":"Sanity check: elements of `IntervalNonTernarySets N` are precisely non ternary subsets of\n`{1,...,N}`\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«168»","statement":"∀ (N : ℕ) (S : Finset ℕ), S ∈ Erdos168.IntervalNonTernarySets N ↔ Erdos168.NonTernary S ∧ S ⊆ Finset.Icc 1 N","subjects":["5","11"],"theorem":"Erdos168.mem_IntervalNonTernarySets_iff"},{"answerKinds":[],"category":"research solved","docstring":"The limit $F(N)/N$ as $N \\to \\infty$ exists. (proved by Graham, Spencer, and Witsenhausen) ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«168»","statement":"∃ x, Filter.Tendsto (fun N => ↑(Erdos168.F N) / ↑N) Filter.atTop (nhds x)","subjects":["5","11"],"theorem":"Erdos168.erdos_168.variants.limit_exists"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«168»","statement":"Erdos168.F 0 = 0","subjects":["5","11"],"theorem":"Erdos168.F_0"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the limit $F(N)/N$ as $N \\to \\infty$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«168»","statement":"Filter.Tendsto (fun N => ↑(Erdos168.F N) / ↑N) Filter.atTop (nhds sorry)","subjects":["11"],"theorem":"Erdos168.erdos_168.parts.i"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«168»","statement":"Erdos168.F 3 = 2","subjects":["5","11"],"theorem":"Erdos168.F_3"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«168»","statement":"Erdos168.F 2 = 2","subjects":["5","11"],"theorem":"Erdos168.F_2"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«168»","statement":"Erdos168.F 1 = 1","subjects":["5","11"],"theorem":"Erdos168.F_1"},{"answerKinds":[],"category":"research open","docstring":"Is the limit $F(N)/N$ as $N \\to \\infty$ irrational? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«168»","statement":"True ↔ Irrational (Filter.limsup (fun N => ↑(Erdos168.F N) / ↑N) Filter.atTop)","subjects":["5","11"],"theorem":"Erdos168.erdos_168.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"It follows from the prime number theorem that such a progression has length $\\leq(1+o(1))\\log N$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«200»","statement":"∃ o, ∃ (_ : o =o[Filter.atTop] 1), ∀ (n : ℕ), ↑(Erdos200.longestPrimeArithmeticProgressions n) ≤ (1 + o n) * Real.log ↑n","subjects":["5","11"],"theorem":"Erdos200.erdos_200.variants.upper"},{"answerKinds":[],"category":"research open","docstring":"Does the longest arithmetic progression of primes in $\\{1,\\ldots,N\\}$ have length $o(\\log N)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«200»","statement":"True ↔ (fun n => ↑(Erdos200.longestPrimeArithmeticProgressions n)) =o[Filter.atTop] fun n => Real.log ↑n","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos200.erdos_200"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $f(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there are no\nsolutions to\n$$\\frac{1}{a}= \\frac{1}{b}+\\frac{1}{c}$$\nwith distinct $a,b,c\\in A$? Estimate $f(N)$.\n\nThe colouring version of this is [303], which was solved by Brown and Rödl [BrRo91].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«302»","statement":"∀ (f : ℕ → ℕ),\n  (∀ (N : ℕ), Erdos302.IsMaxNoTripleCard N (f N)) → Filter.Tendsto (fun N => ↑(f N) / ↑N) Filter.atTop (nhds sorry)","subjects":["11"],"theorem":"Erdos302.erdos_302.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"One can take either $A$ to be all odd integers in $[1,N]$ or all integers in $[N/2,N]$ to show\n$f(N)\\geq (1/2+o(1))N$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«302»","statement":"∀ (f : ℕ → ℕ),\n  (∀ (N : ℕ), Erdos302.IsMaxNoTripleCard N (f N)) →\n    ∀ (ε : ℝ), 0 < ε → ∀ᶠ (N : ℕ) in Filter.atTop, (1 / 2 - ε) * ↑N ≤ ↑(f N)","subjects":["11"],"theorem":"Erdos302.erdos_302.variants.lower_half"},{"answerKinds":[],"category":"research solved","docstring":"Wouter van Doorn has proved [va25] that\n$$f(N) \\leq (9/10+o(1))N.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«302»","statement":"∀ (f : ℕ → ℕ),\n  (∀ (N : ℕ), Erdos302.IsMaxNoTripleCard N (f N)) →\n    ∀ (ε : ℝ), 0 < ε → ∀ᶠ (N : ℕ) in Filter.atTop, ↑(f N) ≤ (9 / 10 + ε) * ↑N","subjects":["11"],"theorem":"Erdos302.erdos_302.variants.upper_nine_tenths"},{"answerKinds":[],"category":"research solved","docstring":"In particular, is $f(N)=(\\tfrac{1}{2}+o(1))N$?\n\nThis is false: it is contradicted by Cambie's lower bound of $(5/8+o(1))N$ recorded below,\nsince $5/8 > 1/2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«302»","statement":"∀ (f : ℕ → ℕ),\n  (∀ (N : ℕ), Erdos302.IsMaxNoTripleCard N (f N)) → ¬Filter.Tendsto (fun N => ↑(f N) / ↑N) Filter.atTop (nhds (1 / 2))","subjects":["11"],"theorem":"Erdos302.erdos_302.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Stijn Cambie has observed that\n$$f(N)\\geq (5/8+o(1))N,$$\ntaking $A$ to be all odd integers $\\leq N/4$ and all integers in $[N/2,N]$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«302»","statement":"∀ (f : ℕ → ℕ),\n  (∀ (N : ℕ), Erdos302.IsMaxNoTripleCard N (f N)) →\n    ∀ (ε : ℝ), 0 < ε → ∀ᶠ (N : ℕ) in Filter.atTop, (5 / 8 - ε) * ↑N ≤ ↑(f N)","subjects":["11"],"theorem":"Erdos302.erdos_302.variants.lower_five_eighths"},{"answerKinds":[],"category":"research solved","docstring":"Let $N\\geq 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,\\ldots,A_k\\subseteq \\{1,\\ldots,N\\}$ with $\\sum_{n\\in A_i}\\frac{1}{n}=1$ for all $i$. Estimate $k(N)$. Is it true that $k(N)=o(\\log N)$?\n\nHunter and Sawhney observed that Bloom's theorem [Bl21], together with the greedy\nargument, gives $k(N)=(1-o(1))\\log N$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos296.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«296»","statement":"(∀ (N k : ℕ), Erdos296.HasDisjointUnitDecomps N k → ↑k ≤ Erdos296.recipSum (Finset.Icc 1 N)) ∧\n  ∀ (ε : ℝ), 0 < ε → ε < 1 → ∀ᶠ (N : ℕ) in Filter.atTop, Erdos296.HasDisjointUnitDecomps N ⌊(1 - ε) * Real.log ↑N⌋₊","subjects":["11"],"theorem":"Erdos296.erdos_296"},{"answerKinds":[],"category":"research open","docstring":"Does there exists an entire non-zero transcendental function `f : ℂ → ℂ` such that for any\nsequence `n₀ < n₁ < ...`, `{ z | ∃ k, iteratedDeriv (n k) f z = 0 }` is dense. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«906»","statement":"True ↔\n  ∃ f,\n    Transcendental (Polynomial ℂ) f ∧\n      Differentiable ℂ f ∧ ∀ (n : ℕ → ℕ), StrictMono n → Dense {z | ∃ k, iteratedDeriv (n k) f z = 0}","subjects":["30"],"theorem":"Erdos906.erdos_906"},{"answerKinds":[],"category":"research open","docstring":"Hickerson conjectured the largest solution the equation `n!=a_1!a_2!···a_k!`, with\n`n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k`, is `16!=14!5!2!`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«373»","statement":"(16, [14, 5, 2]) ∈ Erdos373.S ∧ ∀ s ∈ Erdos373.S, s.1 ≤ 16","subjects":["11"],"theorem":"Erdos373.erdos_373.variants.maximal_solution"},{"answerKinds":[],"category":"research solved","docstring":"Show that if `P(n(n−1)) > 4 log n` for large enough `n`, where `P(m)` denotes the\nlargest prime factor of `m`, then the equation `n!=a_1!a_2!···a_k!`, with\n`n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k`, has only finitely many solutions.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«373»","statement":"(∀ᶠ (n : ℕ) in Filter.atTop, 4 * Real.log ↑n < ↑(n * (n - 1)).maxPrimeFac) → Erdos373.S.Finite","subjects":["11"],"theorem":"Erdos373.erdos_373.variants.of_lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Surányi was the first to conjecture that the only non-trivial solution to `a!b!=n!`\nis `6!7!=10!`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«373»","statement":"{(n, a, b) | n.factorial = a.factorial * b.factorial ∧ 1 < n ∧ 1 < a ∧ 1 < b ∧ b ≤ a ∧ a + 1 ≠ n} = {(10, 7, 6)}","subjects":["11"],"theorem":"Erdos373.erdos_373.variants.suranyi"},{"answerKinds":[],"category":"research open","docstring":"Show that the equation `n!=a_1!a_2!···a_k!`, with `n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k`, has\nonly finitely many solutions.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«373»","statement":"Erdos373.S.Finite","subjects":["11"],"theorem":"Erdos373.erdos_373"},{"answerKinds":[],"category":"research solved","docstring":"Show that if `P(n(n+1)) / log n → ∞` where `P(m)` denotes the largest prime factor of `m`, then\nthe equation `n!=a_1!a_2!···a_k!`, with `n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k`, has only\nfinitely many solutions.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«373»","statement":"Filter.Tendsto (fun n => ↑(n * (n + 1)).maxPrimeFac / Real.log ↑n) Filter.atTop Filter.atTop → Erdos373.S.Finite","subjects":["11"],"theorem":"Erdos373.erdos_373.variants.of_limit"},{"answerKinds":[],"category":"textbook","docstring":"Makowski [Ma83] found, for $k=3$:\n$2 * 3 * 4 * 5 \\equiv 6 * 7 * 8 * 9 * 10 * 11 \\equiv 12 * 13 * 14 * 15 \\equiv 1 \\mod 17$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1056»","statement":"Erdos1056.AllModProdEqualsOne 17 ![2, 6, 12, 16]","subjects":["11"],"theorem":"Erdos1056.erdos_1056.variants.k3"},{"answerKinds":[],"category":"textbook","docstring":"This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979\nErdős observed that $3 * 4 \\equiv 5 * 6 * 7 \\equiv 1 \\mod 11$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1056»","statement":"Erdos1056.AllModProdEqualsOne 11 ![3, 5, 8]","subjects":["11"],"theorem":"Erdos1056.erdos_1056.variants.k2"},{"answerKinds":[],"category":"research open","docstring":"Noll and Simmons asked, more generally, whether there are solutions to\n$q_1! \\equiv \\dots \\equiv q_k! \\mod p$ for arbitrarily large $k$ (with $q_1 < \\dots < q_k$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1056»","statement":"True ↔\n  ∀ᶠ (k : ℕ) in Filter.atTop,\n    ∃ p,\n      ∃ (_ : Nat.Prime p),\n        ∃ Q,\n          ∃ (_ : StrictMono Q) (_ : ∀ (i : Fin k), Q i < p), ∀ (i j : Fin k), (Q i).factorial ≡ (Q j).factorial [MOD p]","subjects":["11"],"theorem":"Erdos1056.erdos_1056.variants.noll_simmons"},{"answerKinds":[],"category":"research open","docstring":"Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\\dots,I_k$\nsuch that $\\prod\\limits_{n{\\in}I_i}n \\equiv 1 \\mod n$ for all $1 \\le i \\le k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1056»","statement":"True ↔\n  ∀ k ≥ 2,\n    ∃ p, ∃ (_ : Nat.Prime p), ∃ boundaries, ∃ (_ : StrictMono boundaries), Erdos1056.AllModProdEqualsOne p boundaries","subjects":["11"],"theorem":"Erdos1056.erdos_1056"},{"answerKinds":[],"category":"research open","docstring":"Is it true that for every $k \\geq 1$ we have\n$$\nF(N + k) \\leq F(N) + 1\n$$\nfor all sufficiently large $N$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«155»","statement":"True ↔ ∀ k ≥ 1, ∀ᶠ (N : ℕ) in Filter.atTop, Erdos155.F (N + k) ≤ Erdos155.F N + 1","subjects":["5"],"theorem":"Erdos155.erdos_155"},{"answerKinds":[],"category":"research open","docstring":"Let $p>q\\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of\nintegers of the form $p^kq^l$, none of which divide each other.\n\nIf $\\{p,q\\}\\neq \\{2,3\\}$ then what can be said about the density of non-representable\nnumbers? Are there infinitely many coprime non-representable numbers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1110»","statement":"True ↔\n  ∀ (p q : ℕ),\n    q < p → 2 ≤ q → p.Coprime q → ¬(p = 3 ∧ q = 2) → {n | n.Coprime (p * q) ∧ ¬Erdos1110.Representable p q n}.Infinite","subjects":["5","11"],"theorem":"Erdos1110.erdos_1110"},{"answerKinds":[],"category":"research solved","docstring":"A simpler case: the set of numbers of the form $2^k 3^l$ ($k, l ≥ 0$) is d-complete.\n\nThis was initially conjectured by Erdős in 1992, who called it a \"nice and difficult\"\nproblem, but it was quickly proven by Jansen and others using a simple inductive argument:\n- If $n = 2m$ is even, apply the inductive hypothesis to $m$ and double all summands.\n- If $n$ is odd, let $3^k$ be the largest power of $3$ with $3^k ≤ n$, and apply the\n  inductive hypothesis to $n - 3^k$ (which is even).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«123»","statement":"Erdos123.IsDComplete (↑(Submonoid.powers 2) * ↑(Submonoid.powers 3))","subjects":["11"],"theorem":"Erdos123.erdos_123.variants.powers_2_3"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $a, b, c$ be three integers which are pairwise coprime. Is every large integer\nthe sum of distinct integers of the form $a^k b^l c^m$ ($k, l, m ≥ 0$), none of which\ndivide any other?\n\nEquivalently: is the set $\\{a^k b^l c^m : k, l, m \\geq 0\\}$ d-complete?\n\nNote: For this not to reduce to the two-integer case, we need the integers\nto be greater than one and distinct.\n\nThe prize of \\$250 is offered by Erdős in [Er97] and [Er97e] for a 'proof or disproof'.\n\nThe main problem was resolved in the affirmative by GPT 5.6 (prompted by Snyder).\n\nThis was formalized in Lean by Alexeev.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/a28a04b6b8ce43d5260a7466677c1f23833bfc38/src/latest/ErdosProblems/Erdos123.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«123»","statement":"True ↔\n  ∀ a > 1,\n    ∀ b > 1,\n      ∀ c > 1,\n        Erdos123.PairwiseCoprime a b c →\n          Erdos123.IsDComplete (↑(Submonoid.powers a) * ↑(Submonoid.powers b) * ↑(Submonoid.powers c))","subjects":["11"],"theorem":"Erdos123.erdos_123"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Lewin [ErLe96] proved this conjecture when $a = 3$, $b = 5$, and $c = 7$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«123»","statement":"Erdos123.IsDComplete (↑(Submonoid.powers 3) * ↑(Submonoid.powers 5) * ↑(Submonoid.powers 7))","subjects":["11"],"theorem":"Erdos123.erdos_123.variants.erdos_lewin_3_5_7"},{"answerKinds":[],"category":"research open","docstring":"In [Er92b] Erdős makes the stronger conjecture (for $a=2$, $b=3$, and $c=5$) that, for any\n$\\epsilon>0$, all large integers $n$ can be written as the sum of distinct integers\n$b_1<\\cdots <b_t$ of the form $2^k3^l5^m$ where $b_t<(1+\\epsilon)b_1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«123»","statement":"True ↔\n  ∀ ε > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∃ A,\n        ↑A ⊆ ↑(Submonoid.powers 2) * ↑(Submonoid.powers 3) * ↑(Submonoid.powers 5) ∧\n          Erdos123.IsSnug ε A ∧ ∑ x ∈ A, x = n","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos123.erdos_123.variants.powers_2_3_5_snug"},{"answerKinds":[],"category":"research solved","docstring":"Romanoff [Ro34] showed that the set of odd integers of this form has positive density.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«16»","statement":"have S := {n | Odd n ∧ ∃ k p, Nat.Prime p ∧ n = 2 ^ k + p};\nErdos16.positive_lower_density S","subjects":["11"],"theorem":"Erdos16.erdos_16.variant.romanoff"},{"answerKinds":[],"category":"research solved","docstring":"Using covering congruences Erdős [Er50] proved that the set of odd integers which are not of this\nform contains an infinite arithmetic progression.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«16»","statement":"∃ a, ∃ d > 0, {x | ∃ m, x = a + m * d} ⊆ Erdos16.Erdos16Set","subjects":["11"],"theorem":"Erdos16.erdos_16.variant.erdos"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression\nand a set of density $0$?\n\nErdős called this conjecture \"rather silly\".\n\nChen [Ch23] has proved the answer is no.\n\nThis was formalized in Lean by Chin using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/danielchin/proofs/blob/main/Proofs/ErdosProblems/Erdos16.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«16»","statement":"False ↔ ∃ A B, Erdos16.Erdos16Set = A ∪ B ∧ (∃ a, ∃ d > 0, A = {x | ∃ m, x = a + m * d}) ∧ Erdos16.density_zero B","subjects":["11"],"theorem":"Erdos16.erdos_16"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Sárközy [ErSa87] proved the upper bound $f(N) \\ll N^{3/4}\\log N$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1109»","statement":"(fun N => ↑(Erdos1109.f N)) =O[Filter.atTop] fun N => ↑N ^ (3 / 4) * Real.log ↑N","subjects":["5","11"],"theorem":"Erdos1109.erdos_1109.variants.erdos_sarkozy_upper"},{"answerKinds":[],"category":"research open","docstring":"Is the stronger polylogarithmic bound $f(N) \\leq (\\log N)^{O(1)}$ true?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1109»","statement":"True ↔ ∃ C > 0, (fun N => ↑(Erdos1109.f N)) =O[Filter.atTop] fun N => Real.log ↑N ^ C","subjects":["5","11"],"theorem":"Erdos1109.erdos_1109.variants.polylog"},{"answerKinds":[],"category":"research open","docstring":"Let $f(N)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,N\\}$ such that\nevery $n\\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that\n$f(N)\\leq N^{o(1)}$, or even $f(N) \\leq (\\log N)^{O(1)}$?\n\nThis theorem formalizes the subpolynomial bound as `f(N) = O(N^ε)` for every `ε > 0`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1109»","statement":"True ↔ ∀ ε > 0, (fun N => ↑(Erdos1109.f N)) =O[Filter.atTop] fun N => ↑N ^ ε","subjects":["5","11"],"theorem":"Erdos1109.erdos_1109"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Sárközy [ErSa87] proved the lower bound $\\log N \\ll f(N)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1109»","statement":"(fun N => Real.log ↑N) =O[Filter.atTop] fun N => ↑(Erdos1109.f N)","subjects":["5","11"],"theorem":"Erdos1109.erdos_1109.variants.erdos_sarkozy_lower"},{"answerKinds":[],"category":"research solved","docstring":"Konyagin [Ko04] improved the lower bound to\n$\\log\\log N(\\log N)^2 \\ll f(N)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1109»","statement":"(fun N => Real.log (Real.log ↑N) * Real.log ↑N ^ 2) =O[Filter.atTop] fun N => ↑(Erdos1109.f N)","subjects":["5","11"],"theorem":"Erdos1109.erdos_1109.variants.konyagin_lower"},{"answerKinds":[],"category":"research solved","docstring":"Konyagin [Ko04] improved the upper bound to $f(N) \\ll N^{11/15+o(1)}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1109»","statement":"∀ ε > 0, (fun N => ↑(Erdos1109.f N)) =O[Filter.atTop] fun N => ↑N ^ (11 / 15 + ε)","subjects":["5","11"],"theorem":"Erdos1109.erdos_1109.variants.konyagin_upper"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there an entire non-linear function $f$ such that, for all $x\\in\\mathbb{R}$, $x$ is rational if and only if $f(x)$ is?\n\nBarth and Schneider [BaSc70] proved the stronger result for countable dense subsets of\n$\\mathbb{R}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos226.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«226»","statement":"True ↔\n  ∃ F,\n    Differentiable ℂ F ∧\n      (∀ (x : ℝ), (F ↑x).im = 0) ∧\n        (∀ (g : ℝ →ᵃ[ℝ] ℝ), (fun x => (F ↑x).re) ≠ ⇑g) ∧ Erdos226.PreservesRationality fun x => (F ↑x).re","subjects":["30"],"theorem":"Erdos226.erdos_226"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $(\\epsilon_k)_{k\\geq 0}$ be independently uniformly chosen at random from\n$\\{-1,1\\}$. If $R_n$ counts the number of real roots of\n$f_n(z)=\\sum_{0\\leq k\\leq n}\\epsilon_k z^k$ then is it true that, almost surely,\n$$\\lim_{n\\to \\infty}\\frac{R_n}{\\log n}=\\frac{2}{\\pi}?$$\n\nThe answer is no: this almost-sure limit fails. This result was obtained first by others, who\ndeserve the credit for the problem; the link is to an independent machine-checked proof by\nStar Fleet Math.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-521/Research/LateFourthFinal.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«521»","statement":"False ↔ Erdos521.Claim","subjects":["11","60"],"theorem":"Erdos521.erdos_521"},{"answerKinds":[],"category":"API","docstring":"`fairCoin` gives each of the two signs mass `1/2`. Together with\n`fairCoin_isProbabilityMeasure` this pins the definition down, so a proof stating the\nsame problem with a `Bernoulli(1/2)` measure is stating the same thing. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«521»","statement":"∀ (b : Bool), Erdos521.fairCoin {b} = 2⁻¹","subjects":["60"],"theorem":"Erdos521.fairCoin_apply"},{"answerKinds":[],"category":"API","docstring":"`fairCoin` is a probability measure. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«521»","statement":"MeasureTheory.IsProbabilityMeasure Erdos521.fairCoin","subjects":["60"],"theorem":"Erdos521.fairCoin_isProbabilityMeasure"},{"answerKinds":[],"category":"research open","docstring":"Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with\n$p\\not\\equiv 1\\pmod{m}$ such that $m! + 1 \\equiv 0\\pmod{p}$, then does\n$$\n  \\lim\\frac{|P\\cap[1, x]|}{\\pi(x)}\n$$\nexist? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"True ↔ ∃ c, Erdos1074.PillaiPrimes.HasDensity c {p | Nat.Prime p}","subjects":["11"],"theorem":"Erdos1074.erdos_1074.parts.iii"},{"answerKinds":[],"category":"test","docstring":"Pillai [Pi30] raised the question of whether there exist any primes in $P$. This was answered\nby Chowla, who noted that, for example, $14! + 1 \\equiv 18! + 1 \\equiv 0 \\pmod{23}$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"23 ∈ Erdos1074.PillaiPrimes","subjects":["11"],"theorem":"Erdos1074.erdos_1074.variants.mem_pillaiPrimes"},{"answerKinds":[],"category":"research open","docstring":"Let $S$ be the set of all $m\\geq 1$ such that there exists a prime $p\\not\\equiv 1\\pmod{m}$ such\nthat $m! + 1 \\equiv 0\\pmod{p}$. Does\n$$\n  \\lim\\frac{|S\\cap[1, x]|}{x}\n$$\nexist? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"True ↔ ∃ c, Erdos1074.EHSNumbers.HasDensity c","subjects":["11"],"theorem":"Erdos1074.erdos_1074.parts.i"},{"answerKinds":[],"category":"test","docstring":"The sequence $S$ begins $8, 9, 13, 14, 15, 16, 17, ...$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"Nat.nth Erdos1074.EHSNumbers '' Set.Icc 0 6 = {8, 9, 13, 14, 15, 16, 17}","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos1074.erdos_1074.variants.EHSNumbers_init"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with\n$p\\not\\equiv 1\\pmod{m}$ such that $m! + 1 \\equiv 0\\pmod{p}$, then what is\n$$\n  \\lim\\frac{|P\\cap[1, x]|}{\\pi(x)}?\n$$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"Erdos1074.PillaiPrimes.HasDensity sorry {p | Nat.Prime p}","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos1074.erdos_1074.parts.iv"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Hardy, and Subbarao proved that $P$ is infinite. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"Erdos1074.PillaiPrimes.Infinite","subjects":["11"],"theorem":"Erdos1074.erdos_1074.variants.PillaiPrimes_infinite"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Hardy, and Subbarao proved that $S$ is infinite.\n\nFormal proof linked here provided by AlphaProof.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mzhorvath1/formal-conjectures/blob/3dec597bd1a73778760b761712a1fc5fb24bc5d7/FormalConjectures/ErdosProblems/1074.lean#L99"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"Erdos1074.EHSNumbers.Infinite","subjects":["11"],"theorem":"Erdos1074.erdos_1074.variants.EHSNumbers_infinite"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"23 ∈ Erdos1074.PillaiPrimes","subjects":["11"],"theorem":"Erdos1074.twentyThree_mem_pillaiPrimes"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $S$ be the set of all $m\\geq 1$ such that there exists a prime $p\\not\\equiv 1\\pmod{m}$ such\nthat $m! + 1 \\equiv 0\\pmod{p}$. What is\n$$\n  \\lim\\frac{|S\\cap[1, x]|}{x}?\n$$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"Erdos1074.EHSNumbers.HasDensity sorry","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos1074.erdos_1074.parts.ii"},{"answerKinds":[],"category":"test","docstring":"The sequence $P$ begins $23, 29, 59, 61, 67, 71, ...$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"Nat.nth Erdos1074.PillaiPrimes '' Set.Icc 0 5 = {23, 29, 59, 61, 67, 71}","subjects":["11"],"theorem":"Erdos1074.erdos_1074.variants.PillaiPrimes_init"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"2 ∉ Erdos1074.PillaiPrimes","subjects":["11"],"theorem":"Erdos1074.two_not_mem_pillaiPrimes"},{"answerKinds":[],"category":"research open","docstring":"Regarding the first question, Hardy and Subbarao computed all EHS numbers up to $2^{10}$, and\nwrite \"...if this trend conditions we expect [the limit] to be around 0.5, if it exists.\" ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1074»","statement":"Erdos1074.EHSNumbers.HasDensity (1 / 2)","subjects":["11"],"theorem":"Erdos1074.erdos_1074.variants.EHSNumbers_one_half"},{"answerKinds":[],"category":"research open","docstring":"Erdős asked whether there are infinitely many solutions to `uₙ > uₙ₊₁ > uₙ₊₂`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«968»","statement":"True ↔ {n | Erdos968.u n > Erdos968.u (n + 1) ∧ Erdos968.u (n + 1) > Erdos968.u (n + 2)}.Infinite","subjects":["11"],"theorem":"Erdos968.erdos_968.variants.infinite_decreasingTriples"},{"answerKinds":[],"category":"research open","docstring":"Erdős asked whether there are infinitely many solutions to `uₙ < uₙ₊₁ < uₙ₊₂`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«968»","statement":"True ↔ {n | Erdos968.u n < Erdos968.u (n + 1) ∧ Erdos968.u (n + 1) < Erdos968.u (n + 2)}.Infinite","subjects":["11"],"theorem":"Erdos968.erdos_968.variants.infinite_increasingTriples"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Prachar proved `∑_{pₙ < x} |u (n+1) - u n| ≍ (log x)^2` (see [ErPr61]).\n\nWe encode `∑_{pₙ < x}` as a sum over `n < Nat.primeCounting' x` (the number of primes `< x`).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«968»","statement":"(fun x => ∑ n < x.primeCounting', |Erdos968.u (n + 1) - Erdos968.u n|) =Θ[Filter.atTop] fun x => Real.log ↑x ^ 2","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos968.erdos_968.variants.sum_abs_diff_isTheta_log_sq"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Prachar proved that the set `{n | u n > u (n+1)}` has positive lower density\n(see [ErPr61]).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«968»","statement":"0 < {n | Erdos968.u n > Erdos968.u (n + 1)}.lowerDensity","subjects":["11"],"theorem":"Erdos968.erdos_968.variants.decreasing_steps_pos_lower_density"},{"answerKinds":[],"category":"research open","docstring":"Does the set `{n | u n < u (n+1)}` have positive lower density?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«968»","statement":"True ↔ 0 < {n | Erdos968.u n < Erdos968.u (n + 1)}.lowerDensity","subjects":["11"],"theorem":"Erdos968.erdos_968"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«26»","statement":"∀ {ι : Type u_1} [Finite ι] (A : ι → ℕ), ¬Erdos26.IsThick A","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos26.not_isThick_of_finite"},{"answerKinds":[],"category":"research solved","docstring":"If we allow for $\\sum_{a\\in A} \\frac{1}{a} < \\infty$ then Rusza has found a counter-example.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«26»","statement":"∃ A, StrictMono A ∧ ¬Erdos26.IsThick A ∧ ∀ (k : ℕ), ¬Erdos26.IsBehrend fun x => A x + k","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos26.erdos_26.variants.rusza"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«26»","statement":"∀ {ι : Type u_1} [Infinite ι], ∀ r > 0, Erdos26.IsThick fun x => r","subjects":["11"],"theorem":"Erdos26.isThick_const"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«26»","statement":"∀ {ι : Type u_1} (A : ι → ℕ) {ε : ℝ}, ε < 0 → ¬Erdos26.IsWeaklyBehrend A ε","subjects":["11"],"theorem":"Erdos26.not_isWeaklyBehrend_of_neg"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subset\\mathbb{N}$ be infinite such that $\\sum_{a \\in A} \\frac{1}{a} = \\infty$. Must\nthere exist some $k\\geq 1$ such that almost all integers have a divisor of the form $a+k$\nfor some $a\\in A$?\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos26.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«26»","statement":"False ↔ ∀ (A : ℕ → ℕ), StrictMono A → Erdos26.IsThick A → ∃ k, Erdos26.IsBehrend fun x => A x + k","subjects":["11"],"theorem":"Erdos26.erdos_26"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«26»","statement":"∀ {ι : Type u_1} (A : ι → ℕ) {ε : ℝ}, 1 ≤ ε → Erdos26.IsWeaklyBehrend A ε","subjects":["11"],"theorem":"Erdos26.isWeaklyBehrend_of_ge_one"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Tenenbaum asked the weaker variant where for every $\\epsilon>0$ there is\nsome $k=k(\\epsilon)$ such that at least $1-\\epsilon$ density of all integers have a\ndivisor of the form $a+k$ for some $a\\in A$.\n\nThe DeepMind prover agent has found a formal disproof of this statement.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/09c54540aa51cb40dff73660c94a82e2631386f8/FormalConjectures/ErdosProblems/26.lean#L625"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«26»","statement":"False ↔ ∀ (A : ℕ → ℕ), StrictMono A → Erdos26.IsThick A → ∀ ε > 0, ∃ k, Erdos26.IsWeaklyBehrend (fun x => A x + k) ε","subjects":["11"],"theorem":"Erdos26.erdos_26.variants.tenenbaum"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«26»","statement":"∀ {ι : Type u_1} (A : ι → ℕ), 1 ∈ Set.range A → Erdos26.MultiplesOf A = Set.univ","subjects":["11"],"theorem":"Erdos26.multiplesOf_eq_univ"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«26»","statement":"∀ r > 1, ¬Erdos26.IsThick fun n => r ^ n","subjects":["11"],"theorem":"Erdos26.not_isThick_of_geom_one_lt"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«26»","statement":"∀ {ι : Type u_1} (A : ι → ℕ), 1 ∈ Set.range A → Erdos26.IsBehrend A","subjects":["11"],"theorem":"Erdos26.isBehrend_of_contains_one"},{"answerKinds":[],"category":"research solved","docstring":"If $A\\subseteq \\mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determine an obtuse angle.\n\nThe general case was proved by Danzer and Gr\\\"{u}nbaum [DaGr62].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos224.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«224»","statement":"∀ {d : ℕ} (A : Finset (EuclideanSpace ℝ (Fin d))),\n  A.card = 2 ^ d + 1 → ∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ Erdos224.ObtuseAt x y z","subjects":["52"],"theorem":"Erdos224.erdos_224"},{"answerKinds":[],"category":"research open","docstring":"Does the set $A = \\left\\{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\text{ finite}\\right\\}$ of all finite sums of distinct factorials contain only finitely many powerful numbers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1108»","statement":"True ↔ {a | a ∈ Erdos1108.FactorialSums ∧ Erdos1108.IsPowerful a}.Finite","subjects":["11"],"theorem":"Erdos1108.erdos_1108.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"For each $k \\geq 2$, does the set $A = \\left\\{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\text{ finite}\\right\\}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1108»","statement":"True ↔ ∀ k ≥ 2, {a | a ∈ Erdos1108.FactorialSums ∧ ∃ m, m ^ k = a}.Finite","subjects":["11"],"theorem":"Erdos1108.erdos_1108.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"On the other hand, Pommerenke also proved there always exists a line such that the projection has\nmeasure at most 3.3.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1043»","statement":"∀ (f : Polynomial ℂ),\n  f.Monic →\n    f.degree ≥ 1 → ∃ u, ‖u‖ = 1 ∧ MeasureTheory.volume (⇑(ℝ ∙ u).orthogonalProjectionOnto '' Erdos1043.levelSet f) ≤ 3.3","subjects":["28","30"],"theorem":"Erdos1043.erdos_1043.variants.weak"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 1043**:\nLet $f\\in \\mathbb{C}[x]$ be a monic polynomial.\nMust there exist a straight line $\\ell$ such that the projection of\n$$\\{ z: \\lvert f(z)\\rvert\\leq 1\\}$$\nonto $\\ell$ has measure at most $2$?\n\nPommerenke [Po61] proved that the answer is no.\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/7db17471701f15b125d1c36bc1fa5bb9b702d6be/FormalConjectures/ErdosProblems/1043.lean#L214"},{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos1043.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1043»","statement":"False ↔\n  ∀ (f : Polynomial ℂ),\n    f.Monic →\n      f.degree ≥ 1 → ∃ u, ‖u‖ = 1 ∧ MeasureTheory.volume (⇑(ℝ ∙ u).orthogonalProjectionOnto '' Erdos1043.levelSet f) ≤ 2","subjects":["28","30"],"theorem":"Erdos1043.erdos_1043"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many $n$ such that, for all $k\\geq 1$\n$$\n  \\tau(n + k) \\ll k?\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«826»","statement":"True ↔ ∃ C > 0, {n | ∀ k ≥ 1, ↑((ArithmeticFunction.sigma 0) (n + k)) ≤ C * ↑k}.Infinite","subjects":["11"],"theorem":"Erdos826.erdos_826"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\\geq 0$. Show that $f(n)=o(\\log n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«236»","statement":"(fun n => ↑(Erdos236.f n)) =o[Filter.atTop] fun n => Real.log ↑n","subjects":["5","11"],"theorem":"Erdos236.erdos_236"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 567 (K33)**\n\nIs $K_{3,3}$ Ramsey size linear?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«567»","statement":"True ↔ Erdos567.K33.IsRamseySizeLinear","subjects":["5"],"theorem":"Erdos567.erdos_567.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 567 (H5)**\n\nIs $H_5$ ($C_5$ with two vertex-disjoint chords) Ramsey size linear?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«567»","statement":"True ↔ Erdos567.H5.IsRamseySizeLinear","subjects":["5"],"theorem":"Erdos567.erdos_567.parts.iii"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 567 (Q3)**\n\nIs $Q_3$ (the 3-dimensional hypercube) Ramsey size linear?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«567»","statement":"True ↔ Erdos567.Q3.IsRamseySizeLinear","subjects":["5"],"theorem":"Erdos567.erdos_567.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Mirsky [ErMi52] proved that $\\log F(x) \\ll \\frac{(\\log x)^{1/2}}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«945»","statement":"(fun n => Real.log ↑(Erdos945.F ↑n)) =O[Filter.atTop] fun x => √(Real.log ↑x) / Real.log (Real.log ↑x)","subjects":["11"],"theorem":"Erdos945.erdos_945.variants.upper_bound"},{"answerKinds":[],"category":"textbook","docstring":"The two ways of phrasing the conjecture are equivalent.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«945»","statement":"Erdos945.Erdos945Prop ↔ Erdos945.Erdos945Constant","subjects":["11"],"theorem":"Erdos945.erdos_945.variants.equivalence"},{"answerKinds":[],"category":"research open","docstring":"Is there a constant $C > 0$ such that, for all large $x$, every interval $[x, x+(\\log x)C]$\ncontains two integers with the same number of divisors?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«945»","statement":"True ↔ Erdos945.Erdos945Constant","subjects":["11"],"theorem":"Erdos945.erdos_945.variants.constant"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Mirsky [ErMi52] proved that $\\frac{(\\log x)^{1/2}}{\\log\\log x}\\ll F(x)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«945»","statement":"(fun x => √(Real.log ↑x) / Real.log (Real.log ↑x)) =O[Filter.atTop] fun n => ↑(Erdos945.F ↑n)","subjects":["11"],"theorem":"Erdos945.erdos_945.variants.lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $F(x) \\leq (\\log x)^{O(1)}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«945»","statement":"True ↔ Erdos945.Erdos945Prop","subjects":["11"],"theorem":"Erdos945.erdos_945"},{"answerKinds":[],"category":"research open","docstring":"For each $n \\in \\mathbb{N}$ choose some $X_n \\subseteq \\mathbb{Z}/n\\mathbb{Z}$.\nLet $B = \\{m \\in \\mathbb{N} : \\forall n, m \\not\\equiv x \\pmod{n} \\text{ for all } x \\in X_n\\}$.\nMust $B$ have a logarithmic density?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«486»","statement":"True ↔ ∀ (X : (n : ℕ) → Set (ZMod n)), ∃ d, {m | ∀ (n : ℕ), ↑m ∉ X n}.HasLogDensity d","subjects":["11"],"theorem":"Erdos486.erdos_486"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a graph with no isolated vertices and $m$ edges. Is it true that\n$$R(G) \\leq 2^{O(m^{1/2})}?$$\n\nThis is true, and was proved by Sudakov [Su11].\n\nThis problem is #11 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«546»","statement":"True ↔\n  ∃ C > 0,\n    ∀ (m : ℕ) (V : Type) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n      (∀ (v : V), 0 < G.degree v) → G.edgeSet.ncard = m → ↑G.diagonalGraphRamsey ≤ 2 ^ (C * √↑m)","subjects":["5"],"theorem":"Erdos546.erdos_546"},{"answerKinds":[],"category":"research solved","docstring":"Lower bound for the growth rate of `Erdos975Sum`, shown in [Va39]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«975»","statement":"∀ (f : Polynomial ℤ),\n  Irreducible f →\n    f.natDegree ≠ 0 →\n      (∀ᶠ (n : ℤ) in Filter.atTop, 1 ≤ Polynomial.eval n f) →\n        (fun x => x * Real.log x) =O[Filter.atTop] Erdos975.Erdos975Sum f","subjects":["11"],"theorem":"Erdos975.erdos_975.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"More concrete example for $f(n) = n^2 + 1$, where the asymptote is\n$\\sum_{n \\le x} \\tau(n^2 + 1) \\sim \\frac{3}{\\pi} x \\log x + O(x)$. See Tao's blog [T].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«975»","statement":"(fun x => Erdos975.Erdos975Sum (Polynomial.X ^ 2 + 1) x - 3 / Real.pi * x * Real.log x) =O[Filter.atTop] id","subjects":["11"],"theorem":"Erdos975.erdos_975.variants.n2_plus_1_strong"},{"answerKinds":[],"category":"research solved","docstring":"The correctness of the growth rate is shown in [Va39] (lower bound) and [Er52b] (upper bound).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«975»","statement":"∀ (f : Polynomial ℤ),\n  Irreducible f →\n    (∀ᶠ (n : ℤ) in Filter.atTop, 1 ≤ Polynomial.eval n f) →\n      Erdos975.Erdos975Sum f =O[Filter.atTop] fun x => x * Real.log x","subjects":["11"],"theorem":"Erdos975.erdos_975.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Asymptotics for `Erdos975Sum` with $f(X) = X^2 + 1$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«975»","statement":"∃ c > 0,\n  Filter.Tendsto (fun x => Erdos975.Erdos975Sum (Polynomial.X ^ 2 + 1) x / (x * Real.log x)) Filter.atTop (nhds c)","subjects":["11"],"theorem":"Erdos975.erdos_975.variants.n2_plus_1"},{"answerKinds":[],"category":"research open","docstring":"For an irreducible polynomial $f \\in \\mathbb{Z}[x]$ with $f(n) \\ge 1$ for sufficiently large $n$,\ndoes there exists a constant $c = c(f) > 0$ such that\n$\\sum_{n \\le x} \\tau(f(n)) \\approx c \\cdot x \\log x$?\n\nNote that it is unclear whether the polynomial should have integer coefficients or merely be\ninteger-valued. We assume the former. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«975»","statement":"True ↔\n  ∀ (f : Polynomial ℤ),\n    f.natDegree ≠ 0 →\n      Irreducible f →\n        (∀ᶠ (n : ℤ) in Filter.atTop, 1 ≤ Polynomial.eval n f) →\n          ∃ c > 0, Filter.Tendsto (fun x => Erdos975.Erdos975Sum f x / (x * Real.log x)) Filter.atTop (nhds c)","subjects":["11"],"theorem":"Erdos975.erdos_975"},{"answerKinds":["non-Prop"],"category":"research solved","docstring":"When $f$ is an irreducible quadratic polynomial, the question is answered first by Hooley [Ho63].\nMore compact expression of the constant in terms of Hurwitz class numbers (when $a = 1$)\nis given by McKey in [Mc95], [Mc97], [Mc99].\n\nTODO: formalize Hurwitz class numbers and the expression of the constant in terms of them.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«975»","statement":"∀ (f : Polynomial ℤ),\n  Irreducible f →\n    (∀ᶠ (n : ℕ) in Filter.atTop, 1 ≤ Polynomial.eval (↑n) f) →\n      f.degree = 2 →\n        ∀ (c : ℝ),\n          c = sorry →\n            0 < c ∧ Filter.Tendsto (fun x => Erdos975.Erdos975Sum f x / (x * Real.log x)) Filter.atTop (nhds c)","subjects":["11"],"theorem":"Erdos975.erdos_975.variants.quadratic"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $k,r\\geq 2$. Does there exist a set $A\\subseteq \\mathbb{N}$ that contains no non-trivial\narithmetic progression of length $k+1$, yet in any $r$-colouring of $A$ there must exist a\nmonochromatic non-trivial arithmetic progression of length $k$?\n\nErdős [Er75b] reported that 'Spencer has recently shown that such a sequence exists', but gives\nno reference.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos966.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«966»","statement":"True ↔\n  ∀ (k r : ℕ),\n    2 ≤ k → 2 ≤ r → ∃ A, A.IsAPOfLengthFree (↑k + 1) ∧ ∀ (coloring : ↑A → Fin r), ContainsMonoAPofLength coloring k","subjects":["5","11"],"theorem":"Erdos966.erdos_966"},{"answerKinds":[],"category":"research solved","docstring":"Erdős (unpublished)\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/erdos-979-k3/blob/930fcc877fa4a2b4fc7c59614cbbab04a6d834ca/lean/K3Lean/Erdos979K3Final.lean#L46-L47"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«979»","statement":"Filter.limsup (fun n => (Erdos979.solutionSet n 3).encard) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos979.erdos_979.variants.k3"},{"answerKinds":[],"category":"research open","docstring":"Let $k ≥ 2$, and let $f_k(n)$ count the number of solutions to $n = p_1^k + \\dots + p_k^k$,\nwhere the $p_i$ are prime numbers. Is it true that $\\limsup f_k(n) = \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«979»","statement":"True ↔ ∀ k ≥ 2, Filter.limsup (fun n => (Erdos979.solutionSet n k).encard) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos979.erdos_979"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er37b] proved that if $f_2(n)$ counts the number of solutions to $n = p_1^2 + p_2^2$, where $p_1$ and $p_2$ are prime numbers, then $\\limsup f_2(n) = \\infty$.\n\n[Er37b] Erdős, Paul, On the Sum and Difference of Squares of Primes. J. London Math. Soc. (1937), 133--136.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«979»","statement":"Filter.limsup (fun n => (Erdos979.solutionSet n 2).encard) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos979.erdos_979.variants.k2"},{"answerKinds":[],"category":"research open","docstring":"Let `A` be an infinite `B₂[2]` set. Must `liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«158»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    A.Infinite → Erdos158.B2 2 A → Filter.liminf (fun N => ↑(A ∩ Set.Iio N).ncard * ↑N ^ (-1 / 2)) Filter.atTop = 0","subjects":["5"],"theorem":"Erdos158.erdos_158"},{"answerKinds":[],"category":"research solved","docstring":"As a corollary of `erdos_158.isSidon'`, we can prove that\n`liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0` for any infinite Sidon set `A`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«158»","statement":"∀ {A : Set ℕ}, A.Infinite → IsSidon A → Filter.liminf (fun N => ↑(A ∩ Set.Iio N).ncard * ↑N ^ (-1 / 2)) Filter.atTop = 0","subjects":["5"],"theorem":"Erdos158.erdos_158.variants.isSidon"},{"answerKinds":[],"category":"API","docstring":"A set is `B₂[1]` iff it is Sidon. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«158»","statement":"∀ {A : Set ℕ}, Erdos158.B2 1 A ↔ IsSidon A","subjects":["5"],"theorem":"Erdos158.b2_one"},{"answerKinds":[],"category":"research solved","docstring":"Let `A` be an infinite Sidon set. Then\n`liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) * (log N) ^ (1 / 2) < ∞`. This is proved in [ESS94]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«158»","statement":"∀ {A : Set ℕ},\n  A.Infinite →\n    IsSidon A →\n      Filter.liminf (fun N => ENNReal.ofReal (↑(A ∩ Set.Iio N).ncard * ↑N ^ (-1 / 2) * Real.log ↑N ^ (1 / 2)))\n          Filter.atTop <\n        ⊤","subjects":["5"],"theorem":"Erdos158.erdos_158.variants.isSidon'"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \\geq 3$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«273»","statement":"True ↔ ∃ c, ∀ (i : c.ι), ∃ p, Nat.Prime p ∧ 3 ≤ p ∧ c.moduli i = Ideal.span {p - 1}","subjects":["5","11"],"theorem":"Erdos273.erdos_273.variants.three"},{"answerKinds":[],"category":"research open","docstring":"Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \\geq 5$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«273»","statement":"True ↔ ∃ c, ∀ (i : c.ι), ∃ p, Nat.Prime p ∧ 5 ≤ p ∧ c.moduli i = Ideal.span {↑(p - 1)}","subjects":["5","11"],"theorem":"Erdos273.erdos_273"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the size of the largest $A\\subseteq\\{1, \\dots, N\\}$ such that there is a function\n$\\delta : A \\to \\{-1, 1\\}$ such that\n$$\n  \\sum_{n\\in A} \\frac{\\delta n}{n} = 0\n$$\nand\n$$\n  \\sum_{n\\in A'}\\frac{\\delta n}{n} \\neq 0\n$$\nfor all non-empty $A'\\subsetneq A$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«319»","statement":"∀ (N : ℕ),\n  IsGreatest\n    {x |\n      ∃ A,\n        ∃ (_ : A ⊆ Finset.Icc 1 N) (_ :\n          ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), A.card = x}\n    sorry","subjects":["5"],"theorem":"Erdos319.erdos_319"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c(N)$ be the size of the largest $A\\subseteq\\{1, \\dots, N\\}$ such that there is a function\n$\\delta : A \\to \\{-1, 1\\}$ such that\n$$\n  \\sum_{n\\in A} \\frac{\\delta n}{n} = 0\n$$\nand\n$$\n  \\sum_{n\\in A'}\\frac{\\delta n}{n} \\neq 0\n$$\nfor all non-empty $A'\\subsetneq A$. What is $\\Theta(c(N))$?","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«319»","statement":"∀ (N : ℕ) (c : ℕ → ℝ),\n  (∀ (N : ℕ),\n      IsGreatest\n        {x |\n          ∃ A,\n            ∃ (_ : A ⊆ Finset.Icc 1 N) (_ :\n              ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑A.card = x}\n        (c N)) →\n    c =Θ[Filter.atTop] sorry","subjects":["5"],"theorem":"Erdos319.erdos_319.variants.isTheta"},{"answerKinds":[],"category":"research solved","docstring":"Adenwalla has observed that a lower bound (on the maximum size of $A$) of\n$$\n  |A| \\geq (1 - \\frac{1}{e} + o(1))N\n$$\nfollows from the main result of Croot [Cr01].\n\n[Cr01] Croot, III, Ernest S., _On unit fractions with denominators in short intervals_.\nActa Arith. (2001), 99-114.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«319»","statement":"∃ o,\n  o =o[Filter.atTop] 1 ∧\n    ∀ᶠ (N : ℕ) in Filter.atTop,\n      (1 - 1 / Real.exp 1 + o N) * ↑N ≤\n        sSup\n          {x |\n            ∃ A,\n              ∃ (_ : A ⊆ Finset.Icc 1 N) (_ :\n                ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑A.card = x}","subjects":["5"],"theorem":"Erdos319.erdos_319.variants.lb"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c(N)$ be the size of the largest $A\\subseteq\\{1, \\dots, N\\}$ such that there is a function\n$\\delta : A \\to \\{-1, 1\\}$ such that\n$$\n  \\sum_{n\\in A} \\frac{\\delta n}{n} = 0\n$$\nand\n$$\n  \\sum_{n\\in A'}\\frac{\\delta n}{n} \\neq 0\n$$\nfor all non-empty $A'\\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = o(g(N))$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«319»","statement":"∀ (N : ℕ) (c : ℕ → ℝ),\n  (∀ (N : ℕ),\n      IsGreatest\n        {x |\n          ∃ A,\n            ∃ (_ : A ⊆ Finset.Icc 1 N) (_ :\n              ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑A.card = x}\n        (c N)) →\n    c =o[Filter.atTop] sorry","subjects":["5"],"theorem":"Erdos319.erdos_319.variants.isLittleO"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c(N)$ be the size of the largest $A\\subseteq\\{1, \\dots, N\\}$ such that there is a function\n$\\delta : A \\to \\{-1, 1\\}$ such that\n$$\n  \\sum_{n\\in A} \\frac{\\delta n}{n} = 0\n$$\nand\n$$\n  \\sum_{n\\in A'}\\frac{\\delta n}{n} \\neq 0\n$$\nfor all non-empty $A'\\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = O(g(N))$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«319»","statement":"∀ (N : ℕ) (c : ℕ → ℝ),\n  (∀ (N : ℕ),\n      IsGreatest\n        {x |\n          ∃ A,\n            ∃ (_ : A ⊆ Finset.Icc 1 N) (_ :\n              ∃ δ, ∑ n ∈ A, ↑↑(δ n) / ↑n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', ↑↑(δ n) / ↑n ≠ 0), ↑A.card = x}\n        (c N)) →\n    c =O[Filter.atTop] sorry","subjects":["5"],"theorem":"Erdos319.erdos_319.variants.isBigO"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Goodman raises the question of the maximum number of non-convex components that are possible as\na function of the degree of $f$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1047»","statement":"∀ (n : ℕ),\n  IsGreatest\n    {k |\n      ∃ f c,\n        f.Monic ∧\n          f.natDegree = n ∧\n            0 < c ∧\n              (Erdos1047.componentsIn (Erdos1047.sublevelSet f c)).ncard = (f.rootSet ℂ).ncard ∧\n                {t | t ∈ Erdos1047.componentsIn (Erdos1047.sublevelSet f c) ∧ ¬Convex ℝ t}.ncard = k}\n    sorry","subjects":["30","52"],"theorem":"Erdos1047.erdos_1047.variants.max_non_convex_components"},{"answerKinds":[],"category":"research solved","docstring":"Goodman [Go66] constructed an example with simple roots, of degree $4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1047»","statement":"∃ f c,\n  f.Monic ∧\n    f.natDegree = 4 ∧\n      (f.rootSet ℂ).ncard = 4 ∧\n        0 < c ∧\n          (Erdos1047.componentsIn (Erdos1047.sublevelSet f c)).ncard = 4 ∧\n            ∃ t ∈ Erdos1047.componentsIn (Erdos1047.sublevelSet f c), ¬Convex ℝ t","subjects":["30","52"],"theorem":"Erdos1047.erdos_1047.variants.goodman_simple_roots"},{"answerKinds":[],"category":"research solved","docstring":"The answer is no, as shown by Pommerenke [Po61], who showed that, if $k$ is sufficiently large,\nand $f(z)=z^k(z-a)$ where $a$ is sufficiently close to $(1+\\frac{1}{k})k^{\\frac{1}{k+1}}$, then\n$\\{ z: \\lvert f(z)\\rvert\\leq 1\\}$ has two components, and the component which contains $0$ is\nnot convex.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1047»","statement":"∀ᶠ (k : ℕ) in Filter.atTop,\n  ∃ δ,\n    0 < δ ∧\n      ∀ (a : ℝ),\n        (1 + 1 / ↑k) * ↑k ^ (1 / (↑k + 1)) < a →\n          a < (1 + 1 / ↑k) * ↑k ^ (1 / (↑k + 1)) + δ →\n            (Erdos1047.componentsIn\n                    (Erdos1047.sublevelSet (Polynomial.X ^ k * (Polynomial.X - Polynomial.C ↑a)) 1)).ncard =\n                2 ∧\n              ¬Convex ℝ\n                  (connectedComponentIn (Erdos1047.sublevelSet (Polynomial.X ^ k * (Polynomial.X - Polynomial.C ↑a)) 1)\n                    0)","subjects":["30","52"],"theorem":"Erdos1047.erdos_1047.variants.pommerenke"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f\\in \\mathbb{C}[x]$ be a monic polynomial with $m$ distinct roots, and let $c>0$ be a\nconstant small enough such that $\\{ z: \\lvert f(z)\\rvert\\leq c\\}$ has $m$ distinct connected\ncomponents.\n\nMust all these components be convex?\n\nA question of Grunsky, which was reported by Erdős, Herzog, and Piranian [EHP58].\n\nThe answer is no, as shown by Pommerenke [Po61], who showed that, if $k$ is sufficiently large,\nand $f(z)=z^k(z-a)$ where $a$ is sufficiently close to $(1+\\frac{1}{k})k^{\\frac{1}{k+1}}$, then\n$\\{ z: \\lvert f(z)\\rvert\\leq 1\\}$ has two components, and the component which contains $0$ is\nnot convex.\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/latest/ErdosProblems/Erdos1047.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1047»","statement":"False ↔\n  ∀ (f : Polynomial ℂ) (m : ℕ) (c : ℝ),\n    f.Monic →\n      (f.rootSet ℂ).ncard = m →\n        0 < c →\n          (Erdos1047.componentsIn (Erdos1047.sublevelSet f c)).ncard = m →\n            ∀ t ∈ Erdos1047.componentsIn (Erdos1047.sublevelSet f c), Convex ℝ t","subjects":["30","52"],"theorem":"Erdos1047.erdos_1047"},{"answerKinds":[],"category":"research solved","docstring":"The referee of the paper [Go66] also gave the example of $\\lvert z(z^5-1)\\rvert< 5\\cdot 6^{-6/5}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1047»","statement":"(Erdos1047.componentsIn\n        (Erdos1047.strictSublevelSet (Polynomial.X * (Polynomial.X ^ 5 - 1)) (5 * 6 ^ (-6 / 5)))).ncard =\n    6 ∧\n  ∃ t ∈ Erdos1047.componentsIn (Erdos1047.strictSublevelSet (Polynomial.X * (Polynomial.X ^ 5 - 1)) (5 * 6 ^ (-6 / 5))),\n    ¬Convex ℝ t","subjects":["30","52"],"theorem":"Erdos1047.erdos_1047.variants.referee"},{"answerKinds":[],"category":"research solved","docstring":"Goodman [Go66] proved that one of the three components of\n$$\\{ z: \\lvert (z^2+1)(z-2)^2\\rvert < 5^{3/2}/4\\}$$\nis not convex.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1047»","statement":"(Erdos1047.componentsIn\n        (Erdos1047.strictSublevelSet ((Polynomial.X ^ 2 + 1) * (Polynomial.X - Polynomial.C 2) ^ 2)\n          (5 ^ (3 / 2) / 4))).ncard =\n    3 ∧\n  ∃\n    t ∈\n      Erdos1047.componentsIn\n        (Erdos1047.strictSublevelSet ((Polynomial.X ^ 2 + 1) * (Polynomial.X - Polynomial.C 2) ^ 2) (5 ^ (3 / 2) / 4)),\n    ¬Convex ℝ t","subjects":["30","52"],"theorem":"Erdos1047.erdos_1047.variants.goodman"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $\\max(x^2,y^2,z^2)\\leq n$?\n\nThis was proved affirmatively by Chojecki [Ch26], using a Duke-type equidistribution theorem.\nA Lean formalisation of the reduction (conditional on a Duke-type equidistribution theorem) exists;\nsee the [forum discussion](https://www.erdosproblems.com/forum/thread/1148#post-4849).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1148»","statement":"True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, Erdos1148.Erdos1148Prop n","subjects":["11"],"theorem":"Erdos1148.erdos_1148"},{"answerKinds":[],"category":"textbook","docstring":"The integer $6563$ cannot be written as $x^2 + y^2 - z^2$ with $\\max(x^2, y^2, z^2) \\leq 6563$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1148»","statement":"¬Erdos1148.Erdos1148Prop 6563","subjects":["11"],"theorem":"Erdos1148.erdos_1148.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"[Va99] reports this is 'obvious' if we replace $\\leq n$ with $\\leq n+2\\sqrt{n}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1148»","statement":"∀ (n : ℕ), Erdos1148.erdos_1148_weaker_prop n","subjects":["11"],"theorem":"Erdos1148.erdos_1148.variants.weaker"},{"answerKinds":[],"category":"research solved","docstring":"Erdos and Selfridge [ErSe75] proved that the product of\nconsecutive integers is never a power (establishing the case $r=1$).\n\nTheorem 1 from [ErSe75].\n\nIt is implied from `erdos_930.variants.consecutive_strong`.\n\n[ErSe75] Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«930»","statement":"∀ (n k : ℕ), 0 ≤ n → 2 ≤ k → ¬Erdos930.IsPower (∏ m ∈ Finset.Icc (n + 1) (n + k), m)","subjects":["11"],"theorem":"Erdos930.erdos_930.variants.consecutive_integers"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for every $r$, there is a $k$ such that\nif $I_1,\\ldots,I_r$ are disjoint intervals of consecutive integers,\nall of length at least $k$, then\n$$\n  \\prod_{1\\leq i\\leq r}\\prod_{m\\in I_i}m\n$$\nis not a perfect power?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«930»","statement":"True ↔\n  ∀ r > 0,\n    ∃ k,\n      ∀ (I₁ I₂ : Fin r → ℕ),\n        (∀ (i : Fin r), 0 < I₁ i ∧ I₁ i + k ≤ I₂ i + 1) →\n          (∀ (i j : Fin r), i < j → I₂ i < I₁ j) → ¬Erdos930.IsPower (∏ i, ∏ m ∈ Finset.Icc (I₁ i) (I₂ i), m)","subjects":["11"],"theorem":"Erdos930.erdos_930"},{"answerKinds":[],"category":"research solved","docstring":"Let $k$, $l$, $n$ be integers such that $k \\ge 3$, $l \\ge 2$ and $n + k \\ge p^{(k)}$,\nwhere $p^{(k)}$ is the least prime satisfying $p^{(k)} \\ge k$.\nThen there is a prime $p \\ge k$ for which $l$ does not divide\nthe multiplicity of the prime factor $p$ in $(n + 1) \\ldots (n + k)$.\n\nTheorem 2 from [ErSe75].\n\n[ErSe75] Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«930»","statement":"∀ (k l n : ℕ),\n  3 ≤ k →\n    2 ≤ l →\n      Erdos930.nextPrime k ≤ n + k →\n        ∃ p, k ≤ p ∧ Nat.Prime p ∧ ¬l ∣ (∏ m ∈ Finset.Icc (n + 1) (n + k), m).factorization p","subjects":["11"],"theorem":"Erdos930.erdos_930.variants.consecutive_strong"},{"answerKinds":[],"category":"research solved","docstring":"Abbott and Hanson [AbHa70] improved Erdős's upper bound to $f_r(N) \\leq N^{1/2+o(1)}$\nfor all $r \\geq 3$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«535»","statement":"∀ {r : ℕ}, 3 ≤ r → ∀ ε > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos535.f r N) ≤ ↑N ^ (1 / 2 + ε)","subjects":["5","11"],"theorem":"Erdos535.erdos_535.variants.abbott_hanson"},{"answerKinds":[],"category":"research open","docstring":"Erdős [Er73] records that Abbott pointed out the ordinary sunflower conjecture\ndoes not seem to suffice here. The stronger auxiliary conjecture uses $Ω(n)=k$,\ni.e. prime factors counted with multiplicity; this stronger statement would imply\nthe conjectured upper bound for $f_r(N)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«535»","statement":"∀ {r : ℕ},\n  3 ≤ r →\n    ∃ c_r > 0,\n      ∀ (k : ℕ) (A : Finset ℕ),\n        Erdos535.AllBigOmega k A → Erdos535.NoConstantPairwiseGcdCoprimeSubsets r A → ↑A.card ≤ c_r ^ k","subjects":["5","11"],"theorem":"Erdos535.erdos_535.variants.sunflower_strong"},{"answerKinds":[],"category":"research solved","docstring":"For the stronger $Ω(n)=k$ variant above, the Erdős–Rado method gives the weaker\nbound $c_r^k \\cdot k!$; see Erdős [Er73].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«535»","statement":"∀ {r : ℕ},\n  3 ≤ r →\n    ∃ c_r > 0,\n      ∀ (k : ℕ) (A : Finset ℕ),\n        Erdos535.AllBigOmega k A → Erdos535.NoConstantPairwiseGcdCoprimeSubsets r A → ↑A.card ≤ c_r ^ k * ↑k.factorial","subjects":["5","11"],"theorem":"Erdos535.erdos_535.variants.sunflower_erdos_rado"},{"answerKinds":[],"category":"research open","docstring":"The first open case of Erdős Problem 535 is $r = 3$: there should exist $c > 0$ such that\n$f_3(N) \\leq N^{c/\\log\\log N}$ for all sufficiently large $N$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«535»","statement":"∃ c > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos535.f 3 N) ≤ ↑N ^ (c / Real.log (Real.log ↑N))","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos535.erdos_535.variants.first_open_case"},{"answerKinds":[],"category":"research open","docstring":"Let $r \\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\\{1,\\ldots,N\\}$\nsuch that no subset of size $r$ has the same pairwise greatest common divisor between all\nelements. Erdős [Er64] proved that $f_3(N) > N^{c/\\log\\log N}$ for some constant $c > 0$, and\nconjectured this should also be an upper bound; here we state the conjectural upper bound\nfor all $r \\geq 3$.\n\nSee also [536].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«535»","statement":"∀ r ≥ 3, ∃ c > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos535.f r N) ≤ ↑N ^ (c / Real.log (Real.log ↑N))","subjects":["5","11"],"theorem":"Erdos535.erdos_535"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er64] proved that $f_3(N) > N^{c/\\log\\log N}$ for some constant $c > 0$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«535»","statement":"∃ c > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑N ^ (c / Real.log (Real.log ↑N)) ≤ ↑(Erdos535.f 3 N)","subjects":["5","11"],"theorem":"Erdos535.erdos_535.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er64] proved that $f_r(N) \\leq N^{3/4+o(1)}$ for all $r \\geq 3$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«535»","statement":"∀ {r : ℕ}, 3 ≤ r → ∀ ε > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos535.f r N) ≤ ↑N ^ (3 / 4 + ε)","subjects":["5","11"],"theorem":"Erdos535.erdos_535.variants.erdos_upper_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does there exist $B \\subset \\mathbb{N}$ which is not an additive basis,\nbut is such that for every set $A \\subseteq \\mathbb{N}$ of Schnirelmann density $\\alpha$\nand every $N$ there exists $b \\in B$ such that\n$$\n  \\lvert (A \\cup (A+b)) \\cap \\{1, \\ldots, N\\} \\rvert \\geq (\\alpha + f(\\alpha)) N\n$$\nwhere $f(\\alpha) > 0$ for $0 < \\alpha < 1$?\n\nNote: here Erdős seems to use a slightly weaker notion of an additive basis (see [Er56] at the top\nof page 135). In particular, for this problem, a set is an additive basis of order $k$ if every\nnatural number can be written as a sum of _at most_ $k$ elements of the set, rather than as a sum of\n_precisely_ $k$ elements.\n\nA positive [solution](https://github.com/spicylemonade/erdos-38) was given by GPT 5.5 Pro\n(prompted by gebyjaff, cleanup by Liam Price); in fact a sparse random set $B$ has this property,\nwith $f(\\alpha)\\gg \\alpha (1-\\alpha)^2$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://www.erdosproblems.com/forum/thread/38#post-6131"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«38»","statement":"True ↔\n  ∃ B,\n    ¬B.IsWeakAddBasis ∧\n      ∃ f,\n        (∀ (α : ℝ), 0 < α → α < 1 → f α > 0) ∧\n          ∀ (A : Set ℕ) (N : ℕ),\n            have α := schnirelmannDensity A;\n            ∃ b ∈ B, ↑(Set.Ioc 0 N ∩ (A ∪ (A + {b}))).ncard ≥ (α + f α) * ↑N","subjects":["11"],"theorem":"Erdos38.erdos_38"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there some $\\epsilon > 0$ such that there are infinitely\nmany $n$ where all primes $p \\le (2 + \\epsilon) \\log n$ divide\n$$\n  \\prod_{1 \\le i \\le \\log n} (n + i)?\n$$\n\nThis was formalized in Lean by Baretto and van Doorn using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem457.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«457»","statement":"True ↔\n  ∃ ε > 0,\n    {n | ∀ (p : ℕ), ↑p ≤ (2 + ε) * Real.log ↑n → Nat.Prime p → p ∣ ∏ i ∈ Finset.Icc 1 ⌊Real.log ↑n⌋₊, (n + i)}.Infinite","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos457.erdos_457"},{"answerKinds":[],"category":"research open","docstring":"Taking $n$ to be the product of primes\nbetween $\\log n$ and $(2 + o(1)) \\log n$ gives an example where\n$$\n  q(n, \\log n) \\ge (2 + o(1)) \\log n.\n$$\nCan one prove that $q(n, \\log n) < (1 - \\epsilon) (\\log n)^2$\nfor all large $n$ and some $\\epsilon > 0$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«457»","statement":"True ↔ ∃ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos457.q n (Real.log ↑n)) < (1 - ε) * Real.log ↑n ^ 2","subjects":["11"],"theorem":"Erdos457.erdos_457.variants.one_sub"},{"answerKinds":[],"category":"research open","docstring":"More generally, let $q(n, k)$ denote the least prime which\ndoes not divide $\\prod_{1 \\le i \\le k}(n + i)$. This\nproblem asks whether $q(n, \\log n) \\ge (2 + \\epsilon) \\log n$\ninfinitely often.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«457»","statement":"True ↔ ∃ ε > 0, {n | (2 + ε) * Real.log ↑n ≤ ↑(Erdos457.q n (Real.log ↑n))}.Infinite","subjects":["11"],"theorem":"Erdos457.erdos_457.variants.qnk"},{"answerKinds":[],"category":"research open","docstring":"Does every finite graph with minimum degree at least $3$\ncontain a cycle of length $2^k$ for some $k \\geq 2$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«64»","statement":"True ↔\n  ∀ (V : Type u_1) (G : SimpleGraph V) [inst : Fintype V] [inst_1 : DecidableRel G.Adj],\n    G.minDegree ≥ 3 → ∃ k v c, k ≥ 2 ∧ c.IsCycle ∧ c.length = 2 ^ k","subjects":["5"],"theorem":"Erdos64.erdos_64"},{"answerKinds":[],"category":"research solved","docstring":"This is known only for $r=2$ (see [erdosproblems.com/30]).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«241»","statement":"Erdos241.BoseChowlaConjecture 2","subjects":["5"],"theorem":"Erdos241.erdos_241.variants.r_eq_2"},{"answerKinds":[],"category":"research solved","docstring":"Bose and Chowla [BoCh62] provided a construction proving one half of this, namely\n$(1+o(1))N^{1/3}\\leq f(N)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«241»","statement":"∃ ε, (ε =o[Filter.atTop] fun x => 1) ∧ ∀ᶠ (N : ℕ) in Filter.atTop, (1 + ε N) * ↑N ^ (1 / 3) ≤ ↑(Erdos241.f N 3)","subjects":["5"],"theorem":"Erdos241.erdos_241.variants.lower_bound"},{"answerKinds":[],"category":"research open","docstring":"More generally, Bose and Chowla [BoCh62] conjectured that the maximum size of\n$A\\subseteq \\{1,\\ldots,N\\}$ with all $r$-fold sums distinct (aside from the trivial coincidences)\nthen $\\lvert A\\rvert \\sim N^{1/r}.$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«241»","statement":"∀ r ≥ 2, Erdos241.BoseChowlaConjecture r","subjects":["5"],"theorem":"Erdos241.erdos_241.variants.generalization"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $f(N)\\sim N^{1/3}$?\n\nOriginally asked to Erdős by Bose.\n\nThis is discussed in problem C11 of Guy's collection [Gu04].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«241»","statement":"True ↔ Asymptotics.IsEquivalent Filter.atTop (fun N => ↑(Erdos241.f N 3)) fun N => ↑N ^ (1 / 3)","subjects":["5"],"theorem":"Erdos241.erdos_241"},{"answerKinds":[],"category":"research solved","docstring":"The best upper bound known to date is due to Green [Gr01], $f(N) \\leq ((7/2)^{1/3}+o(1))N^{1/3}$.\n(note that $(7/2)^{1/3}\\approx 1.519$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«241»","statement":"∃ ε,\n  (ε =o[Filter.atTop] fun x => 1) ∧\n    ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos241.f N 3) ≤ ((7 / 2) ^ (1 / 3) + ε N) * ↑N ^ (1 / 3)","subjects":["5"],"theorem":"Erdos241.erdos_241.variants.upper_bound"},{"answerKinds":[],"category":"research open","docstring":"Let $\\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\\mathfrak{m}$.\nLet $r\\geq 1$. Must $G$ contain a subgraph of chromatic number $\\mathfrak{m}$ which does not contain\nany odd cycle of length $\\leq r$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«740»","statement":"True ↔\n  ∀ (V : Type u_1) (G : SimpleGraph V),\n    Cardinal.aleph0 ≤ G.chromaticCardinal →\n      ∀ (r : ℕ), ∃ H, H.coe.chromaticCardinal = G.chromaticCardinal ∧ Erdos740.NoShortOddCycle H.coe r","subjects":["5"],"theorem":"Erdos740.erdos_740"},{"answerKinds":[],"category":"research solved","docstring":"Ruzsa and Turjányi do prove (under the same hypotheses) that\n$$\n\\lim_{N\\to \\infty}\\frac{\\lvert (A+A+A)\\cap \\{1,\\ldots,3N\\}\\rvert}\n{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}=\\infty,\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«337»","statement":"∀ (A : Set ℕ),\n  A.IsAsymptoticAddBasis →\n    ((fun N => ↑(A ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => ↑N) →\n      Filter.Tendsto (fun N => ↑((A + A + A) ∩ Set.Icc 1 (3 * N)).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop\n        Filter.atTop","subjects":["5","11"],"theorem":"Erdos337.erdos_337.variants.three_fold"},{"answerKinds":[],"category":"research open","docstring":"Ruzsa and Turjányi do prove (under the same hypotheses) that\n$$\n\\lim_{N\\to \\infty}\\frac{\\lvert (A+A+A)\\cap \\{1,\\ldots,3N\\}\\rvert}\n{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}=\\infty,\n$$\nand conjecture that the same should be true with $(A+A)\\cap \\{1,\\ldots,2N\\}$ in the numerator.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«337»","statement":"∀ (A : Set ℕ),\n  A.IsAsymptoticAddBasis →\n    ((fun N => ↑(A ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => ↑N) →\n      Filter.Tendsto (fun N => ↑((A + A) ∩ Set.Icc 1 (2 * N)).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop\n        Filter.atTop","subjects":["5","11"],"theorem":"Erdos337.erdos_337.variants.ruzsa_turjanyi"},{"answerKinds":[],"category":"research solved","docstring":"This was generalised (to the replacement of $A+A$ by the $h$-fold sumset $hA$ for any $h\\geq 2$)\nby Ruzsa and Turjányi [RT85].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«337»","statement":"∀ (h : ℕ),\n  2 ≤ h →\n    ∃ A,\n      A.IsAsymptoticAddBasis ∧\n        ((fun N => ↑(A ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => ↑N) ∧\n          ¬Filter.Tendsto (fun N => ↑(h • A ∩ Set.Icc 1 N).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop Filter.atTop","subjects":["5","11"],"theorem":"Erdos337.erdos_337.variants.h_fold"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A\\subseteq \\mathbb{N}$ be an additive basis (of any finite order) such that\n$\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert=o(N)$. Is it true that\n$$\n\\lim_{N\\to \\infty}\\frac{\\lvert (A+A)\\cap \\{1,\\ldots,N\\}\\rvert}\n{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}=\\infty?\n$$\n\nThe answer is no, and a counterexample was provided by Turjányi [Tu84]. This was generalised (to\nthe replacement of $A+A$ by the $h$-fold sumset $hA$ for any $h\\geq 2$) by Ruzsa and Turjányi\n[RT85].\n\n\"Additive basis\" is `Set.IsAsymptoticAddBasis`: some finite $h$ has $hA$ containing every\nsufficiently large integer. The exact notion `Set.IsAddBasis`, which asks that $hA$ be all of\n$\\mathbb{N}$, would force $0, 1 \\in A$ and is not the class these results are about.\n\nThe linked file states the basis hypothesis as `∃ N₀, Set.Ici N₀ ⊆ iterated_sumset A k` and\nindexes both counting functions by a real $x$ through $\\lfloor x\\rfloor$, where the counting\nfunctions here are indexed by $N : \\mathbb{N}$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos337.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«337»","statement":"False ↔\n  ∀ (A : Set ℕ),\n    A.IsAsymptoticAddBasis →\n      ((fun N => ↑(A ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => ↑N) →\n        Filter.Tendsto (fun N => ↑((A + A) ∩ Set.Icc 1 N).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop Filter.atTop","subjects":["5","11"],"theorem":"Erdos337.erdos_337"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there an absolute constant $C > 0$ such that every integer $n$ with\n$\\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$?\n\nThis has been solved in the affirmative by Larsen - in fact, for any $\\epsilon>0$ there exists $L$\nsuch that if $n$ has only prime divisors $>L$ and $\\sigma(n)>(2+\\epsilon)n$ then $n$ is the distinct\nsum of proper divisors of $n$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos825.lean#L5893"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«825»","statement":"True ↔ ∃ C, ∃ (_ : C > 0), ∀ (n : ℕ), ↑((ArithmeticFunction.sigma 1) n) > C * ↑n → ∃ s ⊆ n.properDivisors, n = s.sum id","subjects":["11"],"theorem":"Erdos825.erdos_825"},{"answerKinds":[],"category":"research solved","docstring":"Show that if the constant $C > 0$ is such that every integer $n$ with\n$\\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$, then we\nmust have $C > 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«825»","statement":"∀ (C : ℝ),\n  0 < C → (∀ (n : ℕ), ↑((ArithmeticFunction.sigma 1) n) > C * ↑n → ∃ s ⊆ n.properDivisors, n = s.sum id) → 2 < C","subjects":["11"],"theorem":"Erdos825.erdos_825.variants.necessary_cond"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is the maximum size of a set $A ⊆ \\{1, \\dots, N\\}$ such that $ab + 1$ is never squarefree\n(for all $a, b ∈ A$) achieved by taking those $n ≡ 7 \\pmod{25}$?\n\nThis asks whether `Erdos848 N` holds for all $N$ (formulated using `A ⊆ Finset.range N`).\n\nThis was solved for all sufficiently large $N$ by Sawhney in this note. In fact, Sawhney proves\nsomething slightly stronger, that there exists some constant $c>0$ such that if\n$\\lvert A\\rvert \\geq (\\frac{1}{25}-c)N$ and $N$ is large then $A$ is contained in either\n$\\{ n\\equiv 7\\pmod{25}\\}$ or $\\{n\\equiv 18\\pmod{25}\\}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«848»","statement":"True ↔ ∀ (N : ℕ), Erdos848.Erdos848For N","subjects":["11"],"theorem":"Erdos848.erdos_848"},{"answerKinds":[],"category":"research solved","docstring":"There exists $N₀$ such that for all $N ≥ N₀$, if $A ⊆ \\{1, \\dots, N\\}$ satisfies that $ab + 1$\nis never squarefree for all $a, b ∈ A$, then $|A| ≤ |\\{n ≤ N : n ≡ 7 \\pmod{25}\\}|$.\n\nMore precisely, Sawhney proves: there exist absolute constants $η > 0$ and $N₀$\nsuch that for all $N ≥ N₀$, if $|A| ≥ (1/25 - η)N$ then $A ⊆ \\{n : n ≡ 7 \\pmod{25}\\}$ or\n$A ⊆ \\{n : n ≡ 18 \\pmod{25}\\}$.\n\nA complete formal Lean 4 proof is available at:\nhttps://github.com/The-Obstacle-Is-The-Way/erdos-banger ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/The-Obstacle-Is-The-Way/erdos-banger/blob/1cc2ac8e9d70516e979733c6ea5c4d2eb652d1f5/formal/lean/Erdos/848.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«848»","statement":"∀ᶠ (N : ℕ) in Filter.atTop, Erdos848.Erdos848For N","subjects":["11"],"theorem":"Erdos848.erdos_848.variants.asymptotic"},{"answerKinds":[],"category":"textbook","docstring":"It is possible to construct a Sidon set with positive density.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«329»","statement":"∃ A, IsSidon A ∧ 0 < Erdos329.sidonUpperDensity A","subjects":["5","11"],"theorem":"Erdos329.exists_sidon_pos_density"},{"answerKinds":[],"category":"research solved","docstring":"Krückeberg ([Kr61]) exhibited an infinite Sidon set `A` with\n`sidonUpperDensity A = 1 / Real.sqrt 2`, improving Erdős’ earlier\n`1 / 2` lower bound.\n\n[Kr61] Krückeberg, Fritz, $B\\sb{2}$-Folgen und verwandte Zahlenfolgen. J. Reine Angew. Math. (1961), 53-60.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«329»","statement":"∃ A, IsSidon A ∧ Erdos329.sidonUpperDensity A = 1 / √2","subjects":["5","11"],"theorem":"Erdos329.erdos_329.variants.kruckeberg_1961"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Turán [ErTu41] proved the upper bound of 1.\n\n[ErTu41] Erdős, P. and Turán, P., On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. (1941), 212-215.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«329»","statement":"∀ (A : Set ℕ), IsSidon A → Erdos329.sidonUpperDensity A ≤ 1","subjects":["5","11"],"theorem":"Erdos329.erdos_329.variants.turan_1941"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"**Erdős Problem 329.**\nLet `A ⊆ ℕ` be a Sidon set. How large can\n`lim sup_{N → ∞} |A ∩ {1,…,N}| / N^{1/2}`\nbe?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«329»","statement":"sSup {x | ∃ A, ∃ (_ : IsSidon A), Erdos329.sidonUpperDensity A = x} = sorry","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos329.erdos_329"},{"answerKinds":[],"category":"research solved","docstring":"Erdős proved that upper density `1 / 2` can be attained; in particular,\nthere exists a Sidon set whose upper density is *at least* `1 / 2`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«329»","statement":"∃ A, IsSidon A ∧ Erdos329.sidonUpperDensity A ≥ 1 / 2","subjects":["5","11"],"theorem":"Erdos329.erdos_329.variants.lower_bound"},{"answerKinds":[],"category":"research open","docstring":"The converse: if the maximum density is 1, then any finite Sidon set\ncan be embedded in a perfect difference set modulo $n > 0$.\n\nSince the consequent is false (due to the counterexamples in [Ha47] and [AlMi25]),\nthis implication is logically equivalent to the statement that the maximum upper\ndensity of Sidon sets is NOT 1.\nBecause the maximum upper density problem is still open, the truth value of this\nimplication is also an open research problem.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«329»","statement":"sSup {x | ∃ A, ∃ (_ : IsSidon A), Erdos329.sidonUpperDensity A = x} = 1 →\n  ∀ (A : Finset ℕ), IsSidon ↑A → ∃ D n, ∃ (_ : n > 0), ↑A ⊆ D ∧ IsPerfectDifferenceSet D n","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos329.erdos_329.variants.converse_implication"},{"answerKinds":[],"category":"research open","docstring":"Let $r \\ge 3$. It is not known if all large integers are the sum of at most $r$-many\n$r$-powerful numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«940»","statement":"True ↔ ∀ r ≥ 3, ∀ᶠ (x : ℕ) in Filter.atTop, ∃ S, S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ x = S.sum","subjects":["11"],"theorem":"Erdos940.erdos_940.variants.large_integers"},{"answerKinds":[],"category":"research open","docstring":"Is it true that the set of integers which are the sum of at most three cubes has density $0$?\n\nThe cubes are those of non-negative integers, which is what `Multiset ℕ` gives. This choice\ndecides the question: over `ℤ` a sum of three cubes is conjectured to represent every integer\nthat is not $\\pm 4 \\bmod 9$, which is density $7/9$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«940»","statement":"True ↔ {n | ∃ S, S.card ≤ 3 ∧ n = (Multiset.map (fun x => x ^ 3) S).sum}.HasDensity 0","subjects":["11"],"theorem":"Erdos940.erdos_940.variants.three_cubes"},{"answerKinds":[],"category":"research solved","docstring":"Heath-Brown [He88] has proved that all large numbers are the sum of at most three\n$2$-powerful numbers.\n\nThis is the case $r = 2$ of `Erdos1107.erdos_1107`, which asks the same question with $r + 1$\nsummands, and it is stated there as `Erdos1107.erdos_1107.variants.two`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«940»","statement":"∀ᶠ (x : ℕ) in Filter.atTop, ∃ S, S.card ≤ 3 ∧ (∀ s ∈ S, Nat.Full 2 s) ∧ x = S.sum","subjects":["11"],"theorem":"Erdos940.erdos_940.variants.three_powerful"},{"answerKinds":[],"category":"research open","docstring":"Let $r \\ge 3$. Is it true that the set of integers which are the sum of at most $r$ $r$-powerful numbers\nhas density $0$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«940»","statement":"True ↔ ∀ r ≥ 3, {n | ∃ S, S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ n = S.sum}.HasDensity 0","subjects":["11"],"theorem":"Erdos940.erdos_940"},{"answerKinds":[],"category":"research solved","docstring":"The set of integers which are the sum of at most two $2$-powerful numbers has density $0$.\n\nErdős called this 'easy'. Baker and Brüdern [BaBr94] gave the first proof in the literature.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«940»","statement":"{n | ∃ S, S.card ≤ 2 ∧ (∀ s ∈ S, Nat.Full 2 s) ∧ n = S.sum}.HasDensity 0","subjects":["11"],"theorem":"Erdos940.erdos_940.variants.two"},{"answerKinds":[],"category":"research solved","docstring":"Erdős proved (as described on p.35 of [Tu84b]) that such a sequence does exist with\n$\\lvert z_i\\rvert\\leq 1$. Indeed, Erdős' construction gives a value of $C\\approx 1.32$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«973»","statement":"∃ C > 1,\n  ∀ n ≥ 2,\n    ∃ z,\n      z 1 = 1 ∧\n        (∀ i ∈ Finset.Icc 1 n, ‖z i‖ ≤ 1) ∧ ∀ k ∈ Finset.Icc 2 (n + 1), ‖∑ i ∈ Finset.Icc 1 n, z i ^ k‖ < C ^ (-↑n)","subjects":["11"],"theorem":"Erdos973.erdos_973.variants.le_one"},{"answerKinds":[],"category":"research solved","docstring":"Tang notes in the comments that Theorem 6.1 of [Tu84b] implies that, if $\\lvert z_i\\rvert \\geq 1$\nfor all $i$, then\n$\\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert \\geq (2e)^{-(1+o(1))n}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«973»","statement":"∃ f,\n  (f =o[Filter.atTop] fun x => 1) ∧\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (z : ℕ → ℂ),\n        (∀ i ∈ Finset.Icc 1 n, 1 ≤ ‖z i‖) →\n          ∃ k ∈ Finset.Icc 2 (n + 1), ‖∑ i ∈ Finset.Icc 1 n, z i ^ k‖ ≥ (2 * Real.exp 1) ^ (-(1 + f n) * ↑n)","subjects":["11"],"theorem":"Erdos973.erdos_973.variants.tang"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a constant $C>1$ such that, for every $n\\geq 2$, there exists a sequence\n$z_i\\in \\mathbb{C}$ with $z_1=1$ and $\\lvert z_i\\rvert \\geq 1$ for all $1\\leq i\\leq n$ with\n$\\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert < C^{-n}$?\n\nThis is Problem 7.3 in [Ha74], where it is attributed to Erdős.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«973»","statement":"True ↔\n  ∃ C > 1,\n    ∀ n ≥ 2,\n      ∃ z,\n        z 1 = 1 ∧\n          (∀ i ∈ Finset.Icc 1 n, 1 ≤ ‖z i‖) ∧ ∀ k ∈ Finset.Icc 2 (n + 1), ‖∑ i ∈ Finset.Icc 1 n, z i ^ k‖ < C ^ (-↑n)","subjects":["11"],"theorem":"Erdos973.erdos_973"},{"answerKinds":[],"category":"research solved","docstring":"In [Er92f] (a different) Erdős refines this analysis, proving that if\n$M_2=\\min_{z_j} \\max_{2\\leq k\\leq n+1} \\left\\lvert \\sum_{1\\leq j\\leq n}z_j^k\\right\\rvert$\nwhere the minimum is taken over all $z_j\\in \\mathbb{C}$ with $\\max \\lvert z_j\\rvert=1$, then\n$(1.746)^{-n} < M_2 < (1.745)^{-n}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«973»","statement":"∀ n ≥ 2,\n  ∀ (M_2 : ℝ),\n    IsGLB\n        {M |\n          ∃ z,\n            (∀ j ∈ Finset.Icc 1 n, ‖z j‖ ≤ 1) ∧\n              (∃ j ∈ Finset.Icc 1 n, ‖z j‖ = 1) ∧\n                ∃ k ∈ Finset.Icc 2 (n + 1),\n                  M = ‖∑ j ∈ Finset.Icc 1 n, z j ^ k‖ ∧ ∀ m ∈ Finset.Icc 2 (n + 1), ‖∑ j ∈ Finset.Icc 1 n, z j ^ m‖ ≤ M}\n        M_2 →\n      1.746 ^ (-↑n) < M_2 ∧ M_2 < 1.745 ^ (-↑n)","subjects":["11"],"theorem":"Erdos973.erdos_973.variants.m2_bounds"},{"answerKinds":[],"category":"research open","docstring":"Given $n$ points in $\\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«104»","statement":"(fun n => ↑(Erdos104.maxUnitCircleCount n)) =o[Filter.atTop] fun n => ↑n ^ 2","subjects":["52"],"theorem":"Erdos104.erdos_104"},{"answerKinds":[],"category":"research open","docstring":"For any $0<\\alpha<1$, let $f(\\alpha,n)=\\frac{1}{\\log n}\\sum_{1\\leq k\\leq n}(\\tfrac{1}{2}-\n\\{ \\alpha k\\})$. Does $f(\\alpha,n)$ have an asymptotic distribution function?\n\nIn other words, is there a non-decreasing function $g$ such that $g(-\\infty)=0$, $g(\\infty)=1$,\nand $\\lim_{n\\to \\infty}\\lvert \\{ \\alpha\\in (0,1): f(\\alpha,n)\\leq c\\}\\rvert=g(c)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1002»","statement":"True ↔\n  ∃ g,\n    Monotone g ∧\n      Filter.Tendsto g Filter.atBot (nhds 0) ∧\n        Filter.Tendsto g Filter.atTop (nhds 1) ∧\n          ∀ (c : ℝ),\n            Filter.Tendsto\n              (fun n =>\n                (MeasureTheory.volume\n                    {α |\n                      α ∈ Set.Ioo 0 1 ∧\n                        (fun α n => 1 / Real.log ↑n * ∑ k ∈ Finset.Icc 1 n, (1 / 2 - Int.fract (α * ↑k))) α n ≤\n                          c}).toReal)\n              Filter.atTop (nhds (g c))","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos1002.erdos_1002"},{"answerKinds":[],"category":"research solved","docstring":"Kesten [Ke60] proved that if $f(\\alpha,\\beta,n)=\\frac{1}{\\log n}\\sum_{1\\leq k\\leq n}(\\tfrac{1}{2}-\n\\{\\beta+\\alpha k\\})$ then $f(\\alpha,\\beta,n)$ has asymptotic distribution function\n$g(c)=\\frac{1}{\\pi}\\int_{-\\infty}^{\\rho c}\\frac{1}{1+t^2}\\mathrm{d}t$, where $\\rho>0$ is an explicit\nconstant.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1002»","statement":"∃ ρ > 0,\n  have g := fun c => 1 / Real.pi * ∫ (t : ℝ) in Set.Iic (ρ * c), 1 / (1 + t ^ 2);\n  ∀ (c : ℝ),\n    Filter.Tendsto\n      (fun n =>\n        (MeasureTheory.volume\n            {(α, β) |\n              α ∈ Set.Icc 0 1 ∧\n                β ∈ Set.Icc 0 1 ∧ 1 / Real.log ↑n * ∑ k ∈ Finset.Icc 1 n, (1 / 2 - Int.fract (β + α * ↑k)) ≤ c}).toReal)\n      Filter.atTop (nhds (g c))","subjects":["11"],"theorem":"Erdos1002.erdos_1002.variants.kesten"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Originally asked by Bollobás and Erdős in 'a colloquium on graph theory at Tihany' with $m^{2/3}$\nreplaced by $m^{3/4}$. Folkman showed this is false with the counterexample $K_{n,n^2}$, which has\n$n^3$ edges, and yet every subgraph with $>n^2+\\binom{n}{2}$ edges contains a $C_4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1008»","statement":"False ↔\n  ∃ c > 0,\n    ∀ (V : Type) [Fintype V] (G : SimpleGraph V),\n      ∃ H ≤ G, (SimpleGraph.cycleGraph 4).Free H ∧ c * ↑G.edgeSet.ncard ^ (3 / 4) ≤ ↑H.edgeSet.ncard","subjects":["5"],"theorem":"Erdos1008.erdos_1008.variants.three_quarters"},{"answerKinds":[],"category":"research solved","docstring":"In [Er71] Erdős revises the conjecture to $m^{2/3}$, and notes $\\gg m^{1/2}$ is trivial.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1008»","statement":"∃ c > 0,\n  ∀ (V : Type) [Fintype V] (G : SimpleGraph V),\n    ∃ H ≤ G, (SimpleGraph.cycleGraph 4).Free H ∧ c * ↑G.edgeSet.ncard ^ (1 / 2) ≤ ↑H.edgeSet.ncard","subjects":["5"],"theorem":"Erdos1008.erdos_1008.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Folkman's counterexample $K_{n,n^2}$, which has $n^3$ edges, and yet every subgraph with\n$>n^2+\\binom{n}{2}$ edges contains a $C_4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1008»","statement":"∀ (n : ℕ),\n  (completeBipartiteGraph (Fin n) (Fin (n ^ 2))).edgeSet.ncard = n ^ 3 ∧\n    ∀ H ≤ completeBipartiteGraph (Fin n) (Fin (n ^ 2)),\n      n ^ 2 + n.choose 2 < H.edgeSet.ncard → (SimpleGraph.cycleGraph 4).IsContained H","subjects":["5"],"theorem":"Erdos1008.erdos_1008.variants.folkman"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does every graph with $m$ edges contain a subgraph with $\\gg m^{2/3}$ edges which contains\nno $C_4$?\n\nThis problem was first solved in the affirmative by Conlon, Fox, and Sudakov [CFS14b]. A simple\nproof is given by Hunter in the comments.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1008.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1008»","statement":"True ↔\n  ∃ c > 0,\n    ∀ (V : Type) [Fintype V] (G : SimpleGraph V),\n      ∃ H ≤ G, (SimpleGraph.cycleGraph 4).Free H ∧ c * ↑G.edgeSet.ncard ^ (2 / 3) ≤ ↑H.edgeSet.ncard","subjects":["5"],"theorem":"Erdos1008.erdos_1008"},{"answerKinds":[],"category":"test","docstring":"$21$ satisfies the Erdős 1142 property: $21 - 2 = 19$, $21 - 4 = 17$, $21 - 8 = 13$,\n$21 - 16 = 5$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"Erdos1142.Erdos1142Prop 21","subjects":["11"],"theorem":"Erdos1142.erdos_1142.test_21"},{"answerKinds":[],"category":"test","docstring":"$75$ satisfies the Erdős 1142 property: $75 - 2 = 73$, $75 - 4 = 71$, $75 - 8 = 67$,\n$75 - 16 = 59$, $75 - 32 = 43$, $75 - 64 = 11$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"Erdos1142.Erdos1142Prop 75","subjects":["11"],"theorem":"Erdos1142.erdos_1142.test_75"},{"answerKinds":[],"category":"test","docstring":"$7$ satisfies the Erdős 1142 property: $7 - 2 = 5$ and $7 - 4 = 3$ are prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"Erdos1142.Erdos1142Prop 7","subjects":["11"],"theorem":"Erdos1142.erdos_1142.test_7"},{"answerKinds":[],"category":"test","docstring":"$105$ satisfies the Erdős 1142 property: the largest known example.\n$105 - 2 = 103$, $105 - 4 = 101$, $105 - 8 = 97$, $105 - 16 = 89$, $105 - 32 = 73$,\n$105 - 64 = 41$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"Erdos1142.Erdos1142Prop 105","subjects":["11"],"theorem":"Erdos1142.erdos_1142.test_105"},{"answerKinds":[],"category":"test","docstring":"$4$ satisfies the Erdős 1142 property: $4 - 2 = 2$ is prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"Erdos1142.Erdos1142Prop 4","subjects":["11"],"theorem":"Erdos1142.erdos_1142.test_4"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many $n > 2$ such that $n - 2^k$ is prime for all $k \\geq 1$ with $2^k < n$?\n\nThe only known such $n$ are $4, 7, 15, 21, 45, 75, 105$ (OEIS [A039669](https://oeis.org/A039669)).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"True ↔ Infinite ↑{n | Erdos1142.Erdos1142Prop n}","subjects":["11"],"theorem":"Erdos1142.erdos_1142"},{"answerKinds":[],"category":"test","docstring":"$45$ satisfies the Erdős 1142 property: $45 - 2 = 43$, $45 - 4 = 41$, $45 - 8 = 37$,\n$45 - 16 = 29$, $45 - 32 = 13$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"Erdos1142.Erdos1142Prop 45","subjects":["11"],"theorem":"Erdos1142.erdos_1142.test_45"},{"answerKinds":[],"category":"test","docstring":"$15$ satisfies the Erdős 1142 property: $15 - 2 = 13$, $15 - 4 = 11$, $15 - 8 = 7$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"Erdos1142.Erdos1142Prop 15","subjects":["11"],"theorem":"Erdos1142.erdos_1142.test_15"},{"answerKinds":[],"category":"research solved","docstring":"Mientka and Weitzenkamp [MiWe69] proved that the only $n \\leq 2^{44}$ such that $n > 2$ and\n$n - 2^k$ is prime for all $k \\geq 1$ with $2^k < n$ are $4, 7, 15, 21, 45, 75, 105$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"{n | n ≤ 2 ^ 44 ∧ Erdos1142.Erdos1142Prop n} = {4, 7, 15, 21, 45, 75, 105}","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos1142.erdos_1142.variants.mientka_weitzenkamp"},{"answerKinds":[],"category":"test","docstring":"$106$ does not satisfy the Erdős 1142 property ($106 - 2 = 104 = 8 \\times 13$). ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1142»","statement":"¬Erdos1142.Erdos1142Prop 106","subjects":["11"],"theorem":"Erdos1142.erdos_1142.test_not_106"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Estimate an upper bound for$\\alpha(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«507»","statement":"have ans := sorry;\nErdos507.α =O[Filter.atTop] ans ∧ ans =o[Filter.atTop] Erdos507.upperBarrier","subjects":["51"],"theorem":"Erdos507.erdos_507.upper"},{"answerKinds":[],"category":"research solved","docstring":"It is trivial that $\\alpha(n) \\ll 1/n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«507»","statement":"Erdos507.α =O[Filter.atTop] fun n => 1 / ↑n","subjects":["51"],"theorem":"Erdos507.erdos_507.variants.upper_trivial"},{"answerKinds":[],"category":"research solved","docstring":"Current best upper bound [CPZ24]: $\\alpha(n) \\ll n^{-7/6 + o(1)}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«507»","statement":"∃ o, Filter.Tendsto o Filter.atTop (nhds 0) ∧ Erdos507.α =O[Filter.atTop] fun n => Erdos507.upperBarrier n * ↑n ^ o n","subjects":["51"],"theorem":"Erdos507.erdos_507.variants.upper_cpz24"},{"answerKinds":[],"category":"research solved","docstring":"Erdős observed that $\\alpha(n) \\gg 1/n^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«507»","statement":"(fun n => 1 / ↑n ^ 2) =O[Filter.atTop] Erdos507.α","subjects":["51"],"theorem":"Erdos507.erdos_507.variants.lower_erdos"},{"answerKinds":[],"category":"research solved","docstring":"Current best lower bound [KPS82].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«507»","statement":"Erdos507.lowerBest =O[Filter.atTop] Erdos507.α","subjects":["51"],"theorem":"Erdos507.erdos_507.variants.lower_kps82"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $\\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which\ndetermine a triangle of area at most $\\alpha(n)$. Estimate $\\alpha(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«507»","statement":"Asymptotics.IsEquivalent Filter.atTop Erdos507.α sorry","subjects":["51"],"theorem":"Erdos507.erdos_507.equivalent"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Estimate a lower bound for$\\alpha(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«507»","statement":"have ans := sorry;\nErdos507.lowerBest =o[Filter.atTop] ans ∧ ans =O[Filter.atTop] Erdos507.α","subjects":["51"],"theorem":"Erdos507.erdos_507.lower"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 598:**\nLet $m$ be an infinite cardinal and $\\kappa$ be the successor cardinal of $2^{\\aleph_0}$.\nCan one colour the countable subsets of $m$ using $\\kappa$ many colours so that every\n$X \\subseteq m$ with $|X| = \\kappa$ contains subsets of all possible colours?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«598»","statement":"∀ (m : Type u_1) [Infinite m], True ↔ ∃ c, ∀ (X : Set m), Cardinal.mk ↑X = Erdos598.κ → c '' {s | ↑s ⊆ X} = Set.univ","subjects":["3","5"],"theorem":"Erdos598.erdos_598"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Find the value of the limit of `MinOverlapQuotient`!\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Filter.Tendsto Erdos36.MinOverlapQuotient Filter.atTop (nhds sorry)","subjects":["5","11"],"theorem":"Erdos36.erdos_36"},{"answerKinds":[],"category":"research solved","docstring":"A lower bound of $1 - frac{1}{\\sqrt 2}$.\nScherk (written communication), see\n[On the minimal overlap problem of Erdös](https://eudml.org/doc/206397)\nby *Leo Moser*, Аста Аrithmetica V, p. 117-119, 1959\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"1 - (√2)⁻¹ < Filter.liminf Erdos36.MinOverlapQuotient Filter.atTop","subjects":["5","11"],"theorem":"Erdos36.minimum_overlap.variants.lower.scherk_1955"},{"answerKinds":[],"category":"research solved","docstring":"The example (with $N$ even), $A = \\{\\frac N 2 + 1, \\dots, \\frac{3N}{2}\\}$\nshows an upper bound of $\\frac 1 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Filter.limsup Erdos36.MinOverlapQuotient Filter.atTop ≤ 1 / 2","subjects":["5","11"],"theorem":"Erdos36.minimum_overlap.variants.upper.erdos_1955"},{"answerKinds":[],"category":"research solved","docstring":"A lower bound of $0.379005$.\nSee [Erdős' minimum overlap problem](https://arxiv.org/abs/2201.05704)\nby *Ethan Patrick White*, 2022\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"0.379005 < Filter.liminf Erdos36.MinOverlapQuotient Filter.atTop","subjects":["5","11"],"theorem":"Erdos36.minimum_overlap.variants.lower.white_2022"},{"answerKinds":[],"category":"test","docstring":"For `n = 4` the best splitting of `{1, …, 8}` still has maximum overlap `2`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Erdos36.M 4 = 2","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos36.M_four"},{"answerKinds":[],"category":"test","docstring":"This example calculates the value of $M 1$. The set is $\\{1, 2\\}$, so the only partition is\n$A = \\{1\\}, B = \\{2\\}$ (or vice versa). The possible differences are $1 - 2 = -1$ and $2 - 1 = 1$.\nThe `Overlap` for $k=-1$ is 1 (if $A=\\{1\\}, B=\\{2\\}$) and for $k=1$ also 1 (if $A=\\{2\\}, B=\\{1\\}$ ).\nThe `MaxOverlap` is $1$, since the `Overlap` is $0$ for other $k$.\nThus, $M 1 = 1$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Erdos36.M 1 = 1","subjects":["5","11"],"theorem":"Erdos36.M_one"},{"answerKinds":[],"category":"research solved","docstring":"An upper bound of $0.38200298812318988$.\nSee [Advances in the Minimum Overlap Problem](https://doi.org/10.1006%2Fjnth.1996.0064)\nby *Jan Kristian Haugland*, Journal of Number Theory Volume 58, Issue 1, p 71-78, 1996\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Filter.limsup Erdos36.MinOverlapQuotient Filter.atTop ≤ 0.38200298812318988","subjects":["5","11"],"theorem":"Erdos36.minimum_overlap.variants.upper.haugland_1996"},{"answerKinds":[],"category":"research solved","docstring":"A lower bound of $\\sqrt{4 - \\sqrt{15}}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"(4 - 15 ^ (1 / 2)) ^ (1 / 2) < Filter.liminf Erdos36.MinOverlapQuotient Filter.atTop","subjects":["5","11"],"theorem":"Erdos36.minimum_overlap.variants.lower.haugland_1996"},{"answerKinds":[],"category":"research solved","docstring":"An upper bound of $\\frac 2 5$.\nSee [Minimal overlapping under translation.](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-62/issue-6)\nby *T. S. Motzkin*, *K. E. Ralston* and *J. L. Selfridge*,\nin \"The summer meeting in Seattle\" by *V. L. Klee Jr.*, Bull. Amer. Math. Soc.62, p. 558, 1956\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Filter.limsup Erdos36.MinOverlapQuotient Filter.atTop ≤ 2 / 5","subjects":["5","11"],"theorem":"Erdos36.minimum_overlap.variants.upper.MRS_1956"},{"answerKinds":[],"category":"research solved","docstring":"An upper bound of $0.3809268534330870$.\nSee [The minimum overlap problem](https://www.neutreeko.net/mop/index.htm)\nby *Jan Kristian Haugland*\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Filter.limsup Erdos36.MinOverlapQuotient Filter.atTop ≤ 0.3809268534330870","subjects":["5","11"],"theorem":"Erdos36.minimum_overlap.variants.upper.haugland_2022"},{"answerKinds":[],"category":"test","docstring":"For `n = 5` the best splitting of `{1, …, 10}` has maximum overlap `3`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Erdos36.M 5 = 3","subjects":["5","11"],"theorem":"Erdos36.M_five"},{"answerKinds":[],"category":"textbook","docstring":"A lower bound of $\\frac 1 4$.\nSee [Some remarks on number theory (in Hebrew)](https://users.renyi.hu/~p_erdos/1955-13.pdf)\nby *Paul Erdős*, Riveon Lematematika 9, p.45-48,1955\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"1 / 4 < Filter.liminf Erdos36.MinOverlapQuotient Filter.atTop","subjects":["5","11"],"theorem":"Erdos36.minimum_overlap.variants.lower.erdos_1955"},{"answerKinds":[],"category":"research solved","docstring":"The limit of `MinOverlapQuotient` exists and it is less than $0.385694$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"∃ c, Filter.Tendsto Erdos36.MinOverlapQuotient Filter.atTop (nhds c) ∧ c < 0.385694","subjects":["5","11"],"theorem":"Erdos36.erdos_36.variants.exists"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Find a better lower bound!\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"∃ c, 0.379005 < c ∧ c ≤ Filter.liminf Erdos36.MinOverlapQuotient Filter.atTop ∧ c = sorry","subjects":["5","11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos36.erdos_36.variants.lower"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Find a better upper bound!\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"∃ c < 0.380926853433087, Filter.limsup Erdos36.MinOverlapQuotient Filter.atTop ≤ c ∧ c = sorry","subjects":["5","11"],"theorem":"Erdos36.erdos_36.variants.upper"},{"answerKinds":[],"category":"research solved","docstring":"A lower bound of $\\frac{4 - \\sqrt{6}}{5}$.\nSee [On the intersection of a linear set with the translation of its complement](https://bibliotekanauki.pl/articles/969027)\nby *Stanisław Świerczkowski1*, Colloquium Mathematicum 5(2), p. 185-197, 1958\n\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«36»","statement":"(4 - 6 ^ (1 / 2)) / 5 < Filter.liminf Erdos36.MinOverlapQuotient Filter.atTop","subjects":["5","11"],"theorem":"Erdos36.minimum_overlap.variants.lower.swierczkowski_1958"},{"answerKinds":[],"category":"test","docstring":"For `n = 3` the best splitting of `{1, …, 6}` has maximum overlap `2`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Erdos36.M 3 = 2","subjects":["5","11"],"theorem":"Erdos36.M_three"},{"answerKinds":[],"category":"test","docstring":"For $n = 2$, the set is $\\{1, 2, 3, 4\\}$. The balanced partition $A = \\{1, 4\\}, B = \\{2, 3\\}$\nhas all four pairwise differences ($\\pm 1, \\pm 2$) distinct, so `MaxOverlap = 1`.\nAny balanced partition has both pieces nonempty, so `MaxOverlap \\geq 1`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«36»","statement":"Erdos36.M 2 = 1","subjects":["5","11"],"theorem":"Erdos36.M_two"},{"answerKinds":[],"category":"research open","docstring":"Let $S \\subseteq \\mathbb{Z}^3$ be a finite set and let $A = \\lbrace a_1, a_2, \\ldots \\rbrace$ be\nan infinite $S$-walk, so that $a_{i+1} - a_i \\in S$ for all $i$. Must $A$ contain three collinear\npoints?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«193»","statement":"True ↔\n  ∀ (S : Set (Fin 3 → ℤ)),\n    S.Finite →\n      ∀ (a : ℕ → Fin 3 → ℤ),\n        Erdos193.IsSWalk S a →\n          (Set.range a).Infinite → Erdos193.HasCollinearTriple ℚ (Set.range fun n => Int.cast ∘ a n)","subjects":["5"],"theorem":"Erdos193.erdos_193"},{"answerKinds":[],"category":"research solved","docstring":"[GeRa79] showed that the answer is yes for $\\mathbb{Z}^2$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«193»","statement":"∀ (S : Set (Fin 2 → ℤ)),\n  S.Finite →\n    ∀ (a : ℕ → Fin 2 → ℤ),\n      Erdos193.IsSWalk S a → (Set.range a).Infinite → Erdos193.HasCollinearTriple ℚ (Set.range fun n => Int.cast ∘ a n)","subjects":["5"],"theorem":"Erdos193.erdos_193_z2"},{"answerKinds":[],"category":"research solved","docstring":"Tao resolved erdos_442 in the negative in Theorem 1 of https://arxiv.org/pdf/2407.04226.\nThe following is a formalisation of that theorem with $C_0 = 1$.\n\nLet $\\operatorname{Log} x := \\max\\{\\log x, 1\\}$,\n$\\operatorname{Log}_2x = \\operatorname{Log} (\\operatorname{Log} x)$, and\n$\\operatorname{Log}_3x = \\operatorname{Log}(\\operatorname{Log}(\\operatorname{Log} x)).$\nThere exists a set $A$ of natural numbers such that\n$$\n\\sum_{n\\in A: n\\leq x} \\frac{1}{n} =\n  \\exp\\left(\\left(\\left(\\frac{1}{2} + o(1)\\right)\\operatorname{Log}_2^{1/2}x \\operatorname{Log}_3x\\right)\\right)\n$$\nand\n$$\n\\sum_{n, m\\in A: n, m\\leq x} \\frac{1}{\\operatorname{lcm}(n, m)}\\ll\\left(\\sum_{n\\in A: n\\leq x} \\frac{1}{n}\\right)^2\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«442»","statement":"∃ A f C,\n  ∃ (_ : 0 < C) (_ : f =o[Filter.atTop] 1),\n    ∀ᶠ (x : ℝ) in Filter.atTop,\n      ∑ n ∈ (A ∩ Set.Icc 1 ⌊x⌋₊).toFinset, 1 / ↑n =\n          Real.exp\n            ((1 / 2 + f x) * √(Erdos442.Real.maxLogOne (Erdos442.Real.maxLogOne x)) *\n              Erdos442.Real.maxLogOne (Erdos442.Real.maxLogOne (Erdos442.Real.maxLogOne x))) ∧\n        |∑ nm ∈ ((A ∩ Set.Icc 1 ⌊x⌋₊) ×ˢ (A ∩ Set.Icc 1 ⌊x⌋₊)).toFinset, 1 / ↑(nm.1.lcm nm.2)| ≤\n          C * (∑ n ∈ (A ∩ Set.Icc 1 ⌊x⌋₊).toFinset, 1 / ↑n) ^ 2","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos442.erdos_442.variants.tao"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $\\operatorname{Log} x := \\max\\{\\log x, 1\\}$,\n$\\operatorname{Log}_2x = \\operatorname{Log} (\\operatorname{Log} x)$, and\n$\\operatorname{Log}_3x = \\operatorname{Log}(\\operatorname{Log}(\\operatorname{Log} x)).$\nIs it true that if $A\\subseteq\\mathbb{N}$ is such that\n$$\n\\frac{1}{\\operatorname{Log}_2 x} \\sum_{n\\in A: n\\leq x} \\frac{1}{n}\\to\\infty\n$$\nthen\n$$\n\\left(\\sum_{n\\in A: n\\leq x} \\frac{1}{n}\\right)^2 \\sum_{n, m \\in A: n < m \\leq x}\n\\frac{1}{\\operatorname{lcm}(n, m)}\\to\\infty\n$$\nas $x\\to\\infty$?\n\nTao [Ta24b] has shown this is false.\n\n[Ta24b] Tao, T., _Dense sets of natural numbers with unusually large least common multiples_.\narXiv:2407.04226 (2024).\n\nNote: the informal and formal statements follow the solution paper https://arxiv.org/pdf/2407.04226\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«442»","statement":"False ↔\n  ∀ (A : Set ℕ),\n    Filter.Tendsto\n        (fun x => 1 / Erdos442.Real.maxLogOne (Erdos442.Real.maxLogOne x) * ∑ n ∈ (A ∩ Set.Icc 1 ⌊x⌋₊).toFinset, 1 / ↑n)\n        Filter.atTop Filter.atTop →\n      Filter.Tendsto\n        (fun x =>\n          1 / (∑ n ∈ (A ∩ Set.Icc 1 ⌊x⌋₊).toFinset, 1 / ↑n) ^ 2 *\n            ∑ nm ∈ (Erdos442.Set.bddProdUpper A x).toFinset, 1 / ↑(nm.1.lcm nm.2))\n        Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos442.erdos_442"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $A\\subset \\mathbb{R}$ does not contain a 3-term arithmetic progression then must $\\mathbb{R}\\backslash A$ contain an infinite arithmetic progression?\n\nBaumgartner [Ba75] answered this in the negative.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos199.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«199»","statement":"False ↔ ∀ (A : Set ℝ), ThreeAPFree A → ∃ S, S.IsAPOfLength ⊤ ∧ S ⊆ Aᶜ","subjects":["5"],"theorem":"Erdos199.erdos_199"},{"answerKinds":[],"category":"textbook","docstring":"**The empty hypergraph is trivially obligatory**: The 3-uniform hypergraph on `PEmpty` (no\nvertices, no edges) appears in every hypergraph via the empty injection.\n\nThis degenerate case confirms the definition is well-formed.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«593»","statement":"IsObligatory { edges := ∅, uniform := ⋯ }","subjects":["5"],"theorem":"Erdos593.erdos_593.variants.empty_hypergraph_obligatory"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 593 — Sufficient direction**: Every finite 2-colorable 3-uniform\nhypergraph is obligatory.\n\nThis is the converse direction of the `erdos_593` characterization: if 2-colorability\nmatches the graph-case characterization (bipartite ⇔ obligatory), then every 2-colorable\nfinite 3-uniform hypergraph must appear in every 3-uniform hypergraph of chromatic number\n$> \\aleph_0$.\n\nTogether with `erdos_593.variants.obligatory_implies_two_colorable`, this implies `erdos_593`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«593»","statement":"True ↔ ∀ (W : Type) [inst : Fintype W] (F : ThreeUniformHypergraph W), F.IsTwoColorable → IsObligatory F","subjects":["5"],"theorem":"Erdos593.erdos_593.variants.two_colorable_implies_obligatory"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 593 — Necessary direction**: Every obligatory finite 3-uniform\nhypergraph is 2-colorable.\n\nThis is the natural necessary condition for the conjectural characterization in `erdos_593`:\nif a finite 3-uniform hypergraph `F` is not 2-colorable, one expects to construct a\nhypergraph with large chromatic number that contains no copy of `F`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«593»","statement":"True ↔ ∀ (W : Type) [inst : Fintype W] (F : ThreeUniformHypergraph W), IsObligatory F → F.IsTwoColorable","subjects":["5"],"theorem":"Erdos593.erdos_593.variants.obligatory_implies_two_colorable"},{"answerKinds":[],"category":"textbook","docstring":"**Monotonicity of the obligatory property**: If `F₁` appears in `F₂` and `F₂` is obligatory,\nthen `F₁` is also obligatory.\n\n**Proof:** For any `H` with $\\chi(H) > \\aleph_0$, since `F₂` is obligatory, `F₂` appears\nin `H` via some injection `φ₂`. Since `F₁` appears in `F₂` via `φ₁`, the composition\n`φ₂ ∘ φ₁` witnesses that `F₁` appears in `H`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«593»","statement":"∀ {W₁ W₂ : Type} [inst : Fintype W₁] [inst_1 : Fintype W₂] [inst_2 : DecidableEq W₂] {F₁ : ThreeUniformHypergraph W₁}\n  {F₂ : ThreeUniformHypergraph W₂}, F₁.Appears F₂ → IsObligatory F₂ → IsObligatory F₁","subjects":["5"],"theorem":"Erdos593.erdos_593.variants.obligatory_monotone"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Graph analogue — no odd cycle is obligatory (Erdős–Galvin–Hajnal [EGH75])**:\nFor every odd $k \\geq 3$, there exists a graph with chromatic cardinal $\\aleph_1$ that\ncontains no cycle of length $k$. This shows the class of obligatory graphs is strictly\nsmaller than all finite graphs.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«593»","statement":"True ↔\n  ∀ (k : ℕ), Odd k → 3 ≤ k → ∃ V G, G.chromaticCardinal = Cardinal.aleph 1 ∧ IsEmpty (SimpleGraph.cycleGraph k →g G)","subjects":["5"],"theorem":"Erdos593.erdos_593.variants.graph_case_no_odd_cycle"},{"answerKinds":[],"category":"test","docstring":"**Conjunction of the two open implications gives the conjectured characterization**: if both\n`obligatory_implies_two_colorable` and `two_colorable_implies_obligatory` hold, then the\ncharacterization conjectured in `erdos_593` (`IsObligatory F ↔ F.IsTwoColorable`) follows by\nelementary `Iff` manipulation.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«593»","statement":"(∀ (W : Type) [inst : Fintype W] (F : ThreeUniformHypergraph W), IsObligatory F → F.IsTwoColorable) →\n  (∀ (W : Type) [inst : Fintype W] (F : ThreeUniformHypergraph W), F.IsTwoColorable → IsObligatory F) →\n    ∀ (W : Type) [inst : Fintype W] (F : ThreeUniformHypergraph W), IsObligatory F ↔ F.IsTwoColorable","subjects":["5"],"theorem":"Erdos593.erdos_593.variants.implications_combine"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 593 (\\$500)**: Characterize those finite 3-uniform hypergraphs which appear\nin every 3-uniform hypergraph of chromatic number $> \\aleph_0$.\n\nA natural conjectural characterization, recorded here, is that the obligatory finite 3-uniform\nhypergraphs are exactly the 2-colorable ones (Property B). The forward direction\n(`IsObligatory → IsTwoColorable`) and converse (`IsTwoColorable → IsObligatory`) are stated as\nseparate variants below; in the graph case ($r = 2$), Erdős–Galvin–Hajnal [EGH75] proved the\nanalogous result (obligatory ⇔ bipartite).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«593»","statement":"True ↔ ∀ (W : Type) [inst : Fintype W] (F : ThreeUniformHypergraph W), IsObligatory F ↔ F.IsTwoColorable","subjects":["5"],"theorem":"Erdos593.erdos_593"},{"answerKinds":[],"category":"textbook","docstring":"**No hyperedges implies chromatic cardinal ≤ 1**: A 3-uniform hypergraph with no edges can\nbe properly colored with a single color, so its chromatic cardinal is at most 1. In\nparticular, $\\chi(H) > \\aleph_0$ implies `H` has at least one hyperedge.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«593»","statement":"∀ {V : Type} (H : ThreeUniformHypergraph V), Cardinal.aleph0 < H.chromaticCardinal → H.edges.Nonempty","subjects":["5"],"theorem":"Erdos593.erdos_593.variants.nonempty_edges_if_large_chromatic"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Graph analogue — bipartite graphs are obligatory (Erdős–Galvin–Hajnal [EGH75])**:\nFor the 2-uniform (graph) case, a graph of chromatic cardinal $> \\aleph_0$ must contain all\nfinite bipartite graphs. Specifically, for every finite bipartite graph `F` and every graph\n`G` with chromatic cardinal $> \\aleph_0$, there is a graph embedding from `F` into `G`.\n\nThis uses `Nonempty (F ↪g G)` (graph embedding), aligned with the injective vertex map\nused in the hypergraph `Appears` definition.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«593»","statement":"True ↔\n  ∀ (V : Type u_1) (G : SimpleGraph V),\n    Cardinal.aleph0 < G.chromaticCardinal →\n      ∀ (W : Type u_2) [Fintype W] (F : SimpleGraph W), F.IsBipartite → Nonempty (F ↪g G)","subjects":["5"],"theorem":"Erdos593.erdos_593.variants.graph_case_bipartite_obligatory"},{"answerKinds":[],"category":"textbook","docstring":"**Vertices must be uncountable**: Every 3-uniform hypergraph with chromatic cardinal\n$> \\aleph_0$ must have an uncountable vertex set.\n\n**Proof:** If `V` is countable, there exists an injection `φ : V → ℕ`. Using distinct natural\nnumbers as colors gives a proper coloring, so $\\chi(H) \\leq \\#\\mathbb{N} = \\aleph_0$,\ncontradicting $\\chi(H) > \\aleph_0$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«593»","statement":"∀ {V : Type} (H : ThreeUniformHypergraph V), Cardinal.aleph0 < H.chromaticCardinal → ¬Countable V","subjects":["5"],"theorem":"Erdos593.erdos_593.variants.uncountable_vertices_if_large_chromatic"},{"answerKinds":[],"category":"research open","docstring":"Erdős conjectured that the triangular lattice is best possible in 2D, in particular that\n$f_2(3n^2 + 3n + 1) < 9n^2 + 3n$.\n\nNote: in [Er75f] is read $9n^2 + 6n$, but this seems to be a typo.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1084»","statement":"∀ {n : ℕ}, Erdos1084.f 2 (3 * n ^ 2 + 3 * n + 1) = 9 * n ^ 2 + 3 * n","subjects":["52"],"theorem":"Erdos1084.erdos_1084.variants.triangular_optimal_d2"},{"answerKinds":[],"category":"research solved","docstring":"It is easy to check that $f_2(n) < 3n$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1084»","statement":"∀ {n : ℕ}, n ≠ 0 → Erdos1084.f 2 n < 3 * n","subjects":["52"],"theorem":"Erdos1084.erdos_1084.variants.easy_upper_d2"},{"answerKinds":[],"category":"research solved","docstring":"Erdős showed that there is some constant $c > 0$ such that $f_2(n) < 3n - c n^{1/2}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1084»","statement":"∃ c > 0, ∀ n > 0, ↑(Erdos1084.f 2 n) < 3 * ↑n - c * √↑n","subjects":["52"],"theorem":"Erdos1084.erdos_1084.variants.upper_d2"},{"answerKinds":[],"category":"research solved","docstring":"It is easy to check that $f_1(n) = n - 1$. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/Sanexxxx777/formal-conjectures/blob/9e4f3845be122a8fa3190d38543ebdd0a6f25605/FormalConjectures/ErdosProblems/1084.lean#L232"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1084»","statement":"∀ {n : ℕ}, Erdos1084.f 1 n = n - 1","subjects":["52"],"theorem":"Erdos1084.erdos_1084.variants.upper_d1"},{"answerKinds":[],"category":"research solved","docstring":"Erdős claims the existence of two constants $c_1, c_2 > 0$\nsuch that $6n - c_1 n^{2/3} ≤ f_3(n) \\le 6n - c_2 n^{2/3}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1084»","statement":"∃ c₁,\n  ∃ c₂ > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      6 * ↑n - c₁ * ↑n ^ (2 / 3) ≤ ↑(Erdos1084.f 3 n) ∧ ↑(Erdos1084.f 3 n) ≤ 6 * ↑n - c₂ * ↑n ^ (2 / 3)","subjects":["52"],"theorem":"Erdos1084.erdos_1084.variants.upper_lower_d3"},{"answerKinds":[],"category":"research solved","docstring":"Let $a_1,\\ldots,a_r,b_1,\\ldots,b_r\\in \\mathbb{N}$ such that $\\sum_{i}\\frac{1}{a_i}>1$. For any\nfinite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\\ldots,x_n)$ let $T(A)$ denote\nthe sequence of length $rn$ given by\n$$(a_ix_j+b_i)_{1\\leq j\\leq n, 1\\leq i\\leq r}.$$\nProve that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated\nelements.\n\nThis is true. This appears to have first been shown by Klarner [Kl82], with a generalisation\ngiven by Kolpakov and Talambutsa [KoTa22]. Essentially the same proof was found independently by\nBarreto in the comment section.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«481»","statement":"∀ {r : ℕ} (a b : Fin r → ℕ), (∀ (i : Fin r), 0 < a i) → 1 < ∑ i, 1 / ↑(a i) → ∃ k, ¬((Erdos481.T a b)^[k] [1]).Nodup","subjects":["5","11"],"theorem":"Erdos481.erdos_481"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that there are no solutions to `n! = x^k ± y^k` with `x,y,n ∈ ℕ`, `x*y > 1`, and\n`k > 2`?\n\nThe answer is no: Jonas Barfield found the counterexample `10! = 48^4 - 36^4` (equivalently,\n`10! + 36^4 = 48^4`).\n\nThis is discussed in problem D2 of Guy's collection [Gu04].\n\nThis was formalized in Lean by Lu using Codex.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«399»","statement":"False ↔ ¬∃ n x y k, 1 < x * y ∧ 2 < k ∧ (n.factorial = x ^ k + y ^ k ∨ n.factorial + y ^ k = x ^ k)","subjects":["11"],"theorem":"Erdos399.erdos_399"},{"answerKinds":[],"category":"research solved","docstring":"Cambie has also observed that considerations modulo $8$ rule out any solutions to $n!=x^4+y^4$ with\n$(x,y)=1$ and $xy>1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«399»","statement":"∀ {n x y : ℕ}, x.Coprime y → 1 < x * y → n.factorial ≠ x ^ 4 + y ^ 4","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos399.erdos_399.variants.cambie"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Obláth observed that the Bertrand-style fact (first proved by Breusch [Br32]) that, if\n$q_i$ is the sequence of primes congruent to $3\\pmod{4}$ then $q_{i+1}<2q_i$ except for $q_1=3$,\ntogether with Fermat's theorem on the sums of two squares implies that the only solution to\n$n!=x^2+y^2$ is $6!=12^2+24^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«399»","statement":"∀ {n x y : ℕ}, 1 < x * y → n.factorial = x ^ 2 + y ^ 2 → n = 6 ∧ (x = 12 ∧ y = 24 ∨ x = 24 ∧ y = 12)","subjects":["11"],"theorem":"Erdos399.erdos_399.variants.sum_two_squares"},{"answerKinds":[],"category":"research solved","docstring":"Pollack and Shapiro [PoSh73] proved there are no solutions to $n!=x^4-1$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«399»","statement":"∀ (n x : ℕ), n.factorial + 1 ≠ x ^ 4","subjects":["11"],"theorem":"Erdos399.erdos_399.variants.pollack_shapiro"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Obláth [ErOb37] proved this is true when $(x,y)=1$ and $k\\neq 4$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«399»","statement":"∀ {n x y k : ℕ}, x.Coprime y → 1 < x * y → 2 < k → k ≠ 4 → n.factorial ≠ x ^ k + y ^ k ∧ n.factorial + y ^ k ≠ x ^ k","subjects":["11"],"theorem":"Erdos399.erdos_399.variants.erdos_oblath"},{"answerKinds":[],"category":"research open","docstring":"Let $h_1(n) = h(n)$ and $h_k(n) = h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist\n$i$ and $j$ such that $h_i(m) = h_j(n)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«414»","statement":"True ↔ ∀ m > 0, ∀ n > 0, ∃ i j, Erdos414.h^[i] m = Erdos414.h^[j] n","subjects":["11"],"theorem":"Erdos414.erdos_414"},{"answerKinds":[],"category":"research open","docstring":"Let $A\\subseteq \\mathbb{N}$ be an infinite set and consider the following\ngreedy algorithm for a rational $x\\in (0,1)$: choose the minimal $n\\in A$ such\nthat $n\\geq 1/x$ and repeat with $x$ replaced by $x-\\frac{1}{n}$. If this\nterminates after finitely many steps then this produces a representation of\n$x$ as the sum of distinct unit fractions with denominators from $A$.\n\nDoes this process always terminate if $x$ has odd denominator and $A$ is the\nset of odd numbers? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«282»","statement":"∀ {x : ℚ}, x ∈ Set.Ioo 0 1 → Odd x.den → Erdos282.greedyUnitFractionRem {n | Odd n} x =ᶠ[Filter.atTop] 0","subjects":["5"],"theorem":"Erdos282.erdos_282"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"More generally, for which pairs $x$ and $A$ does this process terminate? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«282»","statement":"∀ (x : ℚ) (A : Set ℕ), Erdos282.greedyUnitFractionRem A x =ᶠ[Filter.atTop] 0 ↔ (x, A) ∈ sorry","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"Erdos282.erdos_282.variants.general"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«282»","statement":"∀ (n : ℕ), Erdos282.greedyUnitFractionRem Set.univ (1 / ↑n) 1 = 0","subjects":["5"],"theorem":"Erdos282.greedyUnitFractionRem_one"},{"answerKinds":[],"category":"research open","docstring":"Graham has shown that $\\frac{m}{n}$ is the sum of distinct unit fractions\nwith denominators $\\equiv a\\pmod{d}$ if and only if\n$$\\left(\\frac{n}{(n,a,d)},\\frac{d}{(a,d)}\\right)=1.$$\nDoes the greedy algorithm always\nterminate in such cases?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«282»","statement":"∀ {x : ℚ},\n  x ∈ Set.Ioo 0 1 →\n    ∀ {a d : ℕ},\n      1 < d →\n        (x.den / x.den.gcd (a.gcd d)).gcd (d / a.gcd d) = 1 →\n          (Erdos282.greedyUnitFractionRem {n | n ≡ a [MOD d]} x =ᶠ[Filter.atTop] 0 ↔ True)","subjects":["5"],"theorem":"Erdos282.erdos_282.variants.graham"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«282»","statement":"Erdos282.greedyUnitFractionRem {n | IsSquare n} 1 0 = 0","subjects":["5"],"theorem":"Erdos282.greedyUnitFractionRem_sq_one"},{"answerKinds":[],"category":"textbook","docstring":"In 1202 Fibonacci observed that this process terminates for any $x$ when $A=\\mathbb{N}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«282»","statement":"∀ {x : ℚ}, x ∈ Set.Ioo 0 1 → Erdos282.greedyUnitFractionRem Set.univ x =ᶠ[Filter.atTop] 0","subjects":["5"],"theorem":"Erdos282.erdos_282.variants.fibonacci"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«282»","statement":"∀ (n : ℕ), Erdos282.greedyUnitFractionRem Set.univ (1 / ↑n) 0 = 0","subjects":["5"],"theorem":"Erdos282.greedyUnitFractionRem_zero"},{"answerKinds":[],"category":"research open","docstring":"Graham has also shown that $x$ is the sum of distinct unit fractions with\nsquare denominators if and only if $x\\in [0,\\pi^2/6-1)\\cup [1,\\pi^2/6)$. Does the\ngreedy algorithm for this always terminate? Erdős and Graham believe not - indeed, perhaps it\nfails to terminate almost always.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«282»","statement":"True ↔\n  ∀ (x : ℚ),\n    ↑x ∈ Set.Ico 0 (Real.pi ^ 2 / 6 - 1) ∪ Set.Ico 1 (Real.pi ^ 2 / 6) →\n      Erdos282.greedyUnitFractionRem {n | IsSquare n} x =ᶠ[Filter.atTop] 0","subjects":["5"],"theorem":"Erdos282.erdos_282.variants.sq"},{"answerKinds":[],"category":"research open","docstring":"Let $A=\\{n_1 < n_2 < \\cdots\\}$ be the sequence of powerful numbers (if $p\\mid n$ then $p^2\\mid n$).\nAre there only finitely many three-term progressions of consecutive terms $n_k,n_{k+1},n_{k+2}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«938»","statement":"True ↔\n  {P |\n      (↑P).IsAPOfLength 3 ∧\n        ∃ k, P = {Nat.nth Nat.Powerful k, Nat.nth Nat.Powerful (k + 1), Nat.nth Nat.Powerful (k + 2)}}.Finite","subjects":["11"],"theorem":"Erdos938.erdos_938"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does there exist a $k$ such that every sufficiently large integer can be written in the form\n$$\\prod_{i=1}^k a_i - \\sum_{i=1}^k a_i$$\nfor some integers $a_i\\geq 2$?\n\nErdős attributes this question to Schinzel. Eli Seamans has observed that the answer is yes (with $k=2$) for a very simple reason: $n = 2(n+2)-(2+(n+2))$. There may well have been some additional constraint in the problem as Schinzel posed it, but [Er61] does not record what this is.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos493.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«493»","statement":"True ↔ ∃ k N, ∀ (n : ℤ), N ≤ n → ∃ a, (∀ (i : Fin k), 2 ≤ a i) ∧ ∏ i, a i - ∑ i, a i = n","subjects":["11"],"theorem":"Erdos493.erdos_493"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $f_{k,k}(x) \\gg_\\epsilon x^{1-\\epsilon}$ for all $\\epsilon>0$?\n\nThis would have significant applications to Waring's problem. Erdős and Graham describe this as\n'unattackable by the methods at our disposal'.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«323»","statement":"True ↔ ∀ k ≥ 1, ∀ ε > 0, (fun x => ↑x ^ (1 - ε)) =O[Filter.atTop] fun x => ↑(Erdos323.f k k x)","subjects":["11"],"theorem":"Erdos323.erdos_323.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is it true that if $m < k$ then $f_{k,m}(x) \\gg x^{m/k}$ for sufficiently large $x$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«323»","statement":"True ↔ ∀ (k m : ℕ), 1 ≤ m → m < k → (fun x => ↑x ^ (↑m / ↑k)) =O[Filter.atTop] fun x => ↑(Erdos323.f k m x)","subjects":["11"],"theorem":"Erdos323.erdos_323.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"The case $k=2$ was resolved by Landau, who showed $f_{2,2}(x) \\sim \\frac{cx}{\\sqrt{\\log x}}$ for\nsome constant $c>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«323»","statement":"∃ c > 0, Asymptotics.IsEquivalent Filter.atTop (fun x => ↑(Erdos323.f 2 2 x)) fun x => c * ↑x / √(Real.log ↑x)","subjects":["11"],"theorem":"Erdos323.erdos_323.variants.k_eq_2"},{"answerKinds":[],"category":"research open","docstring":"For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«323»","statement":"True ↔ ∀ k > 2, (fun x => ↑(Erdos323.f k k x)) =o[Filter.atTop] fun x => ↑x","subjects":["11"],"theorem":"Erdos323.erdos_323.variants.k_gt_2"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The main conjecture: for any finite set of integers $A$ with $|A| = n$, the number of distinct\ncommon differences in three-term arithmetic progressions is $O(n^{3/2})$.\n\nThis conjecture was resolved negatively by showing that the problem is exactly equivalent to\nBourgain's sums-differences question [Bo99], which was introduced as an arithmetic path towards\nthe Kakeya conjecture. Under this equivalence:\n- The greatest achievable exponent for this problem is equal to the smallest constant $c$\n  achievable for Bourgain's sums-differences question:\n  $$|A -_G B| \\ll \\max(|A|, |B|, |A +_G B|)^c$$\n- The $O(n^{3/2})$ prediction is disproved because the lower bound has been shown to satisfy\n  $c \\ge 1.77898$ (due to Zheng and AlphaEvolve [GGTW25], improving on Lemm [Le15]), which is\n  strictly greater than $3/2 = 1.5$.\n- The best known upper bound is $c \\le 11/6 \\approx 1.833$ (due to Katz and Tao [KaTa99]).\n- While the specific $O(n^{3/2})$ prediction is resolved negatively, the general question of\n  determining the exact optimal exponent $c$ remains open.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/f13dd54b520cdf2136fdd3a04f0f9fa50e311358/FormalConjectures/ErdosProblems/1097.lean#L306"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1097»","statement":"False ↔ ∃ C > 0, ∀ (A : Finset ℤ), ↑(Erdos1097.CommonDifferencesThreeTermAP A).ncard ≤ C * ↑A.card ^ (3 / 2)","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos1097.erdos_1097"},{"answerKinds":[],"category":"textbook","docstring":"A weaker bound has been proven: there are always at most $n^2$ such values of $d$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1097»","statement":"∀ (A : Finset ℤ), (Erdos1097.CommonDifferencesThreeTermAP A).ncard ≤ A.card ^ 2","subjects":["11"],"theorem":"Erdos1097.erdos_1097.variants.weaker"},{"answerKinds":[],"category":"textbook","docstring":"A trivial lower bound: for sufficiently large `n` there exist sets $A$ with $|A| = n$ that contain at least $\\Omega(n)$\ndistinct common differences of three-term arithmetic progressions.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1097»","statement":"∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∃ A, A.card = n ∧ c * ↑n ≤ ↑(Erdos1097.CommonDifferencesThreeTermAP A).ncard","subjects":["11"],"theorem":"Erdos1097.erdos_1097.variants.lower_bound"},{"answerKinds":[],"category":"API","docstring":"If `a` has property `Erdos951Prop` and `1 < a 0`, then `a` is a set of Beurling\nprime numbers. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«951»","statement":"∀ {a : ℕ → ℝ}, 1 < a 0 → StrictMono a → Erdos951.Erdos951Prop a → IsBeurlingPrimes a","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos951.erdos_951.variants.isBeurlingPrimes"},{"answerKinds":[],"category":"research solved","docstring":"Beurling conjectured that if the number of Beurling integer in `[1, x]`\nis `x + o(log x)`, then `a` must be the sequence of primes. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«951»","statement":"∀ (a : ℕ → ℝ),\n  IsBeurlingPrimes a →\n    (fun x => ↑(BeurlingIntegers a ∩ Set.Iic x).ncard - x) =o[Filter.atTop] Real.log → a = Nat.cast ∘ Nat.nth Nat.Prime","subjects":["11"],"theorem":"Erdos951.erdos_951.variants.beurling"},{"answerKinds":[],"category":"research open","docstring":"If `1 < a 0 < ...` has property `Erdos951Prop`, is it true that `#{a i ≤ x} ≤ π x`? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«951»","statement":"True ↔\n  ∀ (a : ℕ → ℝ),\n    1 < a 0 →\n      StrictMono a → Erdos951.Erdos951Prop a → ∀ᶠ (x : ℝ) in Filter.atTop, {i | a i ≤ x}.ncard ≤ ⌊x⌋₊.primeCounting","subjects":["11"],"theorem":"Erdos951.erdos_951"},{"answerKinds":[],"category":"research solved","docstring":"Let $a_1 < a_2 < \\dotsc$ be an infinite sequence of positive integers such that\n$\\frac{a_{i+1}}{a_i} \\to 1$. If every arithmetic progression contains infinitely many\nintegers which are the sum of distinct $a_i$ then every sufficiently large integer is\nthe sum of distinct $a_i$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«253»","statement":"¬∀ (a : ℕ → ℕ),\n    0 < a 0 →\n      Erdos253.RepresentsAPs a →\n        Filter.Tendsto (fun n => ↑(a (n + 1)) / ↑(a n)) Filter.atTop (nhds 1) →\n          subsetSums (Set.range a) ∈ Filter.cofinite","subjects":["11"],"theorem":"Erdos253.erdos_253"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does there exist a minimal basis $A \\subset \\mathbb{N}$ with positive density\nsuch that, for any $n \\in A$, the (upper) density of integers which\ncannot be represented without using $n$ is positive?\n\nNeither set is asked to have a density, only to have positive upper density, so\n`Set.upperDensity` is used for both rather than `Set.HasPosDensity`. As with many of Erdős'\nquestions \"positive density\" here most likely means positive upper density, and in [Er80] he\nconsiders either the lower or the upper density for the integers not representable without a\nfixed `n`. Requiring the density to exist would ask a strictly harder question than the one\nposed. See #3979.\n\nSuch a set exists, so the answer is yes. The linked proof gives one of order `2`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/a5ffc87b3d684fc00ea906b9e746c283740da900/problems/330/Erdos330.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«330»","statement":"True ↔\n  ∃ A h,\n    Erdos330.MinAsymptoticAddBasisOfOrder A h ∧\n      0 < A.upperDensity ∧ ∀ n ∈ A, 0 < (Erdos330.UnrepWithout A n h).upperDensity","subjects":["5","11"],"theorem":"Erdos330.erdos_330_statement"},{"answerKinds":[],"category":"research solved","docstring":"Härtter and Nathanson proved that there exist additive bases which do not contain\nany minimal additive bases. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/hjyuh/formal-conjectures/blob/e0da6ec78953b17618895a093d4bee90fd3f6f67/FormalConjectures/ErdosProblems/868.lean#L132"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«868»","statement":"∀ (o : ℕ),\n  1 < o →\n    ∃ A,\n      A.IsAsymptoticAddBasisOfOrder o ∧\n        ∀ B ⊆ A, B.IsAsymptoticAddBasisOfOrder o → ∃ b ∈ B, (B \\ {b}).IsAsymptoticAddBasisOfOrder o","subjects":["5","11"],"theorem":"Erdos868.erdos_868.variants.Hartter_Nathanson"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which\n$n$ can be written as the sum of two elements from $A$. If $f(n) \\to \\infty$ as $n \\to \\infty$, then\nmust $A$ contain a minimal additive basis of order $2$?\n\nLarsen and Larsen [LaLa26] answered this in the negative.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«868»","statement":"False ↔\n  ∀ (A : Set ℕ),\n    A.IsAsymptoticAddBasisOfOrder 2 →\n      Filter.Tendsto (fun n => Erdos868.ncard_add_repr A 2 n) Filter.atTop Filter.atTop →\n        ∃ B ⊆ A, B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \\ {b}).IsAsymptoticAddBasisOfOrder 2","subjects":["5","11"],"theorem":"Erdos868.erdos_868.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Erdős and Nathanson proved that this is true if $f(n) > (\\log \\frac{4}{3})^{-1} \\log n$ for\nall large $n$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«868»","statement":"True ↔\n  ∀ (A : Set ℕ),\n    A.IsAsymptoticAddBasisOfOrder 2 →\n      (∀ᶠ (n : ℕ) in Filter.atTop, (Real.log (4 / 3))⁻¹ * Real.log ↑n < ↑(Erdos868.ncard_add_repr A 2 n)) →\n        ∃ B ⊆ A, B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \\ {b}).IsAsymptoticAddBasisOfOrder 2","subjects":["5","11"],"theorem":"Erdos868.erdos_868.variants.fixed_ε"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which\n$n$ can be written as the sum of two elements from $A$. If $f(n) > \\epsilon \\log n$ for large $n$\nand an arbitrary fixed $\\epsilon > 0$, then must $A$ contain a minimal additive\nbasis of order $2$?\n\nLarsen and Larsen [LaLa26] constructed a counterexample with $f(n) > c \\log n$ for all large $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«868»","statement":"False ↔\n  ∀ (A : Set ℕ),\n    ∀ ε > 0,\n      A.IsAsymptoticAddBasisOfOrder 2 →\n        (∀ᶠ (n : ℕ) in Filter.atTop, ε * Real.log ↑n < ↑(Erdos868.ncard_add_repr A 2 n)) →\n          ∃ B ⊆ A, B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \\ {b}).IsAsymptoticAddBasisOfOrder 2","subjects":["5","11"],"theorem":"Erdos868.erdos_868.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Let $S(n)$ denote the largest integer such that, for all $1 ≤ k < n$, the binomial coefficient\n$\\binom{n}{k}$ is divisible by $p^S(n)$ for some prime $p$ (depending on $k$).Then\n$\\limsup S(n) = \\infty$.\n\nThis was formalized in Lean by Tao.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/80a965e9a85d3f3dabd0a398a49adab6742ea6e0/FormalConjectures/ErdosProblems/379.lean#L123"},{"conditions":[],"kind":"lean4","link":"https://github.com/teorth/analysis/blob/4f623b0f4cacdb967f1f8132db0becaee0f1fb3d/Analysis/Misc/erdos_379.lean#L90"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«379»","statement":"Filter.limsup (fun n => ↑(Erdos379.S n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos379.erdos_379"},{"answerKinds":[],"category":"research solved","docstring":"The interval `[⌊n/3⌋, n]` is fork-free, and therefore `f n` is at least `⌈2n / 3⌉`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1062»","statement":"∀ (n : ℕ), ⌈2 * ↑n / 3⌉₊ ≤ Erdos1062.f n","subjects":["11"],"theorem":"Erdos1062.erdos_1062.variants.lower_bound"},{"answerKinds":[],"category":"research open","docstring":"Erdős asked whether the limiting density `f n / n` exists and, if so, whether it is\nirrational. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1062»","statement":"(∃ l, Filter.Tendsto (fun n => ↑(Erdos1062.f n) / ↑n) Filter.atTop (nhds l) ∧ Irrational l) ↔ True","subjects":["11"],"theorem":"Erdos1062.erdos_1062.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Lebensold proved that for large `n`, the function `f n` lies between `0.6725 n` and\n`0.6736 n`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1062»","statement":"∀ᶠ (n : ℕ) in Filter.atTop, 0.6725 * ↑n ≤ ↑(Erdos1062.f n) ∧ ↑(Erdos1062.f n) ≤ 0.6736 * ↑n","subjects":["11"],"theorem":"Erdos1062.erdos_1062.variants.lebensold_bounds"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $G$ is a graph on $n$ vertices which has no two adjacent vertices of degree $\\geq 3$ then\n$$R(G)\\ll n,$$\nwhere the implied constant is absolute.\n\nA problem of Burr and Erdős. Solved in the affirmative by Alon [Al94].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«800»","statement":"True ↔\n  ∃ C > 0,\n    ∀ (n : ℕ) (V : Type) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n      Fintype.card V = n →\n        (∀ (u v : V), G.Adj u v → ¬(3 ≤ G.degree u ∧ 3 ≤ G.degree v)) → ↑G.diagonalGraphRamsey ≤ C * ↑n","subjects":["5"],"theorem":"Erdos800.erdos_800"},{"answerKinds":[],"category":"research solved","docstring":"Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\\sum_n \\frac 1 {F_{2^n}}$ is irrational.\n\n\nFormal proof provided by AlphaProof\nRef:\n* [Go74] Good, I. J., _A reciprocal series of Fibonacci numbers_\n* [BiHo76] Hoggatt, Jr., V. E. and Bicknell, Marjorie, _A reciprocal series of Fibonacci numbers with subscripts $2\\sp{n}k$_\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/2663234a28260853790aa5752d8d4550ff0ab1ca/FormalConjectures/ErdosProblems/267.lean#L56"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«267»","statement":"Irrational (∑' (k : ℕ), 1 / ↑(Nat.fib (2 ^ k)))","subjects":["11"],"theorem":"Erdos267.erdos_267.variants.specialization_pow_two"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $F_1=F_2=1$ and $F_{n+1} = F_n + F_{n-1}$ be the Fibonacci sequence.\nLet $n_1 < n_2 < \\dots$ be an infinite sequence with $\\frac{n_{k+1}}{n_k} \\ge c > 1$. Must\n$\\sum_k \\frac 1 {F_{n_k}}$ be irrational?\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-267/Research/Basic.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«267»","statement":"True ↔\n  ∀ (n : ℕ → ℕ),\n    ∀ c > 1, StrictMono n → (∀ (k : ℕ), c ≤ ↑(n (k + 1)) / ↑(n k)) → Irrational (∑' (k : ℕ), 1 / ↑(Nat.fib (n k)))","subjects":["11"],"theorem":"Erdos267.erdos_267"},{"answerKinds":[],"category":"research solved","docstring":"The sum $\\sum_n \\frac 1 {F_{n}}$ itself was proved to be irrational by André-Jeannin.\n\nRef: André-Jeannin, Richard, _Irrationalité de la somme des inverses de certaines suites récurrentes_.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«267»","statement":"Irrational (∑' (k : ℕ), 1 / ↑(Nat.fib k))","subjects":["11"],"theorem":"Erdos267.erdos_267.variants.fibonacci_inverse_sum"},{"answerKinds":[],"category":"research open","docstring":"Let $F_1=F_2=1$ and $F_{n+1} = F_n + F_{n-1}$ be the Fibonacci sequence.\nLet $n_1 < n_2 < \\dots$ be an infinite sequence with $\\frac {n_k}{k} \\to \\infty$. Must\n$\\sum_k \\frac 1 {F_{n_k}}$ be irrational?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«267»","statement":"True ↔\n  ∀ (n : ℕ → ℕ),\n    StrictMono n →\n      Filter.Tendsto (fun k => ↑(n (k + 1)) / ↑k.succ) Filter.atTop Filter.atTop →\n        Irrational (∑' (k : ℕ), 1 / ↑(Nat.fib (n k)))","subjects":["11"],"theorem":"Erdos267.erdos_267.variants.generalisation_ratio_limit_to_infinity"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Asked by Erdős and Nešetřil, who also ask whether $c(3m+2)=3^m$.\n\nNote that the lower bound $1.4457\\leq \\alpha$ of Gaspers and Mackenzie [GaMa18] provides a\nnegative answer to the above question of Erdős and Nešetřil.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«150»","statement":"False ↔ ∀ (m : ℕ), Erdos150.maxMinimalCuts (3 * m + 2) = 3 ^ m","subjects":["5"],"theorem":"Erdos150.erdos_150.variants.erdos_nesetril"},{"answerKinds":["Prop"],"category":"research solved","docstring":"A minimal cut of a graph is a minimal set of vertices whose removal disconnects the graph. Let\n$c(n)$ be the maximum number of minimal cuts a graph on $n$ vertices can have.\n\nDoes $c(n)^{1/n}\\to \\alpha$ for some $\\alpha <2$?\n\nIt is unclear in [Er88] whether Erdős knew that the limit existed, which follows from a simple\nargument first given in the literature (to the best of my knowledge) by Bradač [Br24].\n\nThat $\\alpha<2$ was proved by Fomin, Kratsch, Todinca, and Villanger [FKTV08], who proved\n$\\alpha \\leq 1.7087$. This was independently studied by Bradač [Br24] (unaware of this earlier\nwork), who proved that $\\alpha \\leq 2^{H(1/3)}\\approx 1.8899$, where $H(\\cdot)$ is the binary\nentropy function.\n\nThis was formalized in Lean by Monticone using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos150.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«150»","statement":"True ↔ ∃ α < 2, Filter.Tendsto (fun n => ↑(Erdos150.maxMinimalCuts n) ^ (1 / ↑n)) Filter.atTop (nhds α)","subjects":["5"],"theorem":"Erdos150.erdos_150"},{"answerKinds":[],"category":"research solved","docstring":"The current best-known bounds on $\\alpha$ are\n$$1.4457\\leq \\alpha \\leq \\frac{1+\\sqrt{5}}{2}\\approx 1.618.$$\nThe lower bound is due to Gaspers and Mackenzie [GaMa18].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«150»","statement":"∀ (α : ℝ), Filter.Tendsto (fun n => ↑(Erdos150.maxMinimalCuts n) ^ (1 / ↑n)) Filter.atTop (nhds α) → 1.4457 ≤ α","subjects":["5"],"theorem":"Erdos150.erdos_150.variants.lower_bound"},{"answerKinds":[],"category":"research solved","docstring":"Seymour observed that $c(3m+2)\\geq 3^m$, as seen by the graph of $m$ independent paths of length\n$4$ joining two vertices.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«150»","statement":"∀ (m : ℕ), 3 ^ m ≤ Erdos150.maxMinimalCuts (3 * m + 2)","subjects":["5"],"theorem":"Erdos150.erdos_150.variants.seymour"},{"answerKinds":[],"category":"research solved","docstring":"The current best-known bounds on $\\alpha$ are\n$$1.4457\\leq \\alpha \\leq \\frac{1+\\sqrt{5}}{2}\\approx 1.618.$$\nThe upper bound is due to Fomin and Villanger [FoVi12] (with a simpler proof in [GaMa18]).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«150»","statement":"∀ (α : ℝ), Filter.Tendsto (fun n => ↑(Erdos150.maxMinimalCuts n) ^ (1 / ↑n)) Filter.atTop (nhds α) → α ≤ (1 + √5) / 2","subjects":["5"],"theorem":"Erdos150.erdos_150.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"The prime number theorem immediately implies a lower bound of $\\gg N(\\log N)^2$ for the sum of\nsquares of gaps between consecutive primes.\n\nFormal proof linked here provided by AlphaProof.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mzhorvath1/formal-conjectures/blob/032848c62fdf4c422bb0ee6663dc8d009d456c2c/FormalConjectures/ErdosProblems/233.lean#L57"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«233»","statement":"(fun N => ↑N * Real.log ↑N ^ 2) =O[Filter.atTop] fun N => ∑ n ∈ Finset.range N, ↑(primeGap n) ^ 2","subjects":["11"],"theorem":"Erdos233.erdos_233.variants.lower_bound"},{"answerKinds":[],"category":"research open","docstring":"A conjecture by Heath-Brown:\nThe sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«233»","statement":"(fun N => ∑ n ∈ Finset.range N, ↑(primeGap n) ^ 2) =O[Filter.atTop] fun N => ↑N * Real.log ↑N ^ 2","subjects":["11"],"theorem":"Erdos233.erdos_233"},{"answerKinds":[],"category":"research solved","docstring":"Cramér proved an upper bound of $O(N(\\log N)^4)$ conditional on the Riemann hypothesis.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«233»","statement":"RiemannHypothesis → (fun N => ∑ n ∈ Finset.range N, ↑(primeGap n) ^ 2) =O[Filter.atTop] fun N => ↑N * Real.log ↑N ^ 4","subjects":["11"],"theorem":"Erdos233.erdos_233.variants.upper_bound"},{"answerKinds":[],"category":"research open","docstring":"Similarly, prove or disprove that\n$$R(3,k+1)-R(3,k)=o(k).$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«544»","statement":"True ↔\n  (fun k => ↑(SimpleGraph.classicalRamsey 3 (k + 1)) - ↑(SimpleGraph.classicalRamsey 3 k)) =o[Filter.atTop] fun k => ↑k","subjects":["5"],"theorem":"Erdos544.erdos_544.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Show that\n$$R(3,k+1)-R(3,k)\\to\\infty$$\nas $k\\to \\infty$.\n\nA problem of Erdős and Sós.\nThis problem is #8 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«544»","statement":"Filter.Tendsto (fun k => ↑(SimpleGraph.classicalRamsey 3 (k + 1)) - ↑(SimpleGraph.classicalRamsey 3 k)) Filter.atTop\n  Filter.atTop","subjects":["5"],"theorem":"Erdos544.erdos_544.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"`∑ σ 1 n / n!` is irrational. This is proved in [ErSt74]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«252»","statement":"Irrational (Erdos252.erdos_252_sum 1)","subjects":["11"],"theorem":"Erdos252.erdos_252.variants.k_eq_one"},{"answerKinds":[],"category":"research solved","docstring":"`∑ σ 2 n / n!` is irrational. This is proved in [ErKa54]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«252»","statement":"Irrational (Erdos252.erdos_252_sum 2)","subjects":["11"],"theorem":"Erdos252.erdos_252.variants.k_eq_two"},{"answerKinds":[],"category":"research solved","docstring":"`∑ σ 4 n / n!` is irrational. This is proved in [Pr22]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«252»","statement":"Irrational (Erdos252.erdos_252_sum 4)","subjects":["11"],"theorem":"Erdos252.erdos_252.variants.k_eq_four"},{"answerKinds":[],"category":"research solved","docstring":"If Schinzel's conjecture is true, then `∑ σ k n / n!` is irrational for all `k`. This is proved\nin [ScPu06]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«252»","statement":"(∀ (fs : Finset (Polynomial ℤ)),\n    (∀ f ∈ fs, BunyakovskyCondition f) →\n      SchinzelCondition fs → Infinite ↑{n | ∀ f ∈ fs, Prime (Polynomial.eval n f).natAbs}) →\n  ∀ (k : ℕ), Irrational (Erdos252.erdos_252_sum k)","subjects":["11"],"theorem":"Erdos252.erdos_252.variants.schinzel"},{"answerKinds":[],"category":"research solved","docstring":"`∑ σ 3 n / n!` is irrational. This is proved in [ScPu06] and [FLC07]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«252»","statement":"Irrational (Erdos252.erdos_252_sum 3)","subjects":["11"],"theorem":"Erdos252.erdos_252.variants.k_eq_three"},{"answerKinds":[],"category":"research open","docstring":"For a fixed `k ≥ 5`, is `∑ σ k n / n!` irrational?. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«252»","statement":"True ↔ ∀ k ≥ 5, Irrational (Erdos252.erdos_252_sum k)","subjects":["11"],"theorem":"Erdos252.erdos_252.variants.k_ge_five"},{"answerKinds":[],"category":"research open","docstring":"Erdős Problem 252: irrationality of the sum for a given $k$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«252»","statement":"True ↔ ∀ k ≥ 1, Irrational (Erdos252.erdos_252_sum k)","subjects":["11"],"theorem":"Erdos252.erdos_252"},{"answerKinds":[],"category":"research solved","docstring":"If the prime `k`-tuples conjecture is true, then `∑ σ k n / n!` is irrational. This is proved\nin [FLC07]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«252»","statement":"∀ {k : ℕ},\n  4 ≤ k →\n    (∀ (a : Fin k → ℕ+) (b : Fin k → ℕ),\n        (∀ (p : ℕ), Nat.Prime p → ∃ n, ¬p ∣ ∏ i, (↑(a i) * n + b i)) →\n          {n | ∀ (i : Fin k), Nat.Prime (↑(a i) * n + b i)}.Infinite) →\n      Irrational (Erdos252.erdos_252_sum k)","subjects":["11"],"theorem":"Erdos252.erdos_252.variants.prime_tuples"},{"answerKinds":[],"category":"research solved","docstring":"`∑ σ 0 n / n!` is irrational. This is proved in [ErSt71]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«252»","statement":"Irrational (Erdos252.erdos_252_sum 0)","subjects":["11"],"theorem":"Erdos252.erdos_252.variants.k_eq_zero"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a maximal Sidon set $A\\subset \\{1,\\ldots,N\\}$ of size $O(N^{1/3})$?\n\nA question of Erdős, Sárközy, and Sós [ESS94].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«156»","statement":"True ↔ (fun N => ↑(Erdos156.minMaximalSidonSet N)) =O[Filter.atTop] fun N => ↑N ^ (1 / 3)","subjects":["5"],"theorem":"Erdos156.erdos_156"},{"answerKinds":[],"category":"research solved","docstring":"Ruzsa [Ru98b] constructed a maximal Sidon set of size $\\ll (N\\log N)^{1/3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«156»","statement":"(fun N => ↑(Erdos156.minMaximalSidonSet N)) =O[Filter.atTop] fun N => (↑N * Real.log ↑N) ^ (1 / 3)","subjects":["5"],"theorem":"Erdos156.erdos_156.variants.ruzsa_upper_bound"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«156»","statement":"∀ (n : ℕ), IsSidon ↑(Finset.greedySidonBelow n)","subjects":["5"],"theorem":"Erdos156.greedySidonSet_isSidon"},{"answerKinds":[],"category":"research solved","docstring":"It is easy to prove that the greedy construction of a maximal Sidon set in $\\{1,\\ldots,N\\}$ has size\n$\\gg N^{1/3}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«156»","statement":"(fun N => ↑N ^ (1 / 3)) =O[Filter.atTop] fun N => ↑(Finset.greedySidonBelow N).card","subjects":["5"],"theorem":"Erdos156.erdos_156.variants.greedy_lower_bound"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«313»","statement":"(6, {2, 3}) ∈ Erdos313.erdos313Solutions","subjects":["11"],"theorem":"Erdos313.erdos_313.variants.solution_6_2_3"},{"answerKinds":[],"category":"textbook","docstring":"There are at least 8 primary pseudoperfect numbers. The first eight terms of\n[A54377](https://oeis.org/A54377) are exhibited together with their explicit\nprime decompositions.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«313»","statement":"8 ≤ {n | Erdos313.IsPrimaryPseudoperfect n}.encard","subjects":["11"],"theorem":"Erdos313.erdos_313.variants.exists_at_least_eight_primary_pseudoperfect"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many pairs `(m, P)` where `m ≥ 2` is an integer\nand `P` is a set of distinct primes such that the following equation holds:\n$\\sum_{p \\in P} \\frac{1}{p} = 1 - \\frac{1}{m}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«313»","statement":"True ↔ Erdos313.erdos313Solutions.Infinite","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos313.erdos_313"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that the set of primary pseudoperfect numbers is infinite.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«313»","statement":"{n | Erdos313.IsPrimaryPseudoperfect n}.Infinite","subjects":["11"],"theorem":"Erdos313.erdos_313.variants.primary_pseudoperfect_are_infinite"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«313»","statement":"(42, {2, 3, 7}) ∈ Erdos313.erdos313Solutions","subjects":["11"],"theorem":"Erdos313.erdos_313.variants.solution_42_2_3_7"},{"answerKinds":[],"category":"research solved","docstring":"Let `X` be the set of points in `Fin d → ℝ` of the shape\n`fun i : Fin d => ∑' n : A, (1 : ℝ) / (n + i)` for some infinite subset `A ⊆ ℕ` such that\n`1 / n` is summable over `A`. `X` has nonempty interior. This is proved in [KoTa24].\n-","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://gist.githubusercontent.com/madeve-unipi/62a8f68cdb4864b85b81a6752dcb0aa4/raw/5793aaa51089c25c37d8d63f60540367f6abe506/Erdos268.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«268»","statement":"∀ (d : ℕ),\n  (interior {x | ∃ A, A.Infinite ∧ (Summable fun n => 1 / ↑↑n) ∧ x = fun i => ∑' (n : ↑A), 1 / (↑↑n + ↑↑i)}).Nonempty","subjects":["40","54"],"theorem":"Erdos268.erdos_268"},{"answerKinds":[],"category":"test","docstring":"$f_1(\\{0\\}, 1) = 0$: can't represent $1$ with a single element from $\\{0\\}$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1192»","statement":"Erdos1192.f_r {0} 1 1 = 0","subjects":["5","11"],"theorem":"Erdos1192.erdos_1192.f_r_no_rep"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Rényi proved by the probabilistic method that there exists a set $A$ such that\n$$\\sum_{n\\leq x}f_r(n)^2 \\ll x$$ and $$\\lvert A\\cap [1,x]\\rvert\\gg x^{1/r}$$ for all $x$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1192»","statement":"∀ r ≥ 2,\n  ∃ A,\n    ((fun x => ∑ n ∈ Finset.range (x + 1), ↑(Erdos1192.f_r A r n) ^ 2) =O[Filter.atTop] fun x => ↑x) ∧\n      (fun x => ↑x ^ (1 / ↑r)) =O[Filter.atTop] fun x => ↑(Nat.count (fun x => x ∈ A) x)","subjects":["5"],"theorem":"Erdos1192.erdos_1192.variants.renyi"},{"answerKinds":[],"category":"research solved","docstring":"Ruzsa [Ru90] proved that the answer is yes for $r=2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1192»","statement":"∃ A,\n  (∀ᶠ (n : ℕ) in Filter.atTop, Erdos1192.f_r A 2 n > 0) ∧\n    (fun x => ∑ n ∈ Finset.range (x + 1), ↑(Erdos1192.f_r A 2 n) ^ 2) =O[Filter.atTop] fun x => ↑x","subjects":["5"],"theorem":"Erdos1192.erdos_1192.variants.ruzsa"},{"answerKinds":[],"category":"test","docstring":"For $r = 1$, $f_1(\\{n\\}, n) = 1$: the only $1$-tuple from $\\{n\\}$ summing to $n$ is $(n)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1192»","statement":"∀ (n : ℕ), Erdos1192.f_r {n} 1 n = 1","subjects":["5","11"],"theorem":"Erdos1192.erdos_1192.f_r_singleton_self"},{"answerKinds":[],"category":"research open","docstring":"Does there exist, for all $r\\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all\nlarge $n$) such that $$\\sum_{n\\leq x}f_r(n)^2 \\ll x$$ for all $x$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1192»","statement":"True ↔\n  ∀ r ≥ 2,\n    ∃ A,\n      (∀ᶠ (n : ℕ) in Filter.atTop, Erdos1192.f_r A r n > 0) ∧\n        (fun x => ∑ n ∈ Finset.range (x + 1), ↑(Erdos1192.f_r A r n) ^ 2) =O[Filter.atTop] fun x => ↑x","subjects":["5"],"theorem":"Erdos1192.erdos_1192"},{"answerKinds":[],"category":"test","docstring":"The empty sum ($r = 0$) gives exactly one representation of $0$: the empty tuple. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1192»","statement":"∀ (A : Set ℕ), Erdos1192.f_r A 0 0 = 1","subjects":["5","11"],"theorem":"Erdos1192.erdos_1192.f_r_zero_zero"},{"answerKinds":[],"category":"test","docstring":"With an empty set, there are no valid $r$-tuples for $r \\geq 1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1192»","statement":"∀ (r n : ℕ), Erdos1192.f_r ∅ (r + 1) n = 0","subjects":["5","11"],"theorem":"Erdos1192.erdos_1192.f_r_empty"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Countable index set case.**\n\nIf the index set is countable, the answer is yes, and the intersection\ncondition is unnecessary. This is Bernstein's Lemma:\nevery countable system of infinite sets has Property B.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«602»","statement":"True ↔\n  ∀ {α : Type u_1} (A : ℕ → Set α),\n    (∀ (i : ℕ), (A i).Countable ∧ (A i).Infinite) →\n      (∀ (i j : ℕ), i ≠ j → (A i ∩ A j).Finite) →\n        (∀ (i j : ℕ), i ≠ j → (A i ∩ A j).ncard ≠ 1) → Erdos602.HasPropertyB ℕ A","subjects":["3","5"],"theorem":"Erdos602.erdos_602.variants.countable_index"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Trivial case: pairwise disjoint families.**\n\nIf the `A_i` are pairwise disjoint (all intersections are empty, which in\nparticular satisfies `|A_i ∩ A_j| ≠ 1`), then Property B holds trivially.\n\n**Proof sketch:** Since each `A_i` is infinite, it has (at least) two distinct elements\n`a_i` and `b_i`. We can define a colouring that assigns colour 0 to `a_i` and colour 1\nto `b_i` for each `i` (using disjointness, these choices don't conflict), and extend\narbitrarily elsewhere. Then no `A_i` is monochromatic. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«602»","statement":"True ↔\n  ∀ {α : Type u_1} {I : Type u_2} (A : I → Set α),\n    (∀ (i : I), (A i).Infinite) → (∀ (i j : I), i ≠ j → Disjoint (A i) (A j)) → Erdos602.HasPropertyB I A","subjects":["3","5"],"theorem":"Erdos602.erdos_602.variants.disjoint"},{"answerKinds":[],"category":"textbook","docstring":"**Intersections of size ≥ 2 suffice.**\n\nFor a single countably infinite set `A ⊆ α`, there trivially exists a 2-colouring\nof `α` that makes `A` non-monochromatic: since `A` is infinite, it has two distinct\nelements, so any colouring that assigns them different colours works. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«602»","statement":"∀ {α : Type u_1} (A : Set α), A.Infinite → ∃ f, ¬Erdos602.IsMonochromatic f A","subjects":["3"],"theorem":"Erdos602.erdos_602.variants.single_set"},{"answerKinds":[],"category":"research open","docstring":"Does every almost-disjoint family of countably infinite sets whose pairwise\nintersections all have size ≠ 1 have Property B?\n\nFormally: let `α` be any type, let `(A_i)_{i ∈ I}` be a family of countably infinite subsets\nof `α` such that for all `i ≠ j`, the intersection `A_i ∩ A_j` is finite and\n`|A_i ∩ A_j| ≠ 1`. Does there exist a 2-colouring `f : α → Fin 2` such that no `A_i` is\nmonochromatic?\n\nThis is an open question about Property B for almost-disjoint families with a\nforbidden intersection size of 1.\n\n**Note:** This generalises the formulation in which the ground set is `ℕ`. Since every\ncountably infinite set is in bijection with `ℕ`, the two formulations are equivalent, but\nworking over an arbitrary ground type makes the statement apply immediately to, e.g.,\nalmost-disjoint families of countable subsets of an uncountable space. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«602»","statement":"True ↔\n  ∀ {α : Type u_1} {I : Type u_2} (A : I → Set α),\n    (∀ (i : I), (A i).Countable ∧ (A i).Infinite) →\n      (∀ (i j : I), i ≠ j → (A i ∩ A j).Finite) →\n        (∀ (i j : I), i ≠ j → (A i ∩ A j).ncard ≠ 1) → Erdos602.HasPropertyB I A","subjects":["3","5"],"theorem":"Erdos602.erdos_602"},{"answerKinds":[],"category":"textbook","docstring":"**Unique index set.**\n\nIf the index set has exactly one element (i.e., `[Unique I]`), then Property B holds:\nany 2-colouring that makes the single set `A (default : I)` non-monochromatic works.\nThis follows from the single-set case. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«602»","statement":"∀ {α : Type u_1} (I : Type u_2) [Unique I] (A : I → Set α), (∀ (i : I), (A i).Infinite) → Erdos602.HasPropertyB I A","subjects":["3"],"theorem":"Erdos602.erdos_602.variants.unique_index"},{"answerKinds":[],"category":"textbook","docstring":"**Empty index set.**\n\nIf the index set `I` is empty (has no elements), then Property B holds vacuously:\nany 2-colouring works, since there are no sets to be made non-monochromatic. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«602»","statement":"∀ {α : Type u_1} (A : PEmpty.{u_2 + 1} → Set α),\n  (∀ (i : PEmpty.{u_2 + 1}), (A i).Infinite) → Erdos602.HasPropertyB PEmpty.{u_2 + 1} A","subjects":["3"],"theorem":"Erdos602.erdos_602.variants.empty_index"},{"answerKinds":[],"category":"research solved","docstring":"Formal disproof of `disjoint_without_infinite_claim`.\n\n**Counterexample:** Take `α = ℕ`, `I = Fin 2`, with `A 0 = {0}` and `A 1 = {1}`.\nThese are pairwise disjoint, satisfying the only hypothesis. But singleton sets\nare vacuously monochromatic under any colouring: the only pair `(x, y) ∈ {0} × {0}`\nis `(0, 0)`, and `f 0 = f 0` trivially. So any colouring makes `A 0` monochromatic,\nmeaning `HasPropertyB` fails. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«602»","statement":"¬Erdos602.disjoint_without_infinite_claim","subjects":["3"],"theorem":"Erdos602.disjoint_without_infinite_claim.disproof"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Two infinite sets with pairwise intersection of size ≠ 1.**\n\nIf the family consists of exactly two countably infinite sets `A₀` and `A₁` with\n`|A₀ ∩ A₁| ≠ 1` (and finite), then Property B holds.\n\n**Proof sketch:**\n- If `A₀ ∩ A₁ = ∅`: the sets are disjoint. Pick distinct `a, b ∈ A₀` and distinct\n  `c, d ∈ A₁`. Colour `b` and `c` with 1, everything else with 0. Then `A₀` has\n  `a` (colour 0) and `b` (colour 1), and `A₁` has `c` (colour 1) and `d` (colour 0),\n  so neither is monochromatic.\n- If `|A₀ ∩ A₁| ≥ 2`: the intersection contains two distinct points `x` and `y`.\n  Assign `x` colour 0 and `y` colour 1. Both `A₀` and `A₁` contain `x` and `y`,\n  so neither is monochromatic. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«602»","statement":"True ↔\n  ∀ {α : Type u_1} (A : Fin 2 → Set α),\n    (∀ (i : Fin 2), (A i).Infinite) → (A 0 ∩ A 1).Finite → (A 0 ∩ A 1).ncard ≠ 1 → Erdos602.HasPropertyB (Fin 2) A","subjects":["3","5"],"theorem":"Erdos602.erdos_602.variants.two_sets"},{"answerKinds":[],"category":"research open","docstring":"A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810\n\n[Needed to index shift in order to avoid trivial case $n = 0$,\nwhere the conjecture is trivially false.]\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«779»","statement":"∀ n ≥ 1,\n  have P := ∏ i ∈ Finset.range (n + 1), Nat.nth Nat.Prime i;\n  ∃ p, Nat.Prime p ∧ Nat.Prime (P + p) ∧ Nat.nth Nat.Prime n < p ∧ p < P","subjects":["11"],"theorem":"Erdos779.erdos_779"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $2 \\leq a_1 \\leq a_2 \\leq \\cdots$ be a monotone sequence with $a_n \\to \\infty$.\nIs $\\sum_n \\frac{d(n)}{a_1 \\cdots a_n}$ irrational, where $d(n)$ is the number of divisors of $n$?\n\nSolution: True (proved by Erdős and Straus [ErSt71], Lemma 2.2 and Theorem 2.13).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«258»","statement":"True ↔\n  ∀ (a : ℕ → ℕ),\n    (∀ (n : ℕ), 2 ≤ a n) →\n      Monotone a →\n        Filter.Tendsto a Filter.atTop Filter.atTop →\n          Irrational (∑' (n : ℕ), ↑(n + 1).divisors.card / ↑(∏ i ∈ Finset.Icc 1 (n + 1), a i))","subjects":["11"],"theorem":"Erdos258.erdos_258.variants.monotone"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $a_n \\to \\infty$ be a sequence of non-zero natural numbers. Is\n$\\sum_n \\frac{d(n)}{(a_1 ... a_n)}$ irrational, where $d(n)$ is the number of divisors of $n$?\n\nThis was proved affirmatively by Chojecki and GPT-5.4 Pro [Ch26], and formalised in Lean\nby ster-oc [St26].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://live.lean-lang.org/#project=mathlib-v4.28.0&url=https://gist.githubusercontent.com/ster-oc/2b7adcf9d753cf6e29d782f7374cc57e/raw/689a8483895cbe147634dfbf2d7b1db93a3b5b5f/Erdos258.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«258»","statement":"True ↔\n  ∀ (a : ℕ → ℕ),\n    (∀ (n : ℕ), 2 ≤ a n) →\n      Filter.Tendsto a Filter.atTop Filter.atTop →\n        Irrational (∑' (n : ℕ), ↑(n + 1).divisors.card / ↑(∏ i ∈ Finset.Icc 1 (n + 1), a i))","subjects":["11"],"theorem":"Erdos258.erdos_258"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is $\\sum_n \\frac{d(n)}{t^n}$ irrational, where $t ≥ 2$ is an integer.\n\nSolution: True (proved by Erdős, see Erdős Problems website)\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«258»","statement":"True ↔ ∀ t ≥ 2, Irrational (∑' (n : ℕ), ↑(n + 1).divisors.card / ↑t ^ (n + 1))","subjects":["11"],"theorem":"Erdos258.erdos_258.variants.constant"},{"answerKinds":["Prop"],"category":"research open","docstring":"Is it true that $f(n,k) < c_k^n$ for some constant $c_k>0$ and for all $n > 0$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«20»","statement":"sorry ↔ ∃ c, ∀ (n k : ℕ), n > 0 → Erdos20.f n k < c k ^ n","subjects":["5"],"theorem":"Erdos20.erdos_20"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«20»","statement":"Erdos20.f 0 1 = 1","subjects":["5"],"theorem":"Erdos20.f_0_1"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Rado [ErRa60] proved the factorial upper bound for the $k$-sunflower\nthreshold: any family of $n$-uniform sets with more than $(k-1)^n \\, n!$ members\ncontains a $k$-sunflower, hence $f(n,k) \\le (k-1)^n \\, n! + 1$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/HowieHwong/lean-erdos-proofs/blob/b8b641ba2d00dc4d1fe205a078a4159372672459/Erdos/P20.lean#L202"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«20»","statement":"∀ (n k : ℕ), n > 0 → 2 ≤ k → Erdos20.f n k ≤ (k - 1) ^ n * n.factorial + 1","subjects":["5"],"theorem":"Erdos20.erdos_20.variants.erdos_rado_bound"},{"answerKinds":[],"category":"research open","docstring":"Let $1\\leq u_1 < u_2 < \\cdots$ be the sequence of integers with at most $2$ prime factors.\nIs it true that $$\\limsup_{k \\to \\infty} \\frac{u_{k+1}-u_k}{\\log k}=\\infty?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1139»","statement":"True ↔\n  Filter.limsup\n      (fun k =>\n        (↑↑(Nat.nth (fun n => 0 < n ∧ ArithmeticFunction.cardFactors n ≤ 2) (k + 1)) -\n            ↑↑(Nat.nth (fun n => 0 < n ∧ ArithmeticFunction.cardFactors n ≤ 2) k)) /\n          ↑(Real.log (↑k + 1)))\n      Filter.atTop =\n    ⊤","subjects":["11"],"theorem":"Erdos1139.erdos_1139"},{"answerKinds":[],"category":"research open","docstring":"Let $A = \\{a_1 < a_2 < \\dots\\} \\subseteq \\mathbb{N}$ and let $F(A,X,k)$ count the number of $i$\nsuch that $[a_i,a_{i+1}, \\dots ,a_{i+k−1}] < X$, where the left-hand side is the least common\nmultiple. Is it true that, for every $\\epsilon > 0$, there exists some $k$ such that\n$F(A,X,k) < X^\\epsilon$?","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«873»","statement":"True ↔ ∀ (a : ℕ → ℕ), ∀ ε > 0, 0 < a 0 → StrictMono a → ∃ k, ∀ X > 0, ↑↑(Erdos873.F a X k) < ↑(X ^ ε)","subjects":["11"],"theorem":"Erdos873.erdos_873"},{"answerKinds":[],"category":"research solved","docstring":"Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number\nof $n \\leq x$ with $\\phi(n) = \\phi(n+1)$ is at most $$\\frac{x}{\\exp((\\log x)^{1/3})}$$.\n\n[EPS87] Erd\\H os, Paul and Pomerance, Carl and S\\'ark\\\"ozy, Andr\\'as, _On locally repeated values of certain arithmetic functions_. {II}. Proc. Amer. Math. Soc. (1987), 1--7.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1003»","statement":"∀ᶠ (x : ℝ) in Filter.atTop, ↑{n | ↑n ≤ x ∧ n.totient = (n + 1).totient}.ncard ≤ x / Real.exp (Real.log x ^ (1 / 3))","subjects":["11"],"theorem":"Erdos1003.erdos_1003.variants.eps87"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many solutions to $\\phi(n) = \\phi(n+1)$, where $\\phi$ is the Euler totient\nfunction?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1003»","statement":"True ↔ {n | n.totient = (n + 1).totient}.Infinite","subjects":["11"],"theorem":"Erdos1003.erdos_1003"},{"answerKinds":[],"category":"research open","docstring":"Erdős [Er85e] says that, presumably, for every $k \\geq 1$ the equation\n$$\\phi(n) = \\phi(n+1) = \\cdots = \\phi (n+k)$$ has infinitely many solutions.\n\n[Er85e] Erdős, P., _Some problems and results in number theory_. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1003»","statement":"True ↔ ∀ k ≥ 1, {n | ∀ i ∈ Set.Icc 1 k, n.totient = (n + i).totient}.Infinite","subjects":["11"],"theorem":"Erdos1003.erdos_1003.variants.Icc"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f:\\mathbb{R}\\to \\mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is it\ntrue that\n$$f=g+h$$\nfor some continuous $g$ and additive $h$ (i.e. $h(x+y)=h(x)+h(y)$)?\n\nA conjecture of Erdős from the early 1950s. Answered in the affirmative by de Bruijn [dB51].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos907.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«907»","statement":"True ↔\n  ∀ (f : ℝ → ℝ),\n    (∀ (h : ℝ), 0 < h → Continuous fun x => f (x + h) - f x) →\n      ∃ g a, Continuous g ∧ (∀ (x y : ℝ), a (x + y) = a x + a y) ∧ f = g + a","subjects":["26","39"],"theorem":"Erdos907.erdos_907"},{"answerKinds":[],"category":"research solved","docstring":"Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers\n$a_n$ such that $\\sum \\frac{1}{a_n}$ converges and\n$$\n  \\liminf_{n \\to \\infty} (a_n^2 \\sum_{k > n} \\frac{1}{a_k^2}) > 0\n$$\nis not an irrationality sequence.\n\n[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«264»","statement":"∀ {a : ℕ → ℕ},\n  StrictMono a →\n    0 ∉ Set.range a →\n      (Summable fun x => 1 / ↑(a x)) →\n        0 < Filter.liminf (fun n => ↑(a n) ^ 2 * ∑' (k : ↑(Set.Ioi n)), 1 / ↑(a ↑k) ^ 2) Filter.atTop →\n          ¬Erdos264.IsIrrationalitySequence a","subjects":["11"],"theorem":"Erdos264.erdos_264.variants.ko_tao_neg"},{"answerKinds":[],"category":"research open","docstring":"Is $n!$ an example of an irrationality sequence?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«264»","statement":"True ↔ Erdos264.IsIrrationalitySequence Nat.factorial","subjects":["11"],"theorem":"Erdos264.erdos_264.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"One example is $2^{2^n}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«264»","statement":"Erdos264.IsIrrationalitySequence fun n => 2 ^ 2 ^ n","subjects":["11"],"theorem":"Erdos264.erdos_264.variants.example"},{"answerKinds":[],"category":"research solved","docstring":"On the other hand, Kovač and Tao [KoTa24] do prove that for any function $F$ with\n$\\lim_{n \\to \\infty} \\frac{F(n + 1)}{F(n)} = \\infty$ there exists such an irrationality sequence with $a_n \\sim F(n)$.\n\n[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«264»","statement":"∀ {F : ℕ → ℕ},\n  Filter.Tendsto (fun n => ↑(F (n + 1)) / ↑(F n)) Filter.atTop Filter.atTop →\n    ∃ a, Erdos264.IsIrrationalitySequence a ∧ Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(a n)) fun n => ↑(F n)","subjects":["11"],"theorem":"Erdos264.erdos_264.variants.ko_tao_pos"},{"answerKinds":[],"category":"research solved","docstring":"Is $2^n$ an example of an irrationality sequence? Kovač and Tao proved that it is not [KoTa24]\n\n[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«264»","statement":"¬Erdos264.IsIrrationalitySequence fun x => 2 ^ x","subjects":["11"],"theorem":"Erdos264.erdos_264.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Let $(a,b)=1$. The set $\\{a^kb^l: k,l\\geq 0\\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\\geq 0$.\n\nWe state the nontrivial case $a,b\\geq 2$, proved by Birch [Bi59].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos246.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«246»","statement":"∀ (a b : ℕ), 2 ≤ a → 2 ≤ b → a.Coprime b → IsAddComplete (Erdos246.Gamma a b)","subjects":["11"],"theorem":"Erdos246.erdos_246"},{"answerKinds":["Prop"],"category":"research open","docstring":"Are there infinitely many primitive weird numbers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«470»","statement":"sorry ↔ Set.Infinite Erdos470.PrimitiveWeird","subjects":["11"],"theorem":"Erdos470.erdos_470.parts.ii"},{"answerKinds":[],"category":"textbook","docstring":"The smallest weird number is 70.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«470»","statement":"(∀ n < 70, ¬n.Weird) ∧ Nat.Weird 70","subjects":["11"],"theorem":"Erdos470.erdos_470.variants.smallest_weird_eq_70"},{"answerKinds":["Prop"],"category":"research open","docstring":"Are there any odd weird numbers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«470»","statement":"sorry ↔ ∃ n, n.Weird ∧ Odd n","subjects":["11"],"theorem":"Erdos470.erdos_470.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Fang [Fa22](https://arxiv.org/abs/2207.12906) has shown there are no odd weird numbers below $10^{21}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«470»","statement":"∀ n < 10 ^ 21, Odd n → ¬n.Weird","subjects":["11"],"theorem":"Erdos470.erdos_470.variants.odd_weird_10_pow_21"},{"answerKinds":[],"category":"research solved","docstring":"If there are no odd weird numbers then every weird number has abundancy index < 4.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/HowieHwong/lean-erdos-proofs/blob/40216cfee225ec5c9f122a48703e671a9f47db90/Erdos/P470.lean#L88"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«470»","statement":"(∀ (n : ℕ), n.Weird → ¬Odd n) → ∀ (n : ℕ), n.Weird → Erdos470.AbundancyIndex n < 4","subjects":["11"],"theorem":"Erdos470.erdos_470.variants.abundancy_index"},{"answerKinds":[],"category":"research solved","docstring":"Benkoski and Erdős [BeEr74](https://mathscinet.ams.org/mathscinet/relay-station?mr=347726) proved\nthat the set of weird numbers has positive density.\n\n`HasPosDensity` is the right reading here, rather than the positive *lower* density that Erdős'\n\"positive density\" usually abbreviates. Their Theorem 5 is stated as \"the density of weird\nnumbers is positive\", and they first establish that the density exists at all: \"It is clear that\nthe weird numbers have a density since both the abundant numbers and the pseudoperfect numbers\nhave a density. (A weird number is abundant and not pseudoperfect.)\" The work in the paper goes\ninto showing that density is not `0`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«470»","statement":"{n | n.Weird}.HasPosDensity","subjects":["11"],"theorem":"Erdos470.erdos_470.variants.weird_pos_density"},{"answerKinds":[],"category":"research solved","docstring":"Melfi [Me15](https://mathscinet.ams.org/mathscinet/relay-station?mr=3276337) has proved that there\nare infinitely many primitive weird numbers, conditional on the fact that\n$p_{n+1} - p_n < \\frac{1}{10} \\sqrt{p_n}$ for all large $n$, which in turn would follow from\nwell-known conjectures concerning prime gaps.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«470»","statement":"(∀ᶠ (n : ℕ) in Filter.atTop, ↑(primeGap n) < √↑(Nat.nth Nat.Prime n) / 10) → Set.Infinite Erdos470.PrimitiveWeird","subjects":["11"],"theorem":"Erdos470.erdos_470.variants.prime_gap_imp_inf_prim_weird"},{"answerKinds":[],"category":"research solved","docstring":"Liddy and Riedl [LiRi18](https://ideaexchange.uakron.edu/honors_research_projects/728/) have shown\nthat an odd weird number must have at least 6 prime divisors.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«470»","statement":"∀ (n : ℕ), Odd n → n.Weird → 6 ≤ {m | m ∈ n.divisors ∧ Nat.Prime m}.ncard","subjects":["11"],"theorem":"Erdos470.erdos_470.variants.odd_weird_prime_div"},{"answerKinds":[],"category":"research open","docstring":"Does every graph with chromatic number $\\aleph_1$ contain a countable subgraph which is\ninfinitely connected?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1068»","statement":"True ↔\n  ∀ (V : Type) (G : SimpleGraph V),\n    G.chromaticCardinal = Cardinal.aleph 1 → ∃ s, s.Countable ∧ (SimpleGraph.induce s G).InfinitelyConnected","subjects":["5"],"theorem":"Erdos1068.erdos_1068"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $n\\geq 1$ and let $m$ be minimal such that $\\sum_{n\\leq k\\leq m}\\frac{1}{k}\\geq 1$. We define\n$$\\epsilon(n) = \\sum_{n\\leq k\\leq m}\\frac{1}{k}-1.$$\nHow small can $\\epsilon(n)$ be? Is it true that\n$$\\liminf n^2\\epsilon(n)=0?$$\n\nThis is true, and shown by Lim and Steinerberger [LiSt24], who further proved that, for any\n$\\delta>0$, there exist infinitely many $n$ and $m$ such that\n$$n^2\\left\\lvert \\sum_{n\\leq k\\leq m}\\frac{1}{k}-1\\right\\rvert\\ll \\frac{1}{(\\log n)^{5/4-\\delta}}.$$\nErdős and Graham (and also Lim and Steinerberger) believe that the exponent of $2$ is best\npossible here, in that $\\liminf \\epsilon(n) n^{2+\\delta}=\\infty$ for all $\\delta>0$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos314.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«314»","statement":"True ↔ Filter.liminf (fun n => ↑n ^ 2 * Erdos314.epsilon n) Filter.atTop = 0","subjects":["11","40"],"theorem":"Erdos314.erdos_314"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Or even $f(n) < n^{c/\\log\\log n}$ for some constant $c > 0$?\n\nAlso disproved, and by the weak form rather than separately: since $c/\\log\\log n \\to 0$, the bound\n$n^{c/\\log\\log n}$ is of the form $n^{o(1)}$, so this statement implies\n`erdos_92.variants.weak` and is false whenever that one is.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«92»","statement":"False ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos92.f n) ≤ ↑n ^ (c / Real.log (Real.log ↑n))","subjects":["52"],"theorem":"Erdos92.erdos_92.variants.strong"},{"answerKinds":[],"category":"test","docstring":"A sanity check to ensure the set of possible `f(n)` values is bounded above. A trivial bound is\n`n`, since the points equidistant from any `x` form a subset of the other `n - 1` points.\nThis ensures `sSup` is well-defined.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«92»","statement":"∀ (n : ℕ), BddAbove (Erdos92.possible_f_values n)","subjects":["52"],"theorem":"Erdos92.possible_f_values_BddAbove"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that $f(n)\\leq n^{o(1)}$?\n\nThe source records this as disproved: \"This is a stronger form of the unit distance conjecture\n(see [90]). As such the recent disproof of [90] also disproves this.\" That disproof is\n`Erdos90.erdos_90`, which this repository already records as `research solved` with the answer\n`False`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«92»","statement":"False ↔ ∃ o, o =o[Filter.atTop] 1 ∧ ∀ (n : ℕ), ↑(Erdos92.f n) ≤ ↑n ^ o n","subjects":["52"],"subsets":["FC100OpenSet1"],"theorem":"Erdos92.erdos_92.variants.weak"},{"answerKinds":[],"category":"research open","docstring":"Let $k\\geq 3$ and $A\\subset \\mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that $$1_A^{(k)}(n) >n^c?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«322»","statement":"True ↔ ∀ (k : ℕ), 3 ≤ k → ∃ c > 0, {n | ↑n ^ c < ↑(Erdos322.representationCount k n)}.Infinite","subjects":["11"],"theorem":"Erdos322.erdos_322"},{"answerKinds":[],"category":"research open","docstring":"Let $C>0$. There exists $\\epsilon>0$ such that if $n$ is sufficiently large the following holds.\n\nFor any $x_1,\\ldots,x_n\\in [-1,1]$ there exist $y_1,\\ldots,y_n\\in [-1,1]$ such that, if $P$ is a\npolynomial of degree $m<(1+\\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\\epsilon)n$ many\n$1\\leq i\\leq n$, then $$\\max_{x\\in [-1,1]}\\lvert P(x)\\rvert >C.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1133»","statement":"True ↔\n  ∀ C > 0,\n    ∃ ε > 0,\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ (x : Fin n → ↑(Set.Icc (-1) 1)),\n          ∃ y,\n            ∀ (P : Polynomial ℝ),\n              ↑P.natDegree < (1 + ε) * ↑n →\n                ↑{i | Polynomial.eval (↑(x i)) P = ↑(y i)}.card ≥ (1 - ε) * ↑n →\n                  ∃ z ∈ Set.Icc (-1) 1, |Polynomial.eval z P| > C","subjects":["26","41"],"theorem":"Erdos1133.erdos_1133"},{"answerKinds":[],"category":"research solved","docstring":"Erdős proved that, for any $C>0$, there exists $\\epsilon>0$ such that if $n$ is sufficiently\nlarge and $m=\\lfloor (1+\\epsilon)n\\rfloor$ then for any $x_1,\\ldots,x_m\\in [-1,1]$ there is a\npolynomial $P$ of degree $n$ such that $\\lvert P(x_i)\\rvert\\leq 1$ for $1\\leq i\\leq m$ and\n$\\max_{x\\in [-1,1]}\\lvert P(x)\\rvert>C$. The conjectured statement would also imply this, but\nErdős in [Er67] says he could not even prove it for $m=n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1133»","statement":"∀ C > 0,\n  ∃ ε > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      have m := ⌊(1 + ε) * ↑n⌋₊;\n      ∀ (x : Fin m → ↑(Set.Icc (-1) 1)),\n        ∃ P,\n          P.natDegree = n ∧\n            (∀ (i : Fin m), |Polynomial.eval (↑(x i)) P| ≤ 1) ∧ ∃ z ∈ Set.Icc (-1) 1, |Polynomial.eval z P| > C","subjects":["26","41"],"theorem":"Erdos1133.erdos_1133.variants.weaker"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The Burr-Erdős conjecture: For any $d\\geq 1$ if $H$ is a graph such that every subgraph\ncontains a vertex of degree at most $d$ then\n$$R(H)\\ll_d n.$$\n\nSolved by Lee [Le17], who proved that $R(H) \\leq 2^{2^{O(d)}}n$.\n\nThis problem is #9 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«163»","statement":"True ↔\n  ∀ (d : ℕ),\n    1 ≤ d →\n      ∃ C > 0,\n        ∀ (V : Type) [inst : Fintype V] (H : SimpleGraph V),\n          H.IsDegenerate d → ↑H.diagonalGraphRamsey ≤ C * ↑(Fintype.card V)","subjects":["5"],"theorem":"Erdos163.erdos_163"},{"answerKinds":[],"category":"research open","docstring":"Show that for any rational $\\alpha \\in [1,2)$ there exists a bipartite graph $G$ such that $$\\mathrm{ex}(n;G)\\asymp n^{\\alpha}.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«571»","statement":"∀ (α : ℚ),\n  1 ≤ α → α < 2 → ∃ q G, G.IsBipartite ∧ (fun n => ↑(SimpleGraph.extremalNumber n G)) =Θ[Filter.atTop] fun n => ↑n ^ ↑α","subjects":["5"],"theorem":"Erdos571.erdos_571"},{"answerKinds":[],"category":"research open","docstring":"Is there a function $f$ with $f(n)\\to\\infty$ as $n\\to\\infty$ such that,\nfor all large $n$, there is a composite number $m$ such that\n$$\nn + f(n) < m < n + p(m)\n$$\nHere $p(m)$ is the least prime factor of $m$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«463»","statement":"True ↔\n  ∃ f,\n    ∃ (_ : Filter.Tendsto f Filter.atTop Filter.atTop),\n      ∀ᶠ (n : ℕ) in Filter.atTop, ∃ m, m.Composite ∧ n + f n < m ∧ m < n + m.minFac","subjects":["11"],"theorem":"Erdos463.erdos_463"},{"answerKinds":[],"category":"research solved","docstring":"From [GuKa15]: diameter $\\gg n / \\log n$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«100»","statement":"∃ C > 0,\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))),\n      A.card = n → Erdos100.DistancesSeparated A → Metric.diam ↑A ≥ C * ↑n / Real.log ↑n","subjects":["52"],"theorem":"Erdos100.erdos_100.variants.guth_katz"},{"answerKinds":[],"category":"research open","docstring":"Stronger conjecture: diameter $\\geq n - 1$ for sufficiently large $n$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«100»","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), A.card = n → Erdos100.DistancesSeparated A → Metric.diam ↑A ≥ ↑n - 1","subjects":["52"],"theorem":"Erdos100.erdos_100.variants.strong"},{"answerKinds":[],"category":"research solved","docstring":"From [Kanold]: diameter $\\geq n^{3/4}$.\nTODO: find reference ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«100»","statement":"∃ C > 0,\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))),\n      A.card = n → Erdos100.DistancesSeparated A → Metric.diam ↑A ≥ ↑n ^ (3 / 4)","subjects":["52"],"theorem":"Erdos100.erdos_100.variants.kanold"},{"answerKinds":[],"category":"research solved","docstring":"From [Piepmeyer]: 9 points with diameter $< 5$.\nTODO: find reference ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/theaustinhatfield/formal-conjectures/blob/solve-erdos-100-piepmeyer/FormalConjectures/ErdosProblems/100.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«100»","statement":"∃ A, A.card = 9 ∧ Erdos100.DistancesSeparated A ∧ Metric.diam ↑A < 5","subjects":["52"],"theorem":"Erdos100.erdos_100_piepmeyer"},{"answerKinds":[],"category":"research open","docstring":"Is the diameter of $A$ at least $Cn$ for some constant $C > 0$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«100»","statement":"True ↔\n  ∃ C > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ∀ (A : Finset (EuclideanSpace ℝ (Fin 2))), A.card = n → Erdos100.DistancesSeparated A → Metric.diam ↑A > C * ↑n","subjects":["52"],"theorem":"Erdos100.erdos_100"},{"answerKinds":[],"category":"research solved","docstring":"There exists a constant `c > 0` such that for all `n`, if\n`k < c * (log n / (log (log n))) ^ 3 → (∀ i < k, ¬ (n + i + 1).Prime)`, then\nthere are distinct primes `p₁, ... pₖ` such that `pᵢ ∣ n + i` for all `1 ≤ i ≤ k`. This is proved\nin [RST75]. There is no need to only consider sufficiently large `n` because one can always take\n`c` small enough so that `k < c * (log n / (log (log n))) ^ 3` implies that `k = 0` until `n` is\nlarge. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«375»","statement":"∃ c > 0,\n  ∀ (n k : ℕ),\n    ↑k < c * (Real.log ↑n / Real.log (Real.log ↑n)) ^ 3 →\n      (∀ i < k, ¬Nat.Prime (n + i + 1)) → ∃ p, Function.Injective p ∧ ∀ (i : Fin k), Nat.Prime (p i) ∧ p i ∣ n + ↑i + 1","subjects":["11"],"theorem":"Erdos375.erdos_375.variants.log"},{"answerKinds":[],"category":"research solved","docstring":"If `Erdos375Prop` is true, then `(n + 1).nth Prime - n.nth Prime < (n.nth Prime) ^ (1 / 2 - c)`\nfor some `c > 0`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«375»","statement":"Erdos375.Erdos375Prop →\n  ∃ c > 0,\n    ∀ᶠ (n : ℕ) in Filter.atTop,\n      ↑(Nat.nth Nat.Prime (n + 1)) - ↑(Nat.nth Nat.Prime n) < ↑(Nat.nth Nat.Prime n) ^ (1 / 2 - c)","subjects":["11"],"theorem":"Erdos375.erdos_375.variants.bounded_gap"},{"answerKinds":[],"category":"research solved","docstring":"In particular, if `Erdos375Prop` is true, then Legendre's conjecture is asymptotically true. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«375»","statement":"Erdos375.Erdos375Prop → ∀ᶠ (n : ℕ) in Filter.atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p","subjects":["11"],"theorem":"Erdos375.erdos_375.variants.legendre"},{"answerKinds":[],"category":"research solved","docstring":"It is easy to see that for any `n ≥ 1` and `k ≤ 2`, if `n + 1, ..., n + k` are all composite,\nthen there are distinct primes `p₁, ... pₖ` such that `pᵢ ∣ n + i` for all `1 ≤ i ≤ k`. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«375»","statement":"∀ n ≥ 1,\n  ∀ k ≤ 2,\n    (∀ i < k, ¬Nat.Prime (n + i + 1)) → ∃ p, Function.Injective p ∧ ∀ (i : Fin k), Nat.Prime (p i) ∧ p i ∣ n + ↑i + 1","subjects":["11"],"theorem":"Erdos375.erdos_375.variants.le_two"},{"answerKinds":[],"category":"research open","docstring":"Is `Erdos375Prop` true? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«375»","statement":"True ↔ Erdos375.Erdos375Prop","subjects":["11"],"theorem":"Erdos375.erdos_375"},{"answerKinds":[],"category":"research open","docstring":"For all large $N$, there exists a prime $p \\in [N, 2N]$ such that $\\frac{p+1}{2}$ is also prime.\n\nThis is an open conjecture. If true, it would imply `erdos_287` for all but at most finitely\nmany exceptions (see `erdos_287.variants.prime_conjecture_implies`).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«287»","statement":"True ↔ ∃ N₀, ∀ (N : ℕ), N₀ ≤ N → ∃ p, Nat.Prime p ∧ N ≤ p ∧ p ≤ 2 * N ∧ Nat.Prime ((p + 1) / 2)","subjects":["11"],"theorem":"Erdos287.erdos_287.variants.prime_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Let $k\\geq2$. Is it true that, for any distinct integers\n$1 < n_1 < \\cdots < n_k$ such that $\\sum_{i=1}^k \\frac{1}{n_i} = 1$,\nwe must have $\\max(n_{i+1} - n_i) \\geq 3$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«287»","statement":"True ↔\n  ∀ (k : ℕ) (hk : 2 ≤ k) (s : Fin k → ℕ), StrictMono s → 1 < s ⟨0, ⋯⟩ → ∑ i, 1 / ↑(s i) = 1 → 3 ≤ Erdos287.max_gap k s","subjects":["11"],"theorem":"Erdos287.erdos_287"},{"answerKinds":[],"category":"test","docstring":"The example $1 = \\frac{1}{2}+\\frac{1}{3}+\\frac{1}{6}$ shows that $3$ would be best possible here:\nthe sequence $(2, 3, 6)$ is a valid Egyptian fraction representation of $1$ with\n`max_gap = 3`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«287»","statement":"have s := ![2, 3, 6];\nStrictMono s ∧ 1 < s 0 ∧ ∑ i, 1 / ↑(s i) = 1 ∧ Erdos287.max_gap 3 s = 3","subjects":["11"],"theorem":"Erdos287.erdos_287.test.best_possible"},{"answerKinds":[],"category":"research solved","docstring":"The lower bound of $\\geq 2$ is equivalent to saying that $1$ is not the sum of reciprocals of\nconsecutive integers, proved by Erdős [Er32].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«287»","statement":"∀ (k : ℕ) (x : 2 ≤ k) (s : Fin k → ℕ), StrictMono s → 1 < s ⟨0, ⋯⟩ → ∑ i, 1 / ↑(s i) = 1 → 2 ≤ Erdos287.max_gap k s","subjects":["11"],"theorem":"Erdos287.erdos_287.variants.gap_at_least_two"},{"answerKinds":[],"category":"textbook","docstring":"The conjecture `erdos_287` would follow for all but at most finitely many exceptions if it were\nknown that, for all large $N$, there exists a prime $p \\in [N, 2N]$ such that $\\frac{p+1}{2}$\nis also prime.\n\nMore precisely: if the prime conjecture holds, then there exists $k_0$ such that for all\n$k \\geq k_0$, any Egyptian fraction representation of $1$ with $k$ terms and all terms $> 1$\nmust have $\\max(n_{i+1} - n_i) \\geq 3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«287»","statement":"(True ↔ ∃ N₀, ∀ (N : ℕ), N₀ ≤ N → ∃ p, Nat.Prime p ∧ N ≤ p ∧ p ≤ 2 * N ∧ Nat.Prime ((p + 1) / 2)) →\n  ∃ k₀,\n    ∀ (k : ℕ),\n      k₀ ≤ k →\n        ∀ (hk : 2 ≤ k) (s : Fin k → ℕ), StrictMono s → 1 < s ⟨0, ⋯⟩ → ∑ i, 1 / ↑(s i) = 1 → 3 ≤ Erdos287.max_gap k s","subjects":["11"],"theorem":"Erdos287.erdos_287.variants.prime_conjecture_implies"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges\nwhich do not occur in a monochromatic triangle?\n\nSolved by Erdős, Rousseau, and Schelp for large $n$, but unpublished. Alon has observed that this\nalso follows from a result of Pyber [Py86], which states that (for large enough $n$) at most\n$\\lfloor n^2/4\\rfloor+2$ monochromatic cliques cover all edges of a $2$-coloured $K_n$.\n\nThis problem was solved completely by Keevash and Sudakov [KeSu04], who proved that the correct\nthreshold is $\\lfloor n^2/4\\rfloor$ for all $n\\geq 7$, is $\\binom{n}{2}$ for $n\\leq 5$, and is\n$10$ for $n=6$.\n\nSince the bound fails for small $n$ (at $n=6$ the threshold is $10 > 6^2/4$), the statement is\nformalized in the asymptotic reading in which the problem was posed and solved: for all\nsufficiently large $n$, every $2$-colouring of the edges of $K_n$ leaves at most $n^2/4$ edges\nnot occurring in a monochromatic triangle. Edges of $K_n$ are the non-diagonal unordered pairs\n`Sym2 (Fin n)`; an edge $\\{x, y\\}$ occurs in a monochromatic triangle if and only if there is a\nthird vertex $z$ with $C(\\{x, z\\}) = C(\\{y, z\\}) = C(\\{x, y\\})$.\n\nThe linked file proves the bound for every finite vertex type with at least $10$ vertices, which\ngives the `atTop` reading below, and collects the uncovered edges as the edge set of a graph\nrather than as a set of pairs.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos639.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«639»","statement":"True ↔\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∀ (C : Sym2 (Fin n) → Fin 2),\n      {e | ¬e.IsDiag ∧ ∀ (x y : Fin n), e = s(x, y) → ¬∃ z, z ≠ x ∧ z ≠ y ∧ C s(x, z) = C e ∧ C s(y, z) = C e}.ncard ≤\n        n ^ 2 / 4","subjects":["5"],"theorem":"Erdos639.erdos_639"},{"answerKinds":[],"category":"research open","docstring":"Let $\\tau(n)$ count the number of divisors of $n$. Is there some $n > 24$ such that\n$$\n  \\max_{m < n}(m + \\tau(m)) \\leq n + 2?\n$$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«647»","statement":"True ↔ ∃ n > 24, ⨆ m, ↑m + (ArithmeticFunction.sigma 0) ↑m ≤ n + 2","subjects":["11"],"theorem":"Erdos647.erdos_647"},{"answerKinds":[],"category":"research open","docstring":"Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in\nfact suggests that\n$$\n  lim_{n\\to\\infty} \\max_{m < n}(\\tau(m) + m − n) = \\infty.\n$$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«647»","statement":"True ↔ Filter.Tendsto (fun n => ⨆ m, (ArithmeticFunction.sigma 0) ↑m + ↑m - n) Filter.atTop Filter.atTop","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos647.erdos_647.variants.lim"},{"answerKinds":[],"category":"research solved","docstring":"This is true for $n = 24$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«647»","statement":"⨆ m, ↑m + (ArithmeticFunction.sigma 0) ↑m ≤ 26","subjects":["11"],"theorem":"Erdos647.erdos_647.variants.twenty_four"},{"answerKinds":[],"category":"research open","docstring":"Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$\nfor which\n$$\n  \\max_{n−k < m < n}(m + \\tau(m)) ≤ n + 2.\n$$ ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«647»","statement":"True ↔ ∀ (k : ℕ), {n | ⨆ m, ↑m + (ArithmeticFunction.sigma 0) ↑m ≤ n + 2}.Infinite","subjects":["11"],"theorem":"Erdos647.erdos_647.variants.infinite"},{"answerKinds":[],"category":"research open","docstring":"Is there a set $A\\subseteq \\mathbb{N}$ such that, for infinitely many $n$, all of $n-a$\nare prime for all $a\\in A$ with $0 < a < n$ and $$\\liminf\\frac{\\lvert A\\cap [1,x]\\rvert}{\\pi(x)}>0?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«428»","statement":"True ↔\n  ∃ A,\n    (∃ᶠ (n : ℕ) in Filter.atTop, ∀ a ∈ A, 0 < a → a < n → Nat.Prime (n - a)) ∧\n      Filter.liminf (fun n => Erdos428.primeDensityRatio A n) Filter.atTop > 0","subjects":["11"],"theorem":"Erdos428.erdos_428"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\sigma(n) - 1$ before a prime\nis reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ (c : ℕ → ℕ),\n  (∀ n > 1, IsLeast {i | Nat.Prime ((fun x => (ArithmeticFunction.sigma 1) x - 1)^[i] n)} (c n)) →\n    (fun n => ↑(c n)) =O[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos409.erdos_409.variants.sigma_isBigO"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"How many iterations of $n\\mapsto\\phi(n) + 1$ are needed before a prime is reached?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ (n : ℕ), 0 < n → IsLeast {i | Nat.Prime ((fun x => x.totient + 1)^[i] n)} sorry","subjects":["11"],"theorem":"Erdos409.erdos_409.parts.i"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the density of $n$ which reach any fixed prime under the iteration $n\\mapsto\\phi(n) + 1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ (p : ℕ), Nat.Prime p → ∀ (α : ℝ), {n | ∃ i, (fun x => x.totient + 1)^[i] n = p}.HasDensity α → α = sorry","subjects":["11"],"theorem":"Erdos409.erdos_409.parts.iii"},{"answerKinds":[],"category":"research open","docstring":"Is it true that iterates of $n\\mapsto\\sigma(n) - 1$ always reach a prime?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"True ↔ ∀ n > 1, ∃ i, Nat.Prime ((fun x => (ArithmeticFunction.sigma 1) x - 1)^[i] n)","subjects":["11"],"theorem":"Erdos409.erdos_409.variants.sigma_prime_termination"},{"answerKinds":[],"category":"test","docstring":"If $n > 0$, then the iteration $n\\mapsto\\phi(n) + 1$ necessarily\nreaches a prime. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ (n : ℕ), 0 < n → ∃ i, Nat.Prime ((fun x => x.totient + 1)^[i] n)","subjects":["11"],"theorem":"Erdos409.erdos_409.variants.termination"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\phi(n) + 1$ before a prime\nis reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ (c : ℕ → ℕ),\n  (∀ n > 0, IsLeast {i | Nat.Prime ((fun x => x.totient + 1)^[i] n)} (c n)) → (fun n => ↑(c n)) =o[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos409.erdos_409.parts.i.isLittleO"},{"answerKinds":[],"category":"research open","docstring":"Can infinitely many $n$ reach the same prime under the iteration $n\\mapsto\\phi(n) + 1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"True ↔ ∃ p, ∃ (_ : Nat.Prime p), {n | ∃ i, (fun x => x.totient + 1)^[i] n = p}.Infinite","subjects":["11"],"theorem":"Erdos409.erdos_409.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"If $n > 1$ then the iteration $n\\mapsto\\sigma(n) - 1$ necessarily reaches a prime.\nNote: this is open — it is not clear that the σ iteration always terminates,\nsince it is non-decreasing (unlike the φ iteration which is strictly decreasing). ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ n > 1, ∃ i, Nat.Prime ((fun x => (ArithmeticFunction.sigma 1) x - 1)^[i] n)","subjects":["11"],"theorem":"Erdos409.erdos_409.variants.sigma_termination"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\phi(n) + 1$ before a prime\nis reached. What is $\\Theta(c(n))$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ (c : ℕ → ℕ),\n  (∀ n > 0, IsLeast {i | Nat.Prime ((fun x => x.totient + 1)^[i] n)} (c n)) → (fun n => ↑(c n)) =Θ[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos409.erdos_409.parts.i.isTheta"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\phi(n) + 1$ before a prime\nis reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ (c : ℕ → ℕ),\n  (∀ n > 0, IsLeast {i | Nat.Prime ((fun x => x.totient + 1)^[i] n)} (c n)) → (fun n => ↑(c n)) =O[Filter.atTop] sorry","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos409.erdos_409.parts.i.isBigO"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\sigma(n) - 1$ before a prime\nis reached. What is $\\Theta(c(n))$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ (c : ℕ → ℕ),\n  (∀ n > 1, IsLeast {i | Nat.Prime ((fun x => (ArithmeticFunction.sigma 1) x - 1)^[i] n)} (c n)) →\n    (fun n => ↑(c n)) =Θ[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos409.erdos_409.variants.sigma_isTheta"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"How many iterations of $n\\mapsto\\sigma(n) - 1$ are needed before a prime is reached?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ n > 1, IsLeast {i | Nat.Prime ((fun x => (ArithmeticFunction.sigma 1) x - 1)^[i] n)} sorry","subjects":["11"],"theorem":"Erdos409.erdos_409.variants.sigma"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c(n)$ be the minimum number of iterations of $n\\mapsto\\sigma(n) - 1$ before a prime\nis reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«409»","statement":"∀ (c : ℕ → ℕ),\n  (∀ n > 1, IsLeast {i | Nat.Prime ((fun x => (ArithmeticFunction.sigma 1) x - 1)^[i] n)} (c n)) →\n    (fun n => ↑(c n)) =o[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos409.erdos_409.variants.sigma_isLittleO"},{"answerKinds":[],"category":"research open","docstring":"Szabo asks whether the maximal $t$ is given by\n$$\n  \\frac{N^2}{2} + O(N)\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«272»","statement":"(fun N => ↑(Erdos272.maxArithInterCard N) - ↑N ^ 2 / 2) =O[Filter.atTop] fun N => ↑N","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"Erdos272.erdos_272.variants.szabo_strong"},{"answerKinds":[],"category":"research solved","docstring":"Simonovits and Sós have shown that $t\\ll N^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«272»","statement":"(fun N => ↑(Erdos272.maxArithInterCard N)) =O[Filter.atTop] fun N => ↑N ^ 2","subjects":["5"],"theorem":"Erdos272.erdos_272.variants.isBigO_sq"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $N\\geq 1$. What is the largest $t$ such that there are\n$A_1,\\ldots,A_t\\subseteq \\{1,\\ldots,N\\}$ with $A_i\\cap A_j$ a non-empty\narithmetic progression for all $i\\neq j$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«272»","statement":"Asymptotics.IsEquivalent Filter.atTop (fun N => ↑(Erdos272.maxArithInterCard N)) sorry","subjects":["5"],"theorem":"Erdos272.erdos_272"},{"answerKinds":[],"category":"research solved","docstring":"Szabo showed that the maximal $t$ is equal to\n$$\n  \\frac{N^2}{2} + O(N^{5/3}\\log^3N).\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«272»","statement":"(fun N => ↑(Erdos272.maxArithInterCard N) - ↑N ^ 2 / 2) =O[Filter.atTop] fun N => ↑N ^ (5 / 3) * Real.log ↑N ^ 3","subjects":["5"],"theorem":"Erdos272.erdos_272.variants.szabo"},{"answerKinds":[],"category":"research open","docstring":"How many (ordered) solutions are there to `σ(a) + σ(b) = σ(a + b)` with `a + b ≤ x`?\nIs it true that this number is asymptotic to `c * x` for some constant `c > 0`?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1061»","statement":"True ↔ ∃ c, 0 < c ∧ Asymptotics.IsEquivalent Filter.atTop Erdos1061.S fun x => c * x","subjects":["11"],"theorem":"Erdos1061.erdos_1061"},{"answerKinds":[],"category":"research open","docstring":"How large must $y=y(\\epsilon,n)$ be such that the number of integers in\n$(x,x+y)$ with a divisor in $(n,2n)$ is at most $\\epsilon y$?\n\nA **linear** scale is known to suffice (see `erdos_450.linear_scale_suffices`).\nWhether the optimal scale is *sublinear* — a sufficient `Y` with `Y ε n = o(n)` —\nis open.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«450»","statement":"True ↔\n  ∃ Y, Erdos450.IsSufficientScale Y ∧ ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun n => ↑(Y ε n) / ↑n) Filter.atTop (nhds 0)","subjects":["11"],"theorem":"Erdos450.erdos_450"},{"answerKinds":[],"category":"research solved","docstring":"A translate-uniform **linear** scale suffices: there is a sufficient window\nlength `Y` with `Y ε n ≤ C(ε) · n`. This is an upper bound on the optimal scale,\nnot the exact threshold asked for in `erdos_450`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-450/Research/TuranAnswer.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«450»","statement":"∃ Y, (∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (n : ℕ), ↑(Y ε n) ≤ C * ↑n) ∧ Erdos450.IsSufficientScale Y","subjects":["11"],"theorem":"Erdos450.erdos_450.linear_scale_suffices"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 1138.**\nLet $x/2 < y < x$ and $C > 1$. If $d = \\max_{p_n < x}(p_{n+1} - p_n)$,\nwhere $p_n$ denotes the $n$-th prime, then is it true that\n$$\\pi(y + Cd) - \\pi(y) \\sim \\frac{Cd}{\\log y}$$?\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/YanYablonovskiy/formal-conjectures/blob/7c134317104d3b98ecc751afbb79ec0adddf8e7c/FormalConjectures/ErdosProblems/1138a.lean#L496"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1138»","statement":"False ↔\n  ∀ C > 1,\n    Asymptotics.IsEquivalent Erdos1138.snd_gt_half_fst (Erdos1138.primeCount_Ioc_mul_const C) fun x =>\n      match x with\n      | (x, y) => C * ↑(Erdos1138.sup_primeGap x) / Real.log y","subjects":["11"],"theorem":"Erdos1138.erdos_1138"},{"answerKinds":[],"category":"research open","docstring":"2. There is a good sequence with sub-exponential growth. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1101»","statement":"∃ u, Erdos1101.IsGood u ∧ (fun n => Real.log ↑(u n)) =o[Filter.atTop] fun n => ↑n","subjects":["11"],"theorem":"Erdos1101.erdos_1101.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"1. There is NO good sequence with polynomial growth. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1101»","statement":"¬∃ u, Erdos1101.IsGood u ∧ ∃ k, (fun n => ↑(u n)) =O[Filter.atTop] fun n => ↑n ^ k","subjects":["11"],"theorem":"Erdos1101.erdos_1101.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"**Menger's theorem for infinite graphs** (Aharoni–Berger [AhBe09]).\n\nThe theorem actually proved by Aharoni and Berger holds for arbitrary vertex sets $A$\nand $B$: in any (possibly infinite) graph $G$ there is a family $P$ of pairwise\nvertex-disjoint $A$--$B$ paths together with an $A$--$B$ separator $S$ consisting of the\nchoice of exactly one vertex from each path in $P$. The disjointness and independence\nhypotheses of `erdos_599` are not needed.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«599»","statement":"∀ (V : Type) (G : SimpleGraph V) (A B : Set V),\n  ∃ ι a b p S,\n    (∀ (i : ι), a i ∈ A) ∧\n      (∀ (i : ι), b i ∈ B) ∧\n        (∀ (i : ι), (p i).IsPath) ∧\n          (Pairwise fun i j => Disjoint {v | v ∈ (p i).support} {v | v ∈ (p j).support}) ∧\n            S ⊆ {v | ∃ i, v ∈ (p i).support} ∧\n              (∀ (i : ι), ∃! v, v ∈ S ∧ v ∈ (p i).support) ∧\n                ∀ a' ∈ A, ∀ b' ∈ B, ∀ (q : G.Walk a' b'), q.IsPath → ∃ v ∈ q.support, v ∈ S","subjects":["5"],"theorem":"Erdos599.erdos_599.variants.aharoni_berger"},{"answerKinds":[],"category":"test","docstring":"Sanity check: when $A = \\varnothing$ the conclusion of `erdos_599` holds trivially, with\nthe empty family of paths and $S = \\varnothing$ (the covering condition is vacuous since\nthere is no path starting in $\\varnothing$).\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«599»","statement":"∀ (V : Type) (G : SimpleGraph V) (B : Set V),\n  ∃ ι a b p S,\n    (∀ (i : ι), a i ∈ ∅) ∧\n      (∀ (i : ι), b i ∈ B) ∧\n        (∀ (i : ι), (p i).IsPath) ∧\n          (Pairwise fun i j => Disjoint {v | v ∈ (p i).support} {v | v ∈ (p j).support}) ∧\n            S ⊆ {v | ∃ i, v ∈ (p i).support} ∧\n              (∀ (i : ι), ∃! v, v ∈ S ∧ v ∈ (p i).support) ∧\n                ∀ a' ∈ ∅, ∀ b' ∈ B, ∀ (q : G.Walk a' b'), q.IsPath → ∃ v ∈ q.support, v ∈ S","subjects":["5"],"theorem":"Erdos599.erdos_599.test.empty_A"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Erdős Problem 599** (the Erdős–Menger conjecture).\n\nLet $G$ be a (possibly infinite) graph and let $A, B$ be disjoint independent sets of\nvertices. Must there exist a family $P$ of pairwise vertex-disjoint paths from $A$ to $B$,\nand a set $S$ of vertices containing exactly one vertex from each path in $P$, such that\nevery path from $A$ to $B$ contains at least one vertex of $S$?\n\nFor finite $G$ this is equivalent to Menger's theorem. The answer is **yes**, proved by\nAharoni and Berger [AhBe09].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«599»","statement":"True ↔\n  ∀ (V : Type) (G : SimpleGraph V) (A B : Set V),\n    Disjoint A B →\n      G.IsIndepSet A →\n        G.IsIndepSet B →\n          ∃ ι a b p S,\n            (∀ (i : ι), a i ∈ A) ∧\n              (∀ (i : ι), b i ∈ B) ∧\n                (∀ (i : ι), (p i).IsPath) ∧\n                  (Pairwise fun i j => Disjoint {v | v ∈ (p i).support} {v | v ∈ (p j).support}) ∧\n                    S ⊆ {v | ∃ i, v ∈ (p i).support} ∧\n                      (∀ (i : ι), ∃! v, v ∈ S ∧ v ∈ (p i).support) ∧\n                        ∀ a' ∈ A, ∀ b' ∈ B, ∀ (q : G.Walk a' b'), q.IsPath → ∃ v ∈ q.support, v ∈ S","subjects":["5"],"theorem":"Erdos599.erdos_599"},{"answerKinds":[],"category":"test","docstring":"Every triangle-free graph on $5$ vertices can be made bipartite by removing at most $1$ edge.\nThis is the $n = 1$ case of Erdős Problem 23.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«23»","statement":"∀ (G : SimpleGraph (Fin 5)), G.CliqueFree 3 → ∃ H ≤ G, H.IsBipartite ∧ (G.edgeFinset \\ H.edgeFinset).card ≤ 1","subjects":["5"],"theorem":"Erdos23.erdos_23.variants.n1"},{"answerKinds":[],"category":"research open","docstring":"Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«23»","statement":"True ↔\n  ∀ (n : ℕ) (V : Type) [inst : Fintype V],\n    Fintype.card V = 5 * n →\n      ∀ (G : SimpleGraph V), G.CliqueFree 3 → ∃ H ≤ G, H.IsBipartite ∧ (G.edgeFinset \\ H.edgeFinset).card ≤ n ^ 2","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"Erdos23.erdos_23"},{"answerKinds":[],"category":"test","docstring":"There exists a triangle-free graph on $5$ vertices such that at least $1$ edge must be removed\nto make it bipartite. This shows the bound in `erdos_23_n1` is tight.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«23»","statement":"∃ G, G.CliqueFree 3 ∧ ∀ H ≤ G, H.IsBipartite → 1 ≤ (G.edgeFinset \\ H.edgeFinset).card","subjects":["5"],"theorem":"Erdos23.erdos_23.variants.n1_tight"},{"answerKinds":[],"category":"research solved","docstring":"Every triangle-free graph on $25$ vertices can be made bipartite by removing at most $25$\nedges.\n\nThis is the $n = 5$ case of Erdős Problem 23.  It follows from the high-density range of\nBalogh-Clemen-Lidicky together with McKay's complete catalogue of the 23-vertex extremal\ngraphs for bipartization of triangle-free graphs.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«23»","statement":"∀ (G : SimpleGraph (Fin 25)), G.CliqueFree 3 → ∃ H ≤ G, H.IsBipartite ∧ (G.edgeFinset \\ H.edgeFinset).card ≤ 25","subjects":["5"],"theorem":"Erdos23.erdos_23.variants.n5"},{"answerKinds":[],"category":"research solved","docstring":"There exists a triangle-free graph on $25$ vertices such that at least $25$ edges must be\nremoved to make it bipartite.  The balanced blow-up of $C_5$ with five parts of size $5$\nwitnesses this.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«23»","statement":"∃ G, G.CliqueFree 3 ∧ ∀ H ≤ G, H.IsBipartite → 25 ≤ (G.edgeFinset \\ H.edgeFinset).card","subjects":["5"],"theorem":"Erdos23.erdos_23.variants.n5_tight"},{"answerKinds":[],"category":"test","docstring":"The blow-up of $C_5$ shows that the bound $n^2$ in Erdős Problem 23 is tight:\nany bipartite subgraph must omit at least $n^2$ edges.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«23»","statement":"∀ (n : ℕ),\n  0 < n → ∀ H ≤ Erdos23.blowupC5 n, H.IsBipartite → n ^ 2 ≤ ((Erdos23.blowupC5 n).edgeFinset \\ H.edgeFinset).card","subjects":["5"],"theorem":"Erdos23.blowupC5_tight"},{"answerKinds":[],"category":"research open","docstring":"What refuting $c(n)\\gg n^n$ leaves open: give good bounds for $c(n)$. Asked here on the\nscale the problem itself sets, $n^n$: does $c$ have a well-defined order\n$$\\lim_{n\\to\\infty}\\frac{\\log c(n)}{n\\log n}?$$\nThe refuted conjecture would have forced this limit to be at least $1$.\nThe cutoff condition is restricted to positive dimensions: in dimension zero,\nexact coverage permits only one tile, so no cutoff exists.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«769»","statement":"True ↔\n  ∃ c,\n    (∀ (n : ℕ), 0 < n → Erdos769.IsCutoff n (c n)) ∧\n      ∃ γ, Filter.Tendsto (fun n => Real.log ↑(c n) / (↑n * Real.log ↑n)) Filter.atTop (nhds γ)","subjects":["52"],"theorem":"Erdos769.erdos_769.variants.growth_rate"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $c(n)$ be minimal such that if $k\\geq c(n)$ then the $n$-dimensional unit\ncube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good\nbounds for $c(n)$ — in particular, is it true that $c(n)\\gg n^n$?\n\nThe `c(n) \\gg n^n` conjecture is **false**: for odd `n`, one can tile the unit\n`n`-cube into `k` homothetic cubes for every `k ≥ U(n)` with `U(n)/n^n → 0`, so\n`c(n)/n^n → 0` along odd dimensions and no absolute constant lower-bounds it.\n\nDetermining good bounds for `c(n)` in general remains open.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-769/Research/Solution.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«769»","statement":"False ↔ Erdos769.Erdos769LowerBound","subjects":["52"],"theorem":"Erdos769.erdos_769"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Can a finite set of disjoint unit segments in a unit square be maximal?\nSolved affirmatively by [Da85], who gave an explicit construction.\n\nThis was formalized in Lean by Alexeev using Aristotle and ChatGPT.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos1071.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1071»","statement":"True ↔\n  ∃ S,\n    Maximal\n      (fun T =>\n        (∀ seg ∈ T,\n            dist seg.1 seg.2 = 1 ∧\n              seg.1.ofLp 0 ∈ Set.Icc 0 1 ∧\n                seg.1.ofLp 1 ∈ Set.Icc 0 1 ∧ seg.2.ofLp 0 ∈ Set.Icc 0 1 ∧ seg.2.ofLp 1 ∈ Set.Icc 0 1) ∧\n          (↑T).Pairwise Erdos1071.SegmentsDisjoint)\n      S","subjects":["52"],"theorem":"Erdos1071.erdos_1071.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite?\nSolved affirmatively by [Fo99], who gave an explicit construction.\n\nThis was formalized in Lean by Alexeev using Aristotle and ChatGPT.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos1071b.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1071»","statement":"True ↔\n  ∃ R S,\n    IsOpen R ∧\n      IsConnected R ∧\n        S.Countable ∧\n          S.Infinite ∧\n            Maximal\n              (fun T =>\n                (∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧ seg.1 ∈ R ∧ seg.2 ∈ R) ∧ T.Pairwise Erdos1071.SegmentsDisjoint)\n              S","subjects":["52"],"theorem":"Erdos1071.erdos_1071.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let G be a graph with n vertices such that every induced subgraph on ≥ $n/2$\nvertices has more than $n^2/50$ edges. Must G contain a triangle?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«128»","statement":"True ↔\n  ∀ (V : Type) [inst : Fintype V] (G : SimpleGraph V),\n    (∀ (V' : Set V),\n        2 * V'.ncard + 1 ≥ Fintype.card V → 50 * (SimpleGraph.induce V' G).edgeSet.ncard > Fintype.card V ^ 2) →\n      ¬G.CliqueFree 3","subjects":["5"],"theorem":"Erdos128.erdos_128"},{"answerKinds":[],"category":"research solved","docstring":"Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$\nsuch that $ϕ(n)< ϕ(n - ϕ(n))$\nReference: [GLW01] Grytczuk, A. and Luca, F. and W\\'ojtowicz, M., A conjecture of {E}rdős concerning inequalities for the\n{E}uler totient function.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1064»","statement":"{n | n.totient < (n - n.totient).totient}.Infinite","subjects":["11"],"theorem":"Erdos1064.erdos_1064.variants.k2"},{"answerKinds":[],"category":"research solved","docstring":"For any function $f(n)=o(n)$,\nwe have $\\phi(n)>\\phi(n-\\phi(n))+f(n)$ for almost all $n$.\nReference:\n[LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\\polhk akowski-{S}chinzel and {E}rd\\H\nos concerning the arithmetical functions {$\\phi$} and\n{$\\sigma$}. Colloq. Math. (2002), 111--130.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1064»","statement":"∀ (f : ℕ → ℕ),\n  ((fun n => ↑(f n)) =o[Filter.atTop] fun n => ↑n) → {n | (n - n.totient).totient + f n < n.totient}.HasDensity 1","subjects":["11"],"theorem":"Erdos1064.erdos_1064.variants.general_function"},{"answerKinds":[],"category":"research solved","docstring":"Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$\nhave asymptotic density 1.\nReference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\\polhk akowski-{S}chinzel and {E}rd\\H\nos concerning the arithmetical functions {$\\phi$} and\n{$\\sigma$}. Colloq. Math.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1064»","statement":"{n | n.totient > (n - n.totient).totient}.HasDensity 1","subjects":["11"],"theorem":"Erdos1064.erdos_1064"},{"answerKinds":[],"category":"research open","docstring":"Does $f(n)$ miss infinitely many integers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«422»","statement":"True ↔ {n | ∀ (x : ℕ+), Erdos422.f x ≠ n}.Infinite","subjects":["11"],"theorem":"Erdos422.erdos_422"},{"answerKinds":[],"category":"research open","docstring":"Is $f$ surjective?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«422»","statement":"True ↔ Function.Surjective Erdos422.f","subjects":["11"],"theorem":"Erdos422.erdos_422.variants.surjective"},{"answerKinds":[],"category":"research open","docstring":"Does $f$ become stationary at some point?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«422»","statement":"True ↔ Filter.EventuallyConst Erdos422.f Filter.atTop","subjects":["11"],"theorem":"Erdos422.erdos_422.variants.eventually_const"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"How does $f$ grow?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«422»","statement":"(fun n => ↑↑(Erdos422.f n)) =O[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos422.erdos_422.variants.growth_rate"},{"answerKinds":[],"category":"research solved","docstring":"Are there infinitely many $n$ such that $\\omega(n + k) \\ll k$ for all $k \\geq 1$?\nHere $\\omega(n)$ is the number of distinct prime divisors of $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«248»","statement":"∃ C > 0, {n | ∀ k ≥ 1, ↑(ArithmeticFunction.cardDistinctFactors (n + k)) ≤ C * ↑k}.Infinite","subjects":["11"],"theorem":"Erdos248.erdos_248"},{"answerKinds":[],"category":"research open","docstring":"Is it true that for every $n \\geq 1$ there is a $k$ such that\n$$\n  n(n + 1) \\cdots (n + k - 1) \\mid (n + k) \\cdots (n + 2k - 1)?\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«389»","statement":"True ↔ ∀ n ≥ 1, ∃ k ≥ 1, ∏ i ∈ Finset.range k, (n + i) ∣ ∏ i ∈ Finset.range k, (n + k + i)","subjects":["11"],"theorem":"Erdos389.erdos_389"},{"answerKinds":[],"category":"textbook","docstring":"Bhavik Mehta has computed the minimal such $k$ for $1 \\leq n \\leq 18$.\nFor example, the minimal $k$ for $n = 4$ is $207$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«389»","statement":"IsLeast {k | 1 ≤ k ∧ ∏ i ∈ Finset.range k, (4 + i) ∣ ∏ i ∈ Finset.range k, (4 + k + i)} 207","subjects":["11"],"theorem":"Erdos389.erdos_389.variants.mehta_four"},{"answerKinds":[],"category":"research solved","docstring":"Stewart improved Mahler's lower bound to $(1_A \\ast 1_A)(n) \\gg (\\log n)^{11/13}$ for\ninfinitely many $n$, where $A$ is the set of perfect cubes.\n\n[St08] Stewart, C. L., _Cubic Thue equations with many solutions_. Int. Math. Res. Not.\n  IMRN (2008), Art. ID rnn040, 11.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«829»","statement":"∃ C > 0, ∃ᶠ (n : ℕ) in Filter.atTop, C * Real.log ↑n ^ (11 / 13) ≤ ↑(AdditiveCombinatorics.sumRep Erdos829.cubes n)","subjects":["11"],"theorem":"Erdos829.variants.stewart"},{"answerKinds":[],"category":"test","docstring":"There is exactly one ordered pair of cubes summing to $0$, namely $(0, 0)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«829»","statement":"AdditiveCombinatorics.sumRep Erdos829.cubes 0 = 1","subjects":["11"],"theorem":"Erdos829.sumRep_cubes_zero"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 829 (open).**  Let $A \\subseteq \\mathbb{N}$ be the set of perfect cubes.  Is\nit true that $(1_A \\ast 1_A)(n) \\ll (\\log n)^{O(1)}$?  That is, does there exist a natural\nnumber $C$ such that the number of representations of $n$ as a sum of two cubes is\n$O((\\log n)^C)$ as $n \\to \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«829»","statement":"True ↔ ∃ C, (fun n => ↑(AdditiveCombinatorics.sumRep Erdos829.cubes n)) =O[Filter.atTop] fun n => Real.log ↑n ^ C","subjects":["11"],"theorem":"Erdos829.erdos_829"},{"answerKinds":[],"category":"test","docstring":"The only ordered pair of cubes summing to $2$ is $(1, 1)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«829»","statement":"AdditiveCombinatorics.sumRep Erdos829.cubes 2 = 1","subjects":["11"],"theorem":"Erdos829.sumRep_cubes_two"},{"answerKinds":[],"category":"test","docstring":"The integer $3$ is not the sum of two cubes. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«829»","statement":"AdditiveCombinatorics.sumRep Erdos829.cubes 3 = 0","subjects":["11"],"theorem":"Erdos829.sumRep_cubes_three"},{"answerKinds":[],"category":"API","docstring":"Membership in `cubes` can be witnessed by a bounded cube root, which makes it\ndecidable for concrete values. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«829»","statement":"∀ (m : ℕ), m ∈ Erdos829.cubes ↔ ∃ k < m + 1, k ^ 3 = m","subjects":["11"],"theorem":"Erdos829.mem_cubes_iff"},{"answerKinds":[],"category":"research solved","docstring":"Mordell proved $\\limsup_{n \\to \\infty} (1_A \\ast 1_A)(n) = \\infty$, where $A$ is the set of\nperfect cubes.  Equivalently, the number of representations of $n$ as a sum of two cubes is\nunbounded.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«829»","statement":"Filter.limsup (fun n => ↑(AdditiveCombinatorics.sumRep Erdos829.cubes n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos829.variants.mordell"},{"answerKinds":[],"category":"research solved","docstring":"Mahler proved $(1_A \\ast 1_A)(n) \\gg (\\log n)^{1/4}$ for infinitely many $n$, where $A$ is\nthe set of perfect cubes.\n\n[Ma35b] Mahler, K., _On the lattice points on curves of genus 1_. Proc. London Math. Soc.\n  (2) (1935), 431-466.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«829»","statement":"∃ C > 0, ∃ᶠ (n : ℕ) in Filter.atTop, C * Real.log ↑n ^ (1 / 4) ≤ ↑(AdditiveCombinatorics.sumRep Erdos829.cubes n)","subjects":["11"],"theorem":"Erdos829.variants.mahler"},{"answerKinds":[],"category":"test","docstring":"The Hardy-Ramanujan taxicab number satisfies $1729 = 1^3 + 12^3 = 9^3 + 10^3$, giving\nthe four ordered representations $(1, 1728), (1728, 1), (729, 1000), (1000, 729)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«829»","statement":"AdditiveCombinatorics.sumRep Erdos829.cubes 1729 = 4","subjects":["11"],"theorem":"Erdos829.sumRep_cubes_taxicab"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Suppose $A\\subseteq\\mathbb{N}$ and $C>0$ is such that $1_A\\ast 1_A(n)\\leq C$ for all\n$n\\in\\mathbb{N}$. Can $A$ be partitioned into $t$ many subsets $A_1,\\ldots,A_t$ (where\n$t=t(C)$ depends only on $C$) such that $1_{A_i}\\ast 1_{A_i}(n)<C$ for all $1\\leq i\\leq t$\nand $n\\in \\mathbb{N}$?\n\nThe answer is no. Asked by Erdős and Newman. Nešetřil and Rödl [NeRo85] have shown the\nanswer is no for all $C$ (even if $t$ is also allowed to depend on $A$).\n\nErdős [Er80e] had previously shown the answer is no for $C=3,4$ and infinitely many other\nvalues of $C$.\n\nSee also [774].\n\nThe linked proof writes the representation function as\n`Set.ncard {p : ℕ × ℕ | p.1 ∈ A ∧ p.2 ∈ A ∧ p.1 + p.2 = n}`, which counts the same ordered\npairs as `sumRep`, and states the partition condition as a named definition with the same two\nconjuncts used below. Its `∃ t` additionally carries `1 ≤ t`, which costs nothing: `t = 0`\nforces `A = ∅`, and `A = {1}` has all representation counts at most `C` for `C ≥ 1`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/f8a51976fd2e66a52b4928c109fb9ae877a1a507/problems/328/Erdos328.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«328»","statement":"False ↔\n  ∀ (C : ℕ),\n    0 < C →\n      ∃ t,\n        ∀ (A : Set ℕ),\n          (∀ (n : ℕ), AdditiveCombinatorics.sumRep A n ≤ C) →\n            ∃ P,\n              ⋃ i, P i = A ∧\n                Set.univ.PairwiseDisjoint P ∧ ∀ (i : Fin t) (n : ℕ), AdditiveCombinatorics.sumRep (P i) n < C","subjects":["11"],"theorem":"Erdos328.erdos_328"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that there are only finitely many collections of disjoint intervals $I_1,\\ldots,I_n$ of size $\\lvert I_i\\rvert \\geq 4$ for $1\\leq i\\leq n$ such that$$\\prod_{1\\leq i\\leq n}\\prod_{m\\in I_i}m$$is a square?\n\nThis is false: Ulas [Ul05] constructed infinitely many such collections.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos363.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«363»","statement":"False ↔ {S | Erdos363.IsValidCollection S}.Finite","subjects":["11"],"theorem":"Erdos363.erdos_363"},{"answerKinds":[],"category":"research solved","docstring":"An example of an $A$ with the property that there are no distinct $a,b,c \\in A$ such that\n$a \\mid (b+c)$ and $b,c > a$ and such that\n$$\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\log N > 0$$\nis given by the set of $p^2$, where $p\\equiv 3\\pmod{4}$ is prime.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«12»","statement":"∀ (A : Set ℕ),\n  A = {x | ∃ p, ∃ (_ : Nat.Prime p) (_ : p ≡ 3 [MOD 4]), p ^ 2 = x} →\n    Erdos12.IsGood A ∧ 0 < Filter.liminf (fun N => ↑(A ∩ Set.Icc 1 N).ncard * Real.log ↑N / √↑N) Filter.atTop","subjects":["11"],"theorem":"Erdos12.erdos_12.variants.example"},{"answerKinds":[],"category":"textbook","docstring":"The set of $p ^ 2$ where $p \\cong 3 \\mod 4$ is prime is an example of a good set.\nFormal proof provided by AlphaProof\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/2663234a28260853790aa5752d8d4550ff0ab1ca/FormalConjectures/ErdosProblems/12.lean#L39"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«12»","statement":"Erdos12.IsGood {x | ∃ p, ∃ (_ : p ≡ 3 [MOD 4]) (_ : Nat.Prime p), p ^ 2 = x}","subjects":["11"],"theorem":"Erdos12.isGood_example"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Sárközy proved that such an $A$ must have density 0.\n[ErSa70] Erd\\H os, P. and Sárk\\\"ozi, A., On the divisibility properties of sequences of integers.\n    Proc. London Math. Soc. (3) (1970), 97-101\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«12»","statement":"∀ (A : Set ℕ), Erdos12.IsGood A → A.HasDensity 0","subjects":["11"],"theorem":"Erdos12.erdos_12.variants.erdos_sarkozy_density_0"},{"answerKinds":[],"category":"research solved","docstring":"Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \\in A$ such\nthat $a \\mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then\n$$\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll N^{2/3}/\\log N$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«12»","statement":"∀ (A : Set ℕ),\n  Erdos12.IsGood A →\n    A.Pairwise Nat.Coprime → (fun N => ↑(A ∩ Set.Icc 1 N).ncard) =O[Filter.atTop] fun N => ↑N ^ (2 / 3) / Real.log ↑N","subjects":["11"],"theorem":"Erdos12.erdos_12.variants.baier"},{"answerKinds":[],"category":"research solved","docstring":"Given any function $f(x)\\to \\infty$ as $x\\to \\infty$ there exists a set $A$ with the property\nthat there are no distinct $a,b,c \\in A$ such that $a \\mid (b+c)$ and $b,c > a$, such that there are\ninfinitely many $N$ such that $$\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert > \\frac{N}{f(N)}.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«12»","statement":"∀ (f : ℕ → ℕ),\n  Filter.Tendsto f Filter.atTop Filter.atTop →\n    ∃ A, Erdos12.IsGood A ∧ {N | ↑N / ↑(f N) < ↑(A ∩ Set.Icc 1 N).ncard}.Infinite","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos12.erdos_12.variants.erdos_sarkozy"},{"answerKinds":[],"category":"research solved","docstring":"Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \\in A$ such\nthat $a \\mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then\n$$\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll N^{2/3}$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«12»","statement":"∀ (A : Set ℕ),\n  Erdos12.IsGood A → A.Pairwise Nat.Coprime → (fun N => ↑(A ∩ Set.Icc 1 N).ncard) =O[Filter.atTop] fun N => ↑N ^ (2 / 3)","subjects":["11"],"theorem":"Erdos12.erdos_12.variants.schoen"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A$ be an infinite set such that there are no distinct $a,b,c \\in A$\nsuch that $a \\mid (b+c)$ and $b,c > a$. Does there exist some absolute constant $c > 0$\nsuch that there are always infinitely many $N$\nwith $|A \\cap \\{1, \\dotsc, N\\}| < N^{1−c}$?\n\nThe DeepMind prover agent has found a formal disproof of this statement.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/118a6a60df73a9f47d6c89f3cdb3786eaa2e8d0a/FormalConjectures/ErdosProblems/12.lean#L740"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«12»","statement":"False ↔ ∃ c > 0, ∀ (A : Set ℕ), Erdos12.IsGood A → {N | ↑(A ∩ Set.Icc 1 N).ncard < ↑N ^ (1 - c)}.Infinite","subjects":["11"],"theorem":"Erdos12.erdos_12.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $A$ be an infinite set such that there are no distinct $a,b,c \\in A$\nsuch that $a \\mid (b+c)$ and $b,c > a$. Is it true that $∑_{n \\in A} \\frac{1}{n} < \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«12»","statement":"True ↔ ∀ (A : Set ℕ), Erdos12.IsGood A → Summable fun n => 1 / ↑↑n","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos12.erdos_12.parts.iii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A$ be an infinite set such that there are no distinct $a,b,c \\in A$\nsuch that $a \\mid (b+c)$ and $b,c > a$. Is there such an $A$ with\n$\\liminf \\frac{|A \\cap \\{1, \\dotsc, N\\}|}{N^{1/2}} > 0$ ?\n\nThe DeepMind prover agent has found a formal proof of this statement.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/8d872b465955e46e2d28bc165d186ea41fd0da9e/FormalConjectures/ErdosProblems/12.lean#L810"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«12»","statement":"True ↔ ∃ A, Erdos12.IsGood A ∧ 0 < Filter.liminf (fun N => ↑(A ∩ Set.Icc 1 N).ncard / √↑N) Filter.atTop","subjects":["11"],"theorem":"Erdos12.erdos_12.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"Heath-Brown proved that at least one of 2, 3, or 5 is a primitive root for infinitely many primes $p$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«985»","statement":"{p | Nat.Prime p ∧ (orderOf 2 = p - 1 ∨ orderOf 3 = p - 1 ∨ orderOf 5 = p - 1)}.Infinite","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos985.erdos_985.variants.two_three_five_primitive_root"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for every prime $p$, there is a prime $q \\leq p$ which is a primitive root modulo $p$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«985»","statement":"True ↔ ∀ (p : ℕ), Nat.Prime p → p ≠ 2 → ∃ q, Nat.Prime q ∧ q < p ∧ orderOf ↑q = p - 1","subjects":["11"],"theorem":"Erdos985.erdos_985"},{"answerKinds":[],"category":"research open","docstring":"Let$$V'(x)=\\#\\{\\phi(m) : 1\\leq m\\leq x\\}$$and$$V(x)=\\#\\{\\phi(m) \\leq x : 1\\leq m\\}.$$\nDoes $\\lim V(x)/V'(x)$ exist?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«417»","statement":"True ↔\n  ∃ L,\n    Filter.Tendsto\n      (fun x => ↑{k | k ∈ Set.range Nat.totient ∧ ↑k ≤ x}.ncard / ↑(Nat.totient '' {m | 1 ≤ m ∧ ↑m ≤ x}).ncard)\n      Filter.atTop (nhds L)","subjects":["11"],"theorem":"Erdos417.erdos_417.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is it $>1$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«417»","statement":"True ↔\n  ∃ L > 1,\n    Filter.Tendsto\n      (fun x => ↑{k | k ∈ Set.range Nat.totient ∧ ↑k ≤ x}.ncard / ↑(Nat.totient '' {m | 1 ≤ m ∧ ↑m ≤ x}).ncard)\n      Filter.atTop (nhds L)","subjects":["11"],"theorem":"Erdos417.erdos_417.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $r \\ge 2$. Is every large integer the sum of at most $r + 1$ many $r$-powerful numbers?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1107»","statement":"∀ r ≥ 2, ∀ᶠ (n : ℕ) in Filter.atTop, Erdos1107.SumOfRPowerful r n","subjects":["11"],"theorem":"Erdos1107.erdos_1107"},{"answerKinds":[],"category":"research solved","docstring":"Heath-Brown [He88] proved every large integer the sum of at most three $2$-powerful numbers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1107»","statement":"∀ᶠ (n : ℕ) in Filter.atTop, Erdos1107.SumOfRPowerful 2 n","subjects":["11"],"theorem":"Erdos1107.erdos_1107.variants.two"},{"answerKinds":[],"category":"research open","docstring":"Is there an infinite set $A \\subset \\mathbb{N}$ such that for every $a \\in A$,\nthere is an integer n such that $\\phi(n)=a$, and\nyet if $n_a$ is the smallest such integer, then $\\frac{n_a}{a} → \\infty$ as $a → ∞$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«51»","statement":"True ↔\n  ∃ A n,\n    A.Infinite ∧\n      (∀ (a : ↑A), IsLeast (Nat.totient ⁻¹' {↑a}) (n a)) ∧\n        Filter.Tendsto (fun a => ↑(n a) / ↑↑a) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos51.erdos_51"},{"answerKinds":[],"category":"research open","docstring":"Is there a covering system all of whose moduli are odd (and greater than 1)?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«7»","statement":"True ↔ ∃ C, ∀ (i : C.ι), ¬C.moduli i ≤ Ideal.span {2} ∧ C.moduli i ≠ ⊤","subjects":["11"],"theorem":"Erdos7.erdos_7"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is\n$$\n  \\sum_{n=1}^\\infty \\frac{\\sigma(n)}{2^n}\n$$\nirrational? Here $\\sigma(n)$ is the sum of divisors function.\n\nThe answer is yes, as shown by Nesterenko [Ne96].\n\n[Ne96] Nesterenko, Yu V., _Modular functions and transcendence questions_,\nMat. Sb. 187 *9* (1996), 1319--1348.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«250»","statement":"(∀ (x : ℝ), HasSum (fun n => ↑((ArithmeticFunction.sigma 1) n) / 2 ^ n) x → Irrational x) ↔ True","subjects":["11"],"theorem":"Erdos250.erdos_250"},{"answerKinds":["Prop"],"category":"research solved","docstring":"The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to\ncolour the vertices of $G$ such that each colour class induces either a complete graph or\nindependent set.\n\nIf $G$ is a graph with chromatic number $\\chi(G)=m$ then must $G$ contain a subgraph $H$ with\n$$\n\\zeta(H) \\gg \\frac{m}{\\log m}?\n$$\n\nA problem of Erdős and Gimbel, who proved that there must exist a subgraph $H$ with\n$$\n\\zeta(H) \\gg \\left(\\frac{m}{\\log m}\\right)^{1/2}.\n$$\nThe proposed bound would be best possible, as shown by taking $G$ to be a complete graph.\n\nThe answer is yes, proved by Alon, Krivelevich, and Sudakov.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos760.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«760»","statement":"True ↔\n  ∃ c > 0,\n    ∀ (V : Type u_1) [Finite V] (G : SimpleGraph V) (m : ℕ),\n      G.chromaticNumber = ↑m → ∃ H k, ↑k ≤ H.coe.cochromaticNumber ∧ c * ↑m / Real.log ↑m ≤ ↑k","subjects":["5"],"theorem":"Erdos760.erdos_760"},{"answerKinds":[],"category":"research solved","docstring":"A problem of Erdős and Gimbel, who proved that there must exist a subgraph $H$ with\n$$\n\\zeta(H) \\gg \\left(\\frac{m}{\\log m}\\right)^{1/2}.\n$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«760»","statement":"∃ c > 0,\n  ∀ (V : Type u_1) [Finite V] (G : SimpleGraph V) (m : ℕ),\n    G.chromaticNumber = ↑m → ∃ H k, ↑k ≤ H.coe.cochromaticNumber ∧ c * √(↑m / Real.log ↑m) ≤ ↑k","subjects":["5"],"theorem":"Erdos760.erdos_760.variants.erdos_gimbel"},{"answerKinds":[],"category":"research open","docstring":"Erdős Problem 839 (Part 2, stronger) [Er78f][Er92c]:\n\nLet $1 \\leq a_1 < a_2 < \\cdots$ be a strictly increasing sequence of positive integers\nsuch that no $a_i$ is the sum of consecutive $a_j$ for $j < i$.\nIs it true that $\\lim_{x \\to \\infty} \\frac{1}{\\log x} \\sum_{a_n < x} \\frac{1}{a_n} = 0$?\n\nThis is equivalent to asking whether the range $\\{a_1,a_2,\\ldots\\}$ has logarithmic density zero\n(see `Set.HasLogDensity`).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«839»","statement":"True ↔\n  ∀ (a : ℕ → ℕ), (∀ (n : ℕ), 1 ≤ a n) → StrictMono a → Erdos839.SumOfConsecutiveFree a → (Set.range a).HasLogDensity 0","subjects":["11"],"theorem":"Erdos839.erdos_839.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Erdős Problem 839 (Part 1) [Er78f][Er92c]:\n\nLet $1 \\leq a_1 < a_2 < \\cdots$ be a strictly increasing sequence of positive integers\nsuch that no $a_i$ is the sum of consecutive $a_j$ for $j < i$.\nIs it true that $\\limsup a_n / n = \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«839»","statement":"True ↔\n  ∀ (a : ℕ → ℕ),\n    (∀ (n : ℕ), 1 ≤ a n) →\n      StrictMono a → Erdos839.SumOfConsecutiveFree a → Filter.limsup (fun n => ↑(a n) / ↑n) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos839.erdos_839.parts.i"},{"answerKinds":[],"category":"research solved","docstring":"It is a classical folklore fact that if $A\\subseteq \\{1,\\ldots,2N\\}$ has size $\\geq N+2$ then\nthere are distinct $a,b\\in A$ such that $a+b\\in A$, which establishes the $k=2$ case.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«865»","statement":"∀ (N : ℕ), ∀ A ⊆ Finset.Icc 1 (2 * N), A.card ≥ N + 2 → ∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ a + b ∈ A","subjects":["5","11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos865.erdos_865.variants.k2"},{"answerKinds":[],"category":"research solved","docstring":"Choi, Erdős, and Szemerédi [CES75] have proved that, for all $k\\geq 3$, there exists $\\epsilon_k>0$\nsuch that (for large enough $N$) $f_k(N)\\leq \\left(\\frac{2}{3}-\\epsilon_k\\right)N$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«865»","statement":"∀ (k : ℕ), 3 ≤ k → ∃ ε > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos865.f N k) ≤ (2 / 3 - ε) * ↑N","subjects":["5","11"],"theorem":"Erdos865.erdos_865.variants.upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"There exists a constant $C>0$ such that, for all large $N$, if $A\\subseteq \\{1,\\ldots,N\\}$ has\nsize at least $\\frac{5}{8}N+C$ then there are distinct $a,b,c\\in A$ such that $a+b,a+c,b+c\\in A$.\n\nA problem of Erdős and Sós (also earlier considered by Choi, Erdős, and Szemerédi [CES75], but Erdős\nhad forgotten this).\n\nThis problem was solved in the affirmative by Cipollini and GPT Pro [Ci26].\n\nThis is true. The linked proof gives it in the contrapositive and with the constant cleared:\nevery triple-free $A\\subseteq\\{1,\\ldots,N\\}$ satisfies $8\\lvert A\\rvert\\leq 5N+C$ for a fixed\n$C$, for every $N$ rather than only for large $N$. It also shows the threshold is sharp, by\nexhibiting triple-free sets of size $(5N+16)/8$ for every $N$ divisible by $8$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Jayyhk/erdos-lean/blob/f8a51976fd2e66a52b4928c109fb9ae877a1a507/problems/865/Erdos865.lean"},{"conditions":[],"kind":"lean4","link":"https://github.com/mrricky22/erdos-865-lean/blob/f861539107a7adeaa97462ce7c7171127696b63a/RequestProject/Main.lean#L45"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«865»","statement":"∃ C > 0,\n  ∀ᶠ (N : ℕ) in Filter.atTop,\n    ∀ A ⊆ Finset.Icc 1 N,\n      ↑A.card ≥ 5 / 8 * ↑N + C → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, a ≠ b ∧ a ≠ c ∧ b ≠ c ∧ a + b ∈ A ∧ a + c ∈ A ∧ b + c ∈ A","subjects":["5","11"],"theorem":"Erdos865.erdos_865"},{"answerKinds":[],"category":"research open","docstring":"Erdős and Sós conjectured that\n$f_k(N)\\sim \\frac{1}{2}\\left(1+\\sum_{1\\leq r\\leq k-2}\\frac{1}{4^r}\\right) N$,\nwhere $f_k(N)$ is the minimal size of a subset of $\\{1, \\dots, N\\}$ guaranteeing $k$ elements\nhave all pairwise sums in the set.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«865»","statement":"∀ (k : ℕ),\n  2 ≤ k →\n    Asymptotics.IsEquivalent Filter.atTop (fun N => ↑(Erdos865.f N k)) fun N =>\n      1 / 2 * (1 + ∑ r ∈ Finset.Icc 1 (k - 2), (1 / 4) ^ r) * ↑N","subjects":["5","11"],"theorem":"Erdos865.erdos_865.variants.sos"},{"answerKinds":[],"category":"research open","docstring":"A general version asks, for a fixed $r \\in \\mathbb{N}$, if a set\n$A \\subseteq \\{1, ..., N\\}$ has no $a \\in A$ and $b_1, ..., b_r \\in A$ such that\n$a | (b_1 + ... + b_r)$ and $a < \\min(b_1, ..., b_r)$, then is it true that\n$|A| \\le N/(r+1) + O(1)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«13»","statement":"True ↔\n  ∀ (r : ℕ),\n    ∃ C,\n      ∀ (N : ℕ),\n        ∀ A ⊆ Finset.Icc 1 N,\n          (∀ a ∈ A, ∀ (b : Fin r → ℕ), (∀ (i : Fin r), b i ∈ A) → (∀ (i : Fin r), a < b i) → ¬a ∣ ∑ i, b i) →\n            ↑A.card ≤ ↑N / (↑r + 1) + C","subjects":["5","11"],"theorem":"Erdos13.erdos_13.variants.general"},{"answerKinds":[],"category":"research solved","docstring":"If $A \\subseteq \\{1, ..., N\\}$ is a set with no $a, b, c \\in A$ such that $a | (b+c)$ and\n$a < \\min(b,c)$, then $|A| \\le N/3 + O(1)$. This has been solved by Bedert [Be23].\n\n[Be23] Bedert, B., _On a problem of Erdős and Sárközy about sequences with no term dividing\nthe sum of two larger terms_. arXiv:2301.07065 (2023).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«13»","statement":"∃ C, ∀ (N : ℕ), ∀ A ⊆ Finset.Icc 1 N, Erdos13.IsForbiddenTripleFree A → ↑A.card ≤ ↑N / 3 + C","subjects":["5","11"],"theorem":"Erdos13.erdos_13"},{"answerKinds":[],"category":"research solved","docstring":"Let $\\{S_k\\}$ be any sequence of sets in the complex plane, each of which has no finite\nlimit point. Then there exists a sequence $\\{n_k\\}$ of positive integers and a\ntranscendental entire function $f(z)$ such that $f^{(n_k)}(z) = 0$ if $z \\in S_k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«229»","statement":"∀ {S : ℕ → Set ℂ},\n  (∀ (k : ℕ), derivedSet (S k) = ∅) →\n    ∃ f n,\n      Differentiable ℂ f ∧\n        Transcendental (Polynomial ℂ) f ∧ ∀ (k : ℕ), 0 < n k ∧ ∀ {z : ℂ}, z ∈ S k → iteratedDeriv (n k) f z = 0","subjects":["30"],"theorem":"Erdos229.theorem_1"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $(S_n)_{n \\ge 1}$ be a sequence of sets of complex numbers, none of which have a finite\nlimit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n \\ge 1$, there\nexists some $k_n \\ge 0$ such that $f^{(k_n)}(z) = 0$ for all $z \\in S_n$.\n\nThis is Problem 2.30 in [Ha74], where it is attributed to Erdős.\n\nSolved in the affirmative by Barth and Schneider [BaSc72].\n\nThis was formalized in Lean by Alexeev using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos229.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«229»","statement":"True ↔\n  ∀ (S : ℕ → Set ℂ),\n    (∀ (n : ℕ), derivedSet (S n) = ∅) →\n      ∃ f, Transcendental (Polynomial ℂ) f ∧ Differentiable ℂ f ∧ ∀ n ≥ 1, ∃ k, ∀ z ∈ S n, iteratedDeriv k f z = 0","subjects":["30"],"theorem":"Erdos229.erdos_229"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A$ be an additive basis of order $2$, and suppose $1_A\\ast 1_A(n)\\to \\infty$ as $n\\to \\infty$. Can $A$ be partitioned into two disjoint additive bases of order $2$?\n\nA question of Erdős and Nathanson [ErNa88], who proved this is true if $1_A\\ast 1_A(n) > c\\log n$ (for all large $n$) for some constant $c>(\\log\\frac{4}{3})^{-1}$. Erdős and Nathanson [ErNa89] also proved that for every $t$ there exists a basis $A$ of order $2$ such that $1_A\\ast 1_A(n)\\geq t$ for all large $n$ and yet $A$ cannot be partitioned into two disjoint additive bases. This has been disproved by Larsen using Claude Opus 4.5 - in fact only a small modification of the argument of [ErNa89] is required.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos871.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«871»","statement":"False ↔\n  ∀ (A : Set ℕ),\n    ((∀ᶠ (n : ℕ) in Filter.atTop, ∃ a ∈ A, ∃ b ∈ A, a + b = n) ∧\n        ∀ (t : ℕ),\n          ∀ᶠ (n : ℕ) in Filter.atTop,\n            ∃ pairs, pairs.card ≥ t ∧ ∀ p ∈ pairs, p.1 ∈ A ∧ p.2 ∈ A ∧ p.1 + p.2 = n ∧ p.1 ≤ p.2) →\n      ∃ B C,\n        (∀ (x : ℕ), x ∈ A ↔ x ∈ B ∨ x ∈ C) ∧\n          Disjoint B C ∧\n            (∀ᶠ (n : ℕ) in Filter.atTop, ∃ a ∈ B, ∃ b ∈ B, a + b = n) ∧\n              ∀ᶠ (n : ℕ) in Filter.atTop, ∃ a ∈ C, ∃ b ∈ C, a + b = n","subjects":["11"],"theorem":"Erdos871.erdos_871"},{"answerKinds":[],"category":"research open","docstring":"Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is the multiset\n$$\\{ \\lfloor \\alpha\\rfloor,\\lfloor 2\\alpha\\rfloor,\\lfloor 4\\alpha\\rfloor,\\ldots\\}\\cup\n\\{ \\lfloor \\beta\\rfloor,\\lfloor 2\\beta\\rfloor,\\lfloor 4\\beta\\rfloor,\\ldots\\}$$ complete? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«354»","statement":"True ↔ ∀ α > 0, ∀ β > 0, Irrational (α / β) → IsAddCompleteNatSeq' (Erdos354.FloorMultiples.interleave α β 2)","subjects":["11"],"theorem":"Erdos354.erdos_354.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is\n$$\\{ \\lfloor \\alpha\\rfloor,\\lfloor \\gamma\\alpha\\rfloor,\\lfloor \\gamma^2\\alpha\\rfloor,\\ldots\\}\\cup\n\\{ \\lfloor \\beta\\rfloor,\\lfloor \\gamma\\beta\\rfloor,\\lfloor \\gamma^2\\beta\\rfloor,\\ldots\\}$$ complete? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«354»","statement":"True ↔\n  ∃ γ ∈ Set.Ioo 1 2,\n    ∀ α > 0, ∀ β > 0, Irrational (α / β) → IsAddCompleteNatSeq' (Erdos354.FloorMultiples.interleave α β γ)","subjects":["11"],"theorem":"Erdos354.erdos_354.parts.ii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Must `f n ≫ n ^ 2`?\n\nThis stronger quadratic variant was also proved formally by the DeepMind prover agent [DM26b].\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/ff58c933d53bb807bf85d98a47402703f9f14ed3/FormalConjectures/ErdosProblems/152.lean#L496"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«152»","statement":"True ↔ (fun n => ↑n ^ 2) =O[Filter.atTop] fun n => ↑(Erdos152.f n)","subjects":["5"],"theorem":"Erdos152.erdos_152.variants.square"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Must `lim f n = ∞`?\n\nThis was proved formally by the DeepMind prover agent [DM26a].\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/29c60aa79729701905cf9e92517af23f588971f2/FormalConjectures/ErdosProblems/152.lean#L485"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«152»","statement":"True ↔ Filter.Tendsto Erdos152.f Filter.atTop Filter.atTop","subjects":["5"],"theorem":"Erdos152.erdos_152"},{"answerKinds":[],"category":"research open","docstring":"Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even\ninteger $t$ such that $d_n = t$ has no solutions for $n \\le x$.\n\nIs it true that $r(x) \\to \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«853»","statement":"Filter.Tendsto Erdos853.r Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos853.erdos_853.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even\ninteger $t$ such that $d_n = t$ has no solutions for $n \\le x$.\n\nIs it true that $r(x) / \\log x \\to \\infty$? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«853»","statement":"Filter.Tendsto (fun n => ↑(Erdos853.r n) / Real.log ↑n) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos853.erdos_853.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Let $A$ be a finite set and\n$$B=\\{ n \\geq 1 : a\\mid n\\textrm{ for some }a\\in A\\}.$$\nIs it true that, for every $m>n\\geq \\max(A)$,\n$$\\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«488»","statement":"True ↔\n  ∀ (A : Finset ℕ),\n    A.Nonempty →\n      0 ∉ A →\n        1 ∉ A →\n          ∀ (n m : ℕ),\n            m > n →\n              A.max ≤ ↑n →\n                ↑{x ∈ Finset.Icc 1 m | x ∈ {n | n ≥ 1 ∧ ∃ a ∈ A, a ∣ n}}.card / ↑m <\n                  2 * ↑{x ∈ Finset.Icc 1 n | x ∈ {n | n ≥ 1 ∧ ∃ a ∈ A, a ∣ n}}.card / ↑n","subjects":["5","11"],"theorem":"Erdos488.erdos_488"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er51] proved this for all $0\\leq \\alpha\\leq 2$.\n\n[Er51] Erdös, P., Some problems and results in elementary number theory.\nPubl. Math. Debrecen (1951), 103-109.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«145»","statement":"∀ {α : ℝ},\n  α ∈ Set.Icc 0 2 →\n    ∃ β,\n      Filter.Tendsto (fun x => 1 / x * ∑ n ∈ Erdos145.A x, (↑(Erdos145.s (n + 1)) - ↑(Erdos145.s n)) ^ α) Filter.atTop\n        (nhds β)","subjects":["11"],"theorem":"Erdos145.erdos_145.variants.le_two"},{"answerKinds":[],"category":"research solved","docstring":"Greaves, Harman, and Huxley [GHH97] showed that this is true for $0 \\leq \\alpha\\leq 11/3$.\n\n[GHH97] Greaves, G. R. H. and Harman, G. and Huxley, M. N., Sieve Methods, Exponential Sums, and\ntheir Applications in Number Theory. (1997).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«145»","statement":"∀ {α : ℝ},\n  α ∈ Set.Icc 0 (11 / 3) →\n    ∃ β,\n      Filter.Tendsto (fun x => 1 / x * ∑ n ∈ Erdos145.A x, (↑(Erdos145.s (n + 1)) - ↑(Erdos145.s n)) ^ α) Filter.atTop\n        (nhds β)","subjects":["11"],"theorem":"Erdos145.erdos_145.variants.le_eleven_thirds"},{"answerKinds":[],"category":"research open","docstring":"Let $s_1 < s_2 < \\cdots$ be the sequence of squarefree numbers. Is it true that, for any\n$\\alpha\\geq 0$,\n$$\n\\lim_{x\\to\\infty} \\frac{1}{x}\\sum_{s_n\\leq x}(s_{n+1}-s_n)^\\alpha\n$$\nexists?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«145»","statement":"True ↔\n  ∀ α ≥ 0,\n    ∃ β,\n      Filter.Tendsto (fun x => 1 / x * ∑ n ∈ Erdos145.A x, (↑(Erdos145.s (n + 1)) - ↑(Erdos145.s n)) ^ α) Filter.atTop\n        (nhds β)","subjects":["11"],"theorem":"Erdos145.erdos_145"},{"answerKinds":[],"category":"research solved","docstring":"Hooley [Ho73] extended this to all $0 \\leq \\alpha\\leq 3$.\n\n[Ho73] Hooley, Christopher, On the intervals between consecutive terms of sequences. Proc. Symp. Pure Math, vol. 24, pp. 129-140. 1973.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«145»","statement":"∀ {α : ℝ},\n  α ∈ Set.Icc 0 3 →\n    ∃ β,\n      Filter.Tendsto (fun x => 1 / x * ∑ n ∈ Erdos145.A x, (↑(Erdos145.s (n + 1)) - ↑(Erdos145.s n)) ^ α) Filter.atTop\n        (nhds β)","subjects":["11"],"theorem":"Erdos145.erdos_145.variants.le_three"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $\\mathbb{N}$ is 2-coloured then is there some infinite set $A\\subseteq \\mathbb{N}$ such that\nall finite subset sums$$ \\sum_{n\\in S}n$$(as $S$ ranges over all non-empty finite subsets of $A$)\nare monochromatic?\n\nAsked by Graham and Rothschild. Proved by Hindman [Hi74] (for any number of colours).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos532.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«532»","statement":"True ↔ ∀ (c : ℕ → Fin 2), ∃ A, A.Infinite ∧ ∃ color, ∀ (S : Finset ℕ), S.Nonempty → ↑S ⊆ A → c (∑ n ∈ S, n) = color","subjects":["5"],"theorem":"Erdos532.erdos_532"},{"answerKinds":[],"category":"research solved","docstring":"The upper bound $f u ≤ u ^ 2$ is attained exactly when `u` is prime: $f p = p ^ 2$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem459.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«459»","statement":"∀ {p : ℕ}, Nat.Prime p → Erdos459.f p = p ^ 2","subjects":["11"],"theorem":"Erdos459.erdos_459.variants.upper_tight"},{"answerKinds":[],"category":"research solved","docstring":"Let $f(u)$ be the largest $v$ such that no $m\\in (u,v)$ is composed entirely of primes dividing\n$uv$. Estimate $f(u)$.\n\nThe estimate $u + 2 \\le f(u) \\le u^2$ holds for every $u \\ge 2$. The upper bound is attained\nwhen $u$ is prime, and the lower bound when $u = 2^k - 2$ with $k \\ge 2$; Cambie further showed\nthat $f(n) = (1 + o(1))n$ for almost all $n$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem459.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«459»","statement":"∀ {u : ℕ}, 2 ≤ u → u + 2 ≤ Erdos459.f u ∧ Erdos459.f u ≤ u ^ 2","subjects":["11"],"theorem":"Erdos459.erdos_459"},{"answerKinds":[],"category":"test","docstring":"For target $4$ and universe $\\{1, 2, 3\\}$, the maximum is $2$: the full set is invalid because\nits subset $\\{1, 3\\}$ sums to $4$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«361»","statement":"Erdos361.maxSubsetSumAvoidingCard 3 4 = 2","subjects":["11"],"theorem":"Erdos361.maxSubsetSumAvoidingCard_three_four"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Let $c > 0$ and $n$ be some large integer. What is the size of the largest set\n$A \\subseteq \\{1, \\ldots, \\lfloor c n \\rfloor\\}$ such that $n$ is not a sum of a subset of $A$?\nDoes this depend on $n$ in an irregular way?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«361»","statement":"∀ (c : ℝ), 0 < c → Erdos361.subsetSumAvoidanceNumber c = sorry","subjects":["11"],"theorem":"Erdos361.erdos_361"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Asymptotic version of Erdős Problem 361: determine the order of growth of the largest cardinality\nas $n \\to \\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«361»","statement":"∀ (c : ℝ), 0 < c → (fun n => ↑(Erdos361.subsetSumAvoidanceNumber c n)) =Θ[Filter.atTop] sorry","subjects":["11"],"theorem":"Erdos361.erdos_361.asymptotic"},{"answerKinds":[],"category":"research open","docstring":"Must $\\alpha$ be rational?\n\nThe same nondegeneracy condition on $G$ is used as in part (i). Rationality means that the real\nnumber $\\alpha$ lies in the image of the canonical embedding $\\mathbb{Q}\\to\\mathbb{R}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«713»","statement":"True ↔\n  ∀ (q : ℕ) (G : SimpleGraph (Fin q)),\n    G.IsBipartite →\n      2 ≤ G.edgeFinset.card →\n        ∀ (α c : ℝ),\n          α ∈ Set.Ico 1 2 →\n            0 < c →\n              (Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(SimpleGraph.extremalNumber n G)) fun n => c * ↑n ^ α) →\n                α ∈ Set.range Rat.cast","subjects":["5"],"theorem":"Erdos713.erdos_713.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for every bipartite graph $G$, there exists some $\\alpha\\in [1,2)$ and $c>0$ such that $$\\mathrm{ex}(n;G)\\sim cn^\\alpha?$$\n\nThe condition that $G$ have at least two edges excludes degenerate forbidden graphs whose\nextremal number is eventually zero, for which the displayed asymptotic with $c>0$ is impossible.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«713»","statement":"True ↔\n  ∀ (q : ℕ) (G : SimpleGraph (Fin q)),\n    G.IsBipartite →\n      2 ≤ G.edgeFinset.card →\n        ∃ α c,\n          α ∈ Set.Ico 1 2 ∧\n            0 < c ∧\n              Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(SimpleGraph.extremalNumber n G)) fun n => c * ↑n ^ α","subjects":["5"],"theorem":"Erdos713.erdos_713.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Is there an integer $m$ with $(m, 6) = 1$ such that none of $2^k \\cdot 3^\\ell \\cdot m + 1$ are prime,\nfor any $k, \\ell \\ge 0$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«203»","statement":"True ↔ ∃ m, m.Coprime 6 ∧ ∀ (k l : ℕ), ¬Nat.Prime (2 ^ k * 3 ^ l * m + 1)","subjects":["5"],"theorem":"Erdos203.erdos_203"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $n_1<n_2<\\cdots$ be an infinite sequence such that, for any choice of congruence classes\n$a_i\\pmod{n_i}$, the set of integers not satisfying any of the congruences $a_i\\pmod{n_i}$ has\ndensity $0$. Is it true that for every $\\epsilon>0$ there exists some $k$ such that, for every\nchoice of congruence classes $a_i$, the density of integers not satisfying any of the congruences\n$a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ is less than $\\epsilon$?\n\nThe answer is yes; the linked Lean proof formalizes Somani's argument.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos281.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«281»","statement":"True ↔\n  ∀ (n : ℕ → ℕ),\n    StrictMono n →\n      (∀ (i : ℕ), 0 < n i) →\n        (∀ (a : Erdos281.ResidueChoice n), (Erdos281.avoidAll n a).HasIntDensity 0) →\n          ∀ (ε : ℝ),\n            0 < ε → ∃ k, ∀ (a : Erdos281.ResidueChoice n), ∃ d, (Erdos281.avoidPrefix n a k).HasIntDensity d ∧ d < ε","subjects":["11"],"theorem":"Erdos281.erdos_281"},{"answerKinds":[],"category":"research solved","docstring":"According to https://www.erdosproblems.com/672, Euler proved this. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«672»","statement":"Erdos672.Erdos672With 4 2","subjects":["11"],"theorem":"Erdos672.erdos_672.variants.euler"},{"answerKinds":[],"category":"research open","docstring":"Can the product of an arithmetic progression of positive integers $n, n + d, ..., n + (k - 1)d$\nof length ≥ 4, with $(n, d) = 1$, be a perfect power?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«672»","statement":"True ↔ ∀ (k l : ℕ), l > 1 → k ≥ 4 → Erdos672.Erdos672With k l","subjects":["11"],"theorem":"Erdos672.erdos_672"},{"answerKinds":[],"category":"research solved","docstring":"According to https://www.erdosproblems.com/672, Obláth proved this.\n\n[Ob51] Oblath, Richard, Eine Bemerkung über Produkte aufeinander folgender Zahlen.\nJ. Indian Math. Soc. (N.S.) (1951), 135-139. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«672»","statement":"Erdos672.Erdos672With 5 2 ∧ Erdos672.Erdos672With 3 3 ∧ Erdos672.Erdos672With 3 4 ∧ Erdos672.Erdos672With 3 5","subjects":["11"],"theorem":"Erdos672.erdos_672.variants.oblath"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does every graph with chromatic number $\\aleph_1$ contain an infinitely connected subgraph with\nchromatic number $\\aleph_1$?\n\nKomjáth [Ko13] proved that it is consistent that the answer is no. This was improved by\nSoukup [So15], who constructed a counterexample using no extra set-theoretical assumptions. A\nsimpler elementary example was given by Bowler and Pitz [BoPi24].\n\nThis was formalized in Lean by Alexeev using Aristotle and Aleph Prover.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos1067.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1067»","statement":"False ↔\n  ∀ (V : Type) (G : SimpleGraph V),\n    G.chromaticCardinal = Cardinal.aleph 1 → ∃ H, H.coe.chromaticCardinal = Cardinal.aleph 1 ∧ H.coe.InfinitelyConnected","subjects":["5"],"theorem":"Erdos1067.erdos_1067"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Thomassen [Th17] constructed a counterexample to the version which asks for infinite\nedge-connectivity (that is, to disconnect the graph requires deleting infinitely many edges).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1067»","statement":"False ↔\n  ∀ (V : Type) (G : SimpleGraph V),\n    G.chromaticCardinal = Cardinal.aleph 1 →\n      ∃ H, H.coe.chromaticCardinal = Cardinal.aleph 1 ∧ Erdos1067.InfinitelyEdgeConnected H.coe","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos1067.erdos_1067.variants.infinite_edge_connectivity"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f:\\mathbb{N}\\to \\{-1,1\\}$ be a multiplicative function. Is it true that\n$$ \\lim_{N\\to \\infty}\\frac{1}{N}\\sum_{n\\leq N}f(n)$$ always exists?\n\nThe answer is yes, as proved by Wirsing [Wi67], and generalised by Halász [Ha68].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«239»","statement":"True ↔\n  ∀ (f : ℕ → ℝ),\n    (∀ n ≥ 1, f n = 1 ∨ f n = -1) ∧ (∀ (m n : ℕ), m.Coprime n → f (m * n) = f m * f n) ∧ f 1 = 1 →\n      ∃ L, Filter.Tendsto (fun N => (∑ n ∈ Finset.Icc 1 N, f n) / ↑N) Filter.atTop (nhds L)","subjects":["11"],"theorem":"Erdos239.erdos_239"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Determine the Ramsey number\n$$R(C_4, S_n),$$\nwhere $S_n=K_{1,n}$ is the star on $n+1$ vertices.\n\nA problem of Burr, Erdős, Faudree, Rousseau, and Schelp [BEFRS89].\n\nThis problem is #19 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«552»","statement":"∀ (n : ℕ), (SimpleGraph.cycleGraph 4).graphRamsey (completeBipartiteGraph (Fin 1) (Fin n)) = sorry","subjects":["5"],"theorem":"Erdos552.erdos_552.parts.i"},{"answerKinds":[],"category":"research open","docstring":"In particular, is it true that, for any $c > 0$, there are infinitely many $n$ such that\n$$R(C_4, S_n) \\leq n + \\sqrt{n} - c?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«552»","statement":"True ↔\n  ∀ (c : ℝ),\n    0 < c →\n      {n | ↑((SimpleGraph.cycleGraph 4).graphRamsey (completeBipartiteGraph (Fin 1) (Fin n))) ≤ ↑n + √↑n - c}.Infinite","subjects":["5"],"theorem":"Erdos552.erdos_552.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Does $p_n/m_n \\to \\infty$ for almost all $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«456»","statement":"True ↔\n  ∃ A,\n    Filter.Tendsto (fun N => ↑(Nat.count (fun x => x ∈ A) N) / ↑N) Filter.atTop (nhds 1) ∧\n      Filter.Tendsto (fun n => ↑(Erdos456.p n) / ↑(Erdos456.m n)) (Filter.atTop ⊓ Filter.principal A) Filter.atTop","subjects":["11"],"theorem":"Erdos456.erdos_456.parts.ii"},{"answerKinds":[],"category":"textbook","docstring":"It is trivial that $m_n \\leq p_n$ always.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«456»","statement":"∀ (n : ℕ), Erdos456.m n ≤ Erdos456.p n","subjects":["11"],"theorem":"Erdos456.erdos_456.variants.mn_leq_pn"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er79e] writes it is 'easy to show' that for infinitely many $n$ we have $m_n < p_n$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«456»","statement":"{n | Erdos456.m n < Erdos456.p n}.Infinite","subjects":["11"],"theorem":"Erdos456.erdos_456.variants.infinitely_many_n"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er79e] writes it is 'easy to show' that $m_n/n \\to \\infty$ for almost all $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«456»","statement":"∃ A,\n  Filter.Tendsto (fun N => ↑(Nat.count (fun x => x ∈ A) N) / ↑N) Filter.atTop (nhds 1) ∧\n    Filter.Tendsto (fun n => ↑(Erdos456.m n) / ↑n) (Filter.atTop ⊓ Filter.principal A) Filter.atTop","subjects":["11"],"theorem":"Erdos456.erdos_456.variants.m_div_n"},{"answerKinds":[],"category":"research solved","docstring":"Linnik's theorem implies that $p_n\\leq n^{O(1)}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«456»","statement":"∃ L, (fun n => ↑(Erdos456.p n)) =O[Filter.atTop] fun n => ↑n ^ L","subjects":["11"],"theorem":"Erdos456.erdos_456.variants.linniks_theorem"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $m_n<p_n$ for almost all $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«456»","statement":"True ↔ Filter.Tendsto (fun N => ↑(Nat.count (fun n => Erdos456.m n < Erdos456.p n) N) / ↑N) Filter.atTop (nhds 1)","subjects":["11"],"theorem":"Erdos456.erdos_456.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«456»","statement":"True ↔ {q | Nat.Prime q ∧ ∀ (n : ℕ), Erdos456.m n = q ↔ n = q - 1}.Infinite","subjects":["11"],"theorem":"Erdos456.erdos_456.parts.iii"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $\\epsilon>0$ and $N$ be sufficiently large. If $A\\subseteq \\{1,\\ldots,N\\}$ has\n$\\lvert A\\rvert \\geq \\epsilon N$ then must there exist $a_1,a_2,a_3\\in A$ and distinct primes\n$p_1,p_2,p_3$ such that\n$$a_1p_1=a_2p_2=a_3p_3?$$\n\nA positive answer would imply [536].\n\nErdős describes a construction of Ruzsa which disproves this: consider the set of all\nsquarefree numbers of the shape $p_1\\cdots p_r$ where $p_{i+1}>2p_i$ for $1\\leq i<r$. This\nset has positive density, and hence if $A$ is its intersection with $(N/2,N)$ then\n$\\lvert A\\rvert \\gg N$ for all large $N$. Suppose now that $p_1a_1=p_2a_2=p_3a_3$ where\n$a_i\\in A$ and $p_1,p_2,p_3$ are distinct primes. Without loss of generality we may assume\nthat $a_2>a_3$ and hence $p_2<p_3$, and so since $p_2p_3\\mid a_1\\in A$ we must have\n$2<p_3/p_2$. On the other hand $p_3/p_2=a_2/a_3\\in (1,2)$, a contradiction.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos537.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«537»","statement":"False ↔\n  ∀ (ε : ℝ),\n    0 < ε →\n      ∀ᶠ (N : ℕ) in Filter.atTop,\n        ∀ A ⊆ Finset.Icc 1 N,\n          ↑A.card ≥ ε * ↑N →\n            ∃ a₁ ∈ A,\n              ∃ a₂ ∈ A,\n                ∃ a₃ ∈ A,\n                  ∃ p₁ p₂ p₃,\n                    Nat.Prime p₁ ∧\n                      Nat.Prime p₂ ∧ Nat.Prime p₃ ∧ p₁ ≠ p₂ ∧ p₁ ≠ p₃ ∧ p₂ ≠ p₃ ∧ a₁ * p₁ = a₂ * p₂ ∧ a₂ * p₂ = a₃ * p₃","subjects":["11"],"theorem":"Erdos537.erdos_537"},{"answerKinds":[],"category":"research solved","docstring":"But this fails for all $k \\geq 3$, and in fact\n$\\prod_{n \\leq m < n+3} B_2(m) \\gg n^2 \\log n$ infinitely often.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«367»","statement":"∃ c > 0, ∃ᶠ (n : ℕ) in Filter.atTop, c * (↑n ^ 2 * Real.log ↑n) ≤ ↑(∏ m ∈ Finset.Ico n (n + 3), Erdos367.B 2 m)","subjects":["11"],"theorem":"Erdos367.erdos_367.variants.k_ge_three_lower"},{"answerKinds":[],"category":"research open","docstring":"Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all\nprimes that divide $n$ exactly once). Is it true that, for every fixed $k \\geq 1$,\n$\\prod_{n \\leq m < n+k} B_2(m) \\ll n^{2+o(1)}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«367»","statement":"True ↔\n  ∀ (k : ℕ),\n    1 ≤ k →\n      ∃ e,\n        e =o[Filter.atTop] 1 ∧\n          ∀ᶠ (n : ℕ) in Filter.atTop, ↑(∏ m ∈ Finset.Ico n (n + k), Erdos367.B 2 m) ≤ ↑n ^ (2 + e n)","subjects":["11"],"theorem":"Erdos367.erdos_367.parts.i"},{"answerKinds":[],"category":"research open","docstring":"It would also be interesting to find upper and lower bounds for the analogous product with $B_r$\nfor $r \\geq 3$, where $B_r(n)$ is the $r$-full part of $n$ (that is, the product of prime powers\n$p^a \\mid n$ such that $p^{a+1} \\nmid n$ and $a \\geq r$). Is it true that, for every fixed\n$r,k \\geq 2$ and $\\epsilon > 0$,\n$\\limsup \\frac{\\prod_{n \\leq m < n+k} B_r(m)}{n^{1+\\epsilon}} \\to \\infty$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«367»","statement":"True ↔\n  ∀ (r k : ℕ),\n    3 ≤ r →\n      2 ≤ k →\n        ∀ (ε : ℝ),\n          0 < ε →\n            Filter.limsup (fun n => ↑(↑(∏ m ∈ Finset.Ico n (n + k), Erdos367.B r m) / ↑n ^ (1 + ε))) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos367.erdos_367.variants.higher_full_parts"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Or perhaps even $\\prod_{n \\leq m < n+k} B_2(m) \\ll_k n^2$?\n\nvan Doorn notes in the comments that this fails for all $k \\geq 3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«367»","statement":"False ↔ ∀ (k : ℕ), 1 ≤ k → (fun n => ↑(∏ m ∈ Finset.Ico n (n + k), Erdos367.B 2 m)) =O[Filter.atTop] fun n => ↑n ^ 2","subjects":["11"],"theorem":"Erdos367.erdos_367.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"van Doorn notes in the comments that for $k \\leq 2$ we trivially have\n$\\prod_{n \\leq m < n+k} B_2(m) \\ll n^2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«367»","statement":"∀ k ≤ 2, (fun n => ↑(∏ m ∈ Finset.Ico n (n + k), Erdos367.B 2 m)) =O[Filter.atTop] fun n => ↑n ^ 2","subjects":["11"],"theorem":"Erdos367.erdos_367.variants.k_le_two"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is there some function $f(r)$ such that $f(r)\\to \\infty$ as $r\\to\\infty$, such that, for\ninfinitely many $n$, there exist $a_1,a_2$ with\n$$a_1+a_2> n+f(r)\\log n$$\nsuch that $a_1!a_2! \\mid n!2^n3^n\\cdots p_r^n$?\n\nIt is ambiguous in [ErGr80] what the intended quantifiers are on the variables (they write 'is\nit true that we can find $a_1+a_2>n+f(r)\\log n$...'). Comparing to previous problems such as\n[728] and [729] it seems most likely that they intended to ask the formulation in the problem\nstatement.\n\nThe answer is yes: Barreto and Leeham have used ChatGPT to provide a proof of the stated problem\n(in fact essentially the same construction as their solution to [729]).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos401.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«401»","statement":"True ↔\n  ∃ f,\n    Filter.Tendsto f Filter.atTop Filter.atTop ∧\n      ∀ (r : ℕ),\n        1 ≤ r →\n          {n |\n              ∃ a₁ a₂,\n                0 < a₁ ∧\n                  0 < a₂ ∧\n                    ↑a₁ + ↑a₂ > ↑n + f r * Real.log ↑n ∧\n                      a₁.factorial * a₂.factorial ∣\n                        n.factorial * (∏ i ∈ Finset.range r, Nat.nth Nat.Prime i) ^ n}.Infinite","subjects":["11"],"theorem":"Erdos401.erdos_401"},{"answerKinds":[],"category":"research open","docstring":"The density of the divisor sum set is asymptotically equivalent to $c_1 / \\log(t)^{c_2}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«859»","statement":"∃ c₁ > 0,\n  ∃ c₂ > 0,\n    ∃ d,\n      (∀ t > 0, (Erdos859.DivisorSumSet t).HasDensity (d t)) ∧\n        Asymptotics.IsEquivalent Filter.atTop (fun t => d t) fun t => c₁ / Real.log ↑t ^ c₂","subjects":["11"],"theorem":"Erdos859.erdos_859"},{"answerKinds":[],"category":"textbook","docstring":"A case where we can easily calculate the density of `DivisorSumSet t` is that of `t=0`.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«859»","statement":"Erdos859.DivisorSumSet 0 = Set.univ","subjects":["11"],"theorem":"Erdos859.erdos_859.variants.trivial_case"},{"answerKinds":[],"category":"research solved","docstring":"A weaker version of the problem proved by Erdos:\nThe density `dₜ` of `DivisorSumSet (t : ℕ)` is bounded from below by `1 / log (t) ^ c₃` and\nfrom above by `1 / log (t) ^ c₄` for some positive constants `c₃` and `c₄`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«859»","statement":"∃ c₃ > 0,\n  ∃ c₄ > 0,\n    ∃ t₀,\n      ∀ᶠ (t : ℕ) in Filter.atTop,\n        ∃ dₜ, (Erdos859.DivisorSumSet t).HasDensity dₜ ∧ 1 / Real.log ↑t ^ c₃ < dₜ ∧ dₜ < 1 / Real.log ↑t ^ c₄","subjects":["11"],"theorem":"Erdos859.erdos_859.variants.erdos_upper_lower_bounds"},{"answerKinds":[],"category":"textbook","docstring":"An easy sanity check is to prove that for every natural number `t` the density `dₜ` is\na positive number.\nHint: investigate some multiplicative structure of `DivisorSumSet t`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«859»","statement":"∀ (t : ℕ), (Erdos859.DivisorSumSet t).HasPosDensity","subjects":["11"],"theorem":"Erdos859.erdos_859.variants.positive_density"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A \\subset \\mathbb{N}$ be an infinite set for which there exists some $\\epsilon > 0$ such that\nin any subset of $A$ of size $n$ there is a subset of size at least $\\epsilon n$ which contains no\nthree-term arithmetic progression.\n\nIs it true that $A$ is the union of a finite number of sets which contain no three-term arithmetic\nprogression?\n\nA negative answer was given by Reiher, Rödl, and Sales [RRS24], who proved that, for any\n$0<\\mu<1/2$, there exists $A\\subseteq \\mathbb{N}$ such that every finite colouring of $A$ contains\na three-term arithmetic progression, and yet every subset of $A$ of size $n$ contains a subset of\nsize $\\geq \\mu n$ without a three-term arithmetic progression.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos847.lean#L113"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«847»","statement":"False ↔ ∀ (A : Set ℕ), Infinite ↑A → Erdos847.HasFew3APs A → ∃ n S, (∀ (i : Fin n), ThreeAPFree (S i)) ∧ A = ⋃ i, S i","subjects":["11"],"theorem":"Erdos847.erdos_847"},{"answerKinds":["non-Prop"],"category":"research solved","docstring":"Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in\\mathbb{C}[x]$ where $\\lvert z_i\\rvert\\leq 1$ for all $i$.\nIf $\\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of\n$$\n\\{ z: \\lvert f(z)\\rvert<1\\}\n$$\nthen determine the infimum of $\\Lambda(f)$.\n\nA problem of Erdős, Herzog, and Piranian [EHP58].\n\nThis has been resolved by Tang, who proved that the infimum of $\\Lambda(f)$ over all such $f$\nis $2$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1044.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1044»","statement":"IsGLB {L | ∃ f, Erdos1044.IsAdmissible f ∧ Erdos1044.maxBoundaryLength f = L} 2","subjects":["30"],"theorem":"Erdos1044.erdos_1044"},{"answerKinds":[],"category":"research open","docstring":"Tang also suggests that, if the degree $n$ is fixed, then the infimum over all such $f$ of\ndegree $n$ is attained by $f_n(z)=z^n-1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1044»","statement":"∀ (n : ℕ),\n  0 < n →\n    IsLeast {L | ∃ f, Erdos1044.IsAdmissible f ∧ f.natDegree = n ∧ Erdos1044.maxBoundaryLength f = L}\n      (Erdos1044.maxBoundaryLength (Polynomial.X ^ n - 1))","subjects":["30"],"theorem":"Erdos1044.erdos_1044.variants.fixed_degree"},{"answerKinds":[],"category":"research solved","docstring":"Tang also suggests that, if the degree $n$ is fixed, then the infimum over all such $f$ of\ndegree $n$ is attained by $f_n(z)=z^n-1$ (and proves this for $n=1$ and $n=2$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1044»","statement":"∀ (n : ℕ),\n  n = 1 ∨ n = 2 →\n    IsLeast {L | ∃ f, Erdos1044.IsAdmissible f ∧ f.natDegree = n ∧ Erdos1044.maxBoundaryLength f = L}\n      (Erdos1044.maxBoundaryLength (Polynomial.X ^ n - 1))","subjects":["30"],"theorem":"Erdos1044.erdos_1044.variants.fixed_degree_of_le_two"},{"answerKinds":[],"category":"research solved","docstring":"This has been resolved by Tang, who proved that the infimum of $\\Lambda(f)$ over all such $f$\nis $2$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1044.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1044»","statement":"IsGLB {L | ∃ f, Erdos1044.IsAdmissible f ∧ Erdos1044.maxBoundaryLength f = L} 2","subjects":["30"],"theorem":"Erdos1044.erdos_1044.variants.infimum_eq_two"},{"answerKinds":[],"category":"research solved","docstring":"Prove that there exists an absolute constant $c>0$ such that, whenever $\\{1,\\ldots,N\\}$ is\n$k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many\nintegers in $\\{1,\\ldots,N\\}$ which are representable as a monochromatic sum (that is, $a+b$\nwhere $a,b\\in \\{1,\\ldots,N\\}$ are in the same colour class and $a\\neq b$).\n\nA conjecture of Roth. Solved by Erdős, Sárközy, and Sós [ESS89], who in fact prove that\nthere are at least $\\frac{N}{2}-O(N^{1-1/2^{k+1}})$ many even numbers which are of this form.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos484.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«484»","statement":"∃ c,\n  0 < c ∧\n    ∀ (k : ℕ),\n      0 < k →\n        ∃ N₀,\n          ∀ (N : ℕ),\n            N₀ ≤ N →\n              ∀ (f : ℕ → Fin k),\n                c * ↑N ≤\n                  ↑{n ∈ Finset.Icc 1 N | ∃ a ∈ Finset.Icc 1 N, ∃ b ∈ Finset.Icc 1 N, a ≠ b ∧ f a = f b ∧ a + b = n}.card","subjects":["5","11"],"theorem":"Erdos484.erdos_484"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for every $k \\geq 1$, there exist integers $N_1 < \\dots < N_k$ such that\n$|\\cap_i D(N_i)| \\geq k$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«885»","statement":"True ↔ ∀ k ≥ 1, ∃ Ns, (∀ n ∈ Ns, 1 ≤ n) ∧ Ns.card = k ∧ (⋂ n ∈ Ns, Erdos885.factorDifferenceSet n).ncard ≥ k","subjects":["11"],"theorem":"Erdos885.erdos_885"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Rosenfeld [ErRo97] proved this is true for $k=2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«885»","statement":"∃ Ns, (∀ n ∈ Ns, 1 ≤ n) ∧ Ns.card = 2 ∧ (⋂ n ∈ Ns, Erdos885.factorDifferenceSet n).ncard ≥ 2","subjects":["11"],"theorem":"Erdos885.erdos_885.variants.k_eq_2"},{"answerKinds":[],"category":"research solved","docstring":"Bremner [Br19] proved this for $k=4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«885»","statement":"∃ Ns, (∀ n ∈ Ns, 1 ≤ n) ∧ Ns.card = 4 ∧ (⋂ n ∈ Ns, Erdos885.factorDifferenceSet n).ncard ≥ 4","subjects":["11"],"theorem":"Erdos885.erdos_885.variants.k_eq_4"},{"answerKinds":[],"category":"research solved","docstring":"Jiménez-Urroz [Ji99] proved this for $k=3$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«885»","statement":"∃ Ns, (∀ n ∈ Ns, 1 ≤ n) ∧ Ns.card = 3 ∧ (⋂ n ∈ Ns, Erdos885.factorDifferenceSet n).ncard ≥ 3","subjects":["11"],"theorem":"Erdos885.erdos_885.variants.k_eq_3"},{"answerKinds":[],"category":"research open","docstring":"Is there a graph of infinite chromatic number such that every finite subgraph on $n$\nvertices can be made bipartite by deleting at most $\\sqrt{n}$ edges?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«74»","statement":"True ↔ ∃ V G, G.chromaticNumber = ⊤ ∧ ∀ (n : ℕ), ↑(Erdos74.SimpleGraph.maxSubgraphEdgeDistToBipartite G n) ≤ √↑n","subjects":["5"],"theorem":"Erdos74.erdos_74.variants.sqrt"},{"answerKinds":[],"category":"research open","docstring":"Let $f(n)\\to \\infty$ possibly very slowly.\nIs there a graph of infinite chromatic number such that every finite subgraph on $n$\nvertices can be made bipartite by deleting at most $f(n)$ edges?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«74»","statement":"True ↔\n  ∀ (f : ℕ → ℕ),\n    Filter.Tendsto f Filter.atTop Filter.atTop →\n      ∃ V G, G.chromaticNumber = ⊤ ∧ ∀ (n : ℕ), Erdos74.SimpleGraph.maxSubgraphEdgeDistToBipartite G n ≤ f n","subjects":["5"],"theorem":"Erdos74.erdos_74"},{"answerKinds":[],"category":"test","docstring":"The set of edge distances to a bipartite graph is always non-empty because deleting all edges\nfrom a graph makes it bipartite.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«74»","statement":"∀ {V : Type u} {G : SimpleGraph V} (A : G.Subgraph), (Erdos74.SimpleGraph.edgeDistancesToBipartite A).Nonempty","subjects":["5"],"theorem":"Erdos74.SimpleGraph.edgeDistancesToBipartite_nonempty"},{"answerKinds":[],"category":"test","docstring":"The set of minimum edge distances to bipartite for subgraphs of size `n` is bounded above.\nA graph on `n` vertices has at most `n choose 2` edges, and deleting all of them\nmakes the graph bipartite, providing a straightforward upper bound.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«74»","statement":"∀ {V : Type u} (G : SimpleGraph V) (n : ℕ), BddAbove (Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite G n)","subjects":["5"],"theorem":"Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite_bddAbove"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does every finite colouring of the integers have a monochromatic solution to\n$1=\\sum \\frac{1}{n_i}$ with $2\\leq n_1<\\cdots <n_k$?\n\nThe answer is yes, as proved by Croot [Cr03] - indeed, there are infinitely many disjoint such\nmonochromatic solutions.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos46.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«46»","statement":"True ↔ ∀ (𝓒 : ℕ → ℕ), (Set.range 𝓒).Finite → ∃ S, (∀ n ∈ S, 2 ≤ n) ∧ ∑ n ∈ S, 1 / ↑n = 1 ∧ (𝓒 '' ↑S).Subsingleton","subjects":["5","11"],"theorem":"Erdos46.erdos_46"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Croot [Cr03] proved more: there are infinitely many disjoint such monochromatic solutions.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«46»","statement":"True ↔\n  ∀ (𝓒 : ℕ → ℕ),\n    (Set.range 𝓒).Finite →\n      ∃ S,\n        (∀ (i j : ℕ), i ≠ j → Disjoint (S i) (S j)) ∧\n          ∀ (i : ℕ), (∀ n ∈ S i, 2 ≤ n) ∧ ∑ n ∈ S i, 1 / ↑n = 1 ∧ (𝓒 '' ↑(S i)).Subsingleton","subjects":["5","11"],"theorem":"Erdos46.erdos_46.variants.infinitely_many_disjoint"},{"answerKinds":["Prop"],"category":"research solved","docstring":"In [ErGr80] they also ask for a monochromatic representation of any $\\frac{a}{b}>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«46»","statement":"True ↔\n  ∀ (𝓒 : ℕ → ℕ),\n    (Set.range 𝓒).Finite → ∀ (q : ℚ), 0 < q → ∃ S, (∀ n ∈ S, 2 ≤ n) ∧ ∑ n ∈ S, 1 / ↑n = q ∧ (𝓒 '' ↑S).Subsingleton","subjects":["5","11"],"theorem":"Erdos46.erdos_46.variants.positive_rat"},{"answerKinds":[],"category":"research solved","docstring":"Jensen [Je02] gave an construction for $k$-critical graphs without any critical edges for all $k ≥ 5$.\n\n[Je02] Jensen, Tommy R., Dense critical and vertex-critical graphs. Discrete Math. (2002), 63--84.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«944»","statement":"∀ (k : ℕ), 5 ≤ k → ∃ V G, Erdos944.SimpleGraph.IsErdos944 G k 1","subjects":["11"],"theorem":"Erdos944.erdos_944.variants.dirac_conjecture.k_ge_five"},{"answerKinds":[],"category":"research solved","docstring":"Lattanzio [La02] proved there exist $k$-critical graphs without critical edges for all $k$ such that\n$k - 1$ is not prime.\n\n[La02] Lattanzio, John J., A note on a conjecture of {D}irac. Discrete Math. (2002), 323--330\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«944»","statement":"∀ (k : ℕ), 4 ≤ k → ¬Nat.Prime (k - 1) → ∃ V G, Erdos944.SimpleGraph.IsErdos944 G k 1","subjects":["11"],"theorem":"Erdos944.erdos_944.variants.dirac_conjecture.k_sub_one_not_prime"},{"answerKinds":[],"category":"research open","docstring":"Let $k \\ge 4$ and $r\\ge 1$. Must there exist a graph $G$ with chromatic number $k$\nsuch that every vertex is critical, yet every critical set of edges has size $>r$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«944»","statement":"True ↔ ∀ k ≥ 4, ∀ r ≥ 1, ∃ V G, Erdos944.SimpleGraph.IsErdos944 G k r","subjects":["11"],"theorem":"Erdos944.erdos_944"},{"answerKinds":[],"category":"research solved","docstring":"Martinsson and Steiner [MaSt25] proved for every $r \\ge 1$ if $k$ is sufficiently large, depending\non $r$, there exist a graph $G$ with chromatic number $k$ such that every vertex is critical,\nyet every critical set of edges has size $>r$.\n\n[MaSt25] Martinsson, Anders and Steiner, Raphael, Vertex-critical graphs far from edge-criticality. Combin. Probab. Comput. (2025), 151--157\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«944»","statement":"∀ (r : ℕ), 1 ≤ r → ∀ᶠ (k : ℕ) in Filter.atTop, ∃ V G, Erdos944.SimpleGraph.IsErdos944 G k r","subjects":["11"],"theorem":"Erdos944.erdos_944.variants.large_k_for_any_r"},{"answerKinds":[],"category":"research open","docstring":"Let $k \\ge 4$. Must there exist a graph $G$ with chromatic number $k$\nsuch that every vertex is critical, yet every critical set of edges has size $>1$?\n\nThis was conjectured by Dirac in 1970.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«944»","statement":"True ↔ ∀ k ≥ 4, ∃ V G, Erdos944.SimpleGraph.IsErdos944 G k 1","subjects":["11"],"theorem":"Erdos944.erdos_944.variants.dirac_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"Dirac's conjecture was proved, for $k=5$: There exists a graph $G$ with chromatic number $5$, such\nthat every vertex is critical, yet every critical set of edges has size $>1$, or in other words:\nhas no critical edge.\n\n[Br92] Brown, Jason I., A vertex critical graph without critical edges. Discrete Math. (1992), 99--101\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«944»","statement":"∃ V G, Erdos944.SimpleGraph.IsErdos944 G 5 1","subjects":["11"],"theorem":"Erdos944.erdos_944.variants.dirac_conjecture.k_eq_5"},{"answerKinds":[],"category":"research open","docstring":"The case $k=4$ and $r=1$ remains open: Are there $4$-critical graphs without any critical edges?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«944»","statement":"True ↔ ∃ V G, Erdos944.SimpleGraph.IsErdos944 G 4 1","subjects":["11"],"theorem":"Erdos944.erdos_944.variants.dirac_conjecture.k_eq_four"},{"answerKinds":[],"category":"research solved","docstring":"$\\delta < \\frac{m ^ \\alpha + 1}{m}$. This shows that\n$lim_{m\\rightarrow\\infty} \\delta (m, \\alpha) = 0$ for $\\alpha < 1$.\n#TODO: prove this theorem. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«697»","statement":"∀ (m : ℕ) (α : ℝ), Erdos697.δ m α < (↑m ^ α + 1) / ↑m","subjects":["11"],"theorem":"Erdos697.erdos_697.variants.delta_lt"},{"answerKinds":[],"category":"research solved","docstring":"For each $m$ and $\\alpha$, the density of the set of integers which are divisible by\nsome $d \\equiv 1 \\pmod{m}$ with $1 < d < \\exp (m ^ \\alpha)$ exists. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«697»","statement":"∀ (m : ℕ) (α : ℝ), ∃ δ, {n | ∃ d, d ≡ 1 [MOD m] ∧ ↑d ∈ Set.Ioo 1 (Real.exp (↑m ^ α)) ∧ d ∣ n}.HasDensity δ","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos697.density_exists"},{"answerKinds":[],"category":"research solved","docstring":"$lim_{m\\rightarrow\\infty} \\delta (m, \\alpha) = 1$ if $\\beta < \\alpha$.\nThis is proved in [Ha92]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«697»","statement":"∀ {α : ℝ}, α < 1 / Real.log 2 → Filter.Tendsto (fun x => Erdos697.δ x α) Filter.atTop (nhds 1)","subjects":["11"],"theorem":"Erdos697.erdos_697.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Let $\\beta = \\frac{1}{\\log 2}$. Then $lim_{m\\rightarrow\\infty} \\delta (m, \\alpha) = 0$ if\n$\\alpha < \\beta$. This is proved in [Ha92]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«697»","statement":"∀ {α : ℝ}, 1 / Real.log 2 < α → Filter.Tendsto (fun x => Erdos697.δ x α) Filter.atTop (nhds 0)","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos697.erdos_697.parts.i"},{"answerKinds":[],"category":"test","docstring":"A sanity check for `erdos_615`: the empty graph on $n \\geq 3$ vertices contains an independent\nset on at least $n/\\log n$ vertices (namely the whole vertex set), so it satisfies the\nconclusion of the implication in the problem statement.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«615»","statement":"∀ (n : ℕ), 3 ≤ n → ↑n / Real.log ↑n ≤ ↑⊥.indepNum","subjects":["5"],"theorem":"Erdos615.erdos_615.variants.test_bot"},{"answerKinds":[],"category":"research solved","docstring":"The complementary result of Sudakov [Su03]: if $f(n)/\\sqrt{\\log n} \\to \\infty$ then\n$\\mathrm{rt}(n; 4, ne^{-f(n)}) = o(n^2)$; that is, for every $\\epsilon > 0$ and all\nsufficiently large $n$, every $K_4$-free graph on $n$ vertices with independence number at\nmost $ne^{-f(n)}$ has at most $\\epsilon n^2$ edges.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«615»","statement":"∀ (f : ℕ → ℝ),\n  Filter.Tendsto (fun n => f n / √(Real.log ↑n)) Filter.atTop Filter.atTop →\n    ∀ (ε : ℝ),\n      0 < ε →\n        ∀ᶠ (n : ℕ) in Filter.atTop,\n          ∀ (G : SimpleGraph (Fin n)),\n            G.CliqueFree 4 → ↑G.indepNum ≤ ↑n * Real.exp (-f n) → ↑G.edgeFinset.card ≤ ε * ↑n ^ 2","subjects":["5"],"theorem":"Erdos615.erdos_615.variants.sudakov"},{"answerKinds":[],"category":"research solved","docstring":"The result of Fox, Loh, and Zhao [FLZ15] disproving the problem: if $f(n) \\geq 0$ satisfies\n$f(n) = o(\\sqrt{\\log n/\\log\\log n})$, then for every $\\epsilon > 0$ and all sufficiently\nlarge $n$ there is a $K_4$-free graph on $n$ vertices with independence number at most\n$ne^{-f(n)}$ and at least $(1/8 - \\epsilon)n^2$ edges; that is,\n$\\mathrm{rt}(n; 4, ne^{-f(n)}) \\geq (1/8 - o(1))n^2$. Applied with\n$f(n) = \\log\\log n$, this disproves the headline problem, since\n$ne^{-f(n)} = n/\\log n = o(n)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«615»","statement":"∀ (f : ℕ → ℝ),\n  (∀ (n : ℕ), 0 ≤ f n) →\n    Filter.Tendsto (fun n => f n / √(Real.log ↑n / Real.log (Real.log ↑n))) Filter.atTop (nhds 0) →\n      ∀ (ε : ℝ),\n        0 < ε →\n          ∀ᶠ (n : ℕ) in Filter.atTop,\n            ∃ G, G.CliqueFree 4 ∧ ↑G.indepNum ≤ ↑n * Real.exp (-f n) ∧ (1 / 8 - ε) * ↑n ^ 2 ≤ ↑G.edgeFinset.card","subjects":["5"],"theorem":"Erdos615.erdos_615.variants.fox_loh_zhao"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does there exist some constant $c > 0$ such that for all sufficiently large $n$, if $G$ is a\ngraph with $n$ vertices and at least $(1/8 - c)n^2$ edges then $G$ must contain either a $K_4$\nor an independent set on at least $n/\\log n$ vertices?\n\nThe answer is no, as shown by Fox, Loh, and Zhao [FLZ15].\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/dfe2d78128b493c572cf525b1b8edf4897fb7664/src/latest/ErdosProblems/Erdos615.lean#L492"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«615»","statement":"False ↔\n  ∃ c,\n    0 < c ∧\n      ∀ᶠ (n : ℕ) in Filter.atTop,\n        ∀ (G : SimpleGraph (Fin n)),\n          (1 / 8 - c) * ↑n ^ 2 ≤ ↑G.edgeFinset.card → ¬G.CliqueFree 4 ∨ ↑n / Real.log ↑n ≤ ↑G.indepNum","subjects":["5"],"theorem":"Erdos615.erdos_615"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $m,n\\geq 1$. What is\n$$\\# \\{ k(m-k) : 1\\leq k\\leq m/2\\} \\cap \\{ l(n-l) : 1\\leq l\\leq n/2\\}?$$\nCan it be arbitrarily large?\n\nThis was solved independently by Hegyvári [He25] and Cambie (unpublished), who show that if\n$m>n$ then the set in question has size\n$$\\leq m^{O(1/\\log\\log m)},$$\nand that for any integer $s$ there exist infinitely many pairs $(m,n)$ such that the set in\nquestion has size $s$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos443.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«443»","statement":"True ↔ ∀ (s : ℕ), ∃ m, ∃ n < m, s ≤ (Erdos443.A n ∩ Erdos443.A m).card","subjects":["11"],"theorem":"Erdos443.erdos_443.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $m,n\\geq 1$. What is\n$$\\# \\{ k(m-k) : 1\\leq k\\leq m/2\\} \\cap \\{ l(n-l) : 1\\leq l\\leq n/2\\}?$$\nIs it $\\leq (mn)^{o(1)}$ for all sufficiently large $m,n$?\n\nThis was solved independently by Hegyvári [He25] and Cambie (unpublished), who show that if\n$m>n$ then the set in question has size\n$$\\leq m^{O(1/\\log\\log m)},$$\nand that for any integer $s$ there exist infinitely many pairs $(m,n)$ such that the set in\nquestion has size $s$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos443.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«443»","statement":"True ↔ ∀ (ε : ℝ), 0 < ε → ∃ n₀, ∀ (m n : ℕ), n₀ < n → n < m → ↑(Erdos443.A n ∩ Erdos443.A m).card < (↑m * ↑n) ^ ε","subjects":["11"],"theorem":"Erdos443.erdos_443.parts.ii"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for every $\\epsilon>0$, there exist infinitely many $n$ such that\n$g(n) > n^{1-\\epsilon}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«821»","statement":"True ↔ ∀ ε > 0, {n | ↑(Erdos821.g n) > ↑n ^ (1 - ε)}.Infinite","subjects":["11"],"theorem":"Erdos821.erdos_821"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er35b] proved that there exists some constant $c>0$ such that $g(n) > n^c$ for infinitely\nmany $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«821»","statement":"∃ c > 0, {n | ↑n ^ c < ↑(Erdos821.g n)}.Infinite","subjects":["11"],"theorem":"Erdos821.erdos_821.variants.erdos"},{"answerKinds":[],"category":"research solved","docstring":"Pillai proved that $\\limsup g(n)=\\infty$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«821»","statement":"Filter.limsup (fun n => ↑(Erdos821.g n)) Filter.atTop = ⊤","subjects":["11"],"theorem":"Erdos821.erdos_821.variants.pillai"},{"answerKinds":[],"category":"research solved","docstring":"The best known bound is that there are infinitely many $n$ such that $g(n) > n^{0.71568\\cdots}$,\nobtained by Lichtman [Li22] as a consequence of proving that there are\n$\\geq \\frac{x}{(\\log x)^{O(1)}}$ many primes $p\\leq x$ such that all prime factors of $p-1$ are\n$\\leq x^{0.2843\\cdots}$ (which improves a number of previous exponents, most recently Baker and\nHarman [BaHa98]).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«821»","statement":"∃ c > 0.71568, {n | ↑n ^ c < ↑(Erdos821.g n)}.Infinite","subjects":["11"],"theorem":"Erdos821.erdos_821.variants.lichtman"},{"answerKinds":[],"category":"research open","docstring":"Let $0 < \\epsilon < 1$. Is it true that, if $k$ is sufficiently large, then\n$$R(G) > (1-\\epsilon)^k R(k)$$\nfor every graph $G$ with chromatic number $\\chi(G)=k$?\n\nThe restriction $\\epsilon < 1$ excludes negative bases in $(1-\\epsilon)^k$.\n\nThis problem is #12 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«87»","statement":"True ↔\n  ∀ ε > 0,\n    ε < 1 →\n      ∀ᶠ (k : ℕ) in Filter.atTop,\n        ∀ (V : Type) [inst : Fintype V] (G : SimpleGraph V),\n          G.chromaticNumber = ↑k → ↑G.diagonalGraphRamsey > (1 - ε) ^ k * ↑(SimpleGraph.diagonalRamsey k)","subjects":["5"],"theorem":"Erdos87.erdos_87.parts.i"},{"answerKinds":[],"category":"research open","docstring":"Even stronger, is there some $c > 0$ such that, for all large $k$,\n$$R(G) > c R(k)$$\nfor every graph $G$ with chromatic number $\\chi(G)=k$?\n\nThis problem is #13 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«87»","statement":"True ↔\n  ∃ c > 0,\n    ∀ᶠ (k : ℕ) in Filter.atTop,\n      ∀ (V : Type) [inst : Fintype V] (G : SimpleGraph V),\n        G.chromaticNumber = ↑k → ↑G.diagonalGraphRamsey > c * ↑(SimpleGraph.diagonalRamsey k)","subjects":["5"],"theorem":"Erdos87.erdos_87.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"For any fixed $s\\geq 3$,\n$$R(s,k) \\gg \\frac{k^{s-1}}{(\\log k)^c}$$\nfor some constant $c=c(s)>0$.\n\nAccording to Chung and Graham [ChGr98] this was first conjectured by Erdős in 1947.\n\nProved by Bradač [Br26], with $c=2s-4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«986»","statement":"∀ (s : ℕ),\n  3 ≤ s →\n    ∃ c C,\n      0 < c ∧\n        0 < C ∧ ∀ᶠ (k : ℕ) in Filter.atTop, ↑(SimpleGraph.classicalRamsey s k) ≥ C * ↑k ^ (s - 1) / Real.log ↑k ^ c","subjects":["5"],"theorem":"Erdos986.erdos_986"},{"answerKinds":[],"category":"research solved","docstring":"Let $p(x) = x ^ 2 \\in \\mathbb{Q}[x]$. It has been shown that\n$$A=\\{ p(n)+1/n : n \\in \\mathbb{N}\\}$$\nis strongly complete, in the sense that, for any finite set $B$,\n$$\\left\\{\\sum_{a \\in X} a : X \\subseteq A \\setminus B, X \\textrm{ is finite}\\right\\}$$\ncontains all sufficiently large integers. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«351»","statement":"Erdos351.HasCompleteImage (Polynomial.X ^ 2)","subjects":["11"],"theorem":"Erdos351.erdos_351.variants.X_sq"},{"answerKinds":[],"category":"research solved","docstring":"Let $p(x) = x \\in \\mathbb{Q}[x]$. It has been shown that\n$$A=\\{ p(n)+1/n : n \\in \\mathbb{N}\\}$$\nis strongly complete, in the sense that, for any finite set $B$,\n$$\\left\\{\\sum_{a \\in X} a : X \\subseteq A \\setminus B, X \\textrm{ is finite}\\right\\}$$\ncontains all sufficiently large integers.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«351»","statement":"Erdos351.HasCompleteImage Polynomial.X","subjects":["11"],"theorem":"Erdos351.erdos_351.variants.X"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $p(x) \\in \\mathbb{Q}[x]$ be a non-constant rational polynomial with positive leading\ncoefficient. Is it true that $$A=\\{ p(n)+1/n : n \\in \\mathbb{N}\\}$$ is strongly complete,\nin the sense that, for any finite set $B$,\n$$\\left\\{\\sum_{a \\in X} a : X \\subseteq A \\setminus B, X \\textrm{ is finite}\\right\\}$$\ncontains all sufficiently large integers? ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos351.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«351»","statement":"True ↔ ∀ (P : Polynomial ℚ), 0 < P.natDegree → 0 < P.leadingCoeff → Erdos351.HasCompleteImage P","subjects":["11"],"theorem":"Erdos351.erdos_351"},{"answerKinds":[],"category":"research solved","docstring":"[Cambie observed](https://www.erdosproblems.com/1063) the improved bound\n$n_k \\le k \\cdot \\operatorname{lcm}(1, \\dotsc, k - 1)$.\n\nThe hypothesis `3 ≤ k` is necessary here too. At $k = 2$ the right hand side is\n$2 \\cdot \\operatorname{lcm}(1) = 2$, while $n_2 = 4$.\n\nThe source writes the bound as $k[2, 3, \\dotsc, k-1]$. That agrees with the range used here,\nbecause including $1$ does not change a least common multiple. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1063»","statement":"∀ {k : ℕ}, 3 ≤ k → Erdos1063.n k ≤ k * (Finset.Icc 1 (k - 1)).lcm id","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos1063.erdos_1063.variants.cambie_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"The initial values satisfy $n_2 = 4$, $n_3 = 6$, $n_4 = 9$, and $n_5 = 12$ ([Gu04], Problem B31). ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«1063»","statement":"Erdos1063.n 2 = 4 ∧ Erdos1063.n 3 = 6 ∧ Erdos1063.n 4 = 9 ∧ Erdos1063.n 5 = 12","subjects":["11"],"theorem":"Erdos1063.erdos_1063.variants.small_values"},{"answerKinds":[],"category":"research solved","docstring":"Erdős and Selfridge noted that, for $n \\ge 2k$ with $k \\ge 2$, at least one of the numbers\n$n - i$ for $0 \\le i < k$ fails to divide $\\binom{n}{k}$ ([ErSe83]).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1063»","statement":"∀ {n k : ℕ}, 2 ≤ k → 2 * k ≤ n → ∃ i < k, ¬n - i ∣ n.choose k","subjects":["11"],"theorem":"Erdos1063.erdos_1063.variants.exists_exception"},{"answerKinds":[],"category":"research solved","docstring":"Monier observed that $n_k \\le k!$ for $k \\ge 3$ ([Mo85]), since $\\binom{k!}{k}$ is divisible\nby $k! - i$ for $1 \\le i < k$.\n\nThe hypothesis `3 ≤ k` is necessary. At $k = 2$ the bound is false: $n_2 = 4$ and $2! = 2$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1063»","statement":"∀ {k : ℕ}, 3 ≤ k → Erdos1063.n k ≤ k.factorial","subjects":["11"],"theorem":"Erdos1063.erdos_1063.variants.monier_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"The least common multiple bound implies $n_k \\le \\exp((1 + o(1))k)$. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1063»","statement":"∃ f, Filter.Tendsto f Filter.atTop (nhds 0) ∧ ∀ (k : ℕ), ↑(Erdos1063.n k) ≤ Real.exp ((1 + f k) * ↑k)","subjects":["11"],"theorem":"Erdos1063.erdos_1063.variants.exp_upper_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Estimate $n_k$ by finding a better upper bound than Cambie's\n$n_k \\leq k \\cdot \\operatorname{lcm}(1, \\dotsc, k-1)$.\n\nThe comparator takes its least common multiple in `ℕ` and casts the result. Writing the\nascription as `((… ).lcm (fun n : ℕ => n) : ℝ)` instead puts it on the `Finset.lcm`\napplication, so the coercion lands on `n` and the `lcm` is taken in `ℝ`, where `lcm` of\nnon-zero elements is `1` and the whole comparator collapses to `k`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1063»","statement":"have upper_bound := sorry;\n(fun k => ↑(Erdos1063.n k)) =O[Filter.atTop] upper_bound ∧\n  upper_bound =o[Filter.atTop] fun k => ↑k * ↑((Finset.Icc 1 (k - 1)).lcm id)","subjects":["11"],"theorem":"Erdos1063.erdos_1063.better_upper"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Write $M(n, k)$ be the least common multiple of $\\{n+1, \\dotsc, n+k\\}$.\nLet $k$ be sufficiently large. Are there infinitely many $m, n$ with $m \\geq n + k$ such that\n$$\nM(n, k) > M(m, k + 1)\n$$?\nThe answer is yes, as proved in a strong form by Cambie [Ca24].\n[Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).\n\nThis was formalized in Lean by Alexeev using Aristotle, on top of the PNT+ project.\n\nFor a fixed $k$ there are only finitely many such pairs, so \"infinitely many\" is read here as\nranging over $k$ as well: for every sufficiently large $k$ at least one pair $(m, n)$ occurs.\nSee `erdos_678.variants.infinitely_many_triples` for the reading in which the infinitude is\nstated directly, and `erdos_678.variants.not_infinitely_many_pairs` for why it cannot be asked\nof a single $k$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos678.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«678»","statement":"True ↔ ∀ᶠ (k : ℕ) in Filter.atTop, {(m, n) | n + k ≤ m ∧ Finset.lcmInterval m (k + 1) < Finset.lcmInterval n k}.Nonempty","subjects":["11"],"theorem":"Erdos678.erdos_678"},{"answerKinds":[],"category":"test","docstring":"The referee of [Er79] found the example $M(96, 7) > M(104, 8)$, showing that there are cases where\n$M(n, k) > M(m, k + 1)$ with $m \\geq n + k$.\n[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«678»","statement":"Finset.lcmInterval 104 8 < Finset.lcmInterval 96 7","subjects":["11"],"theorem":"Erdos678.lcmInterval_lt_example1"},{"answerKinds":[],"category":"research solved","docstring":"For a fixed sufficiently large $k$ only finitely many pairs occur: $M(m, k + 1) \\geq m + 1$\nbounds $m$ by $M(n, k)$, and for large $n$ the inequality reverses. So the question cannot be\nread as asking for infinitely many pairs at a single $k$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos678.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«678»","statement":"¬∀ᶠ (k : ℕ) in Filter.atTop, {(m, n) | n + k ≤ m ∧ Finset.lcmInterval m (k + 1) < Finset.lcmInterval n k}.Infinite","subjects":["11"],"theorem":"Erdos678.erdos_678.variants.not_infinitely_many_pairs"},{"answerKinds":[],"category":"test","docstring":"The referee of [Er79] found the example $M(132, 7) > M(139, 8)$, showing that there are cases where\n$M(n, k) > M(m, k + 1)$ with $m \\geq n + k$.\n[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«678»","statement":"Finset.lcmInterval 139 8 < Finset.lcmInterval 132 7","subjects":["11"],"theorem":"Erdos678.lcmInterval_lt_example2"},{"answerKinds":[],"category":"test","docstring":"Cambie [Ca24] found the example $M(52, 7) > M(62, 8)$.\n[Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«678»","statement":"Finset.lcmInterval 62 8 < Finset.lcmInterval 52 7","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos678.lcmInterval_lt_example3"},{"answerKinds":[],"category":"test","docstring":"Cambie [Ca24] found the example $M(36, 8) > M(48, 9)$.\n[Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«678»","statement":"Finset.lcmInterval 47 9 < Finset.lcmInterval 36 8","subjects":["11"],"theorem":"Erdos678.lcmInterval_lt_example4"},{"answerKinds":[],"category":"research solved","docstring":"The pairs $(m, n)$ with $m \\geq n + k$ and $M(n, k) > M(m, k + 1)$ are infinite in number once\n$k$ is allowed to vary, which is the sense in which Cambie's result answers the question.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos678.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«678»","statement":"{(k, m, n) | 3 ≤ k ∧ n + k ≤ m ∧ Finset.lcmInterval m (k + 1) < Finset.lcmInterval n k}.Infinite","subjects":["11"],"theorem":"Erdos678.erdos_678.variants.infinitely_many_triples"},{"answerKinds":[],"category":"research solved","docstring":"van Doorn [vD25] has investigated the question of what 'sufficiently large' means for $p(x)=x$.\nvan Doorn has also proved the original conjecture for many linear and quadratic polynomials.\nFor example, if $p(x) = x + b$ with $1 \\leq b \\leq 5000$, then the conjecture is true.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«283»","statement":"∀ (b : ℤ), 1 ≤ b → b ≤ 5000 → Erdos283.Condition (Polynomial.X + Polynomial.C b)","subjects":["11"],"theorem":"Erdos283.erdos_283.variants.van_doorn_linear"},{"answerKinds":[],"category":"research solved","docstring":"Graham [Gr63] has proved this when $p(x)=x$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«283»","statement":"Erdos283.Condition Polynomial.X","subjects":["11"],"theorem":"Erdos283.erdos_283.variants.graham"},{"answerKinds":[],"category":"research solved","docstring":"van Doorn [vD25] has proved the original conjecture for many linear and quadratic polynomials.\nFor example, if $p(x) = x^2 + b$ with $1 \\leq b \\leq 800$, then the conjecture is true.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«283»","statement":"∀ (b : ℤ), 1 ≤ b → b ≤ 800 → Erdos283.Condition (Polynomial.X ^ 2 + Polynomial.C b)","subjects":["11"],"theorem":"Erdos283.erdos_283.variants.van_doorn_quadratic"},{"answerKinds":[],"category":"research solved","docstring":"Cassels [Ca60] has proved that these conditions on the polynomial imply every sufficiently large\ninteger is the sum of $p(n_i)$ with distinct $n_i$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«283»","statement":"∀ (p : Polynomial ℤ),\n  0 < p.leadingCoeff →\n    (¬∃ d ≥ 2, ∀ n ≥ 1, d ∣ Polynomial.eval n p) →\n      ∀ᶠ (m : ℕ) in Filter.atTop, ∃ S, (∀ n ∈ S, 1 ≤ n) ∧ ∑ n ∈ S, Polynomial.eval (↑n) p = ↑m","subjects":["11"],"theorem":"Erdos283.erdos_283.variants.cassels"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $p\\colon \\mathbb{Z} \\rightarrow \\mathbb{Z}$ be a polynomial whose leading coefficient is\npositive and such that there exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$. Is it true that,\nfor all sufficiently large $m$, there exist integers $1≤n_1<\\dots < n_k$ such that\n$$1=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}$$\nand\n$$m=p(n_1)+\\cdots+p(n_k)$$?\n\nGPT 5.5 Pro (prompted by Price) has given a proof that the answer is yes, for the stronger version\nwith $1$ replaced by any rational $\\alpha>0$.\n\nThis was formalized in Lean by Ammanamanchi using Opus 4.6 and GPT 5.5 Pro.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P283/Proof_flat.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«283»","statement":"True ↔ ∀ (p : Polynomial ℤ), Erdos283.Condition p","subjects":["11"],"theorem":"Erdos283.erdos_283"},{"answerKinds":[],"category":"research solved","docstring":"Graham also conjectures that this remains true with $1$ replaced by an arbitrary rational $\\alpha>0$\n(provided $m$ is taken sufficiently large depending on $\\alpha$).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«283»","statement":"∀ (p : Polynomial ℤ) (α : ℚ),\n  0 < p.leadingCoeff →\n    (¬∃ d ≥ 2, ∀ n ≥ 1, d ∣ Polynomial.eval n p) →\n      α > 0 →\n        ∀ᶠ (m : ℕ) in Filter.atTop, ∃ S, (∀ n ∈ S, 1 ≤ n) ∧ ∑ n ∈ S, 1 / ↑n = α ∧ ∑ n ∈ S, Polynomial.eval (↑n) p = ↑m","subjects":["11"],"theorem":"Erdos283.erdos_283.variants.graham_alpha"},{"answerKinds":[],"category":"research solved","docstring":"Alekseyev [Al19] has proved this when $p(x)=x^2$, for all $m>8542$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«283»","statement":"∀ m > 8542, ∃ S, (∀ n ∈ S, 1 ≤ n) ∧ ∑ n ∈ S, 1 / ↑n = 1 ∧ ∑ n ∈ S, ↑n ^ 2 = ↑m","subjects":["11"],"theorem":"Erdos283.erdos_283.variants.alekseyev"},{"answerKinds":[],"category":"research solved","docstring":"Burr has proved this if $p(x)=x^k$ with $k\\geq 1$ and if we allow $n_i=n_j$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«283»","statement":"∀ k ≥ 1,\n  ∀ᶠ (m : ℕ) in Filter.atTop,\n    ∃ M,\n      (∀ n ∈ M, 1 ≤ n) ∧\n        (Multiset.map (fun n => 1 / n) do\n                let a ← M\n                pure ↑a).sum =\n            1 ∧\n          (Multiset.map (fun n => n ^ k) do\n                let a ← M\n                pure ↑a).sum =\n            ↑m","subjects":["11"],"theorem":"Erdos283.erdos_283.variants.burr"},{"answerKinds":[],"category":"research solved","docstring":"The finite case of `erdos_751.parts.i`, which is what the Lean proof establishes: a finite graph\nwith chromatic number at least $4$ has two cycles whose lengths differ by $1$ or $2$.\n\nThe proof reaches this through a $4$-critical subgraph, which supplies the $2$-connectivity and\nthe vertex count its Bondy-Vince step needs on top of minimum degree $3$. Those extra hypotheses\nare why it does not also settle `erdos_751.variants.bondy_vince`, which asks for minimum degree\n$3$ alone.\n\nThis was formalized in Lean by SpringSense Innovation Institute using ChatGPT.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/SpringSense-Innovation-Institute/ai-for-math-lean/blob/ae3ead960a494cf81b28541c477e50997cb03999/erdos-problems/erdos751/Erdos751/Main.lean#L40-L69"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«751»","statement":"∀ {V : Type u_1} [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj],\n  4 ≤ G.chromaticNumber → ∃ m ∈ G.cycleLengths, ∃ m' ∈ G.cycleLengths, m < m' ∧ m' ≤ m + 2","subjects":["5"],"theorem":"Erdos751.erdos_751.variants.finite"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a graph with chromatic number $\\chi(G)=4$. If $m_1<m_2<\\cdots$ are the lengths of the\ncycles in $G$ then can $\\min(m_{i+1}-m_i)$ be arbitrarily large?\n\nThe answer is no: Bondy and Vince [BoVi98] proved that every graph with minimum degree at least $3$\nhas two cycles whose lengths differ by at most $2$, and hence the same is true for every graph with\nchromatic number $4$.\n\n`erdos_751.variants.finite` below carries the Lean proof. It assumes a finite vertex type,\nwhere this quantifies over any `V : Type`, and the two are joined by de Bruijn-Erdős: a graph\nthat is not $3$-colourable has a finite subgraph that is not $3$-colourable, and cycles of that\nsubgraph are cycles of the whole. Mathlib does not have de Bruijn-Erdős, so that step is not\nformalised here.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«751»","statement":"False ↔ ∀ (k : ℕ), ∃ V G, G.chromaticNumber = 4 ∧ ∀ m ∈ G.cycleLengths, ∀ m' ∈ G.cycleLengths, m < m' → m + k ≤ m'","subjects":["5"],"theorem":"Erdos751.erdos_751.parts.i"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $G$ be a graph with chromatic number $\\chi(G)=4$. If $m_1<m_2<\\cdots$ are the lengths of the\ncycles in $G$ then can $\\min(m_{i+1}-m_i)$ be arbitrarily large? Can this happen if the girth of\n$G$ is large?\n\nThe answer is no: Bondy and Vince [BoVi98] proved that every graph with minimum degree at least $3$\nhas two cycles whose lengths differ by at most $2$, and hence the same is true for every graph with\nchromatic number $4$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«751»","statement":"False ↔\n  ∀ (k g : ℕ),\n    ∃ V G, G.chromaticNumber = 4 ∧ g ≤ G.girth ∧ ∀ m ∈ G.cycleLengths, ∀ m' ∈ G.cycleLengths, m < m' → m + k ≤ m'","subjects":["5"],"theorem":"Erdos751.erdos_751.parts.ii"},{"answerKinds":[],"category":"research solved","docstring":"Bondy and Vince [BoVi98] proved that every graph with minimum degree at least $3$ has two cycles\nwhose lengths differ by at most $2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«751»","statement":"∀ {V : Type u_1} [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n  3 ≤ G.minDegree → ∃ m ∈ G.cycleLengths, ∃ m' ∈ G.cycleLengths, m < m' ∧ m' ≤ m + 2","subjects":["5"],"theorem":"Erdos751.erdos_751.variants.bondy_vince"},{"answerKinds":[],"category":"research open","docstring":"**Filaseta–Finch–Kozek conjecture (2008).** Every Sierpiński number is either a perfect power\nor possesses a finite covering set of primes.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1113»","statement":"∀ (k : ℕ), k.IsSierpinskiNumber → k.IsPerfectPower ∨ Erdos1113.HasFinitePrimeCoveringSet k","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Erdos1113.erdos_1113.variants.filaseta_finch_kozek"},{"answerKinds":[],"category":"research solved","docstring":"Sierpiński [Si60] proved that there are infinitely many Sierpiński numbers, using covering\nsystems to construct suitable covering sets for any $k$ satisfying a certain congruence.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/HowieHwong/lean-erdos-proofs/blob/b8b641ba2d00dc4d1fe205a078a4159372672459/Erdos/P1113.lean#L22"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1113»","statement":"{k | k.IsSierpinskiNumber}.Infinite","subjects":["11"],"theorem":"Erdos1113.erdos_1113.variants.infinitely_many_sierpinski"},{"answerKinds":[],"category":"research open","docstring":"**Erdős Problem 1113.** Do there exist Sierpiński numbers that possess no finite covering set\nof primes?\n\nErdős and Graham [ErGr80] conjectured that the answer is yes. A negative answer would imply\nthat there are infinitely many Fermat primes.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1113»","statement":"True ↔ ∃ k, k.IsSierpinskiNumber ∧ ¬Erdos1113.HasFinitePrimeCoveringSet k","subjects":["11"],"theorem":"Erdos1113.erdos_1113"},{"answerKinds":[],"category":"research open","docstring":"The Collatz conjecture states that for any positive integer $n$, there exists a natural\nnumber $m$ such that the $m$-th term of the sequence is 1. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1135»","statement":"∀ n > 0, ∃ m, CollatzConjecture.collatzStep^[m] n = 1","subjects":["11","37"],"theorem":"Erdos1135.erdos_1135"},{"answerKinds":[],"category":"textbook","docstring":"Inequality in `erdos_317.variants.claim2` is obvious, the problem is strict inequality.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«317»","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ (δ : Fin n → ℚ),\n    δ '' Set.univ ⊆ {-1, 0, 1} → |∑ k, δ k / (↑↑k + 1)| ≠ 0 → |∑ k, δ k / (↑↑k + 1)| ≥ 1 / ↑((Finset.Icc 1 n).lcm id)","subjects":["11"],"theorem":"Erdos317.claim2_inequality"},{"answerKinds":[],"category":"textbook","docstring":"`erdos_317.variants.claim2` fails for small $n$, for example\n$$\\frac{1}{2}-\\frac{1}{3}-\\frac{1}{4}=-\\frac{1}{12}.$$\n","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«317»","statement":"¬∀ (δ : Fin 4 → ℚ),\n    δ '' Set.univ ⊆ {-1, 0, 1} → |∑ k, δ k / (↑↑k + 1)| ≠ 0 → |∑ k, δ k / (↑↑k + 1)| > 1 / ↑((Finset.Icc 1 4).lcm id)","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Erdos317.erdos_317.variants.counterexample"},{"answerKinds":[],"category":"research open","docstring":"Is it true that for sufficiently large $n$, for any $\\delta_k\\in \\{-1,0,1\\}$,\n$$\\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert > \\frac{1}{[1,\\ldots,n]}$$\nwhenever the left-hand side is not zero?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«317»","statement":"True ↔\n  ∀ᶠ (n : ℕ) in Filter.atTop,\n    ∀ (δ : Fin n → ℚ),\n      δ '' Set.univ ⊆ {-1, 0, 1} → |∑ k, δ k / (↑↑k + 1)| ≠ 0 → |∑ k, δ k / (↑↑k + 1)| > 1 / ↑((Finset.Icc 1 n).lcm id)","subjects":["11"],"theorem":"Erdos317.erdos_317.variants.claim2"},{"answerKinds":[],"category":"research open","docstring":"Is there some constant $c>0$ such that for every $n\\geq 1$ there exists some $\\delta_k\\in \\{-1,0,1\\}$ for $1\\leq k\\leq n$ with\n$$0< \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert < \\frac{c}{2^n}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«317»","statement":"True ↔\n  ∃ c > 0,\n    ∀ n ≥ 1, ∃ δ, Set.range δ ⊆ {-1, 0, 1} ∧ 0 < |∑ k, ↑(δ k) / (↑↑k + 1)| ∧ |∑ k, ↑(δ k) / (↑↑k + 1)| < c / 2 ^ n","subjects":["11"],"theorem":"Erdos317.erdos_317"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«955»","statement":"Erdos955.s 6 = 6","subjects":["11"],"theorem":"Erdos955.s_six"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«955»","statement":"Erdos955.s 28 = 28","subjects":["11"],"theorem":"Erdos955.s_twenty_eight"},{"answerKinds":[],"category":"research solved","docstring":"Pollack, Pomerance, and Thompson [PPT18] prove that if $\\epsilon(x)=o(1)$ and $A\\subset \\mathbb{N}$\nhas size at most $x^{1/2+\\epsilon(x)}$ then $\\#\\{ n\\leq x: s(n)\\in A\\} =o(x)$ as $x\\to \\infty$. It\nfollows that (using $s(n)\\ll n\\log\\log n$) if $A$ grows like\n$\\lvert A\\cap [1,x]\\rvert\\leq x^{1/2+o(1)}$ then $s^{-1}(A)$ has density $0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«955»","statement":"∀ (A : Set ℕ) (ε : ℕ → ℝ),\n  Filter.Tendsto ε Filter.atTop (nhds 0) →\n    (∀ᶠ (n : ℕ) in Filter.atTop, ↑(Nat.count (fun x => x ∈ A) n) ≤ ↑n ^ (1 / 2 + ε n)) →\n      {x | Erdos955.s x ∈ A}.HasDensity 0","subjects":["11"],"theorem":"Erdos955.erdos_955.variants.pollack_pomerance_thompson_bound"},{"answerKinds":[],"category":"research solved","docstring":"Troupe [Tr20] has also shown this is true if $A$ is the set of integers which are the sum of two\nsquares.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«955»","statement":"{x | ∃ a b, Erdos955.s x = a ^ 2 + b ^ 2}.HasDensity 0","subjects":["11"],"theorem":"Erdos955.erdos_955.variants.troupe_sum_of_two_squares"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«955»","statement":"Erdos955.s 2 = 1","subjects":["11"],"theorem":"Erdos955.s_two"},{"answerKinds":[],"category":"research solved","docstring":"It is possible for $s(A)$ to have positive density even if $A$ has zero density (for example\ntaking $A$ to be the product of two distinct primes).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«955»","statement":"∃ A, A.HasDensity 0 ∧ ∃ d > 0, (Erdos955.s '' A).HasDensity d","subjects":["11"],"theorem":"Erdos955.erdos_955.variants.positive_density"},{"answerKinds":[],"category":"research open","docstring":"If $A\\subset \\mathbb{N}$ has density $0$ then $s^{-1}(A)$ must also have density $0$.\n\nA conjecture of Erdős, Granville, Pomerance, and Spiro [EGPS90].\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«955»","statement":"True ↔ ∀ (A : Set ℕ), A.HasDensity 0 → {x | Erdos955.s x ∈ A}.HasDensity 0","subjects":["11"],"theorem":"Erdos955.erdos_955"},{"answerKinds":[],"category":"research solved","docstring":"Troupe [Tr15] has shown that this is true if $A$ is the set of integers with unusually many prime\nfactors.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«955»","statement":"∀ ε > 0,\n  {x |\n        (1 + ε) * Real.log (Real.log ↑(Erdos955.s x)) <\n          ↑(ArithmeticFunction.cardDistinctFactors (Erdos955.s x))}.HasDensity\n    0","subjects":["11"],"theorem":"Erdos955.erdos_955.variants.troupe_unusually_many_prime_factors"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«955»","statement":"Erdos955.s 12 = 16","subjects":["11"],"theorem":"Erdos955.s_twelve"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«955»","statement":"Erdos955.s 1 = 0","subjects":["11"],"theorem":"Erdos955.s_one"},{"answerKinds":[],"category":"research solved","docstring":"Pollack [Po14b] has shown that this is true if $A$ is the set of primes.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«955»","statement":"{x | Nat.Prime (Erdos955.s x)}.HasDensity 0","subjects":["11"],"theorem":"Erdos955.erdos_955.variants.pollack_primes"},{"answerKinds":[],"category":"research solved","docstring":"Erdős [Er73b] proved that there are sets $A$ of positive density such that $s^{-1}(A)$ is empty.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«955»","statement":"∃ A, (∃ d > 0, A.HasDensity d) ∧ {x | Erdos955.s x ∈ A} = ∅","subjects":["11"],"theorem":"Erdos955.erdos_955.variants.empty_preimage"},{"answerKinds":[],"category":"research open","docstring":"Is it true that $\\sum_{n=1}^\\infty(-1)^n\\frac{n}{p_n}$ converges,\nwhere $p_n$ is the sequence of primes?\n\nNote: In the problem statement, $p_n$ is the $n$-th prime, indexed such that $p_1=2, p_2=3, \\ldots$.\nWe 0-index here to reflect how Nat.nth works.\n\nNote: convergence here is convergence of the sequence of partial sums, which is what the\nproblem asks about. `Summable` would be the wrong notion: it is unconditional summability,\nequivalent over $\\mathbb{R}$ to absolute convergence, and $\\sum_n n/p_n$ diverges.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«15»","statement":"True ↔\n  ∃ l,\n    Filter.Tendsto (fun N => ∑ k ∈ Finset.range N, (-1) ^ (k + 1) * (↑k + 1) / ↑(Nat.nth Nat.Prime k)) Filter.atTop\n      (nhds l)","subjects":["11"],"theorem":"Erdos15.erdos_15"},{"answerKinds":[],"category":"research solved","docstring":"Call a set of distinct integers $1<n_1<\\cdots<n_k$ with associated congruence\nclasses $a_i\\pmod{n_i}$ a distinct covering system if every integer satisfies at\nleast one of these congruences. A minimal distinct covering system is one such\nthat no proper subset forms a covering system. Let $F(x)$ count the number of\nminimal distinct covering systems with all moduli in $[1,x]$. Estimate $F(x)$.\n\nThe estimate is `log(log F(x)) / log x → 1`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-1188/Research/SparseAsymptotic.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1188»","statement":"Filter.Tendsto (fun x => Real.log (Real.log ↑(Erdos1188.coveringCount x)) / Real.log ↑x) Filter.atTop (nhds 1)","subjects":["11"],"theorem":"Erdos1188.erdos_1188"},{"answerKinds":[],"category":"test","docstring":"$a(1) = 2$ by definition. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«342»","statement":"∀ (a : ℕ → ℕ), Erdos342.IsUlamSequence a → a 1 = 2","subjects":["5","11","40"],"theorem":"Erdos342.erdos_342.test.a1"},{"answerKinds":["Prop"],"category":"research open","docstring":"Does Ulam's sequence eventually have periodic differences? That is, is $a(n+1) - a(n)$ eventually periodic?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«342»","statement":"sorry ↔\n  ∀ (a : ℕ → ℕ),\n    Erdos342.IsUlamSequence a →\n      have d := fun n => ↑(a (n + 1)) - ↑(a n);\n      ∃ p > 0, ∀ᶠ (m : ℕ) in Filter.atTop, d (m + p) = d m","subjects":["5","11","40"],"theorem":"Erdos342.erdos_342.parts.ii"},{"answerKinds":[],"category":"test","docstring":"$a(3) = 4$: among sums $> 3$ with a unique representation from $\\{1,2,3\\}$,\nthe smallest is $4 = 1 + 3$. The candidate $5 = 2 + 3$ is ruled out by minimality since\n$4$ has a unique representation. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«342»","statement":"∀ (a : ℕ → ℕ), Erdos342.IsUlamSequence a → a 3 = 4","subjects":["5","11","40"],"theorem":"Erdos342.erdos_342.test.a3"},{"answerKinds":[],"category":"test","docstring":"$a(2) = 3$: the only pair $(i,j)$ with $i < j < 2$ is $(0,1)$, giving $1 + 2 = 3$. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«342»","statement":"∀ (a : ℕ → ℕ), Erdos342.IsUlamSequence a → a 2 = 3","subjects":["5","11","40"],"theorem":"Erdos342.erdos_342.test.a2"},{"answerKinds":["Prop"],"category":"research open","docstring":"Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«342»","statement":"sorry ↔ ∀ (a : ℕ → ℕ), Erdos342.IsUlamSequence a → {n | ∃ m, a m = a n + 2}.Infinite","subjects":["5","11","40"],"theorem":"Erdos342.erdos_342.parts.i"},{"answerKinds":["Prop"],"category":"research open","docstring":"Part (iii), is the density of the sequence 0?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«342»","statement":"sorry ↔ ∀ (a : ℕ → ℕ), Erdos342.IsUlamSequence a → (Set.range a).upperDensity = 0","subjects":["5","11","40"],"theorem":"Erdos342.erdos_342.parts.iii"},{"answerKinds":[],"category":"test","docstring":"$a(0) = 1$ by definition. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«342»","statement":"∀ (a : ℕ → ℕ), Erdos342.IsUlamSequence a → a 0 = 1","subjects":["5","11","40"],"theorem":"Erdos342.erdos_342.test.a0"},{"answerKinds":[],"category":"research open","docstring":"Is it true that, for all large $n$, $f(n + 1) \\ge f(n)$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«85»","statement":"True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, Erdos85.f n ≤ Erdos85.f (n + 1)","subjects":["5"],"theorem":"Erdos85.erdos_85"},{"answerKinds":[],"category":"research open","docstring":"Let $m$ be sufficiently large and let $G$ be a graph with $m$ edges and no isolated vertices.\nIs the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'?\nThat is, if $m=\\binom{n}{2}+t$ edges with $0\\leq t < n$\nthen is\n$$R(G)\\leq R(H),$$\nwhere $H$ is the graph formed by connecting a new vertex to $t$ of the vertices of $K_n$?\n\nA question of Erdős and Graham. The restriction to sufficiently large $m$ excludes the\nsmall counterexamples recorded on the source page.\n\nThis problem is #10 in Ramsey Theory in the graphs problem collection.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«545»","statement":"True ↔\n  ∀ᶠ (m : ℕ) in Filter.atTop,\n    ∀ (n t : ℕ),\n      t < n →\n        m = n.choose 2 + t →\n          ∀ (V : Type) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n            (∀ (v : V), 0 < G.degree v) →\n              G.edgeSet.ncard = m → G.diagonalGraphRamsey ≤ (Erdos545.knPlusTEdges n t).diagonalGraphRamsey","subjects":["5"],"theorem":"Erdos545.erdos_545"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«961»","statement":"Erdos961.Erdos961Prop 1 1","subjects":["11"],"theorem":"Erdos961.erdos_961.variants.sylvester_schur_1_1"},{"answerKinds":["Prop"],"category":"research open","docstring":"It is conjectured that $f(k) \\ll (\\log k)^O(1)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«961»","statement":"sorry ↔ ∃ C, ∀ᶠ (k : ℕ) in Filter.atTop, ↑(Erdos961.f k) < Real.log ↑k ^ C","subjects":["11"],"theorem":"Erdos961.erdos_961"},{"answerKinds":[],"category":"research solved","docstring":"There exists $n$ such that `Erdos961Prop k n` holds. ","hasSorryFreeProof":true,"module":"FormalConjectures.ErdosProblems.«961»","statement":"∀ (k : ℕ), 0 < k → ∃ n, Erdos961.Erdos961Prop k n","subjects":["11"],"theorem":"Erdos961.erdos_961.variants.well_defined"},{"answerKinds":[],"category":"research solved","docstring":"Jutila [Ju74], and Ramachandra--Shorey [RaSh73] proved a stronger upper bound\n$f(k) \\ll \\frac{\\log \\log \\log k}{\\log \\log k} \\frac{k}{\\log k}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«961»","statement":"(fun k => ↑(Erdos961.f k)) =O[Filter.atTop] fun k =>\n  Real.log (Real.log (Real.log ↑k)) / Real.log (Real.log ↑k) * (↑k / Real.log ↑k)","subjects":["11"],"theorem":"Erdos961.erdos_961.variants.jutila_ramachandra_shorey_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Sylvester and Schur [Er34] proved that every set of $k$ consecutive integers greater than $k$\ncontains an integer divisible by a prime greater than $k$, i.e. not $(k+1)$-smooth.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«961»","statement":"∀ (k : ℕ), 0 < k → Erdos961.Erdos961Prop k k","subjects":["11"],"theorem":"Erdos961.erdos_961.sylvester_schur"},{"answerKinds":[],"category":"research solved","docstring":"Erdos [Er55d] proved $f(k) < 3 \\frac{k}{\\log k}$ for sufficiently large $k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«961»","statement":"∀ᶠ (k : ℕ) in Filter.atTop, ↑(Erdos961.f k) < 3 * ↑k / Real.log ↑k","subjects":["11"],"theorem":"Erdos961.erdos_961.variants.erdos_upper_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the supremum of the set of admissible numbers? ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«757»","statement":"∀ {A : Set ℝ}, sorry = sSup {c | Erdos757.IsAdmissible c}","subjects":["5"],"theorem":"Erdos757.erdos_757"},{"answerKinds":[],"category":"research solved","docstring":"The supremum is strictly larger than `1 / 2`, which is proved in [GyLe95]. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«757»","statement":"∀ {A : Set ℝ}, 1 / 2 < sSup {c | Erdos757.IsAdmissible c}","subjects":["5"],"theorem":"Erdos757.erdos_757.variants.lowerBound"},{"answerKinds":[],"category":"research solved","docstring":"In [GyLe95], the authors also prove that the supremum is smaller than `3 / 5`. ","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«757»","statement":"∀ {A : Set ℝ}, sSup {c | Erdos757.IsAdmissible c} < 3 / 5","subjects":["5"],"theorem":"Erdos757.erdos_757.variants.upperBound"},{"answerKinds":[],"category":"research solved","docstring":"The counterexample: a connected bipartite `2`-degenerate `H` whose extremal number exceeds\n$n^{3/2+\\epsilon}$ infinitely often, so the `r = 2` case of `erdos_146` fails.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/openai/ten-proofs/blob/94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6/CompactnessAndDegeneracy.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«146»","statement":"∃ q H,\n  H.Connected ∧\n    H.IsBipartite ∧\n      H.IsDegenerate 2 ∧\n        ∃ c ε, 0 < c ∧ 0 < ε ∧ ∀ᶠ (n : ℕ) in Filter.atTop, c * ↑n ^ (3 / 2 + ε) ≤ ↑(SimpleGraph.extremalNumber n H)","subjects":["5"],"theorem":"Erdos146.erdos_146.variants.two_degenerate_counterexample"},{"answerKinds":["Prop"],"category":"research solved","docstring":"If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum\ndegree $\\leq r$, then\n$$\\mathrm{ex}(n;H) \\ll n^{2-1/r}.$$\n\nThe answer is no. OpenAI [OpenAI26] give a connected bipartite `2`-degenerate `H` and constants\n`c, ε > 0` with $\\mathrm{ex}(n;H)\\geq cn^{3/2+\\epsilon}$ for all large `n`, which exceeds the\nconjectured $n^{2-1/2}=n^{3/2}$. See `erdos_146.variants.two_degenerate_counterexample`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«146»","statement":"False ↔\n  ∀ (r q : ℕ) (H : SimpleGraph (Fin q)),\n    0 < r →\n      H.IsBipartite →\n        H.IsDegenerate r → (fun n => ↑(SimpleGraph.extremalNumber n H)) =O[Filter.atTop] fun n => ↑n ^ (2 - 1 / ↑r)","subjects":["5"],"theorem":"Erdos146.erdos_146"},{"answerKinds":[],"category":"research solved","docstring":"Upper bound (Davies–Illingworth 2022).\nThere exists a constant $c_2 \\ge 2$ such that, for sufficiently large $n$,\n$$\nf(n) \\le c_2 \\sqrt{\\frac{n}{\\log n}},\n$$\nwhere $f(n)$ denotes the maximum chromatic number of a triangle-free graph on\n$n$ vertices, formalized as `triangleFreeMaxChromatic n`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1104»","statement":"∃ c₂, 2 ≤ c₂ ∧ ∀ᶠ (n : ℕ) in Filter.atTop, ↑(Erdos1104.triangleFreeMaxChromatic n) ≤ c₂ * √↑n / √(Real.log ↑n)","subjects":["5"],"theorem":"Erdos1104.erdos_1104.variants.upper"},{"answerKinds":[],"category":"research solved","docstring":"Lower bound (Hefty–Horn–King–Pfender 2025).\nThere exists a constant $c_1 \\in (0,1]$ such that, for sufficiently large $n$,\n$$\nc_1 \\sqrt{\\frac{n}{\\log n}} \\le f(n),\n$$\nwhere $f(n)$ denotes the maximum chromatic number of a triangle-free graph on\n$n$ vertices, formalized as `triangleFreeMaxChromatic n`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«1104»","statement":"∃ c₁, 0 < c₁ ∧ c₁ ≤ 1 ∧ ∀ᶠ (n : ℕ) in Filter.atTop, c₁ * √↑n / √(Real.log ↑n) ≤ ↑(Erdos1104.triangleFreeMaxChromatic n)","subjects":["5"],"theorem":"Erdos1104.erdos_1104.variants.lower"},{"answerKinds":[],"category":"research solved","docstring":"Narkiewicz [Na59] proved that, under the given assumptions (and perhaps swapping $A$ and $B$)\nwe must have $A(2x)/A(x)\\to 1$ and $B(2x)/B(x)\\to 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"∀ (A B : Set ℕ),\n  A.Infinite →\n    B.Infinite →\n      Erdos785.IsExactAdditiveComplement A B →\n        Filter.Tendsto (fun x => ↑(Erdos785.counting A (2 * x)) / ↑(Erdos785.counting A x)) Filter.atTop (nhds 1) ∧\n            Filter.Tendsto (fun x => ↑(Erdos785.counting B (2 * x)) / ↑(Erdos785.counting B x)) Filter.atTop (nhds 2) ∨\n          Filter.Tendsto (fun x => ↑(Erdos785.counting B (2 * x)) / ↑(Erdos785.counting B x)) Filter.atTop (nhds 1) ∧\n            Filter.Tendsto (fun x => ↑(Erdos785.counting A (2 * x)) / ↑(Erdos785.counting A x)) Filter.atTop (nhds 2)","subjects":["11"],"theorem":"Erdos785.erdos_785.variants.narkiewicz"},{"answerKinds":[],"category":"research solved","docstring":"This is sharp, as Chen and Fang [ChFa11] also proved that there exist such $A$ and $B$ with\n$$\\limsup_{x\\to \\infty}\\frac{A(x)B(x)}{x}=\\frac{3}{2}$$\nfor which $A(x)B(x)-x=1$ for infinitely many $x$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"∃ A B,\n  A.Infinite ∧\n    B.Infinite ∧\n      IsAdditiveComplement A B ∧\n        Filter.limsup (fun x => ↑(↑(Erdos785.counting A x) * ↑(Erdos785.counting B x)) / ↑↑x) Filter.atTop = ↑(3 / 2) ∧\n          ∃ᶠ (x : ℕ) in Filter.atTop, ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x = 1","subjects":["11"],"theorem":"Erdos785.erdos_785.variants.chen_fang_sharp"},{"answerKinds":[],"category":"research open","docstring":"Chen conjectures that this should be true with $3/2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"True ↔\n  ∀ (A B : Set ℕ),\n    A.Infinite →\n      B.Infinite →\n        IsAdditiveComplement A B →\n          Filter.limsup (fun x => ↑(↑(Erdos785.counting A x) * ↑(Erdos785.counting B x)) / ↑↑x) Filter.atTop <\n              ↑(3 / 2) →\n            Filter.Tendsto (fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos785.erdos_785.variants.chen_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"Sárközy and Szemerédi [SaSz94] proved that it is impossible for\n$$A(x)B(x)-x=o(A(x)).$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"∀ (A B : Set ℕ),\n  A.Infinite →\n    B.Infinite →\n      Erdos785.IsExactAdditiveComplement A B →\n        ¬(fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) =o[Filter.atTop] fun x =>\n            ↑(Erdos785.counting A x)","subjects":["11"],"theorem":"Erdos785.erdos_785.variants.sarkozy_szemeredi"},{"answerKinds":[],"category":"research solved","docstring":"Chen and Fang [ChFa15] proved $A(x)B(x)-x\\ll A(x)^c$ cannot hold for any constant $c>0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"∀ (A B : Set ℕ),\n  A.Infinite →\n    B.Infinite →\n      Erdos785.IsExactAdditiveComplement A B →\n        ∀ (c : ℝ),\n          0 < c →\n            ¬(fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) =O[Filter.atTop] fun x =>\n                ↑(Erdos785.counting A x) ^ c","subjects":["11"],"theorem":"Erdos785.erdos_785.variants.chen_fang"},{"answerKinds":[],"category":"research solved","docstring":"Danzer [Da64] proved that exact additive complements exist (Hanani had earlier conjectured they\ndo not exist, as reported in [Er57] and [Er61]).\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"∃ A B, A.Infinite ∧ B.Infinite ∧ Erdos785.IsExactAdditiveComplement A B","subjects":["11"],"theorem":"Erdos785.erdos_785.variants.danzer"},{"answerKinds":[],"category":"research solved","docstring":"Ruzsa [Ru17] proves that, if $a^*(x)=\\max A \\cap [1,x]$ and $A$ and $B$ satisfy the conditions\nin the problem then (after possibly changing the roles of $A$ and $B$)\n$$A(x)B(x)-x > (1-o(1))\\frac{a^*(x)}{A(x)}.$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"∀ (A B : Set ℕ),\n  A.Infinite →\n    B.Infinite →\n      Erdos785.IsExactAdditiveComplement A B →\n        (∀ ε > 0,\n            ∀ᶠ (x : ℕ) in Filter.atTop,\n              (1 - ε) * ↑(Erdos785.aStar A x) / ↑(Erdos785.counting A x) <\n                ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) ∨\n          ∀ ε > 0,\n            ∀ᶠ (x : ℕ) in Filter.atTop,\n              (1 - ε) * ↑(Erdos785.aStar B x) / ↑(Erdos785.counting B x) <\n                ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x","subjects":["11"],"theorem":"Erdos785.erdos_785.variants.ruzsa_lower_bound"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $A,B\\subseteq \\mathbb{N}$ be infinite sets such that $A+B$ contains all large integers.\nLet $A(x)=\\lvert A\\cap [1,x]\\rvert$ and similarly for $B(x)$. Is it true that if\n$A(x)B(x)\\sim x$ then\n$$A(x)B(x)-x\\to \\infty$$\nas $x\\to \\infty$?\n\nA conjecture of Erdős and Danzer. The answer is yes, proved by Sárközy and Szemerédi [SaSz94],\nwho actually proved that it is impossible for\n$$A(x)B(x)-x=o(A(x)).$$\n\nThis was formalized in Lean by van Doorn using Aristotle.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem785.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"True ↔\n  ∀ (A B : Set ℕ),\n    A.Infinite →\n      B.Infinite →\n        Erdos785.IsExactAdditiveComplement A B →\n          Filter.Tendsto (fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos785.erdos_785"},{"answerKinds":[],"category":"research solved","docstring":"Ruzsa [Ru17] has constructed, for any function $w(x)\\to \\infty$, such a pair of sets with\n$$A(x)B(x)-x<w(x)$$\nfor infinitely many $x$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"∀ (w : ℕ → ℝ),\n  Filter.Tendsto w Filter.atTop Filter.atTop →\n    ∃ A B,\n      A.Infinite ∧\n        B.Infinite ∧\n          Erdos785.IsExactAdditiveComplement A B ∧\n            ∃ᶠ (x : ℕ) in Filter.atTop, ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x < w x","subjects":["11"],"theorem":"Erdos785.erdos_785.variants.ruzsa_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Chen and Fang [ChFa10] proved the stronger statement that $A(x)B(x)-x\\to \\infty$ if $A$ and $B$\nare infinite sets such that $A+B$ contains all large integers and\n$$\\limsup_{x\\to \\infty}\\frac{A(x)B(x)}{x}<\\frac{5}{4}.$$\nThey later [ChFa14] improved $5/4$ to $3-\\sqrt{3}\\approx 1.268$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.ErdosProblems.«785»","statement":"∀ (A B : Set ℕ),\n  A.Infinite →\n    B.Infinite →\n      IsAdditiveComplement A B →\n        Filter.limsup (fun x => ↑(↑(Erdos785.counting A x) * ↑(Erdos785.counting B x)) / ↑↑x) Filter.atTop < ↑(3 - √3) →\n          Filter.Tendsto (fun x => ↑(Erdos785.counting A x) * ↑(Erdos785.counting B x) - ↑x) Filter.atTop Filter.atTop","subjects":["11"],"theorem":"Erdos785.erdos_785.variants.chen_fang_limsup"},{"answerKinds":[],"category":"textbook","docstring":"The Generalized Poincaré Conjecture holds for surfaces.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.Poincare","statement":"PoincareConjecture.ConjectureFor 2","subjects":["54","57"],"theorem":"PoincareConjecture.poincare_conjecture.variants.dimension_two"},{"answerKinds":[],"category":"textbook","docstring":"The Generalized Poincaré Conjecture holds in dimension 4.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.Poincare","statement":"PoincareConjecture.ConjectureFor 4","subjects":["54","57"],"theorem":"PoincareConjecture.poincare_conjecture.variants.dimension_four"},{"answerKinds":[],"category":"textbook","docstring":"A reformulation of the Millennium Problem in terms of smooth 3-folds. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.Poincare","statement":"PoincareConjecture.SmoothConjectureFor 3","subjects":["54","57"],"theorem":"PoincareConjecture.poincare_conjecture.variants.smooth_for_three"},{"answerKinds":[],"category":"research solved","docstring":"The smooth version of the Poincaré conjecture is known to hold in dimensions\n$1, 2, 3, 5, 6, 12, 56, 61$. See [Wang2017]. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.Poincare","statement":"∀ n ∈ PoincareConjecture.SmoothTrueValues, PoincareConjecture.SmoothConjectureFor n","subjects":["54","57"],"theorem":"PoincareConjecture.poincare_conjecture.variants.smooth_known_cases"},{"answerKinds":[],"category":"research open","docstring":"It is conjectured that the only values of $n > 4$ for which the smooth version of the\nconjecture holds are $n = 5, 6, 12, 56, 61$. See Conjecture 1.17 in [Wang2017]. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.Poincare","statement":"∀ n > 4, n ∉ PoincareConjecture.SmoothTrueValues → ¬PoincareConjecture.SmoothConjectureFor n","subjects":["54","57"],"theorem":"PoincareConjecture.poincare_conjecture.variants.smooth_other_cases"},{"answerKinds":[],"category":"textbook","docstring":"The smooth formulation of the Millennium Problem implies the general case. This follows from\nthe fact that every topological 3-fold admits a smooth structure [mo296171]. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.Poincare","statement":"PoincareConjecture.SmoothConjectureFor 3 → PoincareConjecture.ConjectureFor 3","subjects":["54","57"],"theorem":"PoincareConjecture.poincare_conjecture.variants.smooth_implication"},{"answerKinds":[],"category":"research solved","docstring":"The Millennium Problem, solved by Grigori Perelman in 2003: the Poincaré Conjecture holds.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.Poincare","statement":"PoincareConjecture.ConjectureFor 3","subjects":["54","57"],"theorem":"PoincareConjecture.poincare_conjecture"},{"answerKinds":[],"category":"research open","docstring":"The four dimensional case of the smooth version of the conjecture is still open.\nSee [Wang2017]. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.Poincare","statement":"PoincareConjecture.SmoothConjectureFor 4","subjects":["54","57"],"subsets":["FC100OpenSet1"],"theorem":"PoincareConjecture.poincare_conjecture.variants.smooth_dimension_four"},{"answerKinds":[],"category":"textbook","docstring":"The Generalized Poincaré Conjecture holds for dimensions at least 5.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.Poincare","statement":"∀ (n : ℕ), 5 ≤ n → PoincareConjecture.ConjectureFor n","subjects":["54","57"],"theorem":"PoincareConjecture.poincare_conjecture.variants.dimension_ge_five"},{"answerKinds":[],"category":"research open","docstring":"**Hasse--Weil conjecture**: the $L$-function of an elliptic curve over a number field extends\nto the whole plane. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.BSD","statement":"∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K] (E : WeierstrassCurve K) [E.IsElliptic],\n  ∃ L, BirchSwinnertonDyer.IsLFunction E L","subjects":["11","14"],"theorem":"BirchSwinnertonDyer.exists_isLFunction"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Millenium.BSD","statement":"∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K] {E : WeierstrassCurve K} {L L' : ℂ → ℂ},\n  BirchSwinnertonDyer.IsLFunction E L → BirchSwinnertonDyer.IsLFunction E L' → ∀ (x : ℂ), L =ᶠ[nhdsWithin x {x}ᶜ] L'","subjects":["11","14"],"theorem":"BirchSwinnertonDyer.IsLFunction.unique"},{"answerKinds":[],"category":"research solved","docstring":"The **Hasse--Weil conjecture** over $\\mathbb{Q}$, a consequence of the modularity theorem. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.BSD","statement":"∀ (E : WeierstrassCurve ℚ) [E.IsElliptic], ∃ L, BirchSwinnertonDyer.IsLFunction E L","subjects":["11","14"],"theorem":"BirchSwinnertonDyer.exists_isLFunction_rat"},{"answerKinds":[],"category":"API","docstring":"The divergence of a vector field is $0$ at points where `fderiv` has its junk value. ","hasSorryFreeProof":true,"module":"FormalConjectures.Millenium.NavierStokes","statement":"∀ {n : ℕ} {v : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n)} {x : EuclideanSpace ℝ (Fin n)},\n  ¬DifferentiableAt ℝ v x → ∇⬝ v x = 0","subjects":["35"],"theorem":"NavierStokes.divergence_of_not_differentiableAt"},{"answerKinds":[],"category":"research open","docstring":"(A) Existence and smoothness of Navier–Stokes solutions on ℝ³. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.NavierStokes","statement":"∀ nu > 0,\n  ∀ (u₀ : EuclideanSpace ℝ (Fin 3) → EuclideanSpace ℝ (Fin 3)),\n    NavierStokes.InitialVelocityConditionDecay u₀ → ∃ v p, NavierStokes.NavierStokesExistenceAndSmoothnessRn nu u₀ 0 v p","subjects":["35"],"theorem":"NavierStokes.navier_stokes_existence_and_smoothness_R3"},{"answerKinds":[],"category":"API","docstring":"Divergence commutes with scalar multiplication at differentiability points. ","hasSorryFreeProof":true,"module":"FormalConjectures.Millenium.NavierStokes","statement":"∀ {n : ℕ} (c : ℝ) {v : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n)} {x : EuclideanSpace ℝ (Fin n)},\n  DifferentiableAt ℝ v x → ∇⬝ (fun y => c • v y) x = c * ∇⬝ v x","subjects":["35"],"theorem":"NavierStokes.divergence_smul"},{"answerKinds":[],"category":"API","docstring":"The divergence of the zero vector field is zero. ","hasSorryFreeProof":true,"module":"FormalConjectures.Millenium.NavierStokes","statement":"∀ {n : ℕ} (x : EuclideanSpace ℝ (Fin n)), ∇⬝ 0 x = 0","subjects":["35"],"theorem":"NavierStokes.divergence_zero"},{"answerKinds":[],"category":"research open","docstring":"(C) Breakdown of Navier–Stokes solutions on ℝ³. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.NavierStokes","statement":"∀ nu > 0,\n  ∃ u₀ f,\n    NavierStokes.InitialVelocityConditionDecay u₀ ∧\n      NavierStokes.ForceConditionDecay f ∧ ¬∃ v p, NavierStokes.NavierStokesExistenceAndSmoothnessRn nu u₀ f v p","subjects":["35"],"theorem":"NavierStokes.navier_stokes_breakdown_R3"},{"answerKinds":[],"category":"research open","docstring":"(B) Existence and smoothness of Navier–Stokes solutions in ℝ³/ℤ³. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.NavierStokes","statement":"∀ nu > 0,\n  ∀ (u₀ : EuclideanSpace ℝ (Fin 3) → EuclideanSpace ℝ (Fin 3)),\n    NavierStokes.InitialVelocityConditionPeriodic u₀ →\n      ∃ v p, NavierStokes.NavierStokesExistenceAndSmoothnessPeriodic nu u₀ 0 v p","subjects":["35"],"theorem":"NavierStokes.navier_stokes_existence_and_smoothness_periodic"},{"answerKinds":[],"category":"research open","docstring":"(D) Breakdown of Navier–Stokes Solutions on ℝ³/ℤ³. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.NavierStokes","statement":"∀ nu > 0,\n  ∃ u₀ f,\n    NavierStokes.InitialVelocityConditionPeriodic u₀ ∧\n      NavierStokes.ForceConditionPeriodic f ∧\n        ¬∃ v p, NavierStokes.NavierStokesExistenceAndSmoothnessPeriodic nu u₀ f v p","subjects":["35"],"theorem":"NavierStokes.navier_stokes_breakdown_periodic"},{"answerKinds":[],"category":"API","docstring":"Divergence is additive at points where both vector fields are differentiable. ","hasSorryFreeProof":true,"module":"FormalConjectures.Millenium.NavierStokes","statement":"∀ {n : ℕ} {v w : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n)} {x : EuclideanSpace ℝ (Fin n)},\n  DifferentiableAt ℝ v x → DifferentiableAt ℝ w x → ∇⬝ (fun y => v y + w y) x = ∇⬝ v x + ∇⬝ w x","subjects":["35"],"theorem":"NavierStokes.divergence_add"},{"answerKinds":[],"category":"API","docstring":"The divergence of a constant vector field is zero. ","hasSorryFreeProof":true,"module":"FormalConjectures.Millenium.NavierStokes","statement":"∀ {n : ℕ} (c x : EuclideanSpace ℝ (Fin n)), ∇⬝ (fun x => c) x = 0","subjects":["35"],"theorem":"NavierStokes.divergence_const"},{"answerKinds":[],"category":"test","docstring":"GRH for $\\chi = 1$ is `RiemannHypothesis`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Millenium.RiemannHypothesis","statement":"(DirichletCharacter.IsPrimitive 1 →\n    ∀ (s : ℂ), DirichletCharacter.LFunction 1 s = 0 → s ∉ Int.cast '' GRH.trivialZeros 1 → s.re = 1 / 2) ↔\n  RiemannHypothesis","subjects":["11"],"theorem":"GRH.implies_riemannHypothesis"},{"answerKinds":[],"category":"research open","docstring":"The **Riemann Hypothesis**: all non-trivial zeros of the Riemann zeta function have real\npart $\\frac{1}{2}$. That is, if $\\zeta(s) = 0$, $s \\neq 1$, and $s$ is not a trivial zero\n$-2(n+1)$ for some $n \\in \\mathbb{N}$, then $\\operatorname{Re}(s) = \\frac{1}{2}$.\n\nThis is the official Millennium Prize Problem as posed by the\n[Clay Mathematics Institute](https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf).\n\nThis uses the `RiemannHypothesis` type from Mathlib, which is defined as\n`∀ (s : ℂ), riemannZeta s = 0 → (¬∃ n : ℕ, s = -2 * (n + 1)) → s ≠ 1 → s.re = 1 / 2`. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.RiemannHypothesis","statement":"RiemannHypothesis","subjects":["11"],"theorem":"RiemannHypothesis.riemannHypothesis"},{"answerKinds":[],"category":"research open","docstring":"The **Generalized Riemann Hypothesis** asserts that all the non-trivial zeros of the\nDirichlet $L$-function $L(\\chi, s)$ of a primitive Dirichlet character $\\chi$ have real part\n$\\frac{1}{2}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.RiemannHypothesis","statement":"∀ (q : ℕ) [inst : NeZero q] (χ : DirichletCharacter ℂ q),\n  χ.IsPrimitive → ∀ (s : ℂ), DirichletCharacter.LFunction χ s = 0 → s ∉ Int.cast '' GRH.trivialZeros χ → s.re = 1 / 2","subjects":["11"],"theorem":"GRH.generalized_riemann_hypothesis"},{"answerKinds":[],"category":"textbook","docstring":"The theorem that P is a subset of NP.\n\nThis can be proven by observing that for any language in P,\nwe can construct a verifier that ignores the witness and simply runs the poly-time decider for the\nlanguage.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.PvsNP","statement":"ComplexityTheory.P ⊆ ComplexityTheory.NP","subjects":["68"],"theorem":"ComplexityTheory.P_subset_NP"},{"answerKinds":[],"category":"research open","docstring":"**P ≠ NP**:\n\nThe conjecture that the complexity classes P and NP are not equal.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.PvsNP","statement":"ComplexityTheory.P ≠ ComplexityTheory.NP","subjects":["68"],"theorem":"ComplexityTheory.P_ne_NP"},{"answerKinds":[],"category":"textbook","docstring":"The theorem that the set of complements of languages in P is itself P.\n\nThis can be proven by observing that the boolean negation function is computable in polynomial time,\nand that compositions of poly-time computable functions are also poly-time computable.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.PvsNP","statement":"{L | Lᶜ ∈ ComplexityTheory.P} = ComplexityTheory.P","subjects":["68"],"theorem":"ComplexityTheory.coP_eq_P"},{"answerKinds":[],"category":"textbook","docstring":"The theorem that P is a subset of coNP.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Millenium.PvsNP","statement":"ComplexityTheory.P ⊆ ComplexityTheory.coNP","subjects":["68"],"theorem":"ComplexityTheory.P_subset_coNP"},{"answerKinds":[],"category":"research open","docstring":"**NP ≠ coNP**:\n\nThe conjecture that the complexity classes NP and coNP are not equal.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Millenium.PvsNP","statement":"ComplexityTheory.NP ≠ ComplexityTheory.coNP","subjects":["68"],"theorem":"ComplexityTheory.NP_ne_coNP"},{"answerKinds":[],"category":"research open","docstring":"The negation of `Finite.Equation677_implies_Equation255`.\n\nProbably this is true. It would be a stronger form of\n`Equation677_not_implies_Equation255`.\n\nDiscussion thread here:\nhttps://leanprover.zulipchat.com/#narrow/channel/458659-Equational/topic/FINITE.3A.20677.20-.3E.20255 ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.EquationalTheories_677_255","statement":"∃ G x, Finite G ∧ EquationalTheories_677_255.Equation677 G ∧ ¬EquationalTheories_677_255.Equation255 G","subjects":["8"],"theorem":"EquationalTheories_677_255.Finite.Equation677_not_implies_Equation255"},{"answerKinds":[],"category":"research solved","docstring":"Equation 255 does not imply Equation 677. ","hasSorryFreeProof":true,"module":"FormalConjectures.Other.EquationalTheories_677_255","statement":"∃ G x, EquationalTheories_677_255.Equation255 G ∧ ¬EquationalTheories_677_255.Equation677 G","subjects":["8"],"theorem":"EquationalTheories_677_255.Equation255_not_implies_Equation677"},{"answerKinds":[],"category":"research solved","docstring":"Equation 677 does not imply Equation 255. ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.EquationalTheories_677_255","statement":"∃ G x, EquationalTheories_677_255.Equation677 G ∧ ¬EquationalTheories_677_255.Equation255 G","subjects":["8"],"theorem":"EquationalTheories_677_255.Equation677_not_implies_Equation255"},{"answerKinds":[],"category":"research open","docstring":"The negation of `Finite.Equation677_not_implies_Equation255`.\n\nProbably this is false. ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.EquationalTheories_677_255","statement":"∀ (G : Type) [inst : EquationalTheories_677_255.Magma G] [Finite G],\n  EquationalTheories_677_255.Equation677 G → EquationalTheories_677_255.Equation255 G","subjects":["8"],"theorem":"EquationalTheories_677_255.Finite.Equation677_implies_Equation255"},{"answerKinds":[],"category":"research solved","docstring":"Note that this is a stronger form of `Equation255_not_implies_Equation677`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Other.EquationalTheories_677_255","statement":"∃ G x, Finite G ∧ EquationalTheories_677_255.Equation255 G ∧ ¬EquationalTheories_677_255.Equation677 G","subjects":["8"],"theorem":"EquationalTheories_677_255.Finite.Equation255_not_implies_Equation677"},{"answerKinds":[],"category":"research solved","docstring":"$A$ and $B$ are sets of words of length $n$ over alphabet with $q \\geq 1$ letters.\nNo suffix of a word in $A$ coincides with a prefix of a word in $B$.\nThen $|A| \\cdot |B|$ is at most $\\frac{q^{2n}}{en}$.\n\nThis problem is from *Maximal sets of strings with no prefix-suffix overlap* and was proved in\n*An isoperimetric inequality for word overlap*.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.SuffixPrefixAvoidance","statement":"∀ {n q : ℕ} (A B : Finset (Fin n → Fin q)),\n  0 < q →\n    0 < n → SuffixPrefixAvoidance.IsSuffixPrefixAvoiding A B → ↑A.card * ↑B.card ≤ ↑q ^ (2 * n) / (Real.exp 1 * ↑n)","subjects":["5"],"theorem":"SuffixPrefixAvoidance.suffix_prefix_avoidance_bound"},{"answerKinds":[],"category":"test","docstring":"$A$ and $B$ are sets of words of length $n$ over alphabet with $q$ letters.\nTrivially then $|A| \\cdot |B|$ is at most $q^{2n}$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Other.SuffixPrefixAvoidance","statement":"∀ {n q : ℕ} (A B : Finset (Fin n → Fin q)), A.card * B.card ≤ q ^ (2 * n)","subjects":["5"],"theorem":"SuffixPrefixAvoidance.words_naive_bound"},{"answerKinds":[],"category":"research solved","docstring":"$A$ and $B$ are sets of words of length $n$ over alphabet with $q \\geq 1$ letters.\nNo suffix of a word in $A$ coincides with a prefix of a word in $B$.\nThen $|A| \\cdot |B|$ is at most $\\frac{q^{2n}}{n}$.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/102e47fee802d461946e3a4e0b47fdbe7db4c1ed/FormalConjectures/Other/SuffixPrefixAvoidance.lean#L157"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Other.SuffixPrefixAvoidance","statement":"∀ {n q : ℕ} (A B : Finset (Fin n → Fin q)),\n  0 < q → 0 < n → SuffixPrefixAvoidance.IsSuffixPrefixAvoiding A B → ↑A.card * ↑B.card ≤ ↑q ^ (2 * n) / ↑n","subjects":["5"],"theorem":"SuffixPrefixAvoidance.suffix_prefix_avoidance_weaker_bound"},{"answerKinds":[],"category":"research solved","docstring":"**Gerstenhaber's theorem** [Ger61]: if $A$ and $B$ are commuting $n \\times n$\nmatrices over a field $K$, then the unital $K$-algebra $K[A, B]$ they generate\nhas dimension at most $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.GerstenhaberThreeMatrices","statement":"∀ {K : Type u_1} [inst : Field K] {n : ℕ} (A B : Matrix (Fin n) (Fin n) K), Commute A B → Module.finrank K ↥K[A, B] ≤ n","subjects":["15","16"],"theorem":"Gerstenhaber.finrank_adjoin_pair_le"},{"answerKinds":[],"category":"research solved","docstring":"The analogue of Gerstenhaber's theorem fails for four pairwise commuting matrices:\nover any field there are four pairwise commuting $4 \\times 4$ matrices generating a unital\nalgebra of dimension greater than $4$. The standard example is\n$e_{13}, e_{14}, e_{23}, e_{24}$, whose pairwise products all vanish, so the algebra\nthey generate has dimension $5$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.GerstenhaberThreeMatrices","statement":"∀ (K : Type u_2) [inst : Field K], ∃ A B C D, {A, B, C, D}.Pairwise Commute ∧ 4 < Module.finrank K ↥K[A, B, C, D]","subjects":["15","16"],"theorem":"Gerstenhaber.exists_finrank_adjoin_quadruple_gt"},{"answerKinds":[],"category":"research open","docstring":"**The Gerstenhaber problem**: if $A$, $B$, and $C$ are pairwise commuting\n$n \\times n$ matrices over a field $K$, is the dimension of the unital\n$K$-algebra $K[A, B, C]$ they generate always at most $n$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.GerstenhaberThreeMatrices","statement":"True ↔\n  ∀ (K : Type u_2) [inst : Field K] (n : ℕ) (A B C : Matrix (Fin n) (Fin n) K),\n    {A, B, C}.Pairwise Commute → Module.finrank K ↥K[A, B, C] ≤ n","subjects":["15","16"],"theorem":"Gerstenhaber.finrank_adjoin_triple_le"},{"answerKinds":[],"category":"research open","docstring":"[BMO#5](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#5._1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE_(bbch))\n\nLet $(a_n)_{n \\ge 1}$ and $(b_n)_{n \\ge 1}$ be two sequences such that $(a_1, b_1) = (0, 5)$ and\n\n$$(a_{n+1}, b_{n+1}) = \\begin{cases}\n(a_n+1, b_n-f(a_n)) & \\text{if } b_n \\ge f(a_n) \\cr\n(a_n, 3b_n+a_n+5) & \\text{if } b_n < f(a_n)\n\\end{cases}$$\n\nwhere $f(x)=10\\cdot 2^x-1$ for all non-negative integers $x$.\n\nDoes there exist a positive integer $i$ such that $b_i = f(a_i)-1$?\n\n[BMO#5](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#5._1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE_(bbch)) is equivalent to asking whether the 6-state Turing machine\n[`1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE`](https://wiki.bbchallenge.org/wiki/1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE) halts or not.\n\nThere is presently no consensus on whether the machine halts or not, hence the problem is formulated\nusing `answer(sorry) ↔`.\n\nThe machine was discovered by [bbchallenge.org](https://bbchallenge.org) contributor mxdys\non August 7th 2024.\n\nThe correspondence between the machine's halting problem and the below reformulation has been proven\nin [Rocq](https://github.com/ccz181078/busycoq/blob/BB6/verify/1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE.v).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.BeaverMathOlympiad","statement":"True ↔\n  ∀ (a b f : ℕ → ℕ),\n    (f = fun x => 10 * 2 ^ x - 1) →\n      a 0 = 0 →\n        b 0 = 5 →\n          (∀ (n : ℕ), a (n + 1) = if f (a n) ≤ b n then a n + 1 else a n) →\n            (∀ (n : ℕ), b (n + 1) = if f (a n) ≤ b n then b n - f (a n) else 3 * b n + a n + 5) → ∃ i, b i = f (a i) - 1","subjects":["5","11","68"],"theorem":"BeaverMathOlympiad.beaver_math_olympiad_problem_5"},{"answerKinds":[],"category":"research solved","docstring":"[BMO#3](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#3._1RB0RB3LA4LA2RA_2LB3RA---3RA4RB_(bbch)_and_1RB1RB3LA4LA2RA_2LB3RA---3RA4RB_(bbch))\n\nLet $v_2(n)$ be the largest integer $k$ such that $2^k$ divides $n$.\nLet $(a_n)_{n \\ge 0}$ be a sequence such that\n\n$$a_n = \\begin{cases}\n2 & \\text{if } n=0 \\cr\na_{n-1}+2^{v_2(a_{n-1})+2}-1 & \\text{if } n \\ge 1\n\\end{cases}$$\n\nfor all non-negative integers $n$. Is there an integer $n$ such that $a_n=4^k$ for\nsome positive integer $k$?\n\n[BMO#3](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#3._1RB0RB3LA4LA2RA_2LB3RA---3RA4RB_(bbch)_and_1RB1RB3LA4LA2RA_2LB3RA---3RA4RB_(bbch)) is equivalent to the non-termination of 2-state 5-symbol Turing machine [`1RB0RB3LA4LA2RA_2LB3RA---3RA4RB`](https://wiki.bbchallenge.org/wiki/1RB0RB3LA4LA2RA_2LB3RA---3RA4RB) (from all-0 tape).\n\nThe machine was found and informally proven not to halt by [bbchallenge.org](https://bbchallenge.org)\ncontributor Daniel Yuan on June 18th 2024; see [Discord discussion](https://discord.com/channels/960643023006490684/1084047886494470185/1252634913220591728).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.BeaverMathOlympiad","statement":"∀ (a : ℕ → ℕ), a 0 = 2 → (∀ (n : ℕ), a (n + 1) = a n + 2 ^ (padicValNat 2 (a n) + 2) - 1) → ¬∃ n k, a n = 4 ^ k","subjects":["5","11","68"],"theorem":"BeaverMathOlympiad.beaver_math_olympiad_problem_3"},{"answerKinds":[],"category":"research open","docstring":"[BMO#1](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#1._1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE_(bbch))\n\nLet $(a_n)_{n \\ge 1}$ and $(b_n)_{n \\ge 1}$ be two sequences such that $(a_1, b_1) = (1, 2)$ and\n\n$$(a_{n+1}, b_{n+1}) = \\begin{cases}\n(a_n-b_n, 4b_n+2) & \\text{if }a_n \\ge b_n \\cr\n(2a_n+1, b_n-a_n) & \\text{if }a_n < b_n\n\\end{cases}$$\n\nfor all positive integers $n$. Does there exist a positive integer $i$ such that $a_i = b_i$?\n\nThe first 10 values of $(a_n, b_n)$ are $(1, 2), (3, 1), (2, 6), (5, 4), (1, 18), (3, 17),\n(7, 14), (15, 7), (8, 30), (17, 22)$.\n\n[BMO#1](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#1._1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE_(bbch)) is equivalent to asking whether the 6-state Turing machine\n[`1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE`](https://wiki.bbchallenge.org/wiki/1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE) halts or not.\n\nThere is presently no consensus on whether the machine halts or not, hence the problem is formulated\nusing `answer(sorry) ↔`.\n\nThe machine was discovered by [bbchallenge.org](https://bbchallenge.org) contributor Jason Yuen on\nJune 25th 2024.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.BeaverMathOlympiad","statement":"True ↔\n  ∀ (a b : ℕ → ℕ),\n    a 0 = 1 →\n      (∀ (n : ℕ), a (n + 1) = if b n ≤ a n then a n - b n else 2 * a n + 1) →\n        b 0 = 2 → (∀ (n : ℕ), b (n + 1) = if b n ≤ a n then 4 * b n + 2 else b n - a n) → ∃ i, a i = b i","subjects":["5","11","68"],"theorem":"BeaverMathOlympiad.beaver_math_olympiad_problem_1"},{"answerKinds":[],"category":"research open","docstring":"[BMO#2](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#2._Hydra_and_Antihydra) formulation variant\n\nAlternative statement of beaver_math_olympiad_problem_2_antihydra\nusing set size comparison instead of a recurrent sequence b.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.BeaverMathOlympiad","statement":"∀ (a : ℕ → ℕ),\n  a 0 = 8 →\n    (∀ (n : ℕ), a (n + 1) = 3 * a n / 2) →\n      ∀ (n : ℕ), {x ∈ Finset.Ico 0 n | Odd (a x)}.card ≤ 2 * {x ∈ Finset.Ico 0 n | Even (a x)}.card","subjects":["5","11","68"],"theorem":"BeaverMathOlympiad.beaver_math_olympiad_problem_2_antihydra.variants.set"},{"answerKinds":[],"category":"research open","docstring":"[BMO#8](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#8._1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA_(bbch))\n\nLet $(a_n)_{n \\ge 1}$ and $(b_n)_{n \\ge 1}$ be two sequences such that $(a_1, b_1) = (10, 12)$ and\n\n$$(a_{n+1}, b_{n+1}) = \\begin{cases}\n(a_n - \\lfloor b_n/2 \\rfloor - 3, 3 \\lfloor (b_n+1)/2 \\rfloor + 6) & \\text{if } a_n > \\lfloor b_n/2 \\rfloor \\cr\n(3 a_n + 5, b_n - 2 a_n) & \\text{if } a_n \\le \\lfloor b_n/2 \\rfloor\n\\end{cases}$$\n\nfor all positive integers $n$.  Does there exist a positive integer $i$ such that\n$a_i = \\lfloor b_i/2 \\rfloor + 1$?\n\n[BMO#8](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#8._1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA_(bbch)) is equivalent to asking whether the 6-state Turing machine\n[`1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA`](https://wiki.bbchallenge.org/wiki/1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA) halts or not.\n\nThere is presently no consensus on whether the machine halts or not, hence the problem is formulated\nusing `answer(sorry) ↔`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.BeaverMathOlympiad","statement":"True ↔\n  ∀ (a b : ℕ → ℤ),\n    a 0 = 10 →\n      (∀ (n : ℕ), a (n + 1) = if b n / 2 < a n then a n - b n / 2 - 3 else 3 * a n + 5) →\n        b 0 = 12 →\n          (∀ (n : ℕ), b (n + 1) = if b n / 2 < a n then 3 * ((b n + 1) / 2) + 6 else b n - 2 * a n) →\n            ∃ i, a i = b i / 2 + 1","subjects":["5","11","68"],"theorem":"BeaverMathOlympiad.beaver_math_olympiad_problem_8"},{"answerKinds":[],"category":"research open","docstring":"[BMO#2](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#2._Hydra_and_Antihydra)\n\nAntihydra is a sequence starting at 8, and iterating the function\n$$H(n) = \\left\\lfloor \\frac {3n}2 \\right\\rfloor.$$\nThe conjecture states that the cumulative number of odd values in this sequence\nis never more than twice the cumulative number of even values. It is a relatively new open problem\nwith, so it might be solvable, although seems quite hard because of its Collatz-like flavor.\nThe underlying Collatz-like map has been studied independently in the past,\nsee doi:[10.1017/S0017089508004655](https://doi.org/10.1017/S0017089508004655) (Corollary 4).\n\nIt is equivalent to non-termination of the [`1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA`](https://wiki.bbchallenge.org/wiki/Antihydra) 6-state Turing machine (from all-0 tape). Note that the conjecture\nthat the machine does not halt is based on [a probabilistic argument](https://wiki.bbchallenge.org/wiki/Antihydra#Trajectory).\n\nThis machine and its mathematical reformulations were found by [bbchallenge.org](https://bbchallenge.org)\ncontributors mxdys and Rachel Hunter on June 28th 2024.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.BeaverMathOlympiad","statement":"∀ (a : ℕ → ℕ) (b : ℕ → ℤ),\n  a 0 = 8 →\n    (∀ (n : ℕ), a (n + 1) = 3 * a n / 2) →\n      b 0 = 0 → (∀ (n : ℕ), b (n + 1) = if a n % 2 = 0 then b n + 2 else b n - 1) → ∀ (n : ℕ), b n ≥ 0","subjects":["5","11","68"],"theorem":"BeaverMathOlympiad.beaver_math_olympiad_problem_2_antihydra"},{"answerKinds":[],"category":"research solved","docstring":"[BMO#4](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#4._1RB3RB---1LB0LA_2LA4RA3LA4RB1LB_(bbch))\n\nBonnie the beaver was bored, so she tried to construct a sequence of integers $\\{a_n\\}_{n \\ge 0}$.\nShe first defined $a_0=2$, then defined $a_{n+1}$ depending on $a_n$ and $n$\nusing the following rules:\n\n* If $a_n \\equiv 0\\text{ (mod 3)}$, then $a_{n+1}=\\frac{a_n}{3}+2^n+1$.\n* If $a_n \\equiv 2\\text{ (mod 3)}$, then $a_{n+1}=\\frac{a_n-2}{3}+2^n-1$.\n\nWith these two rules alone, Bonnie calculates the first few terms in the sequence: $2, 0, 3, 6, 11,\n18, 39, 78, 155, 306, \\dots$. At this point, Bonnie plans to continue writing terms until a term\nbecomes $1\\text{ (mod 3)}$. If Bonnie sticks to her plan, will she ever finish?\n\n[BMO#4](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#4._1RB3RB---1LB0LA_2LA4RA3LA4RB1LB_(bbch))\nis equivalent to the non-termination of 2-state 5-symbol Turing machine\n[`1RB3RB---1LB0LA_2LA4RA3LA4RB1LB`](https://wiki.bbchallenge.org/wiki/1RB3RB---1LB0LA_2LA4RA3LA4RB1LB) (from all-0 tape).\n\nThe machine was informally proven not to halt [bbchallenge.org](https://bbchallenge.org)\ncontributor Daniel Yuan on July 19th 2024; see [sketched proof](https://wiki.bbchallenge.org/wiki/1RB3RB---1LB0LA_2LA4RA3LA4RB1LB) and [Discord discussion](https://discord.com/channels/960643023006490684/960643023530762343/1263666591900631210).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.BeaverMathOlympiad","statement":"∀ (a : ℕ → ℕ),\n  a 0 = 2 →\n    (∀ (n : ℕ), a (n + 1) = if a n % 3 = 0 then a n / 3 + 2 ^ n + 1 else (a n - 2) / 3 + 2 ^ n - 1) → ¬∃ n, a n % 3 = 1","subjects":["5","11","68"],"theorem":"BeaverMathOlympiad.beaver_math_olympiad_problem_4"},{"answerKinds":[],"category":"research solved","docstring":"**Schur's Theorem (1924):**\nLet `f_n(x) = ∑_{j=0}^n x^j/j!` be the `n`-th truncated\nexponential polynomial over `ℚ`. Then for `n ≥ 2`:\n\n- If `n ≡ 0 (mod 4)`, the Galois group of `f_n` is isomorphic to the alternating group `A_n`\n- If `n ≢ 0 (mod 4)`, the Galois group of `f_n` is isomorphic to the symmetric group `S_n`\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.SchurTruncatedExponential","statement":"∀ n ≥ 2,\n  if n % 4 = 0 then Nonempty ((SchurTruncatedExponential.truncatedExp n).Gal ≃* ↥(alternatingGroup (Fin n)))\n  else Nonempty ((SchurTruncatedExponential.truncatedExp n).Gal ≃* Equiv.Perm (Fin n))","subjects":["12"],"theorem":"SchurTruncatedExponential.schur_truncatedExp_galoisGroup_equiv"},{"answerKinds":[],"category":"research open","docstring":"If $n \\ge 2$, every convex set in $\\mathbb R^{n + 1}$ has $\\mathrm{VC}_n$ dimension at most 1.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Other.VCDimConvex","statement":"∀ {n : ℕ}, 2 ≤ n → ∀ {C : Set (Fin (n + 1) → ℝ)}, Convex ℝ C → HasAddVCNDimAtMost C n 1","subjects":["5","52"],"theorem":"VCDimConvex.hasAddVCNDimAtMost_n_one_of_convex_rn_add_one"},{"answerKinds":[],"category":"research solved","docstring":"There exists a set in $\\mathbb R^3$ shattering an infinite set. ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.VCDimConvex","statement":"∃ A C, A.Infinite ∧ Convex ℝ C ∧ Shatters {x | ∃ t, t +ᵥ C = x} A","subjects":["5","52"],"theorem":"VCDimConvex.exists_infinite_convex_r3_shatters"},{"answerKinds":[],"category":"research solved","docstring":"Every convex set in $\\mathbb R^2$ has VC dimension at most 3. ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.VCDimConvex","statement":"∀ {C : Set (EuclideanSpace ℝ (Fin 2))}, Convex ℝ C → HasAddVCDimAtMost C 3","subjects":["5","52"],"theorem":"VCDimConvex.hasAddVCDimAtMost_three_of_convex_r2"},{"answerKinds":[],"category":"research solved","docstring":"Not every convex set in $\\mathbb R^3$ has\n$\\mathrm{VC}_2$ dimension at most 1. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/vcdim-convex-counterexample/blob/ad7ffff1514843c886633f5408c6456dfb2e2a49/formal-conjectures-v4.27.0/VCDimConvexCounterexample.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Other.VCDimConvex","statement":"¬∀ {C : Set (EuclideanSpace ℝ (Fin 3))}, Convex ℝ C → HasAddVCNDimAtMost C 2 1","subjects":["5","52"],"theorem":"VCDimConvex.hasAddVCNDimAtMost_two_one_of_convex_r3"},{"answerKinds":[],"category":"research solved","docstring":"There exists a set of infinite $\\mathrm{VC}_n$ dimension in $\\mathbb R^{n + 2}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.VCDimConvex","statement":"∀ (n : ℕ), ∃ C, Convex ℝ C ∧ ∀ (d : ℕ), ¬HasAddVCNDimAtMost C n d","subjects":["5","52"],"theorem":"VCDimConvex.exists_convex_rn_add_two_vc_n_forall_not_hasAddVCNDimAtMost"},{"answerKinds":[],"category":"research open","docstring":"Every convex set in $\\mathbb R^3$ has\n$\\mathrm{VC}_2$ dimension at most 2. ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.VCDimConvex","statement":"∀ {C : Set (EuclideanSpace ℝ (Fin 3))}, Convex ℝ C → HasAddVCNDimAtMost C 2 2","subjects":["5","52"],"theorem":"VCDimConvex.hasAddVCNDimAtMost_two_two_of_convex_r3"},{"answerKinds":[],"category":"research open","docstring":"For every $n \\ge 1$ there exists some $d$ such that every convex set in $\\mathbb R^{n + 1}$ has\n$\\mathrm{VC}_n$ dimension at most $d$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.VCDimConvex","statement":"∀ (n : ℕ), 1 ≤ n → ∃ d, ∀ (C : Set (Fin (n + 1) → ℝ)), Convex ℝ C → HasAddVCNDimAtMost C n d","subjects":["5","52"],"theorem":"VCDimConvex.exists_hasAddVCNDimAtMost_n_of_convex_rn_add_one"},{"answerKinds":[],"category":"research open","docstring":"**Rule 30 Prize, Problem 1 (non-periodicity).** The center column of Rule 30 is not\neventually periodic: there is no positive period $p$ and threshold $N$ past which the column\nrepeats with period $p$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.Rule30","statement":"True ↔ ¬∃ p, 0 < p ∧ ∃ N, ∀ (t : ℕ), N ≤ t → Rule30.centerColumn (t + p) = Rule30.centerColumn t","subjects":["37","68"],"theorem":"Rule30.centerColumn_not_eventually_periodic"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Other.Rule30","statement":"∀ (t : ℕ), Rule30.state (t + 1) = Rule30.step (Rule30.state t)","subjects":["37","68"],"theorem":"Rule30.state_succ"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Other.Rule30","statement":"∀ (i : ℤ), Rule30.state 0 i = decide (i = 0)","subjects":["37","68"],"theorem":"Rule30.state_zero"},{"answerKinds":[],"category":"research open","docstring":"**Rule 30 Prize, Problem 2 (equal frequency).** Each color occurs on average equally often\nin the center column: the set of times at which it is black has natural density $1/2$. This is\nWolfram's phrasing that the discrete limit of $\\mathrm{Total}[c[t]]/t$ as $t \\to \\infty$ is\n$1/2$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Other.Rule30","statement":"True ↔ {t | Rule30.centerColumn t = true}.HasDensity (1 / 2)","subjects":["37","68"],"theorem":"Rule30.centerColumn_frequency_half"},{"answerKinds":[],"category":"test","docstring":"Sanity check: the first center-column values reproduce the known Rule 30 center column\n([OEIS A051023](https://oeis.org/A051023)). Unlike `state_zero`/`state_succ`, this exercises the\ncomposed evolution, guarding against a wrong seed, a swapped `xor`/`||`, or an off-by-one\nneighbour index. ","hasSorryFreeProof":true,"module":"FormalConjectures.Other.Rule30","statement":"List.map Rule30.centerColumn (List.range 8) = [true, true, false, true, true, true, false, false]","subjects":["37","68"],"theorem":"Rule30.centerColumn_prefix"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ {m n : ℕ},\n  Mathoverflow75792.Reachable m n → (∀ n' < n, ¬Mathoverflow75792.Reachable m n') → Mathoverflow75792.complexity m = n","subjects":["11"],"theorem":"Mathoverflow75792.Reachable.complexity_eq"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ (m n : ℕ), 0 < m → 0 < n → Mathoverflow75792.Reachable (m ^ n) (m * n)","subjects":["11"],"theorem":"Mathoverflow75792.Reachable.pow"},{"answerKinds":[],"category":"research open","docstring":"Is `2n` the complexity of `2^n` for `0 < n`? ","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"True ↔ ∀ (n : ℕ), 0 < n → Mathoverflow75792.complexity (2 ^ n) = 2 * n","subjects":["11"],"theorem":"Mathoverflow75792.complexity_two_pow"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"Mathoverflow75792.complexity 1 = 1","subjects":["11"],"theorem":"Mathoverflow75792.complexity_one"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ {m n : ℕ}, Mathoverflow75792.Reachable m n → Mathoverflow75792.complexity m ≤ n","subjects":["11"],"theorem":"Mathoverflow75792.Reachable.complexity_le"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ (m n : ℕ+), Mathoverflow75792.Reachable (↑m ^ ↑n) (↑m * ↑n)","subjects":["11"],"theorem":"Mathoverflow75792.Reachable.pow'"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ (m n : ℕ),\n  2 ≤ m →\n    (Mathoverflow75792.Reachable m n ↔\n      ∃ m₁,\n        ∃ (_ : m₁ < m),\n          ∃ m₂,\n            ∃ (_ : m₂ < m),\n              ∃ n₁,\n                ∃ (_ : n₁ < n),\n                  ∃ n₂,\n                    ∃ (_ : n₂ < n),\n                      n₁ + n₂ = n ∧\n                        Mathoverflow75792.Reachable m₁ n₁ ∧\n                          Mathoverflow75792.Reachable m₂ n₂ ∧ (m₁ + m₂ = m ∨ m₁ * m₂ = m))","subjects":["11"],"theorem":"Mathoverflow75792.reachable_iff_of_two_le"},{"answerKinds":[],"category":"test","docstring":"`5^6 = 15625 = 1 + 2^3 * 3^2 * (1 + 2^3 * 3^3)`! ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"Mathoverflow75792.Reachable (5 ^ 6) 29","subjects":["11"],"theorem":"Mathoverflow75792.Reachable.five_pow_six"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"Mathoverflow75792.complexity 2 = 2","subjects":["11"],"theorem":"Mathoverflow75792.complexity_two"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"Mathoverflow75792.complexity 0 = 0","subjects":["11"],"theorem":"Mathoverflow75792.complexity_zero"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ (n : ℕ), 0 < n → Mathoverflow75792.Reachable n n","subjects":["11"],"theorem":"Mathoverflow75792.Reachable.self"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ {n : ℕ}, 0 < n → Mathoverflow75792.Reachable n (Mathoverflow75792.complexity n)","subjects":["11"],"subsets":["FC100SolvedSet1"],"theorem":"Mathoverflow75792.Reachable.complexity"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ (n : ℕ), ¬Mathoverflow75792.Reachable 0 n","subjects":["11"],"theorem":"Mathoverflow75792.not_reachable_zero_fst"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ {m n : ℕ}, Mathoverflow75792.Reachable m n → ∃ m' n', m' + 1 = m ∧ n' + 1 = n","subjects":["11"],"theorem":"Mathoverflow75792.Reachable.dec"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is `5n` the complexity of `5^n` for `0 < n`? Answer: No. ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"False ↔ ∀ (n : ℕ), 0 < n → Mathoverflow75792.complexity (5 ^ n) = 5 * n","subjects":["11"],"theorem":"Mathoverflow75792.complexity_five_pow"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ {m n₁ n₂ : ℕ}, n₁ ≤ n₂ → Mathoverflow75792.Reachable m n₁ → Mathoverflow75792.Reachable m n₂","subjects":["11"],"theorem":"Mathoverflow75792.Reachable.le"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is `3n` the complexity of `3^n` for `0 < n`? Answer: Yes, by John Selfridge.\n\nReference: https://arxiv.org/abs/1207.4841\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"True ↔ ∀ (n : ℕ), 0 < n → Mathoverflow75792.complexity (3 ^ n) = 3 * n","subjects":["11"],"theorem":"Mathoverflow75792.complexity_three_pow"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«75792»","statement":"∀ (m : ℕ), ¬Mathoverflow75792.Reachable m 0","subjects":["11"],"theorem":"Mathoverflow75792.not_reachable_zero_snd"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f : \\mathbb R^n \\to \\mathbb R,  n \\geq 2$ be a $C^1$ function. Is it true that\n$$\\sup_{x \\in \\mathbb R^n}f(x) = \\sup_{x\\in \\mathbb R^n} f(x+\\nabla f(x))$$?\n\nAnswer: No. A counterexample in $\\mathbb R^2$ is recorded in the linked formal proof.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/commit/fc20c0b55eab6fc26e2bb5b24fb3005303a0910b"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«347178»","statement":"False ↔\n  ∀ n ≥ 2,\n    ∀ (f : EuclideanSpace ℝ (Fin n) → ℝ),\n      ContDiff ℝ 1 f →\n        (BddAbove (Set.range f) ↔ BddAbove (Set.range fun x => f (x + gradient f x))) ∧\n          ⨆ x, ↑(f x) = ⨆ x, ↑(f (x + gradient f x))","subjects":["26"],"theorem":"Mathoverflow347178.mathoverflow_347178"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Let $f : \\mathbb R^n \\to \\mathbb R,  n \\geq 2$ be a $C^1$ function. Is the boundedness of\n$\\sup_{x \\in \\mathbb R^n}f(x)$ and $\\sup_{x\\in \\mathbb R^n} f(x+\\nabla f(x))$ equivalent?\n\nAnswer: No. The same counterexample is recorded in the linked formal proof.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/commit/fc20c0b55eab6fc26e2bb5b24fb3005303a0910b"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«347178»","statement":"False ↔\n  ∀ n ≥ 2,\n    ∀ (f : EuclideanSpace ℝ (Fin n) → ℝ),\n      ContDiff ℝ 1 f → (BddAbove (Set.range f) ↔ BddAbove (Set.range fun x => f (x + gradient f x)))","subjects":["26"],"theorem":"Mathoverflow347178.mathoverflow_347178.variants.bounded_iff"},{"answerKinds":[],"category":"research open","docstring":"Let $f : \\mathbb R^n \\to \\mathbb R,  n \\geq 2$ be a $C^1$ function. Does the equality\n$$\\sup_{x \\in \\mathbb R^n}f(x) = \\sup_{x\\in \\mathbb R^n} f(x+\\nabla f(x))$$\nhold when both suprema are finite?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«347178»","statement":"True ↔\n  ∀ n ≥ 2,\n    ∀ (f : EuclideanSpace ℝ (Fin n) → ℝ),\n      ContDiff ℝ 1 f →\n        BddAbove (Set.range f) →\n          BddAbove (Set.range fun x => f (x + gradient f x)) → ⨆ x, f x = ⨆ x, f (x + gradient f x)","subjects":["26"],"theorem":"Mathoverflow347178.mathoverflow_347178.variants.bounded_only"},{"answerKinds":[],"category":"test","docstring":"The Lebesgue measure of the a rectangle `r` is `r.width * r.height` ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"∀ (r : Mathoverflow34145.Rectangle), Mathoverflow34145.lbMeasure r.toSet = ENNReal.ofReal |r.width * r.height|","subjects":["51"],"theorem":"Mathoverflow34145.lbMeasure_rectangle_toSet"},{"answerKinds":[],"category":"test","docstring":"`lbMeasure` is invariant under `rigidMotion start θ`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"∀ (start : ℝ × ℝ) (θ : Real.Angle) (s : Set (ℝ × ℝ)),\n  Mathoverflow34145.lbMeasure (Mathoverflow34145.rigidMotion start θ '' s) = Mathoverflow34145.lbMeasure s","subjects":["51"],"theorem":"Mathoverflow34145.lbMeasure_rigidMotion"},{"answerKinds":[],"category":"research solved","docstring":"It is known that packing the rectangles into a square of side length `133/132` is possible.\n\nReference: https://www.sciencedirect.com/science/article/pii/0097316594901163\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"∃ c,\n  (∀ (n : ℕ), (c.rect n).toSet ⊆ { width := 133 / 132, height := 133 / 132, start := (0, 0), rotation := 0 }.toSet) ∧\n    c.IsPacking","subjects":["51"],"theorem":"Mathoverflow34145.rectangles_pack_square_133_div_132"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"Mathoverflow34145.rigidMotion (√2, √11) ↑(2 * Real.pi / 3) (√5, √7) = (√2 - √5 / 2 - √21 / 2, -√7 / 2 + √11 + √15 / 2)","subjects":["51"],"theorem":"Mathoverflow34145.rigidMotion_test"},{"answerKinds":[],"category":"research open","docstring":"Equivalently, can a unit square be packed with rectangles of width `1 / (n + 1)` and height\n`1 / (n + 2)`? ","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"True ↔ ∃ c, (∀ (n : ℕ), (c.rect n).toSet ⊆ Mathoverflow34145.unitSquare) ∧ c.IsPacking","subjects":["51"],"theorem":"Mathoverflow34145.rectangles_pack_unit_square"},{"answerKinds":[],"category":"research solved","docstring":"It is known that packing the rectangles into a square of side length `501/500` is possible.\n\nReference: https://www.sciencedirect.com/science/article/pii/S0167506008706009\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"∃ c,\n  (∀ (n : ℕ), (c.rect n).toSet ⊆ { width := 501 / 500, height := 501 / 500, start := (0, 0), rotation := 0 }.toSet) ∧\n    c.IsPacking","subjects":["51"],"theorem":"Mathoverflow34145.rectangles_pack_square_501_div_500"},{"answerKinds":[],"category":"test","docstring":"The Lebesgue measure of the unit square is `1`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"Mathoverflow34145.lbMeasure Mathoverflow34145.unitSquare = 1","subjects":["51"],"theorem":"Mathoverflow34145.lbMeasure_unitSquare"},{"answerKinds":[],"category":"test","docstring":"`lbMeasure` is scaled by `scale`. ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"∀ (x y : ℝ) (s : Set (ℝ × ℝ)),\n  Mathoverflow34145.lbMeasure (Mathoverflow34145.scale x y '' s) =\n    ENNReal.ofReal |x * y| * Mathoverflow34145.lbMeasure s","subjects":["51"],"theorem":"Mathoverflow34145.lbMeasure_scale"},{"answerKinds":[],"category":"test","docstring":"The areas of the required rectangles sum to 1. ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"∑' (n : ℕ), 1 / (↑n + 1) * (1 / (↑n + 2)) = 1","subjects":["51"],"theorem":"Mathoverflow34145.tsum_area_eq_one"},{"answerKinds":[],"category":"research open","docstring":"Can a unit square be covered by rectangles of width `1 / (n + 1)` and height `1 / (n + 2)`? ","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«34145»","statement":"True ↔ ∃ c, ∀ p ∈ Mathoverflow34145.unitSquare, ∃ n, p ∈ (c.rect n).toSet","subjects":["51"],"theorem":"Mathoverflow34145.rectangles_cover_unit_square"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a category that is pretriangulated but not triangulated? ","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«31809»","statement":"True ↔\n  ∀ (C : Type u_1) [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Preadditive C]\n    [inst_2 : CategoryTheory.Limits.HasZeroObject C] [inst_3 : CategoryTheory.HasShift C ℤ]\n    [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [inst_5 : CategoryTheory.Pretriangulated C],\n    CategoryTheory.IsTriangulated C","subjects":["18"],"theorem":"Mathoverflow31809.mathoverflow_31809"},{"answerKinds":[],"category":"research open","docstring":"Is there any polynomial $f(x, y) \\in \\mathbb{Q}[x, y]$ such that\n$f : \\mathbb{Q} \\times \\mathbb{Q} \\rightarrow \\mathbb{Q}$ is a bijection?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«21003»","statement":"True ↔ ∃ f, Function.Bijective fun x => (MvPolynomial.eval x) f","subjects":["12"],"theorem":"Mathoverflow21003.mathoverflow_21003"},{"answerKinds":[],"category":"research solved","docstring":"There exists a semiring with a unique left maximal ideal but more than one right maximal ideals. ","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«486451»","statement":"∃ R x, (∃! I, I.IsMaximal) ∧ ∃ I J, I.IsMaximal ∧ J.IsMaximal ∧ I ≠ J","subjects":["16"],"theorem":"Mathoverflow486451.exists_semiring_unique_left_maximal_not_unique_right_maximal"},{"answerKinds":["Prop"],"category":"research solved","docstring":"There exists a semiring with a unique left maximal ideal and a unique right maximal ideal\nwhich are not the same as sets.\n\nThis has been shown by Goran Žužić and Moritz Firsching using an experimental pipeline:\nAn example is the monoid algebra of the monoid of maps from $\\mathbb{N}$ to $\\mathbb{N}$\nover $\\mathbb{N}$.\n ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/f7502b9ed3e32d193ab8fee53d2e28f7d67f2dc3/FormalConjectures/Mathoverflow/486451.lean#L333"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«486451»","statement":"True ↔\n  ∃ R x,\n    ∃ (hI : ∃! I, I.IsMaximal) (hJ : ∃! J, J.IsMaximal), ↑(Exists.choose hI) ≠ MulOpposite.op ⁻¹' ↑(Exists.choose hJ)","subjects":["16"],"theorem":"Mathoverflow486451.exists_semiring_unique_left_right_maximal_ne"},{"answerKinds":[],"category":"test","docstring":"By a standard result, every continuous map is connected\n","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«235893»","statement":"∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},\n  Continuous f → Mathoverflow235893.IsConnectedMap f","subjects":["54"],"theorem":"Mathoverflow235893.Continuous.isConnectedMap"},{"answerKinds":[],"category":"test","docstring":"A set in $\\mathbb{R}$ is connected if and only if it is order-connected and non-empty.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«235893»","statement":"∀ {s : Set ℝ}, IsConnected s ↔ s.OrdConnected ∧ s.Nonempty","subjects":["54"],"theorem":"Mathoverflow235893.isConnected_iff_ordConnected_and_nonempty"},{"answerKinds":[],"category":"test","docstring":"If $f : \\mathbb{R}^1 \\to \\mathbb{R}^1$ is a connected bijection, then its inverse is also a connected bijection.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«235893»","statement":"∀ (f : EuclideanSpace ℝ (Fin 1) ≃ EuclideanSpace ℝ (Fin 1)),\n  Mathoverflow235893.IsConnectedMap ⇑f → Mathoverflow235893.IsConnectedMap ⇑f.symm","subjects":["54"],"theorem":"Mathoverflow235893.isConnectedMap_symm_of_E1"},{"answerKinds":[],"category":"research open","docstring":"Assume for $n>1$, $f:\\mathbb{R}^n\\to\\mathbb{R}^n$ is a bijection, where $\\mathbb{R}^n$ is equipped\nwith the standard topology. Does the connectedness of (the induced power set map) $f$ imply\nthat of $f^{-1}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«235893»","statement":"True ↔\n  ∀ n > 1,\n    ∀ (f : EuclideanSpace ℝ (Fin n) ≃ EuclideanSpace ℝ (Fin n)),\n      Mathoverflow235893.IsConnectedMap ⇑f → Mathoverflow235893.IsConnectedMap ⇑f.symm","subjects":["26","54"],"theorem":"Mathoverflow235893.mathoverflow_235893"},{"answerKinds":[],"category":"test","docstring":"The composition of two connected maps is a connected map.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«235893»","statement":"∀ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]\n  [inst_2 : TopologicalSpace Z] {f : X → Y} {g : Y → Z},\n  Mathoverflow235893.IsConnectedMap f → Mathoverflow235893.IsConnectedMap g → Mathoverflow235893.IsConnectedMap (g ∘ f)","subjects":["54"],"theorem":"Mathoverflow235893.isConnectedMap_comp"},{"answerKinds":[],"category":"test","docstring":"A homeomorphism is a connected map.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«235893»","statement":"∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (h : X ≃ₜ Y),\n  Mathoverflow235893.IsConnectedMap ⇑h","subjects":["54"],"theorem":"Mathoverflow235893.isConnectedMap_homeomorph"},{"answerKinds":[],"category":"research solved","docstring":"There exists a connected bijection ℝ → ℝ^2 where the inverse is not connected,\nproven in [mathoverflow/260589](https://mathoverflow.net/questions/260589) by user\n[Gro-Tsen](https://mathoverflow.net/users/17064/gro-tsen).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«235893»","statement":"∃ f, Mathoverflow235893.IsConnectedMap ⇑f ∧ ¬Mathoverflow235893.IsConnectedMap ⇑f.symm","subjects":["26","54"],"theorem":"Mathoverflow235893.mathoverflow_260589"},{"answerKinds":[],"category":"test","docstring":"If $f : \\mathbb{R} \\to \\mathbb{R}$ is a connected bijection, then its inverse is also a connected bijection.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«235893»","statement":"∀ (f : ℝ ≃ ℝ), Mathoverflow235893.IsConnectedMap ⇑f → Mathoverflow235893.IsConnectedMap ⇑f.symm","subjects":["54"],"theorem":"Mathoverflow235893.isConnectedMap_symm_of_R"},{"answerKinds":[],"category":"research solved","docstring":"Weaker version proven by Kahn–Kalai: the same conclusion holds when $1000 \\log n$ is replaced by\n$C_\\varepsilon \\, n^\\varepsilon$ for every fixed $\\varepsilon > 0$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ ε > 0,\n  ∃ C > 0,\n    ∀ (n : ℕ),\n      2 ≤ n →\n        ∀ (F : Finset (Finset (Fin n))),\n          Mathoverflow10799.IsMonotoneIncreasing F →\n            ∀ (s t : ℝ),\n              0 < s → s ≤ t → t < 1 → t / s > C * ↑n ^ ε → ∃ p, s ≤ p ∧ p ≤ t ∧ Mathoverflow10799.IsOptimal p F","subjects":["5","60"],"theorem":"Mathoverflow10799.mathoverflow_10799.variants.weak_kahn_kalai"},{"answerKinds":[],"category":"test","docstring":"The boundary count is zero for the full family (every set is in $\\mathcal F$). ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ (n : ℕ) (S : Finset (Fin n)), Mathoverflow10799.boundaryCount n Finset.univ S = 0","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Mathoverflow10799.boundaryCount_univ"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Problem: For every monotone increasing family $F$, given an interval $[s,t]$ of real numbers so\nthat $t/s > 1000 \\log n$ we have some $p$ in the interval $[s,t]$ so that $F$ is optimal with\nrespect to $\\mu_p$.\n\nThis was a \"missing lemma\" in the work of Kahn and Kalai on threshold behavior of monotone\nproperties. The related [Kahn–Kalai conjecture](https://arxiv.org/abs/math/0603218) was\n[settled by Park and Pham](https://arxiv.org/abs/2203.17207).\n\n**This conjecture is false** without the additional assumption $\\mu_t(F) = 1/2$.\nA counterexample was found by Shlomo Perles (April 7, 2026).\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/408f53dc0856c0882a5e77acd24fc83b978f0bc9/FormalConjectures/Mathoverflow/10799.lean#L252"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"False ↔\n  ∀ (n : ℕ),\n    2 ≤ n →\n      ∀ (F : Finset (Finset (Fin n))),\n        Mathoverflow10799.IsMonotoneIncreasing F →\n          ∀ (s t : ℝ),\n            0 < s → s ≤ t → t < 1 → t / s > 1000 * Real.log ↑n → ∃ p, s ≤ p ∧ p ≤ t ∧ Mathoverflow10799.IsOptimal p F","subjects":["5","60"],"theorem":"Mathoverflow10799.mathoverflow_10799"},{"answerKinds":[],"category":"test","docstring":"The edge boundary is zero for the empty family. ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ (n : ℕ) (p : ℝ), Mathoverflow10799.edgeBoundary n p ∅ = 0","subjects":["5"],"theorem":"Mathoverflow10799.edgeBoundary_empty"},{"answerKinds":[],"category":"test","docstring":"The edge boundary is zero for the full family. ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ (n : ℕ) (p : ℝ), Mathoverflow10799.edgeBoundary n p Finset.univ = 0","subjects":["5"],"theorem":"Mathoverflow10799.edgeBoundary_univ"},{"answerKinds":[],"category":"test","docstring":"The boundary count is zero for the empty family (no set is in $\\mathcal F$). ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ (n : ℕ) (S : Finset (Fin n)), Mathoverflow10799.boundaryCount n ∅ S = 0","subjects":["5"],"theorem":"Mathoverflow10799.boundaryCount_empty"},{"answerKinds":[],"category":"test","docstring":"For $p = 1/2$, the $p$-biased measure is the uniform distribution:\n$\\mu_{1/2}(S) = (1/2)^n$ for every $S \\subseteq [n]$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ {n : ℕ} (S : Finset (Fin n)), Mathoverflow10799.μ (1 / 2) S = (1 / 2) ^ n","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Mathoverflow10799.μ_half_eq_uniform"},{"answerKinds":[],"category":"test","docstring":"The measure of the full power set is $1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ (n : ℕ) (p : ℝ), Mathoverflow10799.μFamily p Finset.univ = 1","subjects":["5"],"theorem":"Mathoverflow10799.μFamily_univ"},{"answerKinds":[],"category":"test","docstring":"The $p$-biased measure is a probability distribution: it sums to $1$ over all subsets.\nThis is the binomial identity $(p + (1-p))^n = 1$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ (n : ℕ) (p : ℝ), ∑ S, Mathoverflow10799.μ p S = 1","subjects":["5"],"theorem":"Mathoverflow10799.μ_sum_eq_one"},{"answerKinds":[],"category":"textbook","docstring":"Now a famous isoperimetric relation asserts that\n(IR) $I^p(F) \\ge \\frac{1}{p} \\mu_p(F) \\log_p \\mu_p(F)$\nThis relation is true for every family $F$ and every $p$. It is especially famous and simple when\n$p=1/2$ and $\\mu_p(F)=1/2$. In this case, it says that given a set of half the vertices of the\ndiscrete cube $2^X$, the number of edges between $F$ and its complement is at least $2^{n-1}$.\n\n*Note on translation:* We use `Real.logb p m` to represent the logarithm base $p$ directly.\nThe factor of $p$ in the denominator (equivalent to $1/p$ in front) is consistent with the\ndefinition of `IsOptimal` used in the counterexample proof.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ (n : ℕ) (p : ℝ),\n  0 < p →\n    p < 1 →\n      ∀ (F : Finset (Finset (Fin n))),\n        have m := Mathoverflow10799.μFamily p F;\n        Mathoverflow10799.edgeBoundary n p F ≥ m * Real.logb p m / p","subjects":["5","60"],"theorem":"Mathoverflow10799.discrete_isoperimetric_inequality"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Conjecture 7 from Kahn–Kalai 2006: a fixed-`1000` variant of the original\nconjecture, with the additional assumption that $t$ is the critical probability for $F$,\nnamely $\\mu_t(F) = 1/2$.\n\nThis variant is false by a counterexample due to Sahar Diskin and Uri Kreitner;\nsee the [MathOverflow discussion](https://mathoverflow.net/questions/10799/optimal-monotone-families-for-the-discrete-isoperimetric-inequality)\nand the [counterexample note](https://gilkalai.wordpress.com/wp-content/uploads/2026/06/dual_tribes_more_readable.pdf).\nTheir construction was adapted to this exact Formal Conjectures statement and\n[formalized in Lean](https://github.com/KitaKen1/kahn-kalai-conjecture-7-counterexample)\nby Kenta Kitamura (KitaKen1).\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/kahn-kalai-conjecture-7-counterexample/blob/5446d2f/lean/MO10799CounterexampleFC.lean#L2597-L2604"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"False ↔\n  ∀ (n : ℕ),\n    2 ≤ n →\n      ∀ (F : Finset (Finset (Fin n))),\n        Mathoverflow10799.IsMonotoneIncreasing F →\n          ∀ (s t : ℝ),\n            0 < s →\n              s ≤ t →\n                t < 1 →\n                  Mathoverflow10799.μFamily t F = 1 / 2 →\n                    t / s > 1000 * Real.log ↑n → ∃ p, s ≤ p ∧ p ≤ t ∧ Mathoverflow10799.IsOptimal p F","subjects":["5","60"],"theorem":"Mathoverflow10799.mathoverflow_10799.variants.kahn_kalai_conjecture_7"},{"answerKinds":[],"category":"test","docstring":"Test lemma showing that `boundaryCount` is equivalent to counting subsets $T$\nthat differ from $S$ in exactly one element and exactly one of $S, T$ belongs to $F$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Mathoverflow.«10799»","statement":"∀ (n : ℕ) (F : Finset (Finset (Fin n))) (S : Finset (Fin n)),\n  Mathoverflow10799.boundaryCount n F S = {T | (symmDiff S T).card = 1 ∧ Xor (S ∈ F) (T ∈ F)}.card","subjects":["5"],"theorem":"Mathoverflow10799.boundaryCount_equiv"},{"answerKinds":[],"category":"research open","docstring":"Let $P(x), Q(x) ∈ ℝ[x]$ be two monic polynomials with non-negative coefficients.\nIf $R(x) = P(x)Q(x)$ is a $0,1$ polynomial (coefficients only from $\\{0,1\\}$), then $P(x)$ and $Q(x)$\nare also $0, 1$ polynomials.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«339137»","statement":"∀ (P Q R : Polynomial ℝ),\n  P.Monic →\n    Q.Monic →\n      (∀ c ∈ P.coeffs, 0 ≤ c) →\n        (∀ c ∈ Q.coeffs, 0 ≤ c) →\n          R = P * Q → Mathoverflow339137.IsZeroOne R → Mathoverflow339137.IsZeroOne P ∧ Mathoverflow339137.IsZeroOne Q","subjects":["12"],"theorem":"Mathoverflow339137.mathoverflow_339137"},{"answerKinds":[],"category":"textbook","docstring":"Green's Open Problem 28 is the probabilistic reformulation of Mathoverflow 339137.\n\nSuppose that $X, Y$ are two finitely-supported independent random variables taking integer values,\nand such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly\ndistributed on their ranges?\n\nMathematically, this equivalence is established via Probability Generating Functions (PGFs),\nshifting the support to $\\mathbb{N}$, and appropriately scaling the coefficients.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«339137»","statement":"(∀ (P Q R : Polynomial ℝ),\n    P.Monic →\n      Q.Monic →\n        (∀ c ∈ P.coeffs, 0 ≤ c) →\n          (∀ c ∈ Q.coeffs, 0 ≤ c) →\n            R = P * Q →\n              Mathoverflow339137.IsZeroOne R → Mathoverflow339137.IsZeroOne P ∧ Mathoverflow339137.IsZeroOne Q) ↔\n  (True ↔\n    ∀ (X Y : PMF ℤ),\n      X.support.Finite ∧ Y.support.Finite ∧ Green28.IsUniformOnSupport (Green28.indepSum X Y) →\n        Green28.IsUniformOnSupport X ∧ Green28.IsUniformOnSupport Y)","subjects":["60"],"theorem":"Mathoverflow339137.mathoverflow_339137_probabilistic"},{"answerKinds":[],"category":"textbook","docstring":"If $2^x$, $3^x$ and $5^x$ are integers, then $x$ must be an integer.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«17560»","statement":"∀ {x : ℝ}, (∃ m, 2 ^ x = ↑m) → (∃ m, 3 ^ x = ↑m) → (∃ m, 5 ^ x = ↑m) → ∃ m, x = ↑m","subjects":["11","13"],"theorem":"Mathoverflow17560.mathoverflow_17560.variants.with_5"},{"answerKinds":[],"category":"research open","docstring":"If $2^x$ and $3^x$ are integers, then $x$ must be an integer.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«17560»","statement":"∀ {x : ℝ}, (∃ m, 2 ^ x = ↑m) → (∃ m, 3 ^ x = ↑m) → ∃ m, x = ↑m","subjects":["11","13"],"theorem":"Mathoverflow17560.mathoverflow_17560"},{"answerKinds":[],"category":"textbook","docstring":"If for each natural number $n$ the number $n^x$ is an integer then $x$ must also be an integer.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«17560»","statement":"∀ {x : ℝ}, (∀ (n : ℕ), ∃ m, ↑n ^ x = ↑m) → ∃ m, x = ↑m","subjects":["11","13"],"theorem":"Mathoverflow17560.mathoverflow_17560.variants.all_nats"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does the 6-sphere admit a complex structure, i.e. an atlas of holomorphically compatible charts\nrelating it to `EuclideanSpace ℂ (Fin 3)`? This is known as the Hopf Problem.\n\nThe answer is yes, see [Al26].\nFormalisation of the proof by Boris Alexeev.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/plby/HopfProblem/blob/9ac8a456b526527837d7082ff775213ca8bc9809/Solution.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«1973»","statement":"True ↔ ∃ atlas, IsManifold (modelWithCornersSelf ℂ (EuclideanSpace ℂ (Fin 3))) 1 ↑(Mathoverflow1973.unitSphere 6)","subjects":["32"],"subsets":["FC100OpenSet1"],"theorem":"Mathoverflow1973.mathoverflow_1973"},{"answerKinds":[],"category":"research open","docstring":"The conjecture claims that $\\pi_n\\sim\\frac n{2\\ln(n)}$.\n\nIn other words, primes are distributed among the much sparser sequence $(S_n)_n$\nwith essentially the same density as in the positive integers, up to a factor of $2$.\n\n[MathOverflow 434111](https://mathoverflow.net/questions/434111/are-prime-numbers-among-sums-of-prime-numbers-distributed-as-frac-n2-lnn).\n\n[Me18] Meštrović, R., *Curious Conjectures on the Distribution of Primes\nAmong the Sums of the First `2n` Primes*, [arXiv:1804.04198](https://arxiv.org/abs/1804.04198)\n(2018).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«434111»","statement":"True ↔\n  Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(MathOverflow434111.piRestricted n)) fun n => ↑n / (2 * Real.log ↑n)","subjects":["11"],"theorem":"MathOverflow434111.restricted_prime_number_theorem"},{"answerKinds":[],"category":"research open","docstring":"Meštrović's original formulation [Me18, Conjecture 3.3]: the sequence of sums of the\nfirst $2m$ primes satisfies the Restricted Prime Number Theorem, $\\pi(m, (S_{2m})) \\sim \\frac{m}{\\ln m}$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«434111»","statement":"True ↔\n  Asymptotics.IsEquivalent Filter.atTop (fun m => ↑{k ∈ Finset.Icc 1 m | Nat.Prime (MathOverflow434111.S (2 * k))}.card)\n    fun m => ↑m / Real.log ↑m","subjects":["11"],"theorem":"MathOverflow434111.restricted_prime_number_theorem.variants.even_subsequence"},{"answerKinds":[],"category":"research solved","docstring":"There exists a proper ideal `I` in a (commutative) total ring `R` of fractions that is an invertible module.\nIf `I ⊊ R` is such an example, `I` must have infinite order in the Picard group.\nMoreover, `R` must not be Noetherian (otherwise it must be semi-local and therefore have trivial Picard group).\n\nThe linked formal proof gives the following explicit construction.\n\nLet `D = ℂ[X, Y] / (Y² - X³)` be the coordinate ring of the cuspidal cubic.\nLet `P = (X - 1, Y - 1)`, which is an invertible ideal.\nLet `M` be the direct sum of the evaluation fibres at cusp parameters `r ≠ 1`.\nForm the idealization `R = D ⋉ M`.\nThe desired ideal is the range of the multiplication map `R ⊗[D] P → R`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/mo507128-lean/commit/e9507429c01c4288089e4af1c92a03b7d1e17f74"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Mathoverflow.«507128»","statement":"∃ R x, ∃ (_ : IsFractionRing R R), ∃ I, I ≠ ⊤ ∧ Module.Invertible R ↥I","subjects":["13"],"theorem":"Mathoverflow507128.exists_isFractionRing_self_ideal_ne_top_invertible"},{"answerKinds":[],"category":"research open","docstring":"Do there exist simple pro-orderable groups?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Kourovka.«1_35c»","statement":"True ↔ ∃ G x, IsSimpleGroup G ∧ Kourovka.«1.35c».ProOrderable G","subjects":["20"],"theorem":"Kourovka.«1.35c».kourovka_1_35c"},{"answerKinds":[],"category":"research open","docstring":"Does there exist a Tarski monster group that admits a non-discrete Hausdorff\ngroup topology?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Kourovka.«1_74»","statement":"True ↔ ∃ G x x_1, Kourovka.«1.74».IsTarskiMonster G ∧ IsTopologicalGroup G ∧ T2Space G ∧ ¬DiscreteTopology G","subjects":["20","22"],"theorem":"Kourovka.«1.74».kourovka_1_74"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ and $H$ be finite groups of the same order with\n$\\sum_{g \\in G} \\phi(|g|) = \\sum_{h \\in H} \\phi(|h|)$,\nwhere $\\phi$ is the Euler totient function. Suppose that $G$ is simple. Is\n$H$ necessarily simple?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Kourovka.«19_25»","statement":"True ↔\n  ∀ (G H : Type) [inst : Group G] [inst_1 : Group H] [inst_2 : Fintype G] [inst_3 : Fintype H],\n    Fintype.card G = Fintype.card H →\n      ∑ g, (orderOf g).totient = ∑ h, (orderOf h).totient → IsSimpleGroup G → IsSimpleGroup H","subjects":["20"],"theorem":"Kourovka.«19.25».kourovka_19_25"},{"answerKinds":[],"category":"research open","docstring":"Is a group a nilgroup if it is the product of two normal nilsubgroups?\n\nSince $H$ and $K$ are normal, the product $HK$ coincides with the join\n$H \\sqcup K$, so \"$G$ is the product of $H$ and $K$\" is stated as\n$H \\sqcup K = G$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Kourovka.«1_40»","statement":"True ↔\n  ∀ (G : Type) [inst : Group G] (H K : Subgroup G),\n    H.Normal →\n      K.Normal →\n        Kourovka.«1.40».IsEngelGroup ↥H → Kourovka.«1.40».IsEngelGroup ↥K → H ⊔ K = ⊤ → Kourovka.«1.40».IsEngelGroup G","subjects":["20"],"theorem":"Kourovka.«1.40».kourovka_1_40"},{"answerKinds":[],"category":"research open","docstring":"Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$\nhave order at most $p^k$. Is it true that every abelian subgroup of $G$ has order at most\n$p^{2k}$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Kourovka.«20_76»","statement":"True ↔\n  ∀ (p : ℕ),\n    Nat.Prime p →\n      ∀ (G : Type) (x : Group G),\n        IsPGroup p G →\n          Finite G →\n            ∀ (k : ℕ),\n              (∀ (H : Subgroup G), H.Normal ∧ IsMulCommutative ↥H → Nat.card ↥H ≤ p ^ k) →\n                ∀ (H : Subgroup G), IsMulCommutative ↥H → Nat.card ↥H ≤ p ^ (2 * k)","subjects":["20"],"theorem":"Kourovka.«20.76».kourovka_20_76"},{"answerKinds":[],"category":"API","docstring":"A state built from amplitudes has those amplitudes as its coordinates. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (ψ : OpenQuantumProblem35.Config n d → ℂ) (x : OpenQuantumProblem35.Config n d),\n  (OpenQuantumProblem35.mkStateVector ψ).ofLp x = ψ x","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.mkStateVector_apply"},{"answerKinds":[],"category":"test","docstring":"The identity permutation leaves a configuration unchanged. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (x : OpenQuantumProblem35.Config n d), OpenQuantumProblem35.permuteConfig (Equiv.refl (Fin n)) x = x","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.permuteConfig_refl"},{"answerKinds":[],"category":"research solved","docstring":"The $3$-party GHZ state witnesses the existence of $\\mathrm{AME}(3,d)$ for every local dimension $d \\ge 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {d : ℕ}, 2 ≤ d → OpenQuantumProblem35.ExistsAME 3 d","subjects":["5","15","81","94"],"subsets":["FC100SolvedSet1"],"theorem":"OpenQuantumProblem35.ame_3_exists"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(8,10)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 8 10","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_8_10_open"},{"answerKinds":[],"category":"API","docstring":"The completion criterion gives a maximally mixed reduced state once the coefficient has the correct squared norm. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d m : ℕ} (hm : m ≤ n) (ψ : OpenQuantumProblem35.StateVector n d)\n  (completion : OpenQuantumProblem35.Config m d → OpenQuantumProblem35.Config (n - m) d) (coeff : ℂ),\n  (∀ (x : OpenQuantumProblem35.Config m d) (z : OpenQuantumProblem35.Config (n - m) d),\n      ψ.ofLp (OpenQuantumProblem35.combineFirst m hm x z) = if z = completion x then coeff else 0) →\n    Function.Injective completion →\n      coeff * star coeff = (↑(Fintype.card (OpenQuantumProblem35.Config m d)))⁻¹ →\n        OpenQuantumProblem35.HasMaximallyMixedFirstReduction m hm ψ","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.hasMaximallyMixedFirstReduction_of_completion"},{"answerKinds":[],"category":"test","docstring":"No absolutely maximally entangled state exists in local dimension $0$ once $n \\ge 1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n : ℕ}, 1 ≤ n → ¬OpenQuantumProblem35.ExistsAME n 0","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.not_existsAME_zero_dim"},{"answerKinds":[],"category":"research solved","docstring":"Source-backed benchmark statement: the Bell state witnesses the existence of an $\\mathrm{AME}(2,2)$ state. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"OpenQuantumProblem35.ExistsAME 2 2","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_2_2_exists"},{"answerKinds":[],"category":"API","docstring":"Combining and then restricting to the right block recovers the right input. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d m : ℕ} (hm : m ≤ n) (x : OpenQuantumProblem35.Config m d) (y : OpenQuantumProblem35.Config (n - m) d)\n  (i : Fin (n - m)), OpenQuantumProblem35.combineFirst m hm x y (OpenQuantumProblem35.rightIndex hm i) = y i","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.combineFirst_rightIndex"},{"answerKinds":[],"category":"API","docstring":"The matrix entries of the maximally mixed state are diagonal and equal to the inverse subsystem dimension. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {m d : ℕ} (x y : OpenQuantumProblem35.Config m d),\n  OpenQuantumProblem35.maximallyMixed m d x y =\n    if x = y then (↑(Fintype.card (OpenQuantumProblem35.Config m d)))⁻¹ else 0","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.maximallyMixed_apply"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(8,4)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 8 4","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_8_4_open"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(12,6)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 12 6","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_12_6_open"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(11,10)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 11 10","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_11_10_open"},{"answerKinds":[],"category":"research solved","docstring":"Source-backed benchmark statement: an $\\mathrm{AME}(4,3)$ state exists; see Helwig et al. (2012) and Goyeneche et al. (2015). ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/AllenGrahamHart/FormalConjectures-Bench/blob/8fb9479e9cbfde68d6990ed008b24c883cbd2750/formalizations/openquantum35_ame43/OpenQuantum35AME43Formalization.lean#L333"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"OpenQuantumProblem35.ExistsAME 4 3","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_4_3_exists"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(7,6)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 7 6","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_7_6_open"},{"answerKinds":[],"category":"API","docstring":"The standard $3$-party GHZ state is $\\mathrm{AME}(3,d)$ for every physical local dimension $d \\ge 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {d : ℕ}, 2 ≤ d → OpenQuantumProblem35.IsAME (OpenQuantumProblem35.ghzState d)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ghzState_isAME"},{"answerKinds":[],"category":"research solved","docstring":"Source-backed benchmark statement: an $\\mathrm{AME}(6,2)$ state exists. This is one of the four qubit cases $n=2,3,5,6$; see the OQP page and Scott (2004). ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"OpenQuantumProblem35.ExistsAME 6 2","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_6_2_exists"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(9,6)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 9 6","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_9_6_open"},{"answerKinds":[],"category":"research solved","docstring":"Source-backed benchmark statement: no $\\mathrm{AME}(4,2)$ state exists; see Higuchi--Sudbery (2000) and the OQP page. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"¬OpenQuantumProblem35.ExistsAME 4 2","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_4_2_not_exists"},{"answerKinds":[],"category":"API","docstring":"A configuration on a nonempty index type is constant iff it is equal to some constant configuration. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ},\n  1 ≤ n →\n    ∀ (x : OpenQuantumProblem35.Config n d),\n      OpenQuantumProblem35.IsConstantConfig x ↔ ∃ a, x = OpenQuantumProblem35.constantConfig a","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.isConstantConfig_iff_exists_constantConfig"},{"answerKinds":[],"category":"API","docstring":"The squared norm of the uniform coefficient is the inverse local dimension. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ (d : ℕ), ‖OpenQuantumProblem35.uniformCoeff d‖ ^ 2 = (↑d)⁻¹","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.uniformCoeff_norm_sq"},{"answerKinds":[],"category":"API","docstring":"A configuration obtained by combining one entry with a tail is constant iff the tail is the constant completion of that entry. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (hn : 1 ≤ n) (x : OpenQuantumProblem35.Config 1 d) (z : OpenQuantumProblem35.Config (n - 1) d),\n  OpenQuantumProblem35.IsConstantConfig (OpenQuantumProblem35.combineFirst 1 hn x z) ↔\n    z = OpenQuantumProblem35.constantCompletion x","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.isConstantConfig_combineFirst_one_iff"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(12,5)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 12 5","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_12_5_open"},{"answerKinds":[],"category":"research solved","docstring":"Source-backed benchmark statement: no $\\mathrm{AME}(7,2)$ state exists; see Huber--Gühne--Siewert (2017) and the OQP page. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"¬OpenQuantumProblem35.ExistsAME 7 2","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_7_2_not_exists"},{"answerKinds":[],"category":"API","docstring":"The diagonal state on a split configuration is nonzero exactly on the graph of the constant completion map. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (hn : 1 ≤ n) (x : OpenQuantumProblem35.Config 1 d) (z : OpenQuantumProblem35.Config (n - 1) d),\n  (OpenQuantumProblem35.diagonalState n d).ofLp (OpenQuantumProblem35.combineFirst 1 hn x z) =\n    if z = OpenQuantumProblem35.constantCompletion x then OpenQuantumProblem35.uniformCoeff d else 0","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.diagonalState_combineFirst_one"},{"answerKinds":[],"category":"research solved","docstring":"The Bell state witnesses the existence of $\\mathrm{AME}(2,d)$ for every local dimension $d \\ge 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {d : ℕ}, 2 ≤ d → OpenQuantumProblem35.ExistsAME 2 d","subjects":["5","15","81","94"],"subsets":["FC100SolvedSet1"],"theorem":"OpenQuantumProblem35.ame_2_exists"},{"answerKinds":[],"category":"research solved","docstring":"Source-backed benchmark statement: the three-qubit GHZ state witnesses the existence of an $\\mathrm{AME}(3,2)$ state. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"OpenQuantumProblem35.ExistsAME 3 2","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_3_2_exists"},{"answerKinds":[],"category":"API","docstring":"A uniform superposition over the graph of an injective completion map has reduced density matrix $(c\\overline c) I$ on the first subsystem. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d m : ℕ} (hm : m ≤ n) (ψ : OpenQuantumProblem35.StateVector n d)\n  (completion : OpenQuantumProblem35.Config m d → OpenQuantumProblem35.Config (n - m) d) (coeff : ℂ),\n  (∀ (x : OpenQuantumProblem35.Config m d) (z : OpenQuantumProblem35.Config (n - m) d),\n      ψ.ofLp (OpenQuantumProblem35.combineFirst m hm x z) = if z = completion x then coeff else 0) →\n    Function.Injective completion → OpenQuantumProblem35.reducedDensityFirst m hm ψ = (coeff * star coeff) • 1","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.reducedDensityFirst_of_completion"},{"answerKinds":[],"category":"research solved","docstring":"Source-backed benchmark statement: an $\\mathrm{AME}(5,2)$ state exists. This is one of the four qubit cases $n=2,3,5,6$; see the OQP page and Scott (2004). ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"OpenQuantumProblem35.ExistsAME 5 2","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_5_2_exists"},{"answerKinds":[],"category":"test","docstring":"Every constant configuration is constant. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {m d : ℕ} (a : Fin d), OpenQuantumProblem35.IsConstantConfig (OpenQuantumProblem35.constantConfig a)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.isConstantConfig_constantConfig"},{"answerKinds":[],"category":"test","docstring":"The identity permutation leaves a state vector unchanged. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (ψ : OpenQuantumProblem35.StateVector n d), OpenQuantumProblem35.permuteState (Equiv.refl (Fin n)) ψ = ψ","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.permuteState_refl"},{"answerKinds":[],"category":"research solved","docstring":"Source-backed benchmark statement: an $\\mathrm{AME}(4,6)$ state exists; see Rather et al. (2022). ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"OpenQuantumProblem35.ExistsAME 4 6","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_4_6_exists"},{"answerKinds":[],"category":"API","docstring":"The squared norm of the uniform coefficient is the inverse local dimension. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ (d : ℕ), OpenQuantumProblem35.uniformCoeff d * star (OpenQuantumProblem35.uniformCoeff d) = (↑d)⁻¹","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.uniformCoeff_mul_star"},{"answerKinds":[],"category":"API","docstring":"A state is normalized iff its squared $L^2$ norm is $1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (ψ : OpenQuantumProblem35.StateVector n d), OpenQuantumProblem35.IsNormalized ψ ↔ ‖ψ‖ ^ 2 = 1","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.isNormalized_iff_norm_sq_eq_one"},{"answerKinds":[],"category":"API","docstring":"For $n \\ge 1$ and $d \\ge 1$, the diagonal state is normalized. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ}, 1 ≤ n → 1 ≤ d → OpenQuantumProblem35.IsNormalized (OpenQuantumProblem35.diagonalState n d)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.diagonalState_isNormalized"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Open Quantum Problem 35: classify all pairs $(n,d)$ with $n \\ge 2$ and $d \\ge 2$ for which an $\\mathrm{AME}(n,d)$ state exists. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"{nd | 2 ≤ nd.1 ∧ 2 ≤ nd.2 ∧ OpenQuantumProblem35.ExistsAME nd.1 nd.2} = sorry","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.oqp_35"},{"answerKinds":[],"category":"test","docstring":"Sanity check: the standard GHZ family on $4$ parties is not absolutely maximally entangled for any local dimension $d \\ge 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {d : ℕ}, 2 ≤ d → ¬OpenQuantumProblem35.IsAME (OpenQuantumProblem35.ghzState4 d)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ghzState4_not_ame"},{"answerKinds":[],"category":"test","docstring":"A simple binary two-party configuration with different entries is not constant. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"¬OpenQuantumProblem35.IsConstantConfig fun i => if i = 0 then 0 else 1","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.not_isConstantConfig_example"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(10,6)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 10 6","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_10_6_open"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(12,10)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 12 10","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_12_10_open"},{"answerKinds":[],"category":"API","docstring":"Every index in $\\mathrm{Fin}\\, n$ is either the unique left index or a right index when the left block has size $1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n : ℕ} (hn : 1 ≤ n) (i : Fin n),\n  i = OpenQuantumProblem35.leftIndex hn 0 ∨ ∃ j, i = OpenQuantumProblem35.rightIndex hn j","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.eq_leftIndex_zero_or_eq_rightIndex"},{"answerKinds":[],"category":"API","docstring":"The standard Bell state is $\\mathrm{AME}(2,d)$ for every physical local dimension $d \\ge 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {d : ℕ}, 2 ≤ d → OpenQuantumProblem35.IsAME (OpenQuantumProblem35.bellState d)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.bellState_isAME"},{"answerKinds":[],"category":"API","docstring":"Evaluating the diagonal state returns the uniform coefficient on constant strings and `0` otherwise. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (x : OpenQuantumProblem35.Config n d),\n  (OpenQuantumProblem35.diagonalState n d).ofLp x =\n    if OpenQuantumProblem35.IsConstantConfig x then OpenQuantumProblem35.uniformCoeff d else 0","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.diagonalState_apply"},{"answerKinds":[],"category":"API","docstring":"Combining and then restricting to the left block recovers the left input. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d m : ℕ} (hm : m ≤ n) (x : OpenQuantumProblem35.Config m d) (y : OpenQuantumProblem35.Config (n - m) d)\n  (i : Fin m), OpenQuantumProblem35.combineFirst m hm x y (OpenQuantumProblem35.leftIndex hm i) = x i","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.combineFirst_leftIndex"},{"answerKinds":[],"category":"API","docstring":"Permuting the parties preserves the property of being a constant configuration. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (π : Equiv.Perm (Fin n)) (x : OpenQuantumProblem35.Config n d),\n  OpenQuantumProblem35.IsConstantConfig (OpenQuantumProblem35.permuteConfig π x) ↔\n    OpenQuantumProblem35.IsConstantConfig x","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.isConstantConfig_permute_iff"},{"answerKinds":[],"category":"API","docstring":"On a nonempty index type, different constants give different constant configurations. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ}, 1 ≤ n → Function.Injective OpenQuantumProblem35.constantConfig","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.constantConfig_injective"},{"answerKinds":[],"category":"API","docstring":"The number of computational-basis configurations on $m$ parties of local dimension $d$ is $d^m$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ (m d : ℕ), Fintype.card (OpenQuantumProblem35.Config m d) = d ^ m","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.card_config"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(7,10)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 7 10","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_7_10_open"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(11,6)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 11 6","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_11_6_open"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Open benchmark statement: does an $\\mathrm{AME}(11,4)$ state exist?\n\nAnswer: `AME(11, 4)` exists. The graph state defined by the circulant matrix Gamma\n    over $GF(4)$ with first row $(0, 0, 0, 1, ω, ω, ω, ω, 1, 0, 0)$ is an\n    absolutely maximally entangled state of $11$ ququarts.\n   This result has been found by Moritz Firsching and Goran Žužić using an\nexperimental pipeline\n\n\n Before, it was already known that there is a quantum code for `[11,0]]_5`, which corresponds to an `AME(11,4)` state (which is another approach to a solution).\n      ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/91bed229b434b68d66f5fd35cdcfee19a79985e8/FormalConjectures/OpenQuantumProblems/35.lean#L1861"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"True ↔ OpenQuantumProblem35.ExistsAME 11 4","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_11_4_open"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Open benchmark statement: does an $\\mathrm{AME}(11,5)$ state exist?\n\nThe DeepMind prover agent has shown that such a state exists.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/google-deepmind/formal-conjectures/blob/47383bf7fbe86effc9ac184446e320f26ddbee3a/FormalConjectures/OpenQuantumProblems/35.lean#L2138"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"True ↔ OpenQuantumProblem35.ExistsAME 11 5","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_11_5_open"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(9,10)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 9 10","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_9_10_open"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(11,3)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 11 3","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_11_3_open"},{"answerKinds":[],"category":"API","docstring":"If $\\lfloor n/2 \\rfloor = 1$, then the diagonal state is $\\mathrm{AME}(n,d)$ for every $d \\ge 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ}, 2 ≤ n → n / 2 = 1 → 2 ≤ d → OpenQuantumProblem35.IsAME (OpenQuantumProblem35.diagonalState n d)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.diagonalState_isAME_of_div_two_eq_one"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(10,10)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 10 10","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_10_10_open"},{"answerKinds":[],"category":"API","docstring":"The diagonal state has maximally mixed one-party reductions once $n \\ge 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (hn : 2 ≤ n),\n  OpenQuantumProblem35.HasMaximallyMixedFirstReduction 1 ⋯ (OpenQuantumProblem35.diagonalState n d)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.diagonalState_hasMaximallyMixedFirstReduction_one"},{"answerKinds":[],"category":"API","docstring":"On $4$ parties, the diagonal state vanishes on any split configuration whose first two entries are different. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {d : ℕ} {x z : OpenQuantumProblem35.Config 2 d},\n  x 0 ≠ x 1 → (OpenQuantumProblem35.diagonalState 4 d).ofLp (OpenQuantumProblem35.combineFirst 2 ⋯ x z) = 0","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.diagonalState_combineFirst_two_of_ne"},{"answerKinds":[],"category":"API","docstring":"The completion map for constant configurations is injective once $n \\ge 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ}, 2 ≤ n → Function.Injective OpenQuantumProblem35.constantCompletion","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.constantCompletion_injective"},{"answerKinds":[],"category":"API","docstring":"A tail configuration equals the constant completion of $x$ iff all of its entries agree with the unique entry of $x$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (x : OpenQuantumProblem35.Config 1 d) (z : OpenQuantumProblem35.Config (n - 1) d),\n  z = OpenQuantumProblem35.constantCompletion x ↔ ∀ (i : Fin (n - 1)), z i = x 0","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.constantCompletion_eq_iff"},{"answerKinds":[],"category":"API","docstring":"Evaluating a permuted state vector reads the amplitude at the permuted configuration. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ {n d : ℕ} (π : Equiv.Perm (Fin n)) (ψ : OpenQuantumProblem35.StateVector n d) (x : OpenQuantumProblem35.Config n d),\n  (OpenQuantumProblem35.permuteState π ψ).ofLp x = ψ.ofLp (OpenQuantumProblem35.permuteConfig π x)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.permuteState_apply"},{"answerKinds":["Prop"],"category":"research open","docstring":"Open benchmark statement: does an $\\mathrm{AME}(8,6)$ state exist? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"sorry ↔ OpenQuantumProblem35.ExistsAME 8 6","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.ame_8_6_open"},{"answerKinds":[],"category":"API","docstring":"The diagonal state is invariant under permutations of the parties. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«35»","statement":"∀ (n d : ℕ) (π : Equiv.Perm (Fin n)),\n  OpenQuantumProblem35.permuteState π (OpenQuantumProblem35.diagonalState n d) = OpenQuantumProblem35.diagonalState n d","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem35.diagonalState_permute"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $64$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 64","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_64"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $69$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 69","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_69"},{"answerKinds":[],"category":"test","docstring":"The BB84 family has the right cardinality for a qubit SIC but fails the constant-overlap condition. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"¬OpenQuantumProblem23.IsSICFamily 2 OpenQuantumProblem23.bb84Family","subjects":["15","47","81"],"subsets":["FC100SolvedSet1"],"theorem":"OpenQuantumProblem23.bb84Family_not_isSICFamily"},{"answerKinds":[],"category":"test","docstring":"Every vector in the tetrahedral qubit SIC family is normalized. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"∀ (i : Fin 4), OpenQuantumProblem23.IsNormalized (OpenQuantumProblem23.qubitSICFamily i)","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.qubitSICFamily_normalized"},{"answerKinds":[],"category":"test","docstring":"Any singleton family has constant pairwise squared overlap, vacuously. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"∀ {d : ℕ} (c : ℝ) (ψ : OpenQuantumProblem23.StateVector d), OpenQuantumProblem23.HasConstantOverlapSq c fun x => ψ","subjects":["15","47","81"],"subsets":["FC100SolvedSet1"],"theorem":"OpenQuantumProblem23.hasConstantOverlapSq_singleton"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $75$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 75","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_75"},{"answerKinds":[],"category":"test","docstring":"Dimension $2$ admits a SIC-POVM, witnessed by the tetrahedral qubit SIC. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"OpenQuantumProblem23.HasSICPOVM 2","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_two"},{"answerKinds":[],"category":"test","docstring":"The SIC overlap value in dimension $3$ is $1/4$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"OpenQuantumProblem23.sicOverlapSq 3 = 1 / 4","subjects":["15","47","81"],"subsets":["FC100SolvedSet1"],"theorem":"OpenQuantumProblem23.sicOverlapSq_three"},{"answerKinds":[],"category":"test","docstring":"The Hesse qutrit SIC family has the correct constant pairwise overlap. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"OpenQuantumProblem23.HasConstantOverlapSq (OpenQuantumProblem23.sicOverlapSq 3) OpenQuantumProblem23.hesseFamily","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hesseFamily_pairwise"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $68$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 68","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_68"},{"answerKinds":[],"category":"test","docstring":"Dimension $3$ admits a SIC-POVM, witnessed by the Hesse qutrit SIC. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"OpenQuantumProblem23.HasSICPOVM 3","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_three"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $59$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 59","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_59"},{"answerKinds":[],"category":"test","docstring":"Any normalized state in dimension $1$ yields a SIC family. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"∀ {ψ : OpenQuantumProblem23.StateVector 1},\n  OpenQuantumProblem23.IsNormalized ψ → OpenQuantumProblem23.IsSICFamily 1 fun x => ψ","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.isSICFamily_one_of_normalized"},{"answerKinds":[],"category":"test","docstring":"The tetrahedral qubit SIC family has the correct constant pairwise overlap. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"OpenQuantumProblem23.HasConstantOverlapSq (OpenQuantumProblem23.sicOverlapSq 2) OpenQuantumProblem23.qubitSICFamily","subjects":["15","47","81"],"subsets":["FC100SolvedSet1"],"theorem":"OpenQuantumProblem23.qubitSICFamily_pairwise"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $60$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 60","subjects":["15","47","81"],"subsets":["FC100OpenSet1"],"theorem":"OpenQuantumProblem23.hasSICPOVM_60"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $58$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 58","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_58"},{"answerKinds":[],"category":"test","docstring":"The SIC overlap value is positive in every dimension. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"∀ (d : ℕ), 0 < OpenQuantumProblem23.sicOverlapSq d","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.sicOverlapSq_pos"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $71$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 71","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_71"},{"answerKinds":[],"category":"test","docstring":"Dimension $1$ admits a SIC-POVM. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"OpenQuantumProblem23.HasSICPOVM 1","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_one"},{"answerKinds":[],"category":"test","docstring":"The SIC overlap value in dimension $2$ is $1/3$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"OpenQuantumProblem23.sicOverlapSq 2 = 1 / 3","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.sicOverlapSq_two"},{"answerKinds":["Prop"],"category":"research open","docstring":"Do SIC-POVMs exist in every finite dimension? ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ ∀ (d : ℕ), 1 ≤ d → OpenQuantumProblem23.HasSICPOVM d","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.sicPOVMs"},{"answerKinds":[],"category":"test","docstring":"Every vector in the BB84 family is normalized. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"∀ (i : Fin 4), OpenQuantumProblem23.IsNormalized (OpenQuantumProblem23.bb84Family i)","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.bb84Family_normalized"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $72$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 72","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_72"},{"answerKinds":[],"category":"test","docstring":"The SIC overlap value in dimension $1$ is $1/2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"OpenQuantumProblem23.sicOverlapSq 1 = 1 / 2","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.sicOverlapSq_one"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $56$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 56","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_56"},{"answerKinds":[],"category":"test","docstring":"Every vector in the Hesse qutrit SIC family is normalized. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"∀ (i : Fin 9), OpenQuantumProblem23.IsNormalized (OpenQuantumProblem23.hesseFamily i)","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hesseFamily_normalized"},{"answerKinds":["Prop"],"category":"research open","docstring":"Benchmark open subproblem: existence of a SIC-POVM in dimension $70$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"sorry ↔ OpenQuantumProblem23.HasSICPOVM 70","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_70"},{"answerKinds":[],"category":"test","docstring":"In dimension $1$, a singleton family is SIC exactly when its vector is normalized. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"∀ {ψ : OpenQuantumProblem23.StateVector 1},\n  (OpenQuantumProblem23.IsSICFamily 1 fun x => ψ) ↔ OpenQuantumProblem23.IsNormalized ψ","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.isSICFamily_singleton_iff"},{"answerKinds":[],"category":"test","docstring":"The empty family witnesses the degenerate dimension-$0$ case. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«23»","statement":"OpenQuantumProblem23.HasSICPOVM 0","subjects":["15","47","81"],"theorem":"OpenQuantumProblem23.hasSICPOVM_zero"},{"answerKinds":[],"category":"API","docstring":"The squared norm of `ω` is $1/2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"‖OpenQuantumProblem13.Qubit.ω‖ ^ 2 = 2⁻¹","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.omega_norm_sq"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Special case in dimension $12$ (not a prime power): determine the maximal number of\nmutually unbiased orthonormal bases in $\\mathbb{C}^{12}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.IsMaxMUBCount 12 sorry","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.mutuallyUnbiasedBases_dim12"},{"answerKinds":[],"category":"API","docstring":"The relative product of two scaled phase matrices is obtained by scaling the corresponding\nrelative product of phase matrices. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (ζ η : ℂ),\n  star (OpenQuantumProblem13.Qubit.ω • OpenQuantumProblem13.Qubit.phaseMatrix ζ) *\n      OpenQuantumProblem13.Qubit.ω • OpenQuantumProblem13.Qubit.phaseMatrix η =\n    (star OpenQuantumProblem13.Qubit.ω * OpenQuantumProblem13.Qubit.ω) •\n      !![1 + star ζ * η, 1 - star ζ * η; 1 - star ζ * η, 1 + star ζ * η]","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.star_phaseBasis_mul_phaseBasis"},{"answerKinds":[],"category":"API","docstring":"Multiplying $\\omega$ by a unit-modulus phase preserves the squared norm $1/2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ {ζ : ℂ}, star ζ * ζ = 1 → ‖OpenQuantumProblem13.Qubit.ω * ζ‖ ^ 2 = 2⁻¹","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.omega_mul_phase_norm_sq"},{"answerKinds":[],"category":"API","docstring":"The $(0,0)$ entry of the relative unitary is the overlap of the first columns. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (U V : OpenQuantumProblem13.UMat 2),\n  OpenQuantumProblem13.relativeUnitary U V 0 0 =\n    star (OpenQuantumProblem13.Qubit.u0 U) * OpenQuantumProblem13.Qubit.u0 V +\n      star (OpenQuantumProblem13.Qubit.u1 U) * OpenQuantumProblem13.Qubit.u1 V","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.relativeUnitary_apply_zero_zero"},{"answerKinds":[],"category":"API","docstring":"Every qubit Bloch vector has squared Euclidean norm $1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (U : OpenQuantumProblem13.UMat 2),\n  inner ℝ (OpenQuantumProblem13.Qubit.bloch U) (OpenQuantumProblem13.Qubit.bloch U) = 1","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.bloch_inner_self"},{"answerKinds":[],"category":"API","docstring":"The relative product of two phase matrices has the expected $2 \\times 2$ form. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (ζ η : ℂ),\n  star (OpenQuantumProblem13.Qubit.phaseMatrix ζ) * OpenQuantumProblem13.Qubit.phaseMatrix η =\n    !![1 + star ζ * η, 1 - star ζ * η; 1 - star ζ * η, 1 + star ζ * η]","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.star_phaseMatrix_mul_phaseMatrix"},{"answerKinds":[],"category":"API","docstring":"A qubit Bloch vector is never the zero vector. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (U : OpenQuantumProblem13.UMat 2), OpenQuantumProblem13.Qubit.bloch U ≠ 0","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.bloch_ne_zero"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Special case in dimension $10$ (not a prime power): determine the maximal number of\nmutually unbiased orthonormal bases in $\\mathbb{C}^{10}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.IsMaxMUBCount 10 sorry","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.mutuallyUnbiasedBases_dim10"},{"answerKinds":[],"category":"API","docstring":"If $\\overline{\\zeta}\\,\\eta = i$, then the corresponding phase bases are mutually unbiased. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ {ζ η : ℂ} (hζ : star ζ * ζ = 1) (hη : star η * η = 1),\n  star ζ * η = Complex.I →\n    OpenQuantumProblem13.IsUnbiased (OpenQuantumProblem13.Qubit.phaseU ζ hζ) (OpenQuantumProblem13.Qubit.phaseU η hη)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.isUnbiased_phaseU_phaseU_of_mul_eq_I"},{"answerKinds":[],"category":"API","docstring":"No family of mutually unbiased bases in dimension $2$ has size greater than $3$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (m : ℕ), OpenQuantumProblem13.HasMUBs 2 m → m ≤ 3","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.qubit_upper_bound"},{"answerKinds":[],"category":"research solved","docstring":"Known general bounds in dimension $6$: the maximal number of mutually unbiased bases\nsatisfies $3 \\le \\mu(6) \\le 7$. ","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/XC0R/formal-conjectures/blob/c8733543568e8011288a9fa7ef33375f5e5907d3/FormalConjectures/OpenQuantumProblems/13.lean#L1168"}],"hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.HasMUBs 6 3 ∧ ∀ (m : ℕ), OpenQuantumProblem13.HasMUBs 6 m → m ≤ 7","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.mutuallyUnbiasedBases_dim6_bounds"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Special case in dimension $6$: determine the maximal number of mutually unbiased\northonormal bases in $\\mathbb{C}^6$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.IsMaxMUBCount 6 sorry","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.mutuallyUnbiasedBases_dim6"},{"answerKinds":[],"category":"API","docstring":"Mutual unbiasedness is symmetric. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ {d : ℕ} {U V : OpenQuantumProblem13.UMat d}, OpenQuantumProblem13.IsUnbiased U V → OpenQuantumProblem13.IsUnbiased V U","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.IsUnbiased.symm"},{"answerKinds":[],"category":"research solved","docstring":"In dimension $2$, the maximum number of mutually unbiased orthonormal bases is $3$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.IsMaxMUBCount 2 3","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.mutuallyUnbiasedBases_dim2"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Open Quantum Problem 13: determine the maximal number of mutually unbiased orthonormal\nbases in $\\mathbb{C}^d$ for $d \\ge 2$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (d : ℕ), 2 ≤ d → OpenQuantumProblem13.IsMaxMUBCount d (sorry d)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.mutuallyUnbiasedBases"},{"answerKinds":[],"category":"API","docstring":"If $\\zeta$ has unit modulus, then the phase matrix is orthogonal up to the scalar factor $2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ {ζ : ℂ},\n  star ζ * ζ = 1 → star (OpenQuantumProblem13.Qubit.phaseMatrix ζ) * OpenQuantumProblem13.Qubit.phaseMatrix ζ = 2 • 1","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.star_phaseMatrix_mul_self_of_unit_phase"},{"answerKinds":[],"category":"API","docstring":"A complex number with $\\overline{\\zeta}\\,\\zeta = 1$ has squared norm $1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ {ζ : ℂ}, star ζ * ζ = 1 → ‖ζ‖ ^ 2 = 1","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.phase_norm_sq_eq_one"},{"answerKinds":[],"category":"API","docstring":"The product $\\overline{\\omega}\\,\\omega$ is $1/2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"star OpenQuantumProblem13.Qubit.ω * OpenQuantumProblem13.Qubit.ω = (↑2)⁻¹","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.conj_omega_mul_omega"},{"answerKinds":[],"category":"API","docstring":"The real part of $z \\overline{w}$ is the Euclidean dot product of the coordinate pairs of\n`z` and `w`. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (z w : ℂ), (z * star w).re = z.re * w.re + z.im * w.im","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.re_mul_conj"},{"answerKinds":[],"category":"test","docstring":"Every dimension admits a family of one mutually unbiased basis. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (d : ℕ), OpenQuantumProblem13.HasMUBs d 1","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.hasMUBs_one"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Special case in dimension $15$ (not a prime power): determine the maximal number of\nmutually unbiased orthonormal bases in $\\mathbb{C}^{15}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.IsMaxMUBCount 15 sorry","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.mutuallyUnbiasedBases_dim15"},{"answerKinds":[],"category":"API","docstring":"If $\\overline{\\zeta}\\,\\eta = i$, then the relative unitary between the corresponding phase\nbases is the qubit mutually unbiased overlap matrix. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ {ζ η : ℂ} (hζ : star ζ * ζ = 1) (hη : star η * η = 1),\n  star ζ * η = Complex.I →\n    OpenQuantumProblem13.relativeUnitary (OpenQuantumProblem13.Qubit.phaseU ζ hζ)\n        (OpenQuantumProblem13.Qubit.phaseU η hη) =\n      !![OpenQuantumProblem13.Qubit.ω, star OpenQuantumProblem13.Qubit.ω;\n        star OpenQuantumProblem13.Qubit.ω, OpenQuantumProblem13.Qubit.ω]","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.relative_phaseU_phaseU_of_mul_eq_I"},{"answerKinds":[],"category":"API","docstring":"The Bloch inner product is determined by the $(0,0)$ entry of the relative unitary. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (U V : OpenQuantumProblem13.UMat 2),\n  inner ℝ (OpenQuantumProblem13.Qubit.bloch U) (OpenQuantumProblem13.Qubit.bloch V) =\n    2 * Complex.normSq (OpenQuantumProblem13.relativeUnitary U V 0 0) - 1","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.bloch_inner_eq_two_normSq_sub_one"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"Special case in dimension $14$ (not a prime power): determine the maximal number of\nmutually unbiased orthonormal bases in $\\mathbb{C}^{14}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.IsMaxMUBCount 14 sorry","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.mutuallyUnbiasedBases_dim14"},{"answerKinds":[],"category":"API","docstring":"The relative unitary of a basis with itself is the identity matrix. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (U : OpenQuantumProblem13.UMat 2), OpenQuantumProblem13.relativeUnitary U U = 1","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.relativeUnitary_self"},{"answerKinds":[],"category":"API","docstring":"Mutually unbiased qubit bases have orthogonal Bloch vectors. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ {U V : OpenQuantumProblem13.UMat 2},\n  OpenQuantumProblem13.IsUnbiased U V →\n    inner ℝ (OpenQuantumProblem13.Qubit.bloch U) (OpenQuantumProblem13.Qubit.bloch V) = 0","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.bloch_inner_eq_zero_of_isUnbiased"},{"answerKinds":[],"category":"API","docstring":"The standard basis is mutually unbiased with any phase basis of unit-modulus phase. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (ζ : ℂ) (hζ : star ζ * ζ = 1),\n  OpenQuantumProblem13.IsUnbiased OpenQuantumProblem13.Qubit.ZU (OpenQuantumProblem13.Qubit.phaseU ζ hζ)","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.isUnbiased_Z_phaseU"},{"answerKinds":[],"category":"API","docstring":"The standard qubit family is a family of mutually unbiased bases. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.IsMUBFamily OpenQuantumProblem13.Qubit.qubitFamily","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.qubitFamily_isMUB"},{"answerKinds":[],"category":"API","docstring":"The maximum number of mutually unbiased bases in dimension $2$ is $3$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.IsMaxMUBCount 2 3","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.qubit_maximal"},{"answerKinds":[],"category":"API","docstring":"Taking the star of a scalar multiple on the left and multiplying by another scalar multiple\ncollects the scalar factor as $\\overline{a} a$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (a : ℂ) (A B : Matrix (Fin 2) (Fin 2) ℂ), star (a • A) * a • B = (star a * a) • (star A * B)","subjects":["5","15","81","94"],"subsets":["FC100SolvedSet1"],"theorem":"OpenQuantumProblem13.Qubit.star_smul_mul_smul"},{"answerKinds":[],"category":"test","docstring":"Every dimension admits the empty family of mutually unbiased bases. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (d : ℕ), OpenQuantumProblem13.HasMUBs d 0","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.hasMUBs_zero"},{"answerKinds":[],"category":"API","docstring":"There exist three mutually unbiased bases in dimension $2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"OpenQuantumProblem13.HasMUBs 2 3","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.qubit_hasThreeMUBs"},{"answerKinds":[],"category":"API","docstring":"The first column of a unitary matrix has squared norm $1$. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ (U : OpenQuantumProblem13.UMat 2),\n  Complex.normSq (OpenQuantumProblem13.Qubit.u0 U) + Complex.normSq (OpenQuantumProblem13.Qubit.u1 U) = 1","subjects":["5","15","81","94"],"subsets":["FC100SolvedSet1"],"theorem":"OpenQuantumProblem13.Qubit.firstCol_normSq"},{"answerKinds":[],"category":"API","docstring":"Scaling a phase matrix by $\\omega$ produces a unitary matrix whenever the phase has unit modulus. ","hasSorryFreeProof":true,"module":"FormalConjectures.OpenQuantumProblems.«13»","statement":"∀ {ζ : ℂ},\n  star ζ * ζ = 1 →\n    OpenQuantumProblem13.Qubit.ω • OpenQuantumProblem13.Qubit.phaseMatrix ζ ∈ Matrix.unitaryGroup (Fin 2) ℂ","subjects":["5","15","81","94"],"theorem":"OpenQuantumProblem13.Qubit.scaled_phaseMatrix_mem_unitary"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"What is the exact value of the constant? ","hasSorryFreeProof":false,"module":"FormalConjectures.OptimizationConstants.«1a»","statement":"Constant1a.C1a = sorry","subjects":["5","11","26"],"theorem":"Constant1a.c1a_eq"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"How can the lower bound be improved? ","hasSorryFreeProof":false,"module":"FormalConjectures.OptimizationConstants.«1a»","statement":"sorry ∈ Set.Ioc 1.2748 Constant1a.C1a","subjects":["5","11","26"],"theorem":"Constant1a.mem_Ioc_c1a"},{"answerKinds":[],"category":"research solved","docstring":"The best known upper bound, proven by Yuksekgonul et al. in [Y2026] ","hasSorryFreeProof":false,"module":"FormalConjectures.OptimizationConstants.«1a»","statement":"Constant1a.C1a ≤ 1.5029","subjects":["5","11","26"],"theorem":"Constant1a.c1a_upper_bound"},{"answerKinds":["non-Prop"],"category":"research open","docstring":"How can the upper bound be improved? ","hasSorryFreeProof":false,"module":"FormalConjectures.OptimizationConstants.«1a»","statement":"sorry ∈ Set.Ico Constant1a.C1a 1.5029","subjects":["5","11","26"],"theorem":"Constant1a.mem_Ico_c1a"},{"answerKinds":[],"category":"research solved","docstring":"The best known lower bound, proven by Matolcsi-Vinuesa in [M2010]","hasSorryFreeProof":false,"module":"FormalConjectures.OptimizationConstants.«1a»","statement":"1.2748 ≤ Constant1a.C1a","subjects":["5","11","26"],"theorem":"Constant1a.c1a_lower_bound"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1601.03081».UniqueCrystalComponents","statement":"Arxiv.«1601.03081».IsCrystalWithComponents 35 5 7","subjects":["11"],"theorem":"Arxiv.«1601.03081».isCrystalWithComponents_35_5_7"},{"answerKinds":[],"category":"research open","docstring":"If $n = ab$ is a crystal, then there are no other pairs of\npositive integers $c, d > 1$, different from the couple $a, b$, such that $n = cd$ and\n$B(c, d) ∈ ℕ$, i.e., the components of the crystals are unique.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1601.03081».UniqueCrystalComponents","statement":"∀ (n a b c d : ℕ),\n  Arxiv.«1601.03081».IsCrystalWithComponents n a b → Arxiv.«1601.03081».IsCrystalWithComponents n c d → {a, b} = {c, d}","subjects":["11","26"],"subsets":["FC100OpenSet1"],"theorem":"Arxiv.«1601.03081».crystals_components_unique"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 1.1 (Dean, 1988).** For every integer $k \\geq 3$, every finite simple graph with\nminimum degree at least $k$ contains a cycle whose length is divisible by $k$.\n\nA cycle has length at least `3`, so the divisor is never `0` and the statement is not\nsatisfied for a trivial reason. `SimpleGraph.minDegree` is `0` on a graph with no vertices and\non a graph with no edges, so the hypothesis excludes both.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.02731».DeanCycles","statement":"True ↔\n  ∀ (k : ℕ),\n    3 ≤ k →\n      ∀ (V : Type) [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n        k ≤ G.minDegree → ∃ m ∈ G.cycleLengths, k ∣ m","subjects":["5"],"theorem":"Arxiv.«2605.02731».dean_conjecture"},{"answerKinds":[],"category":"research open","docstring":"The case $k = 5$. This is the only case of the conjecture that is still open.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.02731».DeanCycles","statement":"True ↔\n  ∀ (V : Type) [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n    5 ≤ G.minDegree → ∃ m ∈ G.cycleLengths, 5 ∣ m","subjects":["5"],"theorem":"Arxiv.«2605.02731».dean_conjecture.variants.five"},{"answerKinds":[],"category":"research solved","docstring":"The case $k = 4$, proved by Dean, Lesniak and Saito [DeLeSa93].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.02731».DeanCycles","statement":"∀ {V : Type} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  4 ≤ G.minDegree → ∃ m ∈ G.cycleLengths, 4 ∣ m","subjects":["5"],"theorem":"Arxiv.«2605.02731».dean_conjecture.variants.four"},{"answerKinds":[],"category":"research solved","docstring":"The cases $k \\geq 6$, proved by Liu, Ma and Zhao (2026). With [ChSa94] and [DeLeSa93] this\nleaves `dean_conjecture.variants.five` as the only open case.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.02731».DeanCycles","statement":"∀ {V : Type} [inst : Fintype V] [DecidableEq V] {k : ℕ},\n  6 ≤ k → ∀ (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj], k ≤ G.minDegree → ∃ m ∈ G.cycleLengths, k ∣ m","subjects":["5"],"theorem":"Arxiv.«2605.02731».dean_conjecture.variants.six_le"},{"answerKinds":[],"category":"test","docstring":"The complete graph on four vertices has minimum degree `3`, so it is one of the graphs the\ncase `k = 3` applies to. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2605.02731».DeanCycles","statement":"⊤.minDegree = 3","subjects":["5"],"theorem":"Arxiv.«2605.02731».minDegree_top_fin_four"},{"answerKinds":[],"category":"test","docstring":"A graph with no edges has minimum degree `0`, so the hypothesis of the conjecture rules it\nout. This is a check that the hypothesis carries weight. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2605.02731».DeanCycles","statement":"⊥.minDegree = 0","subjects":["5"],"theorem":"Arxiv.«2605.02731».minDegree_bot_eq_zero"},{"answerKinds":[],"category":"research solved","docstring":"The case $k = 3$, proved by Chen and Saito [ChSa94].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.02731».DeanCycles","statement":"∀ {V : Type} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  3 ≤ G.minDegree → ∃ m ∈ G.cycleLengths, 3 ∣ m","subjects":["5"],"theorem":"Arxiv.«2605.02731».dean_conjecture.variants.three"},{"answerKinds":[],"category":"research open","docstring":"[KLM2023, Problem 7.2] For every positive dimension $n$, a convex body $A$ and a bounded,\nmeasurable set $B$, must spectrality of $A \\times B$ imply spectrality of $B$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2209.04540».SpectralSetsAndWeakTiling","statement":"True ↔ ∀ (n m : ℕ), 0 < n → 0 < m → SpectralSetProduct.spectralProductImpliesRightSpectral n m","subjects":["42","46"],"theorem":"SpectralSetProduct.isSpectral_right_of_product_of_convexBody"},{"answerKinds":[],"category":"research open","docstring":"[KLM2023, Problem 7.2] For a three-dimensional convex body $A$ and a bounded,\nmeasurable set $B$, must spectrality of $A \\times B$ imply spectrality of $B$?\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2209.04540».SpectralSetsAndWeakTiling","statement":"True ↔ ∀ (m : ℕ), 0 < m → SpectralSetProduct.spectralProductImpliesRightSpectral 3 m","subjects":["42","46"],"theorem":"SpectralSetProduct.isSpectral_right_of_product_three_dimensional"},{"answerKinds":[],"category":"research open","docstring":"[KLM2023, Problem 7.1] asks whether a bounded, measurable, nowhere dense subset\n$\\Omega \\subset \\mathbb{R}^d$ of positive measure can be spectral for every $d \\ge 2$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2209.04540».SpectralSetsAndWeakTiling","statement":"True ↔\n  ∀ (d : ℕ),\n    2 ≤ d → ∃ Ω, Bornology.IsBounded Ω ∧ MeasurableSet Ω ∧ IsNowhereDense Ω ∧ 0 < MeasureTheory.volume Ω ∧ isSpectral Ω","subjects":["42","46"],"theorem":"NowhereDenseSpectralSet.exists_nowhereDense_spectralSet"},{"answerKinds":["Prop"],"category":"research solved","docstring":"[KLM2023, Problem 7.2; GL20] For a two-dimensional convex body $A$ and a bounded,\nmeasurable set $B$, if $A \\times B$ is spectral, then $B$ is spectral.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2209.04540».SpectralSetsAndWeakTiling","statement":"True ↔ ∀ (m : ℕ), 0 < m → SpectralSetProduct.spectralProductImpliesRightSpectral 2 m","subjects":["42","46"],"theorem":"SpectralSetProduct.isSpectral_right_of_product_two_dimensional"},{"answerKinds":["Prop"],"category":"research solved","docstring":"[KLM2023, Problem 7.2; GL16] For a one-dimensional convex body $A$ and a bounded,\nmeasurable set $B$, if $A \\times B$ is spectral, then $B$ is spectral.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2209.04540».SpectralSetsAndWeakTiling","statement":"True ↔ ∀ (m : ℕ), 0 < m → SpectralSetProduct.spectralProductImpliesRightSpectral 1 m","subjects":["42","46"],"theorem":"SpectralSetProduct.isSpectral_right_of_product_one_dimensional"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Does there exist a constant $c > 0$ so that for every graph $G$ and every $\\epsilon$ between\n$0$ and $1$, $V$ contains an $\\epsilon$-light subset $S$ of size at least $c \\epsilon |V|$?\n\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/frenzymath/Archon-FirstProof-Results/blob/main/FirstProof/FirstProof6/Problem6.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof6","statement":"True ↔\n  ∃ c > 0,\n    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (ε : ℝ),\n      0 < ε → ε < 1 → ∃ S, Arxiv.«2602.05192».IsEpsilonLight G ε S ∧ ↑S.card ≥ c * ε * ↑n","subjects":["5"],"theorem":"Arxiv.«2602.05192».epsilon_light_subset_exists"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of\ndegree $2$, then\n$$\\frac{1}{\\Phi_2(p\\boxplus_n q)} \\ge \\frac{1}{\\Phi_2(p)}+\\frac{1}{\\Phi_2(q)}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof4","statement":"True ↔ ∀ (p q : Polynomial ℝ), Arxiv.«2602.05192».FourProp p q 2","subjects":["26"],"theorem":"Arxiv.«2602.05192».four_2"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of\ndegree $3$, then\n$$\\frac{1}{\\Phi_3(p\\boxplus_n q)} \\ge \\frac{1}{\\Phi_3(p)}+\\frac{1}{\\Phi_3(q)}?$$\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof4","statement":"True ↔ ∀ (p q : Polynomial ℝ), Arxiv.«2602.05192».FourProp p q 3","subjects":["26"],"theorem":"Arxiv.«2602.05192».four_3"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof4","statement":"∀ (n : ℕ) (p q : Polynomial ℝ),\n  0 < n → p.degree = ↑n → q.degree = ↑n → p.Monic → q.Monic → (Arxiv.«2602.05192».finiteAdditiveConvolution n p q).Monic","subjects":["26"],"subsets":["FC100SolvedSet1"],"theorem":"Arxiv.«2602.05192».finiteAdditiveConvolution_monic'"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof4","statement":"∀ (n : ℕ) (p q : Polynomial ℝ),\n  p.degree = ↑n → q.degree = ↑n → (Arxiv.«2602.05192».finiteAdditiveConvolution n p q).degree = ↑n","subjects":["26"],"theorem":"Arxiv.«2602.05192».finiteAdditiveConvolution_degree"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof4","statement":"∀ {F : Type} [inst : Field F] (n : ℕ) (p q : Polynomial F),\n  Arxiv.«2602.05192».finiteAdditiveConvolution n p q = Arxiv.«2602.05192».finiteAdditiveConvolution n q p","subjects":["26"],"theorem":"Arxiv.«2602.05192».finiteAdditiveConvolution_comm"},{"answerKinds":["Prop"],"category":"research solved","docstring":"Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of\ndegree $n$, then\n$$\\frac{1}{\\Phi_n(p\\boxplus_n q)} \\ge \\frac{1}{\\Phi_n(p)}+\\frac{1}{\\Phi_n(q)}?$$\n\n[arxiv/2602.05192v2](https://arxiv.org/abs/2602.05192v2) contains a proof.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/frenzymath/Archon-FirstProof-Results/blob/main/FirstProof/FirstProof4/Problem4.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2602.05192».FirstProof4","statement":"True ↔ ∀ (p q : Polynomial ℝ) (n : ℕ), Arxiv.«2602.05192».FourProp p q n","subjects":["26"],"theorem":"Arxiv.«2602.05192».four"},{"answerKinds":[],"category":"research solved","docstring":"**Huang–Shi, Theorem 1.2**\n\nLet `F` be a finite field of characteristic `p ∈ {3, 5, 7, 11}`, and set\n`K = F((t⁻¹))`, `A = F[t]`. Let\n\n* `D` be the diagonal subgroup of `SL₄(K)`,\n* `Γ = SL₄(A)` the lattice subgroup embedded into `SL₄(K)` via the natural inclusion `A →+* K`.\n\nThen there exists `z : SL₄(K)/Γ` such that the `D`-orbit of `z` has compact\nclosure but is not closed.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2504.17644».Margulis","statement":"∀ (F : Type u) [inst : Field F] [Fintype F],\n  ringChar F ∈ {3, 5, 7, 11} →\n    ∃ z,\n      IsCompact\n          (closure (MulAction.orbit (↥(Matrix.SpecialLinearGroup.diagonalSubgroup (Fin 4) (LaurentSeries F))) z)) ∧\n        ¬IsClosed (MulAction.orbit (↥(Matrix.SpecialLinearGroup.diagonalSubgroup (Fin 4) (LaurentSeries F))) z)","subjects":["11","15","22"],"theorem":"Margulis.huang_shi_theorem_1_2"},{"answerKinds":[],"category":"research open","docstring":"Let `D` be the diagonal group of `SL_n(ℝ)` where n ≥ 3.\nThen any relatively compact `D`-orbit in `SL_n(ℝ) / SL_n(ℤ)` is closed. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2504.17644».Margulis","statement":"∀ {n : ℕ},\n  3 ≤ n →\n    ∀ (g : Matrix.SpecialLinearGroup (Fin n) ℝ ⧸ Subgroup.map (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) ⊤),\n      IsCompact (closure (MulAction.orbit (↥(Matrix.SpecialLinearGroup.diagonalSubgroup (Fin n) ℝ)) g)) →\n        IsClosed (MulAction.orbit (↥(Matrix.SpecialLinearGroup.diagonalSubgroup (Fin n) ℝ)) g)","subjects":["11","15","22"],"theorem":"Margulis.conjecture_1_1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2303.01089».FurstenbergTimesPTimesQ","statement":"∀ (n : ℕ), Continuous (Arxiv.id2303_01089.Tn n)","subjects":["28"],"theorem":"Arxiv.id2303_01089.Tn_continuous"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 1.3** (the $\\times p, \\times q$ conjecture): the only atomless Borel probability\nmeasure on $\\mathbb{T}$ which is both $T_p$- and $T_q$-invariant is the Lebesgue measure.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2303.01089».FurstenbergTimesPTimesQ","statement":"∀ {p q : ℕ},\n  2 ≤ p →\n    2 ≤ q →\n      Arxiv.id2303_01089.MultiplicativelyIndependent p q →\n        ∀ {μ : MeasureTheory.Measure 𝕋} [MeasureTheory.IsProbabilityMeasure μ]\n          [Arxiv.id2303_01089.MeasureTheory.IsAtomLess μ],\n          MeasureTheory.MeasurePreserving (Arxiv.id2303_01089.Tn p) μ μ →\n            MeasureTheory.MeasurePreserving (Arxiv.id2303_01089.Tn q) μ μ → μ = MeasureTheory.volume","subjects":["37"],"theorem":"Arxiv.id2303_01089.conjecture_1_3"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Conjecture 1.4**: if $\\mu$ is an atomless $T_p$-invariant Borel probability measure on\n$\\mathbb{T}$, then $T_{q^n}\\mu$ converges weak-star to Lebesgue measure.\nThis paper disproves the conjecture.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2303.01089».FurstenbergTimesPTimesQ","statement":"False ↔\n  ∀ (p q : ℕ),\n    2 ≤ p →\n      2 ≤ q →\n        Arxiv.id2303_01089.MultiplicativelyIndependent p q →\n          ∀ (μ : MeasureTheory.ProbabilityMeasure 𝕋),\n            Arxiv.id2303_01089.MeasureTheory.IsAtomLess ↑μ →\n              MeasureTheory.MeasurePreserving (Arxiv.id2303_01089.Tn p) ↑μ ↑μ →\n                Filter.Tendsto (fun n => μ.map ⋯) Filter.atTop\n                  (nhds Arxiv.id2303_01089.UnitAddCircle.ProbabilityMeasure)","subjects":["37"],"theorem":"Arxiv.id2303_01089.conjecture_1_4"},{"answerKinds":[],"category":"research solved","docstring":"Boxdot Conjecture: every normal modal logic that faithfully interprets KT\nby the boxdot translation is included in KT.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/FormalizedFormalLogic/Foundation"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1308.0994».BoxdotConjecture","statement":"∀ (L : Arxiv.«1308.0994».NormalModalLogic),\n  (∀ (φ : Arxiv.«1308.0994».Formula),\n      Arxiv.«1308.0994».proves L (Arxiv.«1308.0994».t φ) ↔ Arxiv.«1308.0994».proves Arxiv.«1308.0994».KT φ) →\n    L.thms ⊆ Arxiv.«1308.0994».KT.thms","subjects":["3"],"theorem":"Arxiv.«1308.0994».BoxdotConjecture"},{"answerKinds":[],"category":"API","docstring":"If `KProof Γ φ`, then `KTProof Γ φ`. In other words, KT extends K.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1308.0994».BoxdotConjecture","statement":"∀ {Γ : Set Arxiv.«1308.0994».Formula} {φ : Arxiv.«1308.0994».Formula},\n  Arxiv.«1308.0994».KProof Γ φ → Arxiv.«1308.0994».KTProof Γ φ","subjects":["3"],"subsets":["FC100SolvedSet1"],"theorem":"Arxiv.«1308.0994».KTExtendsK"},{"answerKinds":[],"category":"API","docstring":"The all-ones multiset always has the right size and the right sum, so `IsValidMod` is a\nuniqueness statement rather than an existence one. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.08366».MinModulus","statement":"∀ {N : ℕ} (A : Finset (ZMod N)), ∑ _a ∈ A, 1 = A.card ∧ ∑ a ∈ A, ↑1 * a = ∑ a ∈ A, a","subjects":["11"],"theorem":"Arxiv.«2607.08366».one_sum_eq"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 1 (Fonollosa, 2026).** For every $n \\geq 2$ and every\n$N < 2^n - 2^{\\lfloor \\log_2 n\\rfloor}$, no set of $n$ residues mod $N$ is valid.\n\nEquivalently the super-increasing set $\\{2^k - 1 : 0 \\leq k \\leq n-1\\}$ attains the least\nvalid modulus, which is `minModulus n`.\n\n`0 < N` excludes `N = 0`, where `ZMod 0` is `ℤ` rather than a finite modulus and `{1, 2}` is\nvalid, which would make the statement false for a reason unrelated to the question.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.08366».MinModulus","statement":"True ↔\n  ∀ (n N : ℕ),\n    2 ≤ n →\n      0 < N →\n        N < Arxiv.«2607.08366».minModulus n → ∀ (A : Finset (ZMod N)), A.card = n → ¬Arxiv.«2607.08366».IsValidMod A","subjects":["11"],"theorem":"Arxiv.«2607.08366».min_modulus"},{"answerKinds":[],"category":"API","docstring":"A set with fewer than two elements is valid for a silly reason, so the conjecture asks\nabout `2 ≤ n`: with `#A ≤ 1` the only multiset of size `#A` drawn from `A` is the all-ones one. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.08366».MinModulus","statement":"∀ {N : ℕ} {A : Finset (ZMod N)}, A.card ≤ 1 → Arxiv.«2607.08366».IsValidMod A","subjects":["11"],"theorem":"Arxiv.«2607.08366».isValidMod_of_subsingleton"},{"answerKinds":[],"category":"research solved","docstring":"**Theorem A (Fonollosa, 2026).** `minModulus n` admits a valid set of `n` residues, the\nsuper-increasing set $\\{2^k - 1 : 0 \\leq k \\leq n - 1\\}$. This bounds the least valid modulus\nfrom above; that no smaller modulus works is the open half, stated in `min_modulus`.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/jarfo/min-modulus/blob/e7c78dd63955092b5f8d8a5fa826476337c0f4be/MinModulus/UniqueSums.lean#L837-L838"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.08366».MinModulus","statement":"∀ (n : ℕ), 2 ≤ n → ∃ A, A.card = n ∧ Arxiv.«2607.08366».IsValidMod A","subjects":["11"],"theorem":"Arxiv.«2607.08366».exists_isValidMod_minModulus"},{"answerKinds":[],"category":"test","docstring":"For $n = 8$, $2$ is not contained in the base $3$ digits of $n$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2107.12475».CollatzLike","statement":"2 ∉ Nat.digits 3 (2 ^ 8)","subjects":["5","11"],"theorem":"Arxiv.«2107.12475».two_not_in_digits_three_pow_eight"},{"answerKinds":[],"category":"research open","docstring":"For $n > 8$, $2^n$ is not the the sum of distinct powers of $3$. Expressed here in terms of the base $3$ digits of $n$.\n\nThis conjecture is equivalent to the halting of a $15$-state $2$-symbol Turing Machine.\n\nTODO(lezeau): Formalize the Turing Machine version of this problem.\n\nSource: *Hardness of Busy Beaver Value BB(15)*: https://link.springer.com/chapter/10.1007/978-3-031-72621-7_9\nThis is also https://arxiv.org/abs/2107.12475.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2107.12475».CollatzLike","statement":"∀ (n : ℕ), 8 < n → 2 ∈ Nat.digits 3 (2 ^ n)","subjects":["5","11"],"theorem":"Arxiv.«2107.12475».CollatzLike"},{"answerKinds":[],"category":"test","docstring":"**Conjecture 1.1 → Conjecture 4.4**: If conjecture 1.1 holds true, then this implies a special\ncase of conjecture 4.4 where $n = 0$. In this case the lower bound for the odd prime $k$\nwould be $0$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"(∀ (k : ℕ), Nat.Prime k → Odd k → 0 < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑0 * ↑k < Arxiv.«2501.03234».S k","subjects":["11"],"theorem":"Arxiv.«2501.03234».conjecture_4_4_def_0"},{"answerKinds":[],"category":"test","docstring":"**Conjecture 4.3 → Conjecture 4.4**: If conjecture 4.3 holds true, then a special\ncase of conjecture 4.4 for $n = 3$ is obtained, and the lower bound is $3119$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"(∀ (k : ℕ), Nat.Prime k → Odd k → k > 3119 → 3 * ↑k < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑3 * ↑k < Arxiv.«2501.03234».S k","subjects":["11"],"theorem":"Arxiv.«2501.03234».conjecture_4_4_def_3"},{"answerKinds":[],"category":"test","docstring":"**Conjecture 4.1 → Conjecture 4.4**: If conjecture 4.1 holds true, then this implies a special\ncase of conjecture 4.4 where $n = 1$. In this case the lower bound would be $5$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"(∀ (k : ℕ), Nat.Prime k → Odd k → k > 5 → ↑k < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑1 * ↑k < Arxiv.«2501.03234».S k","subjects":["11"],"theorem":"Arxiv.«2501.03234».conjecture_4_4_def_1"},{"answerKinds":[],"category":"test","docstring":"**Conjecture 4.2 → Conjecture 4.4**: If conjecture 4.2 holds true, then this implies a special\ncase of conjecture 4.4 for $n = 2$. For this scenario, the lower bound is now $233$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"(∀ (k : ℕ), Nat.Prime k → Odd k → k > 233 → 2 * ↑k < Arxiv.«2501.03234».S k) →\n  ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑2 * ↑k < Arxiv.«2501.03234».S k","subjects":["11"],"theorem":"Arxiv.«2501.03234».conjecture_4_4_def_2"},{"answerKinds":[],"category":"test","docstring":"Note that in Table 1 in https://arxiv.org/abs/2501.03234v1, there seems to be an error:\n11 appears twice. The first 10 values of $S$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"List.map Arxiv.«2501.03234».S (List.range 10) = [0, 0, 1, 2, 5, 4, 7, 10, 11, 8]","subjects":["11"],"theorem":"Arxiv.«2501.03234».S_fst_10"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 4.4**: Given a natural number $n ∈ ℕ$, for all large enough odd prime $k$ (depending on $n$),\n$nk < S(k)$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"∀ (n : ℕ), ∀ᶠ (k : ℕ) in Filter.atTop, Nat.Prime k → Odd k → ↑n * ↑k < Arxiv.«2501.03234».S k","subjects":["11"],"theorem":"Arxiv.«2501.03234».conjecture_4_4"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 1.1**: For any odd prime $k$, the sum associated with the classical theta function $θ_3$,\n$S(k)$ is positive.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"∀ (k : ℕ), Nat.Prime k → Odd k → 0 < Arxiv.«2501.03234».S k","subjects":["11"],"theorem":"Arxiv.«2501.03234».conjecture_1_1"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 4.3**: For any prime $k$ larger than $3119$, $S(k) > 3k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"∀ (k : ℕ), Nat.Prime k → Odd k → k > 3119 → 3 * ↑k < Arxiv.«2501.03234».S k","subjects":["11"],"theorem":"Arxiv.«2501.03234».conjecture_4_3"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 4.2**: For any prime $k$ larger than $233$, $S(k) > 2k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"∀ (k : ℕ), Nat.Prime k → Odd k → k > 233 → 2 * ↑k < Arxiv.«2501.03234».S k","subjects":["11"],"theorem":"Arxiv.«2501.03234».conjecture_4_2"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 4.1**: For any prime $k$ larger than $5$, $S(k) > k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2501.03234».ArithmeticSumS","statement":"∀ (k : ℕ), Nat.Prime k → Odd k → k > 5 → ↑k < Arxiv.«2501.03234».S k","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Arxiv.«2501.03234».conjecture_4_1"},{"answerKinds":[],"category":"textbook","docstring":"All gaugings of a symmetric matrix share one concentration, so $\\mathrm{con}$ is an\nattribute of the matrix. The source records this after Definition 2.3, as the calculation\n$c = c\\mathbf{1}^T v' = (v^TA^T)v' = v^T(c'\\mathbf{1}) = c'$, and Definition 2.4 rests on it. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.05349».MicroscopicWeighting","statement":"∀ {X : Type u_1} [inst : Fintype X] {M : Matrix X X ℝ},\n  M.IsSymm →\n    ∀ {g g' : X → ℝ} {c c' : ℝ}, Arxiv.«2607.05349».IsGauging M g c → Arxiv.«2607.05349».IsGauging M g' c' → c = c'","subjects":["15"],"theorem":"Arxiv.«2607.05349».concentration_unique"},{"answerKinds":["Prop"],"category":"research solved","docstring":"**Conjecture 3.3 (Roff-Willerton, 2026).** A finite metric space admits a microscopic weighting\nif and only if its distance matrix has finite concentration.\n\n`Nonempty` is needed and not just tidiness. On the empty space every gauging condition fails,\nsince `∑ i, g i` is `0` rather than `1`, while `X → ℝ` is a subsingleton so the weighting\nconverges trivially. The equivalence would be false there for reasons that have nothing to do\nwith the question.\n\nThe answer is false as proved by Kenta Kitamura assisted by ChatGPT 5.6 sol.\n\nThe proof proceeds by constructing an explicit counterexample: a finite metric space\non 10 points that has finite concentration but does not admit a microscopic weighting.\nThe metric space is defined by the following $10 \\times 10$ distance matrix $A$:\n$$\n\\begin{pmatrix}\n0 & 116 & 236 & 231 & 260 & 124 & 64 & 290 & 266 & 64 \\cr\n116 & 0 & 312 & 268 & 296 & 112 & 64 & 280 & 296 & 64 \\cr\n236 & 312 & 0 & 68 & 40 & 236 & 288 & 68 & 36 & 288 \\cr\n231 & 268 & 68 & 0 & 34 & 237 & 288 & 72 & 40 & 288 \\cr\n260 & 296 & 40 & 34 & 0 & 264 & 320 & 40 & 68 & 320 \\cr\n124 & 112 & 236 & 237 & 264 & 0 & 64 & 280 & 264 & 64 \\cr\n64 & 64 & 288 & 288 & 320 & 64 & 0 & 312 & 320 & 120 \\cr\n290 & 280 & 68 & 72 & 40 & 280 & 312 & 0 & 40 & 312 \\cr\n266 & 296 & 36 & 40 & 68 & 264 & 320 & 40 & 0 & 320 \\cr\n64 & 64 & 288 & 288 & 320 & 64 & 120 & 312 & 320 & 0\n\\end{pmatrix}\n$$\n\nThe space has finite concentration, demonstrated by the explicit gauging $g$:\n$$\ng = \\frac{1}{24842973905} \\begin{pmatrix} 3672468740 \\cr 6389133731 \\cr 9124217512 \\cr\n5612262448 \\cr -6875621136 \\cr 2754831248 \\cr 0 \\cr 10758780188 \\cr -6593098826 \\cr 0\n\\end{pmatrix}\n$$\nwhich gives $A g = \\frac{4111107017312}{24842973905} \\mathbf{1}$.\n\nTo prove that it does not admit a microscopic weighting, we provide a vector $v \\in \\ker A$:\n$$\nv = \\begin{pmatrix} -2 \\cr -1 \\cr 3 \\cr 4 \\cr -4 \\cr -2 \\cr 2 \\cr 2 \\cr -4 \\cr 2 \\end{pmatrix}\n$$\nalong with a row-space certificate $q$ for the entrywise square matrix $B = A^{\\circ 2}$ satisfying $q A = v^\\top B$:\n$$\nq^\\top = \\frac{1}{128472094291} \\begin{pmatrix} 43681853675722 \\cr -53873248293642 \\cr\n-66890627544007 \\cr -81187181670120 \\cr 24308499983196 \\cr 40819375894674 \\cr 0 \\cr\n54690260644468 \\cr 35226831040652 \\cr 0\n\\end{pmatrix}\n$$\nThis certificate shows that the obstruction $v^\\top B$ annihilates $\\ker A$, making it impossible to construct a convergent weighting.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/microscopic-weighting-counterexample/blob/eff8979/lean/MicroscopicWeightingCounterexampleFC.lean#L886-L894"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.05349».MicroscopicWeighting","statement":"False ↔\n  ∀ (X : Type) [inst : Fintype X] [inst_1 : DecidableEq X] [Nonempty X] [inst_3 : MetricSpace X],\n    Arxiv.«2607.05349».HasMicroscopicWeighting X ↔\n      Arxiv.«2607.05349».HasFiniteConcentration (Arxiv.«2607.05349».distanceMatrix X)","subjects":["15","51"],"theorem":"Arxiv.«2607.05349».microscopic_weighting_iff_finite_concentration"},{"answerKinds":[],"category":"research solved","docstring":"One direction is known: a microscopic weighting implies finite concentration (Theorem 3.1(3)).\nTogether with the conjecture above this leaves the converse, that finite concentration is enough.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.05349».MicroscopicWeighting","statement":"∀ {X : Type u_1} [inst : Fintype X] [inst_1 : DecidableEq X] [Nonempty X] [inst_3 : MetricSpace X],\n  Arxiv.«2607.05349».HasMicroscopicWeighting X →\n    Arxiv.«2607.05349».HasFiniteConcentration (Arxiv.«2607.05349».distanceMatrix X)","subjects":["15","51"],"theorem":"Arxiv.«2607.05349».hasFiniteConcentration_of_hasMicroscopicWeighting"},{"answerKinds":[],"category":"research solved","docstring":"Theorem 3.8: the conjecture is known when the distance matrix is invertible, and the microscopic\nweighting is then the unique gauging.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.05349».MicroscopicWeighting","statement":"∀ {X : Type u_1} [inst : Fintype X] [inst_1 : DecidableEq X] [Nonempty X] [inst_3 : MetricSpace X],\n  IsUnit (Arxiv.«2607.05349».distanceMatrix X).det →\n    (Arxiv.«2607.05349».HasMicroscopicWeighting X ↔\n      Arxiv.«2607.05349».HasFiniteConcentration (Arxiv.«2607.05349».distanceMatrix X))","subjects":["15","51"],"theorem":"Arxiv.«2607.05349».hasMicroscopicWeighting_iff_of_isUnit"},{"answerKinds":[],"category":"test","docstring":"The alternating group on five letters has order $60 = 2^2 \\cdot 3 \\cdot 5$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2604.08040».Conjecture5_5","statement":"Fintype.card ↥(alternatingGroup (Fin 5)) = 60","subjects":["20"],"theorem":"Arxiv.«2604.08040».card_alternatingGroup_fin_five"},{"answerKinds":[],"category":"test","docstring":"The trivial group has exactly one cyclic subgroup (itself).\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2604.08040».Conjecture5_5","statement":"Arxiv.«2604.08040».cyc PUnit.{u_2 + 1} = 1","subjects":["20"],"theorem":"Arxiv.«2604.08040».cyc_punit"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 5.5** (Das, Dey, Sharma 2026):\nIf a finite group `G` satisfies $\\mathrm{cyc}(G) < 2^{t+2}$, where $t = \\pi(G)$ is the\nnumber of distinct prime divisors of $|G|$, then `G` is solvable.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2604.08040».Conjecture5_5","statement":"True ↔\n  ∀ (G : Type) [inst : Group G] [inst_1 : Fintype G],\n    Arxiv.«2604.08040».cyc G < 2 ^ (Arxiv.«2604.08040».numPrimeFactors G + 2) → Group.IsSolvable G","subjects":["20"],"theorem":"Arxiv.«2604.08040».solvable_of_cyc_lt"},{"answerKinds":[],"category":"test","docstring":"$A_5$ is the sharpness witness: it has $\\pi(A_5) = 3$ and $\\mathrm{cyc}(A_5) = 32 = 2^{3+2}$,\nso it misses the strict inequality $\\mathrm{cyc}(G) < 2^{t+2}$ by exactly $1$.\nThis confirms $A_5$ is consistent with Conjecture 5.5 despite being non-solvable.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2604.08040».Conjecture5_5","statement":"¬Arxiv.«2604.08040».cyc ↥(alternatingGroup (Fin 5)) <\n    2 ^ (Arxiv.«2604.08040».numPrimeFactors ↥(alternatingGroup (Fin 5)) + 2)","subjects":["20"],"theorem":"Arxiv.«2604.08040».not_cyc_alternatingGroup_five_lt"},{"answerKinds":[],"category":"research solved","docstring":"**Theorem 3.1.** Below $5 \\cdot 2^{t-2}$ cyclic subgroups, the group is nilpotent.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2604.08040».Conjecture5_5","statement":"∀ (G : Type u_1) [inst : Group G] [inst_1 : Fintype G],\n  Arxiv.«2604.08040».cyc G < 5 * 2 ^ (Arxiv.«2604.08040».numPrimeFactors G - 2) → Group.IsNilpotent G","subjects":["20"],"theorem":"Arxiv.«2604.08040».nilpotent_of_cyc_lt"},{"answerKinds":[],"category":"research solved","docstring":"**Theorem 4.2.** Below $2^{t+1}$ cyclic subgroups, the group is supersolvable.\n\nStated here as solvability, which supersolvability implies, since Mathlib does not\ncurrently have a standalone `IsSupersolvable` class.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2604.08040».Conjecture5_5","statement":"∀ (G : Type u_1) [inst : Group G] [inst_1 : Fintype G],\n  Arxiv.«2604.08040».cyc G < 2 ^ (Arxiv.«2604.08040».numPrimeFactors G + 1) → Group.IsSolvable G","subjects":["20"],"theorem":"Arxiv.«2604.08040».solvable_of_cyc_lt_two_pow_succ"},{"answerKinds":[],"category":"test","docstring":"The trivial group has zero prime divisors.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2604.08040».Conjecture5_5","statement":"Arxiv.«2604.08040».numPrimeFactors PUnit.{u_2 + 1} = 0","subjects":["20"],"theorem":"Arxiv.«2604.08040».numPrimeFactors_trivial"},{"answerKinds":[],"category":"test","docstring":"The alternating group on five letters has exactly $32$ cyclic subgroups: the trivial one,\n$15$ of order $2$, $10$ of order $3$ and $6$ of order $5$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2604.08040».Conjecture5_5","statement":"Arxiv.«2604.08040».cyc ↥(alternatingGroup (Fin 5)) = 32","subjects":["20"],"theorem":"Arxiv.«2604.08040».cyc_alternatingGroup_five"},{"answerKinds":[],"category":"research open","docstring":"The sequence will eventually reach $1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«0912.2382».CurlingNumberConjecture","statement":"∀ (S₀ : List ℤ), S₀ ≠ [] → ∃ m, Arxiv.«0912.2382».k (Arxiv.«0912.2382».S S₀ m) = 1","subjects":["11"],"subsets":["FC100OpenSet1"],"theorem":"Arxiv.«0912.2382».curling_number_conjecture"},{"answerKinds":[],"category":"research open","docstring":"Every circulant Hadamard matrix has order at most four. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2402.13202».CirculantHadamard","statement":"∀ {n : ℕ} (v : Fin n → ℝ), Hadamard.IsHadamard' (Matrix.circulant v) → n ≤ 4","subjects":["15"],"theorem":"CirculantHadamard.circulant_hadamard_conjecture"},{"answerKinds":[],"category":"test","docstring":"The order-four generator gives a circulant Hadamard matrix. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2402.13202».CirculantHadamard","statement":"Hadamard.IsHadamard' (Matrix.circulant CirculantHadamard.orderFourGenerator)","subjects":["15"],"theorem":"CirculantHadamard.orderFourGenerator_isHadamard"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 4 (Alon-Tarsi, 1985).** Every bridgeless graph has a list of cycles covering\nevery edge, with $\\sum_{C} |E(C)| \\leq \\frac{7}{5}|E(G)|$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.06396».AlonTarsi","statement":"True ↔\n  ∀ (V : Type) [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n    G.IsBridgeless →\n      ∃ C, Arxiv.«2607.06396».IsCycleCover G C ∧ ↑(Arxiv.«2607.06396».totalLength C) ≤ 7 / 5 * ↑G.edgeFinset.card","subjects":["5"],"theorem":"Arxiv.«2607.06396».alon_tarsi_short_cycle_cover"},{"answerKinds":[],"category":"test","docstring":"Acyclic bridgeless graphs satisfy the conjecture, with the empty cover. In a forest every\nedge is a bridge, so such a graph has no edges at all. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.06396».AlonTarsi","statement":"∀ {V : Type u_1} [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n  G.IsAcyclic →\n    G.IsBridgeless →\n      ∃ C, Arxiv.«2607.06396».IsCycleCover G C ∧ ↑(Arxiv.«2607.06396».totalLength C) ≤ 7 / 5 * ↑G.edgeFinset.card","subjects":["5"],"theorem":"Arxiv.«2607.06396».exists_cover_of_isAcyclic"},{"answerKinds":[],"category":"test","docstring":"The empty cover works when there are no edges, so the bound is attained with room to spare\non edgeless graphs. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.06396».AlonTarsi","statement":"∀ {V : Type u_1} [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],\n  G.edgeFinset = ∅ →\n    ∃ C, Arxiv.«2607.06396».IsCycleCover G C ∧ ↑(Arxiv.«2607.06396».totalLength C) ≤ 7 / 5 * ↑G.edgeFinset.card","subjects":["5"],"theorem":"Arxiv.«2607.06396».exists_cover_of_edgeFinset_eq_empty"},{"answerKinds":[],"category":"API","docstring":"A cycle has at least three edges, so any cover of a graph with an edge has total length at\nleast three. This is what makes the `7/5` bound a real constraint rather than a formality. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.06396».AlonTarsi","statement":"∀ {V : Type u_1} {G : SimpleGraph V} (c : G.Cycle), 3 ≤ c.length","subjects":["5"],"theorem":"Arxiv.«2607.06396».three_le_length"},{"answerKinds":[],"category":"research solved","docstring":"An integer value is divisible in a way that forces it to be large: if $x_n = m$ then\n$K_n \\mid 1 + m^2$, and $|m| \\geq \\sqrt{K_n - 1}$ once $K_n > 1$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/AxiomMath/TanArctan/blob/5382d3c20ee3f30e2cbd84362eb07a7e93250348/output/solution.lean#L176"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.05739».TanArctanSum","statement":"∀ {n : ℕ},\n  1 ≤ n →\n    Arxiv.«2607.05739».A n ≠ 0 →\n      ∀ {m : ℤ},\n        Arxiv.«2607.05739».x n = ↑m →\n          ↑(Arxiv.«2607.05739».kernel n) ∣ 1 + m ^ 2 ∧\n            (1 < Arxiv.«2607.05739».kernel n → √(↑(Arxiv.«2607.05739».kernel n) - 1) ≤ |↑m|)","subjects":["11"],"theorem":"Arxiv.«2607.05739».kernel_dvd_of_eq_intCast"},{"answerKinds":[],"category":"test","docstring":"The conjecture holds at the first few values it covers. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.05739».TanArctanSum","statement":"∀ n ∈ Finset.Icc 5 10, ¬Arxiv.«2607.05739».IsIntegerValue n","subjects":["11"],"theorem":"Arxiv.«2607.05739».not_isIntegerValue_of_mem_Icc_five_ten"},{"answerKinds":[],"category":"research solved","docstring":"Every exceptional index sits close to a multiple of $\\pi/2$ in angle. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/AxiomMath/TanArctan/blob/5382d3c20ee3f30e2cbd84362eb07a7e93250348/output/solution.lean#L453"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.05739».TanArctanSum","statement":"∀ {n : ℕ}, n ∈ Arxiv.«2607.05739».exceptional → ∃ j, |Arxiv.«2607.05739».angleSum n - ↑j * (Real.pi / 2)| < 2 / ↑n","subjects":["11"],"theorem":"Arxiv.«2607.05739».exists_near_half_pi"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture (Amdeberhan-Medina-Moll, 2008).** For every integer $n \\geq 5$, the value\n$$x_n = \\tan(\\arctan 1 + \\arctan 2 + \\cdots + \\arctan n)$$\nis not an integer.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.05739».TanArctanSum","statement":"True ↔ ∀ (n : ℕ), 5 ≤ n → ¬Arxiv.«2607.05739».IsIntegerValue n","subjects":["11"],"theorem":"Arxiv.«2607.05739».tan_arctan_sum_not_integer"},{"answerKinds":[],"category":"research solved","docstring":"The exceptional indices are sparse: $\\#(E \\cap [1,N]) = O(\\log N)$. This is the sense in\nwhich integer values are rare, and it is what [Ono26] proves. ","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/AxiomMath/TanArctan/blob/5382d3c20ee3f30e2cbd84362eb07a7e93250348/output/solution.lean#L975"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.05739».TanArctanSum","statement":"∃ C N₀, 0 < C ∧ ∀ (N : ℕ), N₀ ≤ N → ↑(Arxiv.«2607.05739».exceptional ∩ Set.Icc 1 N).ncard ≤ C * Real.log ↑N","subjects":["11"],"theorem":"Arxiv.«2607.05739».exceptional_ncard_le"},{"answerKinds":[],"category":"test","docstring":"The four integer values the conjecture leaves out. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.05739».TanArctanSum","statement":"Arxiv.«2607.05739».IsIntegerValue 1 ∧\n  Arxiv.«2607.05739».IsIntegerValue 2 ∧ Arxiv.«2607.05739».IsIntegerValue 3 ∧ Arxiv.«2607.05739».IsIntegerValue 4","subjects":["11"],"theorem":"Arxiv.«2607.05739».isIntegerValue_of_le_four"},{"answerKinds":[],"category":"textbook","docstring":"$x_n$ satisfies $x_1 = 1$ and $x_n = \\dfrac{x_{n-1} + n}{1 - n x_{n-1}}$, which is the\ntangent addition formula. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.05739».TanArctanSum","statement":"∀ (n : ℕ),\n  1 ≤ n → Arxiv.«2607.05739».x (n + 1) = (Arxiv.«2607.05739».x n + (↑n + 1)) / (1 - (↑n + 1) * Arxiv.«2607.05739».x n)","subjects":["11"],"theorem":"Arxiv.«2607.05739».x_succ"},{"answerKinds":[],"category":"textbook","docstring":"$x_n$ is the tangent of the partial sum of arctangents it is named for. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.05739».TanArctanSum","statement":"∀ (n : ℕ), 1 ≤ n → ↑(Arxiv.«2607.05739».x n) = Real.tan (∑ k ∈ Finset.Icc 1 n, Real.arctan ↑k)","subjects":["11"],"theorem":"Arxiv.«2607.05739».x_eq_tan_sum_arctan"},{"answerKinds":[],"category":"test","docstring":"$x_1 = 1$, $x_2 = -3$, $x_3 = 0$, $x_4 = 4$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.05739».TanArctanSum","statement":"Arxiv.«2607.05739».x 1 = 1 ∧ Arxiv.«2607.05739».x 2 = -3 ∧ Arxiv.«2607.05739».x 3 = 0 ∧ Arxiv.«2607.05739».x 4 = 4","subjects":["11"],"theorem":"Arxiv.«2607.05739».x_values"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 5 (Fradelizi-Manui-Meyer-Ndiaye, 2026).** Among planar convex bodies with a\ncentre of symmetry containing the origin, for $p > 1$, equality in Corollary 29 holds only for\nparallelograms with a vertex at the origin.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.03582».LpRogersShephard","statement":"True ↔\n  ∀ (K : Set (EuclideanSpace ℝ (Fin 2))),\n    Convex ℝ K →\n      IsCompact K →\n        (interior K).Nonempty →\n          Arxiv.«2607.03582».HasCentreOfSymmetry K →\n            0 ∈ K →\n              ∀ (p q : ℝ),\n                1 < p →\n                  1 / p + 1 / q = 1 →\n                    MeasureTheory.volume (Arxiv.«2607.03582».lpSum p K (-K)) =\n                        ENNReal.ofReal (Arxiv.«2607.03582».rsConstant q) * MeasureTheory.volume K →\n                      Arxiv.«2607.03582».IsParallelogramAtOrigin K","subjects":["52"],"theorem":"Arxiv.«2607.03582».isParallelogramAtOrigin_of_volume_lpSum_eq"},{"answerKinds":[],"category":"API","docstring":"At $p = 2$ the conjugate is also $2$, and $\\Gamma(3/2)^2 / \\Gamma(2) = \\pi/4$, so the\nconstant is $\\pi/2 + 2$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2607.03582».LpRogersShephard","statement":"Arxiv.«2607.03582».rsConstant 2 = Real.pi / 2 + 2","subjects":["52"],"theorem":"Arxiv.«2607.03582».rsConstant_two"},{"answerKinds":[],"category":"research solved","docstring":"**Corollary 29 (Fradelizi-Manui-Meyer-Ndiaye, 2026).** For a planar convex body $K$ with a\ncentre of symmetry containing the origin and $p > 1$,\n$$|K \\oplus_p -K| \\leq \\left(\\frac{2\\Gamma(1+1/q)^2}{\\Gamma(1+2/q)} + 2\\right)|K|,$$\nwhere $q$ is the Hölder conjugate of $p$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.03582».LpRogersShephard","statement":"∀ (K : Set (EuclideanSpace ℝ (Fin 2))),\n  Convex ℝ K →\n    IsCompact K →\n      (interior K).Nonempty →\n        Arxiv.«2607.03582».HasCentreOfSymmetry K →\n          0 ∈ K →\n            ∀ (p q : ℝ),\n              1 < p →\n                1 / p + 1 / q = 1 →\n                  MeasureTheory.volume (Arxiv.«2607.03582».lpSum p K (-K)) ≤\n                    ENNReal.ofReal (Arxiv.«2607.03582».rsConstant q) * MeasureTheory.volume K","subjects":["52"],"theorem":"Arxiv.«2607.03582».volume_lpSum_le"},{"answerKinds":[],"category":"research solved","docstring":"Parallelograms with a vertex at the origin attain the bound, which is the sharpness half of\nCorollary 29.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2607.03582».LpRogersShephard","statement":"∀ (K : Set (EuclideanSpace ℝ (Fin 2))),\n  Arxiv.«2607.03582».IsParallelogramAtOrigin K →\n    ∀ (p q : ℝ),\n      1 < p →\n        1 / p + 1 / q = 1 →\n          MeasureTheory.volume (Arxiv.«2607.03582».lpSum p K (-K)) =\n            ENNReal.ofReal (Arxiv.«2607.03582».rsConstant q) * MeasureTheory.volume K","subjects":["52"],"theorem":"Arxiv.«2607.03582».volume_lpSum_eq_of_isParallelogramAtOrigin"},{"answerKinds":[],"category":"test","docstring":"but $(1, 2, 3)$ is not $2$-less than $(1, 2, 4)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"¬Arxiv.«1609.08688».lt₂ ![1, 2, 3] ![1, 2, 4]","subjects":["5"],"theorem":"Arxiv.«1609.08688».not_lt₂_example"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"Arxiv.«1609.08688».sequenceProduct [![1, 1, 1]] [![1, 1, 1]] = toLex [toLex ![(1, 1), (1, 1), (1, 1)]]","subjects":["5"],"theorem":"Arxiv.«1609.08688».sequenceProduct_example"},{"answerKinds":[],"category":"API","docstring":"In a set of more than $n^2$ triples with coordinates from $\\{1, ..., n\\}$ we must\nhave two triples that are equal in their first two coordinates. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {s : List (Fin 3 → ℕ)} {n : ℕ},\n  2 ≤ n → (∀ a ∈ s, Set.range a ⊆ Set.Icc 1 n) → s.length > n ^ 2 → ∃ i j, i ≠ j ∧ s[i] 0 = s[j] 0 ∧ s[i] 1 = s[j] 1","subjects":["5"],"theorem":"Arxiv.«1609.08688».exists_pair_of_mem_Icc"},{"answerKinds":[],"category":"research solved","docstring":"$F(n) \\leq n^2 / \\exp(\\Omega(\\log^*(n)))$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∃ Ω,\n  (fun n => ↑(↑n).iteratedLog) =O[Filter.atTop] Ω ∧\n    ∀ (n : ℕ), ↑(Arxiv.«1609.08688».maximalLength n) ≤ ↑n ^ 2 / Real.exp (Ω n)","subjects":["5"],"theorem":"Arxiv.«1609.08688».maximalLength_le_isBigO"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {α : Type u_1} (a : α), (Arxiv.«1609.08688».tripleProduct (fun x => a) fun x => a) = toLex fun x => (a, a)","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Arxiv.«1609.08688».tripleProduct_const"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {α : Type u_1} [inst : LinearOrder α] {val : α} {s : List (Fin 3 → α)},\n  Arxiv.«1609.08688».IsIncreasing₂ s → (∀ a ∈ s, ∀ (j : Fin 3), a j = val) → s.length < 2","subjects":["5"],"theorem":"Arxiv.«1609.08688».isIncreasing₂_const_length"},{"answerKinds":[],"category":"research solved","docstring":"Suppose that for some $n$ we have $F(n) = n ^ {\\alpha}$. Then there are arbitrarily\nlarge $m$ such that $F(m) \\geq m^{\\alpha}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {n : ℕ} {e : ℝ},\n  1 < n →\n    ↑(Arxiv.«1609.08688».maximalLength n) = ↑n ^ e →\n      ∀ᶠ (m : ℕ) in Filter.atTop, ↑m ^ e ≤ ↑(Arxiv.«1609.08688».maximalLength m)","subjects":["5"],"theorem":"Arxiv.«1609.08688».maximalLength_pow"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {α : Type u_1} [inst : LinearOrder α] (a : Fin 3 → α), ¬Arxiv.«1609.08688».lt₂ a a","subjects":["5"],"theorem":"Arxiv.«1609.08688».not_lt₂_self"},{"answerKinds":[],"category":"test","docstring":"$(5, 6, 1) <_2 (7, 7, 7)$ ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"Arxiv.«1609.08688».lt₂ ![5, 6, 1] ![7, 7, 7]","subjects":["5"],"theorem":"Arxiv.«1609.08688».lt₂_example_2"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {α : Type u_1} (a : α), Arxiv.«1609.08688».tripleProduct ![a, a, a] ![a, a, a] = toLex ![(a, a), (a, a), (a, a)]","subjects":["5"],"theorem":"Arxiv.«1609.08688».tripleProduct_vecConst_const"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"Arxiv.«1609.08688».maximalLength 1 = 1","subjects":["5"],"theorem":"Arxiv.«1609.08688».maximalLength_one"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"Arxiv.«1609.08688».maximalLength 0 = 0","subjects":["5"],"theorem":"Arxiv.«1609.08688».maximalLength_zero"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"Arxiv.«1609.08688».maximalLength 4 = 8","subjects":["5"],"theorem":"Arxiv.«1609.08688».maximalLength_four"},{"answerKinds":[],"category":"test","docstring":"$(7, 7, 7) <_2 (7, 8, 9)$ ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"Arxiv.«1609.08688».lt₂ ![7, 7, 7] ![7, 8, 9]","subjects":["5"],"theorem":"Arxiv.«1609.08688».lt₂_example_3"},{"answerKinds":[],"category":"research open","docstring":"$F(n) \\leq n^{3/2}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ (n : ℕ), ↑(Arxiv.«1609.08688».maximalLength n) ≤ √↑n ^ 3","subjects":["5"],"theorem":"Arxiv.«1609.08688».maximalLength_le_strong"},{"answerKinds":[],"category":"research solved","docstring":"Moreover, whenever $n$ is a perfect square we have $F(n) \\geq n^{3/2}$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {n : ℕ}, IsSquare n → n.sqrt ^ 3 ≤ Arxiv.«1609.08688».maximalLength n","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"Arxiv.«1609.08688».maximalLength_ge_of_isSquare"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {α : Type u_1} [inst : LinearOrder α] {a b : Fin 3 → α},\n  ¬Arxiv.«1609.08688».lt₂ a b ↔ ∀ (i j : Fin 3), i ≠ j → a i < b i → b j ≤ a j","subjects":["5"],"theorem":"Arxiv.«1609.08688».not_lt₂"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {α : Type u_1} [inst : LT α] (a : Fin 3 → α), Arxiv.«1609.08688».IsIncreasing₂ [a]","subjects":["5"],"theorem":"Arxiv.«1609.08688».isIncreasing₂_singleton"},{"answerKinds":[],"category":"research solved","docstring":"For all $n$ we have $F(n) \\leq n^2$.\n\nThis is the upper bound in [GoLo21, Proposition 1.4], proved by applying the\npigeonhole principle to the first two coordinates.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ (n : ℕ), Arxiv.«1609.08688».maximalLength n ≤ n ^ 2","subjects":["5"],"theorem":"Arxiv.«1609.08688».maximalLength_le"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {α : Type u_1} [inst : LinearOrder α] {a b : Fin 3 → α} (i j : Fin 3),\n  i ≠ j → b i ≤ a i → b j ≤ a j → ¬Arxiv.«1609.08688».lt₂ a b","subjects":["5"],"theorem":"Arxiv.«1609.08688».not_lt₂_of_exists"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {α : Type u_1} [inst : LT α], Arxiv.«1609.08688».IsIncreasing₂ []","subjects":["5"],"theorem":"Arxiv.«1609.08688».isIncreasing₂_nil"},{"answerKinds":[],"category":"test","docstring":"For example, $(3, 3, 9) <_2 (5, 6, 1)$. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"Arxiv.«1609.08688».lt₂ ![3, 3, 9] ![5, 6, 1]","subjects":["5"],"theorem":"Arxiv.«1609.08688».lt₂_example_1"},{"answerKinds":[],"category":"API","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∀ {α : Type u_1} [inst : LinearOrder α] {a b : Fin 3 → α}, (∀ (i : Fin 3), b i ≤ a i) → ¬Arxiv.«1609.08688».lt₂ a b","subjects":["5"],"theorem":"Arxiv.«1609.08688».not_lt₂_of_forall_le"},{"answerKinds":[],"category":"API","docstring":"The $2$-less relation is not transitive on the naturals. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1609.08688».sIncreasingrTuples","statement":"∃ a b c, Arxiv.«1609.08688».lt₂ a b ∧ Arxiv.«1609.08688».lt₂ b c ∧ ¬Arxiv.«1609.08688».lt₂ a c","subjects":["5"],"theorem":"Arxiv.«1609.08688».not_trans_lt₂_nat"},{"answerKinds":[],"category":"research solved","docstring":"For all sufficiently large $n$, the complete graph $K_{2n+1}$ decomposes into\n$2n+1$ edge-disjoint copies of any tree $T$ with $n$ edges.\n\nA \"copy\" of $T$ is the image $T.\\text{map}(f_i)$ of $T$ under a vertex embedding\n$f_i : V \\hookrightarrow \\text{Fin}(2n+1)$; the copies are pairwise edge-disjoint\nand together cover every edge of $K_{2n+1}$.\n\nThis follows from `kotzig_conjecture_large`; see `Paper/RingelConjecture.lean` for the open form.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2001.02665».RingelConjecture","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ {V : Type} [Finite V] (T : SimpleGraph V),\n    T.IsTree →\n      T.edgeSet.ncard = n →\n        ∃ f,\n          (Pairwise fun i j => Disjoint (SimpleGraph.map (⇑(f i)) T).edgeSet (SimpleGraph.map (⇑(f j)) T).edgeSet) ∧\n            ⨆ i, SimpleGraph.map (⇑(f i)) T = ⊤","subjects":["5"],"theorem":"Arxiv.«2001.02665».ringel_conjecture_large"},{"answerKinds":[],"category":"research solved","docstring":"For all sufficiently large $n$, the complete graph $K_{2n+1}$ decomposes into\n$2n+1$ edge-disjoint copies of any tree $T$ with $n$ edges via cyclic shifts of a single\nembedding.\n\nThe $2n+1$ copies are $f_0, f_1, \\dots, f_{2n}$ where $f_i(v) = f_0(v) + i$ for all vertices\n$v$ — that is, each copy is obtained by adding $i \\pmod{2n+1}$ to every vertex of the\nbase copy. This is strictly stronger than `ringel_conjecture_large`.\n\nKotzig conjectured this holds for all $n$; see `Paper/KotzigConjecture.lean`. Montgomery,\nPokrovskiy, and Sudakov prove it for all sufficiently large $n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2001.02665».RingelConjecture","statement":"∀ᶠ (n : ℕ) in Filter.atTop,\n  ∀ {V : Type} [Finite V] (T : SimpleGraph V),\n    T.IsTree →\n      T.edgeSet.ncard = n →\n        ∃ f,\n          (∀ (i : Fin (2 * n + 1)) (v : V), (f i) v = (f 0) v + i) ∧\n            (Pairwise fun i j => Disjoint (SimpleGraph.map (⇑(f i)) T).edgeSet (SimpleGraph.map (⇑(f j)) T).edgeSet) ∧\n              ⨆ i, SimpleGraph.map (⇑(f i)) T = ⊤","subjects":["5"],"theorem":"Arxiv.«2001.02665».kotzig_conjecture_large"},{"answerKinds":[],"category":"test","docstring":"The sequence $(1,1,1,-1)$ is a Barker sequence. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2104.00502».BarkerSequence","statement":"Arxiv.«2104.00502».IsBarkerSequence [1, 1, 1, -1]","subjects":["5","11","94"],"theorem":"Arxiv.«2104.00502».isBarkerSequence_length_four"},{"answerKinds":[],"category":"research open","docstring":"Every Barker sequence has length at most $13$. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2104.00502».BarkerSequence","statement":"∀ (a : List ℤ), Arxiv.«2104.00502».IsBarkerSequence a → a.length ≤ 13","subjects":["5","11","94"],"theorem":"Arxiv.«2104.00502».barker_conjecture"},{"answerKinds":[],"category":"test","docstring":"The known sequence $(1,1,1,1,1,-1,-1,1,1,-1,1,-1,1)$ of length $13$ is a Barker sequence.\n","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2104.00502».BarkerSequence","statement":"Arxiv.«2104.00502».IsBarkerSequence [1, 1, 1, 1, 1, -1, -1, 1, 1, -1, 1, -1, 1]","subjects":["5","11","94"],"theorem":"Arxiv.«2104.00502».isBarkerSequence_length_thirteen"},{"answerKinds":[],"category":"test","docstring":"The constant sequence $(1,1,1,1)$ is not a Barker sequence. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«2104.00502».BarkerSequence","statement":"¬Arxiv.«2104.00502».IsBarkerSequence [1, 1, 1, 1]","subjects":["5","11","94"],"theorem":"Arxiv.«2104.00502».not_isBarkerSequence_constant_four"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 1.6 (Even case).**\nFor a nonempty isolate-free graph $G$ on $n$ vertices,\nif $D$ is even, then $(D + 2)^2 \\cdot i(G) \\leq (D^2 + 4) \\cdot n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2107.00295».IndependentDomination","statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  0 < G.minDegree →\n    Even G.maxDegree →\n      have D := G.maxDegree;\n      have i := G.indepDominationNumber;\n      have n := Fintype.card V;\n      (D + 2) ^ 2 * i ≤ (D ^ 2 + 4) * n","subjects":["5"],"theorem":"Arxiv.«2107.00295».independentDominationEven"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 1.6 (Odd case).**\nFor a nonempty isolate-free graph $G$ on $n$ vertices,\nif $D$ is odd, then $(D + 1)(D + 3) \\cdot i(G) \\leq (D^2 + 3) \\cdot n$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2107.00295».IndependentDomination","statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n  0 < G.minDegree →\n    Odd G.maxDegree →\n      have D := G.maxDegree;\n      have i := G.indepDominationNumber;\n      have n := Fintype.card V;\n      (D + 1) * (D + 3) * i ≤ (D ^ 2 + 3) * n","subjects":["5"],"theorem":"Arxiv.«2107.00295».independentDominationOdd"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1102.4662».AtiyahSutcliffe","statement":"∀ (v : AtiyahSutcliffe.Point), AtiyahSutcliffe.linearFactor (AtiyahSutcliffe.directionLift v) ≠ 0","subjects":["51","70"],"theorem":"AtiyahSutcliffe.linearFactor_directionLift_ne_zero"},{"answerKinds":[],"category":"research open","docstring":"[Atiyah–Sutcliffe Conjecture 1](https://doi.org/10.1098/rspa.2001.0913), stated as\nConjecture 1.1 in [Mazur–Petrenko](https://arxiv.org/abs/1102.4662): the configuration\npolynomials are linearly independent. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1102.4662».AtiyahSutcliffe","statement":"∀ {n : ℕ} (x : Fin n → AtiyahSutcliffe.Point),\n  Function.Injective x → LinearIndependent ℂ (AtiyahSutcliffe.pointPolynomial x)","subjects":["51","70"],"theorem":"AtiyahSutcliffe.conjecture_one"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1102.4662».AtiyahSutcliffe","statement":"AtiyahSutcliffe.directionLift (EuclideanSpace.single 2 1) = (1, 0)","subjects":["51","70"],"theorem":"AtiyahSutcliffe.directionLift_northPole"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1102.4662».AtiyahSutcliffe","statement":"AtiyahSutcliffe.directionLift (EuclideanSpace.single 0 1) = (1, 1)","subjects":["51","70"],"theorem":"AtiyahSutcliffe.directionLift_xAxis"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1102.4662».AtiyahSutcliffe","statement":"∀ (x : Fin 1 → AtiyahSutcliffe.Point), AtiyahSutcliffe.pointPolynomial x 0 = 1","subjects":["51","70"],"theorem":"AtiyahSutcliffe.onePoint_polynomial"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1102.4662».AtiyahSutcliffe","statement":"∀ (x : Fin 1 → AtiyahSutcliffe.Point), LinearIndependent ℂ (AtiyahSutcliffe.pointPolynomial x)","subjects":["51","70"],"theorem":"AtiyahSutcliffe.onePoint_linearIndependent"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1102.4662».AtiyahSutcliffe","statement":"have x := fun i => if i = 0 then 0 else EuclideanSpace.single 0 1;\nAtiyahSutcliffe.pointPolynomial x 0 = MvPolynomial.X 0 - MvPolynomial.X 1","subjects":["51","70"],"theorem":"AtiyahSutcliffe.twoPoint_xAxis_polynomial"},{"answerKinds":[],"category":"research solved","docstring":"The case $k = 1$ is Dirac's theorem. The condition reads $\\delta(G) \\geq n/2$, and no path in\n$G - V(C)$ can hold a vertex, so a longest cycle covers every vertex.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2606.03696».BondyLongestCycles","statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] {G : SimpleGraph V} [inst_2 : DecidableRel G.Adj],\n  G.IsKConnected 1 →\n    ↑(Fintype.card V) / 2 ≤ ↑G.minDegree →\n      ∀ {a : V} (C : G.Walk a a),\n        C.IsCycle →\n          C.length = G.circumference →\n            ∀ (u v : ↑(Arxiv.«2606.03696».offWalk C))\n              (P : (SimpleGraph.induce (Arxiv.«2606.03696».offWalk C) G).Walk u v), P.IsPath → P.support.length + 1 ≤ 1","subjects":["5"],"theorem":"Arxiv.«2606.03696».bondy_conjecture.variants.one"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 1 (Bondy, 1980).** Let $k \\geq 1$ and let $G$ be a $k$-connected graph on $n$\nvertices. If $\\delta(G) \\geq \\frac{n + k(k-1)}{k+1}$, then for every longest cycle $C$ of $G$,\nevery path in $G - V(C)$ has at most $k-1$ vertices.\n\nThe bound on the number of vertices of a path is written as `+ 1 ≤ k` rather than `≤ k - 1`,\nbecause subtraction on `ℕ` is truncated. The two forms agree for `k ≥ 1`.\n\nA longest cycle is a cycle whose length is the circumference of `G`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2606.03696».BondyLongestCycles","statement":"True ↔\n  ∀ (k : ℕ),\n    1 ≤ k →\n      ∀ (V : Type) [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n        G.IsKConnected k →\n          (↑(Fintype.card V) + ↑k * (↑k - 1)) / (↑k + 1) ≤ ↑G.minDegree →\n            ∀ (a : V) (C : G.Walk a a),\n              C.IsCycle →\n                C.length = G.circumference →\n                  ∀ (u v : ↑(Arxiv.«2606.03696».offWalk C))\n                    (P : (SimpleGraph.induce (Arxiv.«2606.03696».offWalk C) G).Walk u v),\n                    P.IsPath → P.support.length + 1 ≤ k","subjects":["5"],"theorem":"Arxiv.«2606.03696».bondy_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"The case $k = 2$ is the theorem of Nash-Williams.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2606.03696».BondyLongestCycles","statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] {G : SimpleGraph V} [inst_2 : DecidableRel G.Adj],\n  G.IsKConnected 2 →\n    (↑(Fintype.card V) + 2) / 3 ≤ ↑G.minDegree →\n      ∀ {a : V} (C : G.Walk a a),\n        C.IsCycle →\n          C.length = G.circumference →\n            ∀ (u v : ↑(Arxiv.«2606.03696».offWalk C))\n              (P : (SimpleGraph.induce (Arxiv.«2606.03696».offWalk C) G).Walk u v), P.IsPath → P.support.length + 1 ≤ 2","subjects":["5"],"theorem":"Arxiv.«2606.03696».bondy_conjecture.variants.two"},{"answerKinds":[],"category":"research solved","docstring":"The case $k = 3$ is proved.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2606.03696».BondyLongestCycles","statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] {G : SimpleGraph V} [inst_2 : DecidableRel G.Adj],\n  G.IsKConnected 3 →\n    (↑(Fintype.card V) + 6) / 4 ≤ ↑G.minDegree →\n      ∀ {a : V} (C : G.Walk a a),\n        C.IsCycle →\n          C.length = G.circumference →\n            ∀ (u v : ↑(Arxiv.«2606.03696».offWalk C))\n              (P : (SimpleGraph.induce (Arxiv.«2606.03696».offWalk C) G).Walk u v), P.IsPath → P.support.length + 1 ≤ 3","subjects":["5"],"theorem":"Arxiv.«2606.03696».bondy_conjecture.variants.three"},{"answerKinds":[],"category":"research solved","docstring":"**Theorem 1.1 (Ma-Ning-Zhao, 2026).** The conjecture holds for every graph with enough\nvertices. The full conjecture, for graphs of every size, stays open.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2606.03696».BondyLongestCycles","statement":"∀ (k : ℕ),\n  1 ≤ k →\n    ∃ N,\n      ∀ (V : Type) [inst : Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst_2 : DecidableRel G.Adj],\n        N ≤ Fintype.card V →\n          G.IsKConnected k →\n            (↑(Fintype.card V) + ↑k * (↑k - 1)) / (↑k + 1) ≤ ↑G.minDegree →\n              ∀ (a : V) (C : G.Walk a a),\n                C.IsCycle →\n                  C.length = G.circumference →\n                    ∀ (u v : ↑(Arxiv.«2606.03696».offWalk C))\n                      (P : (SimpleGraph.induce (Arxiv.«2606.03696».offWalk C) G).Walk u v),\n                      P.IsPath → P.support.length + 1 ≤ k","subjects":["5"],"theorem":"Arxiv.«2606.03696».bondy_conjecture.variants.large"},{"answerKinds":[],"category":"research solved","docstring":"Empirical evidence seems to suggest that Slud's bound does not hold for all $p$, and in fact, as $n\\to\\infty$,\nthe maximal permissible $p$ shrinks to $\\frac{1}{2}$. Also, the following appears to be true:\n\nWhen $p\\in(0,1/2)$ and\n$m = 2k$ is even, and $\\sigma := \\sqrt{p(1-p)}$,\n$$\n  \\mathbb{P}[B(p,m) \\geq m/2] \\geq 1 - \\Phi\\left(\\frac{(1/2-p)\\sqrt{m}}{\\sigma}\\right) + \\frac 1 2\\binom{m}{m/2}\\sigma^{m}.\n$$\n\nA solution of this statement has been put out by Logical Intelligence\nhttps://github.com/logical-intelligence/proofs, see\n[here](https://github.com/logical-intelligence/proofs/blob/main/LI/Conj63_informal_proof.md) for\nand informal sketch of the proof.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/logical-intelligence/proofs/blob/0dbb9215f472c532ca8af1376ed58a7ebca6dec2/LI/Conj63.lean#L8845"}],"hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«0911.2077».Conjecture6_3","statement":"∀ (p : ℝ) (h_p : p ∈ Set.Ioo 0 (1 / 2)) (k : ℕ),\n  0 < k →\n    ∀ (σ : ℝ),\n      σ = √(p * (1 - p)) →\n        1 - ↑(ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)) ((1 / 2 - p) * ↑(NNReal.sqrt (2 * ↑k)) / σ) +\n            1 / 2 * ↑((2 * k).choose k) * σ ^ (2 * k) ≤\n          (ProbabilityTheory.binomial (2 * k) ⟨p, ⋯⟩).real (Set.Icc k (2 * k))","subjects":["60"],"theorem":"Arxiv.«0911.2077».arxiv.id0911_2077.conjecture6_3"},{"answerKinds":[],"category":"research open","docstring":"The Poisson Conjecture in dimension $1$ ($PC_1$) is open. By [AvdE07], Theorem 7, it is\nimplied by the Jacobian conjecture in dimension $2$ (Keller's original problem, open) and\nimplies the Dixmier conjecture for the first Weyl algebra (Problem 1 of Dixmier (1968),\nopen).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«math.0608009».PoissonConjecture","statement":"∀ {K : Type u_1} [inst : Field K] [CharZero K], Arxiv.«math.0608009».PoissonConjectureFor K 1","subjects":["14","17"],"theorem":"Arxiv.«math.0608009».poisson_conjecture.variants.dimension_one"},{"answerKinds":[],"category":"research solved","docstring":"The Poisson Conjecture is false in every dimension $n ≥ 3$, by padding the dimension $3$\ncounterexample with identity coordinates.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«math.0608009».PoissonConjecture","statement":"∀ {K : Type u_1} [inst : Field K] [CharZero K] (n : ℕ), 3 ≤ n → ¬Arxiv.«math.0608009».PoissonConjectureFor K n","subjects":["14","17"],"theorem":"Arxiv.«math.0608009».poisson_conjecture.variants.dimension_ge_three"},{"answerKinds":[],"category":"research solved","docstring":"The Jacobian conjecture in dimension $2n$ implies the Poisson conjecture in dimension `n`\n([AvdE07], Theorem 1 and Theorem 7).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«math.0608009».PoissonConjecture","statement":"∀ {K : Type u_1} [inst : Field K] [CharZero K] (n : ℕ),\n  JacobianConjecture.JacobianConjectureProp K (Fin (2 * n)) → Arxiv.«math.0608009».PoissonConjectureFor K n","subjects":["14","17"],"theorem":"Arxiv.«math.0608009».poisson_conjecture.variants.jacobian_implication"},{"answerKinds":[],"category":"test","docstring":"The Poisson conjecture holds trivially in dimension $0$, where the Poisson algebra is `K`\nitself and the identity is the only endomorphism. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«math.0608009».PoissonConjecture","statement":"∀ {K : Type u_1} [inst : Field K], Arxiv.«math.0608009».PoissonConjectureFor K 0","subjects":["17"],"theorem":"Arxiv.«math.0608009».poissonConjectureFor_zero"},{"answerKinds":[],"category":"research open","docstring":"The Poisson Conjecture in dimension $2$ ($PC_2$) is open. Since the Jacobian conjecture is\nfalse in every dimension $n ≥ 3$ [Alp26], no known implication bounds $PC_2$ from above\nany more; the chain of [AvdE07], Theorem 7 places it as the strongest of the remaining\nopen conjectures $PC_2 \\Longrightarrow DC_2 \\Longrightarrow JC_2 \\Longrightarrow PC_1 \\Longrightarrow DC_1$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«math.0608009».PoissonConjecture","statement":"∀ {K : Type u_1} [inst : Field K] [CharZero K], Arxiv.«math.0608009».PoissonConjectureFor K 2","subjects":["14","17"],"theorem":"Arxiv.«math.0608009».poisson_conjecture.variants.dimension_two"},{"answerKinds":[],"category":"research solved","docstring":"The **Poisson Conjecture** ([AvdE07], the characteristic zero case): for every `n`, every\nendomorphism of the `n`-th canonical Poisson algebra over a field `K` of characteristic\nzero is an automorphism. This is **false**: the Jacobian conjecture fails in dimension $3$\n[Alp26], hence so does $PC_3$ (via [AvdE07], Theorem 7, or directly via the symplectic\nlift of the counterexample map).\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«math.0608009».PoissonConjecture","statement":"∀ {K : Type u_1} [inst : Field K] [CharZero K], ¬∀ (n : ℕ), Arxiv.«math.0608009».PoissonConjectureFor K n","subjects":["14","17"],"theorem":"Arxiv.«math.0608009».poisson_conjecture"},{"answerKinds":[],"category":"test","docstring":"The canonical bracket pairs each position variable with its momentum variable. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«math.0608009».PoissonConjecture","statement":"∀ {K : Type u_1} [inst : Field K] (n : ℕ) (i : Fin n),\n  (MvPolynomial.X (Sum.inl i)).poissonBracket (MvPolynomial.X (Sum.inr i)) = 1","subjects":["17"],"theorem":"Arxiv.«math.0608009».poissonBracket_X_pairing"},{"answerKinds":[],"category":"research solved","docstring":"The Poisson Conjecture is false in dimension $3$: the cotangent (symplectic) lift\n$\\Phi(X_i) = F_i$, $\\Phi(X_{i+3}) = \\sum_j M_{ij} X_{j+3}$ with $M = (JF)^{-\\top}$ of the\ndegree-$6$ counterexample map $F$ of [Alp26] (whose Jacobian determinant is the constant\n$-2$, so $M$ is a polynomial matrix) is a Poisson endomorphism of $P_3(K)$ that is not\ninjective on points — e.g. it identifies $(0, 6, -142, 0, 0, 0)$ and $(1, 0, 2, 0, 0, 0)$\n— and is therefore no automorphism. Alternatively, $PC_3$ fails by [AvdE07], Theorem 7,\nas it implies the Jacobian conjecture in dimension $3$, contradicting [Alp26].\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«math.0608009».PoissonConjecture","statement":"∀ {K : Type u_1} [inst : Field K] [CharZero K], ¬Arxiv.«math.0608009».PoissonConjectureFor K 3","subjects":["14","17"],"theorem":"Arxiv.«math.0608009».poisson_conjecture.variants.dimension_three"},{"answerKinds":[],"category":"test","docstring":"The identity is a Poisson endomorphism. ","hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«math.0608009».PoissonConjecture","statement":"∀ {K : Type u_1} [inst : Field K] (n : ℕ),\n  Arxiv.«math.0608009».IsPoissonEndomorphism (AlgHom.id K (MvPolynomial (Fin n ⊕ Fin n) K))","subjects":["17"],"theorem":"Arxiv.«math.0608009».isPoissonEndomorphism_id"},{"answerKinds":[],"category":"research solved","docstring":"Every finite-dimensional real normed space whose isometry group acts transitively on the\nunit sphere is Euclidean. ","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«math.0110202».BanachMazurRotation","statement":"∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E]\n  [MulAction.IsPretransitive (E ≃ₗᵢ[ℝ] E) ↑(Metric.sphere 0 1)], InnerProductSpaceable E","subjects":["46"],"theorem":"Arxiv.«math.0110202».banach_mazur_rotation_problem.finite_dimensional"},{"answerKinds":[],"category":"research open","docstring":"The Banach--Mazur rotation problem asks whether every separable Banach space whose group of linear\nisometric equivalences acts transitively on the unit sphere is linearly isometric to a Hilbert\nspace.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«math.0110202».BanachMazurRotation","statement":"True ↔\n  ∀ (E : Type u_1) [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E]\n    [TopologicalSpace.SeparableSpace E] [MulAction.IsPretransitive (E ≃ₗᵢ[ℝ] E) ↑(Metric.sphere 0 1)],\n    ∃ H x x_1, Nonempty (E ≃ₗᵢ[ℝ] H)","subjects":["46"],"theorem":"Arxiv.«math.0110202».banach_mazur_rotation_problem"},{"answerKinds":[],"category":"research open","docstring":"The **Zariski Cancellation Problem**: every polynomial ring over a field `k` of characteristic\n`0` is cancellative.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2208.14736».ZariskiCancellation","statement":"∀ {k : Type u_1} [inst : Field k] [CharZero k] {ι : Type u_2} [inst_2 : Fintype ι],\n  Arxiv.«2208.14736».IsCancellative k (MvPolynomial ι k)","subjects":["13","14"],"theorem":"Arxiv.«2208.14736».zariski_cancellation_problem"},{"answerKinds":[],"category":"research solved","docstring":"The positive characteristic case of the Zariski Cancellation Problem is false in dimension `3`\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2208.14736».ZariskiCancellation","statement":"∀ (p : ℕ) [hp : Fact (Nat.Prime p)] {ι : Type u_1} [inst : Fintype ι],\n  Fintype.card ι = 3 → ¬Arxiv.«2208.14736».IsCancellative (ZMod p) (MvPolynomial ι (ZMod p))","subjects":["13","14"],"theorem":"Arxiv.«2208.14736».zariski_cancellation_problem.variants.false_pos_card"},{"answerKinds":[],"category":"research solved","docstring":"The single variable polynomial ring `k[X]` is cancellative in any characteristic\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2208.14736».ZariskiCancellation","statement":"∀ {k : Type u_1} [inst : Field k], Arxiv.«2208.14736».IsCancellative k (Polynomial k)","subjects":["13","14"],"theorem":"Arxiv.«2208.14736».zariski_cancellation_problem.variants.dim_one"},{"answerKinds":[],"category":"research solved","docstring":"The two variable polynomial ring `k[X]` is cancellative in any characteristic\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2208.14736».ZariskiCancellation","statement":"∀ {k : Type u_1} [inst : Field k], Arxiv.«2208.14736».IsCancellative k (MvPolynomial (Fin 2) k)","subjects":["13","14"],"theorem":"Arxiv.«2208.14736».zariski_cancellation_problem.variants.dim_two"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $\\Gamma_{4 \\oplus 3}$ has rank $3$: it is not $2$-generated.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.12342».Conjecture1","statement":"∀ (h₁ h₂ : ↥(Arxiv.«2605.12342».gammaSubgroup 4 3)), Subgroup.closure {h₁, h₂} ≠ ⊤","subjects":["20"],"theorem":"Arxiv.«2605.12342».conjecture_1.variants.rank_4_3"},{"answerKinds":[],"category":"research open","docstring":"**Conjecture 1 (Fernandes, 2026):**\nLet $m \\ge n \\ge 2$ be integers with $(m, n) \\notin \\{(2,2), (3,3), (4,3), (4,4)\\}$.\nThen the group\n$$\n\\Gamma_{m \\oplus n} = \\{(\\sigma_1, \\sigma_2) \\in \\mathrm{S}_m \\times \\mathrm{S}_n :\n  \\mathrm{sgn}(\\sigma_1) = \\mathrm{sgn}(\\sigma_2)\\}\n$$\nhas rank $2$, i.e., minimal generating set of size $2$.\n\nNote: Fernandes states the conjecture for groups of exact rank $2$, which is why $(2,2)$\nis in the exception list: $\\Gamma_{2 \\oplus 2} \\cong C_2$ has rank $1$. The formalised\nconclusion `∃ g₁ g₂, closure {g₁, g₂} = ⊤` encodes 2-generation (at most $2$ generators),\nwhich $\\Gamma_{2 \\oplus 2}$ also satisfies. The other three exceptions $(3,3), (4,3), (4,4)$\nhave rank $3$ and are genuinely not 2-generated.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.12342».Conjecture1","statement":"∀ {m n : ℕ}, 2 ≤ m → 2 ≤ n → n ≤ m → (m, n) ∉ {(2, 2), (3, 3), (4, 3), (4, 4)} → ∃ g₁ g₂, Subgroup.closure {g₁, g₂} = ⊤","subjects":["20"],"theorem":"Arxiv.«2605.12342».conjecture_1"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $\\Gamma_{3 \\oplus 3}$ has rank $3$: it is not $2$-generated.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.12342».Conjecture1","statement":"∀ (h₁ h₂ : ↥(Arxiv.«2605.12342».gammaSubgroup 3 3)), Subgroup.closure {h₁, h₂} ≠ ⊤","subjects":["20"],"theorem":"Arxiv.«2605.12342».conjecture_1.variants.rank_3_3"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $\\Gamma_{2 \\oplus 2} \\cong C_2$ has rank $1$: it is cyclic, generated by a\nsingle element.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.12342».Conjecture1","statement":"∃ g, Subgroup.closure {g} = ⊤","subjects":["20"],"theorem":"Arxiv.«2605.12342».conjecture_1.variants.rank_2_2"},{"answerKinds":[],"category":"research solved","docstring":"It is known that $\\Gamma_{4 \\oplus 4}$ has rank $3$: it is not $2$-generated.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«2605.12342».Conjecture1","statement":"∀ (h₁ h₂ : ↥(Arxiv.«2605.12342».gammaSubgroup 4 4)), Subgroup.closure {h₁, h₂} ≠ ⊤","subjects":["20"],"theorem":"Arxiv.«2605.12342».conjecture_1.variants.rank_4_4"},{"answerKinds":[],"category":"research open","docstring":"Jones's conjecture (first kind): for every positive integer $k$, there are\ninfinitely many primes $p$ that start a first-kind Cunningham chain of\nexactly length $k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1104.1579».CunninghamChain","statement":"∀ (k : ℕ), 0 < k → {p | CunninghamChain.IsFirstKindChainOfLength p k}.Infinite","subjects":["11"],"theorem":"CunninghamChain.infinitely_many_firstKind_chains"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1104.1579».CunninghamChain","statement":"CunninghamChain.IsSecondKindChainOfLength 7 2","subjects":["11"],"theorem":"CunninghamChain.seven_starts_secondKind_length_two"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.Arxiv.«1104.1579».CunninghamChain","statement":"CunninghamChain.IsFirstKindChainOfLength 2 5","subjects":["11"],"theorem":"CunninghamChain.two_starts_firstKind_length_five"},{"answerKinds":[],"category":"research open","docstring":"Jones's conjecture (second kind): for every positive integer $k$, there are\ninfinitely many primes $p$ that start a second-kind Cunningham chain of\nexactly length $k$.\n","hasSorryFreeProof":false,"module":"FormalConjectures.Arxiv.«1104.1579».CunninghamChain","statement":"∀ (k : ℕ), 0 < k → {p | CunninghamChain.IsSecondKindChainOfLength p k}.Infinite","subjects":["11"],"theorem":"CunninghamChain.infinitely_many_secondKind_chains"},{"answerKinds":[],"category":"test","docstring":"The discrete group `ℤ` admits a Lie group structure. ","hasSorryFreeProof":true,"module":"FormalConjectures.HilbertProblems.«5»","statement":"AdmitsLieGroupStructure (Multiplicative ℤ)","subjects":["22"],"theorem":"Hilbert5.admitsLieGroupStructure_multiplicative_int"},{"answerKinds":[],"category":"research open","docstring":"**Hilbert–Smith conjecture**: every locally compact topological group acting continuously\nand faithfully on a connected finite-dimensional topological manifold is a Lie group. ","hasSorryFreeProof":false,"module":"FormalConjectures.HilbertProblems.«5»","statement":"∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] {X : Type u_2} [inst_2 : TopologicalSpace X] [T2Space X]\n  [ConnectedSpace X] [IsTopologicalGroup G] [LocallyCompactSpace G] [inst_7 : MulAction G X] [ContinuousSMul G X]\n  [FaithfulSMul G X], AdmitsLieGroupStructure G","subjects":["22","57","58"],"theorem":"Hilbert5.hilbert_smith_conjecture"},{"answerKinds":[],"category":"research solved","docstring":"**Pardon's theorem** (2013): the Hilbert–Smith conjecture holds for connected 3-manifolds,\nsee [Pardon 2013]. ","hasSorryFreeProof":false,"module":"FormalConjectures.HilbertProblems.«5»","statement":"∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] {X : Type u_3} [inst_2 : TopologicalSpace X] [T2Space X]\n  [ConnectedSpace X] [ChartedSpace (EuclideanSpace ℝ (Fin 3)) X] [IsTopologicalGroup G] [LocallyCompactSpace G]\n  [inst_8 : MulAction G X] [ContinuousSMul G X] [FaithfulSMul G X], AdmitsLieGroupStructure G","subjects":["22","57","58"],"theorem":"Hilbert5.hilbert_smith_conjecture.variants.dimension_three"},{"answerKinds":[],"category":"research solved","docstring":"The Hilbert–Smith conjecture holds for actions by isometries of a connected smooth Riemannian\nmanifold `X`: the isometry group of `X` is a Lie group by the Myers–Steenrod theorem, so `G` has\nno small subgroups and is a Lie group by the Gleason–Montgomery–Zippin theorem.\n\nHere `X` carries its Riemannian distance (`IsRiemannianManifold`), so the metric topology of `X`\nis its manifold topology. ","hasSorryFreeProof":false,"module":"FormalConjectures.HilbertProblems.«5»","statement":"∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] {n : ℕ} {X : Type u_3} [inst_2 : EMetricSpace X]\n  [ConnectedSpace X] [inst_4 : ChartedSpace (EuclideanSpace ℝ (Fin n)) X]\n  [inst_5 : IsManifold (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) (↑⊤) X]\n  [inst_6 : Bundle.RiemannianBundle fun x => TangentSpace (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) x]\n  [IsContMDiffRiemannianBundle (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) (↑⊤) (EuclideanSpace ℝ (Fin n))\n      fun x => TangentSpace (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) x]\n  [IsRiemannianManifold (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) X] [IsTopologicalGroup G]\n  [LocallyCompactSpace G] [inst_11 : MulAction G X] [ContinuousSMul G X] [FaithfulSMul G X],\n  (∀ (g : G), Isometry fun x => g • x) → AdmitsLieGroupStructure G","subjects":["22","53","57","58"],"theorem":"Hilbert5.hilbert_smith_conjecture.variants.riemannian"},{"answerKinds":[],"category":"research solved","docstring":"**Hilbert's fifth problem** (Gleason–Montgomery–Zippin, 1952): every Hausdorff,\nsecond-countable topological group modeled on a finite-dimensional Euclidean space is continuously\nisomorphic to a real-analytic Lie group.\n\nThe input `ChartedSpace` supplies only a topological atlas. The compatible analytic atlas and\nanalytic group operations belong to the output `LieGroupPresentation`. ","hasSorryFreeProof":false,"module":"FormalConjectures.HilbertProblems.«5»","statement":"∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] {n : ℕ} [IsTopologicalGroup G] [T2Space G]\n  [SecondCountableTopology G] [ChartedSpace (EuclideanSpace ℝ (Fin n)) G], Nonempty (LieGroupPresentation G n)","subjects":["22","57"],"theorem":"Hilbert5.hilbert_fifth_problem"},{"answerKinds":[],"category":"test","docstring":"The circle group admits a Lie group structure. ","hasSorryFreeProof":true,"module":"FormalConjectures.HilbertProblems.«5»","statement":"AdmitsLieGroupStructure Circle","subjects":["22"],"theorem":"Hilbert5.admitsLieGroupStructure_circle"},{"answerKinds":[],"category":"research open","docstring":"**p-adic formulation**: the p-adic integers `ℤ_[p]` cannot act continuously and faithfully\non a connected finite-dimensional topological manifold. This is equivalent to\n`hilbert_smith_conjecture` by the Gleason–Yamabe theorem together with Newman's theorem on\nperiodic transformations of manifolds. ","hasSorryFreeProof":false,"module":"FormalConjectures.HilbertProblems.«5»","statement":"∀ {X : Type u_2} [inst : TopologicalSpace X] [T2Space X] [ConnectedSpace X] (p : ℕ) [inst_3 : Fact (Nat.Prime p)]\n  [inst_4 : AddAction ℤ_[p] X] [ContinuousVAdd ℤ_[p] X], ¬FaithfulVAdd ℤ_[p] X","subjects":["22","57","58"],"theorem":"Hilbert5.hilbert_smith_padic_formulation"},{"answerKinds":[],"category":"textbook","docstring":"The Motzkin polynomial cannot be written as a sum of squares of polynomials. ","hasSorryFreeProof":false,"module":"FormalConjectures.HilbertProblems.«17»","statement":"¬∃ n, ∃ (_ : 0 < n), ∃ S, Hilbert17.f = ∑ i, S i ^ 2","subjects":["12"],"theorem":"Hilbert17.f_not_sum_of_squares"},{"answerKinds":[],"category":"research solved","docstring":"Hilbert's 17th problem for homogeneous polynomials: characterization of dimensions and degrees\nwhere non-negative polynomials are sums of squares of polynomials.\n","hasSorryFreeProof":false,"module":"FormalConjectures.HilbertProblems.«17»","statement":"∀ {n d : ℕ}, 0 < n → 0 < d → (Hilbert17.Hilbert17thProblemHomogenousPoly n d ↔ n = 1 ∨ n = 2 ∨ d = 1 ∨ n = 3 ∧ d = 2)","subjects":["12"],"theorem":"Hilbert17.hilbert_17th_problem_poly"},{"answerKinds":[],"category":"research solved","docstring":"Hilbert's 17th problem: every non-negative multivariate polynomial is a sum of\nsquares of rational functions.\n","hasSorryFreeProof":false,"module":"FormalConjectures.HilbertProblems.«17»","statement":"∀ {n : ℕ},\n  0 < n →\n    ∀ (f : MvPolynomial (Fin n) ℝ),\n      (∀ (x : Fin n → ℝ), 0 ≤ (MvPolynomial.eval x) f) →\n        ∃ m g, (algebraMap (MvPolynomial (Fin n) ℝ) (MvRatFunc (Fin n) ℝ)) f = ∑ i, g i ^ 2","subjects":["12"],"theorem":"Hilbert17.hilbert_17th_problem"},{"answerKinds":[],"category":"textbook","docstring":"The Motzkin polynomial is non-negative everywhere. ","hasSorryFreeProof":true,"module":"FormalConjectures.HilbertProblems.«17»","statement":"∀ (x y : ℝ), 0 ≤ (MvPolynomial.eval ![x, y]) Hilbert17.f","subjects":["12"],"theorem":"Hilbert17.f_nonneg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.HilbertProblems.«17»","statement":"∀ (n : ℕ), Hilbert17.Hilbert17thProblemHomogenousPoly n 0","subjects":["12"],"theorem":"Hilbert17.Hilbert17thProblemHomogenousPoly_zero_right"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.HilbertProblems.«17»","statement":"∀ (d : ℕ), Hilbert17.Hilbert17thProblemHomogenousPoly 0 d","subjects":["12"],"theorem":"Hilbert17.Hilbert17thProblemHomogenousPoly_zero_left"},{"answerKinds":[],"category":"research open","docstring":"WOWII [Conjecture 61](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$, the size $f(G)$ of a largest induced forest\nsatisfies $f(G) \\ge \\mathrm{residue}(G) + \\lceil \\mathrm{diam}(G) / 3 \\rceil$,\nwhere $\\mathrm{residue}(G)$ is the Havel-Hakimi residue and $\\mathrm{diam}(G)$\nis the diameter of $G$.\n\nSee: Favaron, Mahéo, Saclé (1991) for the residue; DeLaVina's Graffiti.pc for the conjecture.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture61","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],\n  G.Connected → ↑G.residue + ↑⌈↑G.diam / 3⌉ ≤ ↑G.largestInducedForestSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture61.conjecture61"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 65](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/):\n\nDoes every simple connected graph $G$ satisfy\n$f(G) \\ge \\operatorname{dist\\_min}(A) + \\lceil \\operatorname{dist\\_min}(M) / 3 \\rceil$,\nwhere $A$ is the set of minimum-degree vertices, $M$ is the set of maximum-degree vertices,\nand $\\operatorname{dist\\_min}(S)$ is the minimum distance between two distinct vertices of $S$\n(see `distMin`).\n\nThe answer is no. `Counterexample.graph` has a conjectured lower bound of $16$, but every\ninduced forest in it has at most $15$ vertices.\n","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture65","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α)\n    [inst_3 : DecidableRel G.Adj],\n    G.Connected →\n      have A := {v | G.degree v = G.minDegree};\n      have M := {v | G.degree v = G.maxDegree};\n      ↑(G.distMin A) + ↑⌈↑(G.distMin M) / 3⌉ ≤ ↑G.largestInducedForestSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture65.conjecture65"},{"answerKinds":[],"category":"API","docstring":"The largest induced forest in the counterexample has at most $15$ vertices. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture65","statement":"WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.largestInducedForestSize ≤ 15","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture65.Counterexample.largest_induced_forest_le_fifteen"},{"answerKinds":[],"category":"API","docstring":"The minimum distance between maximum-degree vertices is $10$. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture65","statement":"WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.distMin {1, 11} = 10","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture65.Counterexample.distMin_max_degree_vertices"},{"answerKinds":[],"category":"API","docstring":"The minimum-degree vertices of the counterexample are the two path endpoints. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture65","statement":"{v |\n    WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.degree v =\n      WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.minDegree} =\n  {0, 12}","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture65.Counterexample.min_degree_vertices"},{"answerKinds":[],"category":"API","docstring":"The maximum-degree vertices of the counterexample are the two triangle attachment points. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture65","statement":"{v |\n    WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.degree v =\n      WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.maxDegree} =\n  {1, 11}","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture65.Counterexample.max_degree_vertices"},{"answerKinds":[],"category":"API","docstring":"Every induced forest in the counterexample has at most $15$ vertices. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture65","statement":"∀ (S : Finset (Fin 17)),\n  (SimpleGraph.induce (↑S) WrittenOnTheWallII.GraphConjecture65.Counterexample.graph).IsAcyclic → S.card ≤ 15","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture65.Counterexample.forest_card_le_fifteen"},{"answerKinds":[],"category":"API","docstring":"An acyclic induced subgraph cannot contain all three vertices of a triangle. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture65","statement":"∀ (S : Finset (Fin 17)),\n  (SimpleGraph.induce (↑S) WrittenOnTheWallII.GraphConjecture65.Counterexample.graph).IsAcyclic →\n    ∀ {a b c : Fin 17},\n      WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.Adj a b →\n        WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.Adj b c →\n          WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.Adj c a → ¬{a, b, c} ⊆ S","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture65.Counterexample.triangle_not_subset_of_isAcyclic"},{"answerKinds":[],"category":"API","docstring":"The counterexample is connected. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture65","statement":"WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.Connected","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture65.Counterexample.connected"},{"answerKinds":[],"category":"API","docstring":"The minimum distance between minimum-degree vertices is $12$. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture65","statement":"WrittenOnTheWallII.GraphConjecture65.Counterexample.graph.distMin {0, 12} = 12","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture65.Counterexample.distMin_min_degree_vertices"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 85](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\nasked whether every simple connected graph $G$ satisfies\n$\\operatorname{tree}(G) \\geq\n\\lceil\\sqrt{1 + 2\\min_v \\operatorname{distEven}(v)}\\rceil$.\nThe answer is no, as witnessed by $C_5[K_4]$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Kuberwastaken/wowii-63-85-counterexample/blob/cba739842ec59adf7426c180009175b31935701d/lean/WOWII85.lean#L171-L183"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture85","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α),\n    G.Connected →\n      have minDistEven := (Finset.image (fun x => G.distEven x) Finset.univ).min' ⋯;\n      ↑⌈√(1 + 2 * ↑minDistEven)⌉ ≤ ↑G.largestInducedTreeSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture85.conjecture85"},{"answerKinds":[],"category":"research open","docstring":"WOWII [Conjecture 100](http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj100)\n(status O):\n\nFor a simple connected graph `G`,\n`α(G) ≤ ⌈(max_v l(v) + 0.5 · degreeL2Norm(Gᶜ)) / 2⌉`\nwhere `α(G) = G.indepNum` is the independence number,\n`max_v l(v)` is the maximum over all vertices of the independence number of\nthe neighbourhood (in `G`), and `degreeL2Norm(Gᶜ)` is the square root of the\nsum of the squares of the degrees in the complement `Gᶜ`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture100","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α)\n  [inst_3 : DecidableRel G.Adj],\n  G.Connected →\n    have maxL := (Finset.image G.indepNeighborsCard Finset.univ).max' ⋯;\n    ↑G.indepNum ≤ ↑⌈(↑maxL + 1 / 2 * Gᶜ.degreeL2Norm) / 2⌉","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture100.conjecture100"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 144](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$,\n$\\mathrm{tree}(G) \\ge \\mathrm{girth}(G) - 1 + \\mathrm{ecc}(\\mathrm{Centers})$\nwhere $\\mathrm{tree}(G)$ is the largest induced tree size, $\\mathrm{girth}(G)$ is\nthe length of the shortest cycle ($0$ if acyclic),\n$\\mathrm{Centers} = G.\\mathrm{center}$ is the set of vertices with minimum\neccentricity (the center of $G$), and $\\mathrm{ecc}(\\mathrm{Centers})$ is the\neccentricity of the center set — the maximum distance from any non-center\nvertex to the nearest center vertex.\n\n**Proof sketch.** Let $g = \\mathrm{girth}(G)$ and\n$e = \\mathrm{ecc}(G, \\mathrm{center}(G))$. The acyclic and $e = 0$ cases are\nimmediate. If $g \\le e + 2$, the formalized Bacsó--Tuza induced-path argument,\nwith an elementary $e = 1$ case, produces an induced tree on at least $2e + 1$\nvertices. Since $g - 1 + e \\le 2e + 1$, this proves the result.\n\nSuppose instead that $e + 3 \\le g$, and fix a shortest cycle $C$. Removing one\nvertex of $C$ leaves an induced tree on $g - 1$ vertices. Analyze the connected\ncomponents outside $C$. If one has at least $e$ vertices, it supplies an\nattached rooted induced tree of order $e$. Otherwise, let $D$ be the sum of the\ncomponents' attachment depths. If $D \\ge e$, select pairwise edge-separated\nrooted branches of total order $e$ and attach them to $C$, deleting a suitable\ncycle vertex. If $D < e$, shortest-cycle contact restrictions bound\ncorresponding exclusion arcs. Their complement consists of central cycle\nvertices and $D$-dominates the graph, forcing $e \\le D$, a contradiction.\n\nFinite graph search was used only during discovery to test candidate\nstructural lemmas and reject false proof templates; no finite-search result is\nused in the universal proof.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/beowulf127/wowii144-lean/blob/046429d509b28c90ee2ec38ae27c1ad377c6a5fc/WOWII144/Main.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture144","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α),\n  G.Connected → ↑G.girth - 1 + ↑(G.ecc G.center) ≤ ↑G.largestInducedTreeSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture144.conjecture144"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 142](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/):\n\nFor a simple connected graph $G$,\n$\\mathrm{tree}(G) \\ge (2/3) \\cdot \\mathrm{girth}(G) + \\mathrm{ecc}(B)$\nwhere $\\mathrm{tree}(G)$ is the largest induced tree size, $\\mathrm{girth}(G)$\nis the length of the shortest cycle ($0$ if acyclic), $B$ is the set of\nboundary vertices (those of maximum eccentricity), and $\\mathrm{ecc}(B)$ is\nthe eccentricity of the set $B$.\n\n**Proof sketch.** The proof first strengthens the real-valued target to the\nintegral bound\n$$\n\\mathrm{ecc}(B) + \\mathrm{girth}(G)\n  - \\lfloor \\mathrm{girth}(G) / 3 \\rfloor \\leq \\mathrm{tree}(G).\n$$\nFor a cyclic graph, start with a shortest cycle $K$. It is chordless, so\nremoving one vertex leaves an induced path on $\\mathrm{girth}(G)-1$ vertices.\nShortest paths from selected vertices to $K$ are called *descents*. Pairwise\nnoninteracting descents attach to the retained cycle path as disjoint branches.\nIf two descents interact, the proof cuts at the first interaction encountered\nfrom the outer endpoint toward the cycle and splices the paths there.\nMinimality of that interaction makes the retained prefix disjoint and\nnonadjacent to the first descent; geodesicity excludes chords, while girth at\nleast five rules out extra cross-edges and cycle attachments.\n\nChoose a vertex realizing $\\mathrm{ecc}(B)$ and two endpoints of a diametral\ngeodesic. Either two of their descents interact, directly giving a sufficiently\nlong spliced tree, or all three are separate, in which case a three-point\ndistance inequality forces their total length to be large enough. This proves\nthe integral bound in the main range. Girths three and four, small\n$\\mathrm{ecc}(B)$, and the two remaining congruence cases are handled by\nseparate geodesic/cycle certificates with one or two descents; the acyclic case\nfollows from a diametral path. Finally,\n$$\n\\mathrm{girth}(G)-\\lfloor\\mathrm{girth}(G)/3\\rfloor\n  = \\lceil 2\\,\\mathrm{girth}(G)/3\\rceil\n  \\geq 2\\,\\mathrm{girth}(G)/3,\n$$\nwhich gives the stated real-valued inequality.\n\nThe argument was developed with assistance from ChatGPT Pro. The linked Lean 4\nformalization was produced with assistance from OpenAI Codex and checked by\nLean.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/AlperTheKing/formal-conjectures/blob/46bf39015f5c3c3ba3bfcf9f752b4b1e49b584ac/FormalConjecturesForMathlib/WrittenOnTheWallII/GraphConjecture142Proof.lean#L3927-L3932"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture142","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],\n  G.Connected →\n    have B := G.maxEccentricityVertices;\n    2 / 3 * ↑G.girth + ↑(G.eccSet B) ≤ ↑G.largestInducedTreeSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture142.conjecture142"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 1](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G` the maximum number of leaves of a spanning\ntree satisfies `Ls(G) ≥ n(G) + 1 - 2·m(G)` where `n(G)` counts vertices and\n`m(G)` is the size of a maximum matching.\n\nA formal proof reduces to a spanning tree `T`. If `I` is the set of non-leaves\nof `T`, Hall's theorem applied to a bipartition of `T` gives a matching `M`\nwith `|I| + 1 ≤ 2 * |M|`. Since `|V|` is the sum of the numbers of leaves and\nnon-leaves, and every matching and leaf count constructed in `T` is admissible\nfor the corresponding supremum in `G`, the stated inequality follows.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/MiskinAleksandr23/WOWII-1/blob/eda16f6e96b313bd112351ae9859133b77d537c9/WOWII1/GraphConjecture1.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture1","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],\n  G.Connected → ↑(Fintype.card α) + 1 - 2 * G.matchingNumber ≤ G.Ls","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture1.conjecture1"},{"answerKinds":[],"category":"research open","docstring":"WOWII [Conjecture 141](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`,\n`tree(G) ≥ ⌊girth(G) / 2⌋ - 1 + max_v l(v)`\nwhere `tree(G)` is the number of vertices of a largest induced tree subgraph,\n`girth(G)` is the length of the shortest cycle (0 if acyclic), and\n`l(v) = indepNeighbors G v` is the independence number of the neighbourhood of `v`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture141","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],\n  G.Connected → ↑G.girth / 2 - 1 + ↑(Finset.univ.sup G.indepNeighborsCard) ≤ ↑G.largestInducedTreeSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture141.conjecture141"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 146](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$,\n$\\mathrm{tree}(G) \\ge 2 \\cdot \\mathrm{ecc}(B) / \\mathrm{rad}(G^2)$\nwhere $\\mathrm{tree}(G)$ is the number of vertices in a largest induced subtree,\n$\\mathrm{ecc}(B)$ is the eccentricity of the boundary vertices of $G$ (`eccSet`\nand `maxEccentricityVertices`), and $\\mathrm{rad}(G^2)$ is the radius of the square\ngraph of $G$.\n\nWe state the inequality in the form\n$\\mathrm{tree}(G) \\cdot \\mathrm{rad}(G^2) \\ge 2 \\cdot \\mathrm{ecc}(B)$ to avoid division.\n\n## Informal proof\n\nWrite $t$ for the largest induced-tree order, $r$ and $d$ for the radius and\ndiameter of $G$, and $p$ for the eccentricity of its peripheral set.  First,\nthe distance in the graph square is\n$\\operatorname{dist}_{G^2}(u,v)=\\lceil\\operatorname{dist}_G(u,v)/2\\rceil$,\nso $\\operatorname{rad}(G^2)=\\lceil r/2\\rceil$.  A diametral geodesic is an\ninduced path, giving $t\\ge d+1$, while $p\\le d-1$.  These bounds settle every\ncase except $r=2$, $d=4$, and $p=3$.  In that remaining configuration, choose\na centre and shortest paths to two diametral vertices and to a vertex at\ndistance three from the peripheral set.  A finite case analysis on the two\npossible cross-arm edges (and then the possible chords) always produces an\ninduced tree on at least six vertices.  Hence $t\\,\\operatorname{rad}(G^2)\n\\ge 2p$ in the exceptional case as well.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/akakabrian/WOW-146/blob/f9e0ad75d829170804ce1d8f9fd4c1d4a0085203/WOW146/Conjecture146.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture146","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],\n  G.Connected →\n    0 < WrittenOnTheWallII.GraphConjecture146.graphSquareRadius G →\n      2 * G.eccSet G.maxEccentricityVertices ≤\n        G.largestInducedTreeSize * WrittenOnTheWallII.GraphConjecture146.graphSquareRadius G","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture146.conjecture146"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 16](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, the size `b(G)` of a largest induced bipartite subgraph\nsatisfies `b(G) ≥ 2 * (rad(G) - 1) + max_{v ∈ V} l(v)`, where `rad(G)` is the radius\nof `G` (the minimum eccentricity) and `l(v) = indepNeighborsCard G v` is the independence\nnumber of the neighbourhood of `v`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture16","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α),\n  G.Connected → 2 * (↑G.radius.toNat - 1) + ↑((Finset.image (fun v => G.indepNeighborsCard v) Finset.univ).max' ⋯) ≤ G.b","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"WrittenOnTheWallII.GraphConjecture16.conjecture16"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 143](http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj143):\n\nFor a simple connected graph $G$,\n$\\mathrm{tree}(G) \\ge (\\mathrm{girth}(G) + 1) / \\sigma(G)$,\nwhere $\\mathrm{tree}(G)$ is the largest induced tree size, $\\mathrm{girth}(G)$\nis the length of the shortest cycle, and\n$\\sigma(G) = G.\\mathrm{secondSmallestDegree}$ is the **second-smallest degree**\nof $G$'s degree sequence (per WOWII defEntry 65). We state the inequality in\ndenominator-free form to avoid the $\\sigma = 0$ corner case ($n \\le 1$).\n\nThe proof splits into the acyclic case and two cyclic cases. For an acyclic\ngraph the girth is zero. If $\\sigma \\ge 2$, deleting two consecutive edges from\na shortest cycle leaves an induced tree on $\\mathrm{girth}(G)-1$ vertices, and\nthe factor $\\sigma \\ge 2$ yields the required inequality. If $\\sigma = 1$,\nthere are two degree-one vertices. A maximal induced tree containing them must\nhave an external vertex with two attachments; those attachments create a cycle\nwhose length forces the tree to contain at least $\\mathrm{girth}(G)+1$ vertices.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/DomTheDeveloper/formal-conjectures/blob/693e9aa206a5c6c98598aa4e6e5f3db0994a79b7/FormalConjectures/WrittenOnTheWallII/Proofs/GraphConjecture143.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture143","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],\n  G.Connected → 0 < G.secondSmallestDegree → ↑G.girth + 1 ≤ ↑G.largestInducedTreeSize * ↑G.secondSmallestDegree","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture143.conjecture143"},{"answerKinds":[],"category":"research open","docstring":"WOWII [Conjecture 160](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$,\n$L_s(G) \\ge \\max_v l(v) + \\max_v T(v) \\cdot \\chi_{C_4}(G)$\nwhere:\n\n- $L_s(G) = \\mathrm{SimpleGraph.Ls}\\, G$ is the maximum number of leaves over all\n  spanning trees of $G$,\n- $\\max_v l(v)$ is the maximum local independence number over vertices,\n- $\\max_v T(v)$ is the maximum number of triangles incident to any vertex,\n- $\\chi_{C_4}(G)$ is `1` if $G$ has no cycle of length four and `0` otherwise.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.«160»","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α)\n  [inst_3 : DecidableRel G.Adj],\n  G.Connected →\n    have maxL := (Finset.image G.indepNeighborsCard Finset.univ).max' ⋯;\n    have maxT := WrittenOnTheWallII.GraphConjecture160.maxTrianglesAtVertex G;\n    have cC4 := if ∃ v c, c.IsCycle ∧ c.length = 4 then 0 else 1;\n    ↑maxL + ↑maxT * ↑cC4 ≤ G.Ls","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture160.conjecture160"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 17](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, the size `b(G)` of a largest induced bipartite subgraph\nsatisfies `b(G) ≥ α(G) + ⌈diam(G) / 3⌉`, where `α(G)` is the independence number of `G`\nand `diam(G)` is the diameter of `G`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture17","statement":"∀ {α : Type u_1} [Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α),\n  G.Connected → ↑G.indepNum + ↑⌈↑G.diam / 3⌉ ≤ G.b","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture17.conjecture17"},{"answerKinds":[],"category":"research open","docstring":"WOWII [Conjecture 40](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a nontrivial connected graph `G` the size `f(G)` of a largest induced forest\nsatisfies `f(G) ≥ ceil((p(G) + b(G) + 1)/2)` where `p(G)` is the path cover\nnumber and `b(G)` is the largest induced bipartite subgraph size.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture40","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α),\n  G.Connected → 1 < Fintype.card α → ⌈(↑G.pathCoverNumber + G.b + 1) / 2⌉ ≤ ↑G.largestInducedForestSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture40.conjecture40"},{"answerKinds":[],"category":"research open","docstring":"WOWII [Conjecture 198a](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, if `b(G) ≤ 2 + ecc_avg(G)`, then `G` has a Hamiltonian path.\nHere `b(G)` is the number of vertices in a largest induced bipartite subgraph, and\n`ecc_avg(G)` is the average eccentricity of `G`.\nA Hamiltonian path is a walk visiting every vertex exactly once.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture198a","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α),\n  G.Connected → G.b ≤ 2 + G.averageEccentricity → ∃ a b p, p.IsHamiltonian","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture198a.conjecture198a"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 63](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\nasked whether every simple connected graph $G$ satisfies\n$f(G) \\geq \\lceil(\\min_v \\operatorname{distEven}(v) + b(G) + 1)/3\\rceil$.\nThe answer is no, as witnessed by $C_5[K_4]$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/Kuberwastaken/wowii-63-85-counterexample/blob/cba739842ec59adf7426c180009175b31935701d/lean/WOWII63.lean#L184-L195"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture63","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α),\n    G.Connected →\n      have minDistEven := (Finset.image (fun x => G.distEven x) Finset.univ).min' ⋯;\n      ↑⌈(↑minDistEven + G.b + 1) / 3⌉ ≤ ↑G.largestInducedForestSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture63.conjecture63"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 2](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$,\n$Ls(G) \\ge 2 \\cdot (l(G) - 1)$ where $l(G)$ is the average independence number of\nthe neighbourhoods of the vertices of $G$.\n\nA formal proof has been found with the methods described in\n[arxiv/2605.22763](https://arxiv.org/abs/2605.22763), where an informal proof is also provided.\n\nAnother formal proof combines a spanning-tree leaf bound from connected domination\nwith an ordered-pair double-counting argument for adjacent neighbourhoods.\n\nA third formal proof starts from a triangle-free spanning subgraph of $G$ with the maximum\nnumber of edges. Maximality bounds the neighbourhood independence number of each vertex by its\ndegree in that subgraph; an edge whose endpoint degrees sum to at least $2 \\cdot l(G)$ then\ncarries a double star, which is acyclic and extends to a spanning tree with at least\n$2 \\cdot (l(G) - 1)$ leaves.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/kingcharlezz/formal-conjectures/blob/7d88e8b7946791ff53c322651f73de8d4df0ba53/FormalConjectures/WrittenOnTheWallII/Proofs/GraphConjecture2.lean#L622"},{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/wowii-graph-conjecture-2-lean/blob/fc6cc1f5b857e9c3f9693b98587660eb09606abc/lean/GraphConjecture2.lean#L673"},{"conditions":[],"kind":"lean4","link":"https://github.com/google-deepmind/alphaproof-nexus-results/blob/0647711a71183c1ea492ad60860776617ce1ea88/APNOutputs/AICollaborator/Graphs/GraphConjecture2.lean#L704"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture2","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α),\n  G.Connected → 2 * (G.averageIndepNeighbors - 1) ≤ G.Ls","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture2.conjecture2"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 34](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, `path(G) ≥ ⌈dist_avg(C, V) + dist_avg(M, V)⌉`,\nwhere `path(G)` is the floor of the average distance of `G`, `C` is the set of center\nvertices (those with minimum eccentricity), `M` is the set of maximum-degree vertices,\nand `dist_avg(S, V)` is the average distance from all vertices to the set `S`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture34","statement":"True ↔\n  ∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α)\n    [inst_3 : DecidableRel G.Adj],\n    G.Connected →\n      have C := G.center;\n      have M := {v | G.degree v = G.maxDegree};\n      ⌈G.distavg C + G.distavg M⌉ ≤ ↑G.path","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture34.conjecture34"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 291](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$ with $n > 2$,\n$\\gamma_t(G) \\le k + \\mathrm{frequency}(t_{\\min}(v))$\nwhere:\n\n- $\\gamma_t(G)$ is the total domination number,\n- $k$ is the first step in which a zero appears in the Havel-Hakimi process,\n- $\\mathrm{frequency}(t_{\\min}(v))$ is the number of vertices achieving the\n  minimum triangle count.\n\nThis is false: the source records a 12-vertex counterexample by Zyad Tamimi (July 23, 2026)\nwith $\\gamma_t(G) = 4 > 2 + 1 = k + \\mathrm{frequency}(t_{\\min}(v))$.\n\nThe hypothesis $n > 2$ is stated in the source's full list of conjectures (as for the\nneighbouring Conjectures 290, 292 and 293) but omitted on its list of open conjectures;\nthe counterexample has $12$ vertices, so it refutes both readings.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture291","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α)\n    [inst_3 : DecidableRel G.Adj],\n    G.Connected →\n      2 < Fintype.card α →\n        G.totalDominationNumber ≤\n          WrittenOnTheWallII.GraphConjecture291.havelHakimiZeroStep G +\n            WrittenOnTheWallII.GraphConjecture291.freqMinTriangles G","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture291.conjecture291"},{"answerKinds":[],"category":"research open","docstring":"WOWII [Conjecture 314](http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj314):\n\nFor every finite simple connected graph $G$ with $n > 1$ vertices,\nif $G$ is triangle-free and $\\mathrm{path}(G) \\le 4$, then $G$ is well totally\ndominated.\n\nHere $\\mathrm{path}(G) = \\mathrm{largestInducedPathSize}\\, G$ is the **size of a\nlargest induced path** in $G$, defined locally above.\n\n**Disambiguation.** Earlier revisions of this file used the `SimpleGraph.path`\ninvariant, but that is the *floor of the average distance*, not the size of a\nlargest induced path — a different quantity that makes Conjecture 314 vacuous\nin many cases.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture314","statement":"∀ {α : Type u_1} [Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst : DecidableRel G.Adj],\n  G.Connected →\n    (∀ (a b c : α), G.Adj a b → G.Adj b c → G.Adj c a → False) →\n      WrittenOnTheWallII.GraphConjecture314.largestInducedPathSize G ≤ 4 → G.IsWellTotallyDominated","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture314.conjecture314"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 6](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a connected graph `G` we have\n`Ls(G) ≥ 1 + n(G) - m(G) - a(G)` where `a(G)` is defined via independent sets.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture6","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],\n  G.Connected → 1 + ↑(Fintype.card α) - G.matchingNumber - ↑G.indepNum ≤ G.Ls","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture6.conjecture6"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 109](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$, the independence number $\\alpha(G)$ was\nconjectured to satisfy\n$\\alpha(G) \\le \\lfloor (\\mathrm{residue}(G) + 2 \\cdot b(G)) / 3 \\rfloor$, where\n$\\mathrm{residue}(G)$ is the Havel--Hakimi residue and $b(G)$ is the size of a\nlargest induced bipartite subgraph.\n\nThis is false. A connected graph on 21 vertices has an independent set of size\n15, residue 8, and no induced bipartite subgraph with more than 18 vertices, so\nthe conjectured right-hand side is at most 14.\n\nA smaller counterexample is the connected 13-vertex graph\n$\\overline K_7 \\vee (K_3 \\sqcup K_3)$. It has independence number $7$, residue\n$2$, and largest induced bipartite subgraph size $9$, so its conjectured\nright-hand side is $6$.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/DomTheDeveloper/formal-conjectures/blob/cf59008ef1cd432bf9803275dcf5d62ab1f094a3/FormalConjectures/WrittenOnTheWallII/GraphConjecture109.lean"},{"conditions":[],"kind":"lean4","link":"https://github.com/chelokot/wowii-109-counterexample/blob/543a66ee78565b553f4ba6dc7fd32b5610557913/lean/GraphConjecture109.lean#L29-L139"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture109","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],\n    G.Connected → ↑G.indepNum ≤ ↑⌊(↑G.residue + 2 * G.b) / 3⌋","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture109.conjecture109"},{"answerKinds":[],"category":"research open","docstring":"WOWII [Conjecture 19](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nIf `G` is connected then the size `b(G)` of a largest induced bipartite subgraph\nsatisfies\n`b(G) ≥ FLOOR((∑ ecc(v))/(|V|) + sSup (range (l G)))`, where `ecc(v)` denotes\neccentricity and `l(G)` is the independence number of neighbourhoods.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture19","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] (G : SimpleGraph α) [Nontrivial α],\n  G.Connected → ↑⌊(∑ v, ↑(G.eccent v).toNat) / ↑(Fintype.card α) + sSup (Set.range G.indepNeighbors)⌋ ≤ G.b","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture19.conjecture19"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 4](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nIf `G` is a connected graph then the maximum number of leaves over all spanning\ntrees satisfies `Ls(G) ≥ NG(G) - 1` where `NG(G)` is the minimal neighbourhood\nsize of a non-edge of `G`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture4","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] (G : SimpleGraph α) [inst_2 : DecidableRel G.Adj]\n  [Nontrivial α], G.Connected → G.NG - 1 ≤ G.Ls","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture4.conjecture4"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 23](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, `b(G) ≥ ⌊α(G) + dist_avg(M, V) / 2⌋`, where `b(G)` is\nthe size of a largest induced bipartite subgraph, `α(G)` is the independence number,\nand `M` is the set of maximum-degree vertices, and `dist_avg(M, V)` is the average\ndistance from all vertices to `M`.\n\nThis conjecture is false; there is a counterexample with `b(G) = 19`, `α(G) = 15`,\nand `dist_avg(M, V) = 10`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture23","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],\n    G.Connected →\n      have M := {v | G.degree v = G.maxDegree};\n      ↑⌊↑G.indepNum + G.distavg M / 2⌋ ≤ G.b","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture23.conjecture23"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 194](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, if `α(G) ≤ 1 + l_avg(G)`, then `G` has a Hamiltonian path.\nHere `α(G) = G.indepNum` is the independence number, and\n`l_avg(G) = averageIndepNeighbors G` is the average over all vertices of the independence number\nof the neighbourhood.\nA Hamiltonian path is a walk visiting every vertex exactly once. The answer is no, as witnessed by\nthe 18-vertex graph described above.\n\nCounterexample (Graph6): `Q~~~~~~~~~~~~}~}^~??G??_??_`\n\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/anagnorisis2peripeteia/formal-conjectures/blob/4bff865a14c2cd61fefbffbe9c49cbfc5a89ac45/FormalConjectures/WrittenOnTheWallII/GraphConjecture194.lean#L128-L140"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture194","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α),\n    G.Connected → ↑G.indepNum ≤ 1 + G.averageIndepNeighbors → ∃ a b p, p.IsHamiltonian","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture194.conjecture194"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 322](http://cms.uhd.edu/faculty/delavinae/research/wowII/open.html)\n\nLet `G` be a simple connected graph on `n ≥ 5` vertices. If the maximum over all\nvertices `v` of `l(v)` in the complement graph `Gᶜ` — the independence number of\nthe neighborhood `N(v)` of `v` — is at most 1, then `G` is well totally dominated.\n\nHere `l(v) = α(Gᶜ[N(v)])` is the independence number of the subgraph induced by the\nopen neighborhood of `v` in `Gᶜ`.\n\nThis proof was provided by Samuel Schlesinger.\n","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture322","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] (G : SimpleGraph α) [inst_2 : DecidableRel G.Adj],\n  G.Connected → 5 ≤ Fintype.card α → (∀ (v : α), Gᶜ.indepNeighborsCard v ≤ 1) → G.IsWellTotallyDominated","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture322.conjecture322"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 327](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nLet `G` be a simple connected graph. If `3 · γ(G) = γ_i(G)`, then `G` is well\ntotally dominated, where `γ(G)` is the domination number of `G` and `γ_i(G)` is\nthe independent domination number of `G`.\n\n**Proof Sketch:**\nThe conjecture states that if $3\\gamma(G) = i(G)$ for a connected graph $G$, then $G$ is well totally dominated.\nHowever, this conjecture is **FALSE**.\n\n**Counterexample:**\nConsider a graph $G$ with 12 vertices: $u, v, a_0, a_1, a_2, a_3, a_4, b_0, b_1, b_2, b_3, b_4$.\nThe edges are:\n- $(u, v)$\n- $(u, a_i)$ for all $i \\in \\{0, 1, 2, 3, 4\\}$\n- $(v, b_i)$ for all $i \\in \\{0, 1, 2, 3, 4\\}$\n- $(a_0, b_3), (a_1, b_3), (a_2, b_0), (a_3, b_0), (a_4, b_3), (a_4, b_4)$\n\nProperties of $G$:\n1. **Connected**: Yes, path exists between any two vertices through $u$ and $v$.\n2. **Domination Number $\\gamma(G)$**: The set $\\{u, v\\}$ dominates all vertices. Since there is no universal vertex, $\\gamma(G) = 2$.\n3. **Independent Domination Number $i(G)$**: The minimum independent dominating set has size 6 (e.g., $\\{u, b_0, b_1, b_2, b_3, b_4\\}$). Thus $i(G) = 6$.\n4. **Condition**: $3 \\gamma(G) = 3 \\times 2 = 6 = i(G)$. The condition holds.\n5. **Well Totally Dominated**: A graph is well totally dominated if all minimal total dominating sets have the same size.\n   - $\\{u, v\\}$ is a minimal total dominating set of size 2.\n   - $\\{v, b_0, b_3\\}$ is a minimal total dominating set of size 3.\n   Since $2 \\neq 3$, $G$ is NOT well totally dominated.\n\nThe counterexample has been found by Moritz Firsching and Goran Žužić using an\nexperimental pipeline.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/6e85aabe821e6ddf718d050a5bd8f19a48e4f2d9/FormalConjectures/WrittenOnTheWallII/GraphConjecture327.lean#L233"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture327","statement":"False ↔\n  ∀ (V : Type) [Fintype V] [DecidableEq V] (G : SimpleGraph V) [inst : DecidableRel G.Adj],\n    G.Connected → 3 * G.dominationNumber = G.indepDominationNumber → G.IsWellTotallyDominated","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"WrittenOnTheWallII.GraphConjecture327.conjecture327"},{"answerKinds":[],"category":"test","docstring":"The logarithm of the counterexample's average eccentricity lies strictly between one and two. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture103","statement":"1 < Real.log (30 / 11) ∧ Real.log (30 / 11) < 2","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture103.one_lt_log_thirty_eleven_and_lt_two"},{"answerKinds":[],"category":"test","docstring":"The counterexample has average eccentricity $30/11$. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture103","statement":"WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample.averageEccentricity = 30 / 11","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_averageEccentricity"},{"answerKinds":[],"category":"test","docstring":"The counterexample is connected. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture103","statement":"WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample.Connected","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_connected"},{"answerKinds":[],"category":"test","docstring":"The counterexample has independence number nine. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture103","statement":"WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample.indepNum = 9","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_indepNum"},{"answerKinds":[],"category":"test","docstring":"The largest induced bipartite subgraph of the counterexample has ten vertices. ","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture103","statement":"WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample.largestInducedBipartiteSubgraphSize = 10","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_bipartiteSize"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 103](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$,\n$\\alpha(G) \\le \\lfloor b(G) - \\ln(\\mathrm{ecc\\_avg}(G)) \\rfloor$\nwhere $\\alpha(G) = G.\\mathrm{indepNum}$ is the independence number,\n$b(G)$ is the largest induced bipartite subgraph size, and\n$\\mathrm{ecc\\_avg}(G) = G.\\mathrm{averageEccentricity}$ is the average\neccentricity of $G$. Uses `Real.log` (natural logarithm).\n\nThis conjecture is false. The graph `wowii103Counterexample` has independence number $9$,\nlargest induced bipartite subgraph size $10$, and average eccentricity $30/11$. Since\n$1 < \\ln(30/11) < 2$, the proposed upper bound is $8$.\n","hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture103","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α),\n    G.Connected → ↑G.indepNum ≤ ↑⌊G.b - Real.log G.averageEccentricity⌋","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture103.conjecture103"},{"answerKinds":[],"category":"research open","docstring":"WOWII [Conjecture 133](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/):\n\nFor a simple connected graph $G$,\n$\\operatorname{path}(G) \\ge \\operatorname{rad}(G) +\n\\lfloor \\mathrm{avg}_v\\, l(v) \\rfloor^{cC_4(G)}$,\nwhere $\\operatorname{path}(G)$ is the path number of the graph (number of vertices of a largest induced path),\n$\\operatorname{rad}(G)$ is the radius (minimum eccentricity, as a natural number),\n$\\mathrm{avg}_v\\, l(v) = l(G)$ is the average independence number of vertex\nneighbourhoods, and $cC_4(G)$ is the $C_4$-free characteristic function\n(1 if $G$ is $C_4$-free, not necessarily induced, and 0 otherwise).\n\nWe read DeLaVina's bracket notation `[average of λ(v)]` in the source as the\nfloor (a standard Graffiti.pc convention), hence `⌊l G⌋` in Lean.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture133","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α)\n  [inst_3 : DecidableRel G.Adj],\n  G.Connected →\n    have rad := G.radius.toNat;\n    let hasC4 :=\n      ∃ a b c d, a ≠ b ∧ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ∧ c ≠ d ∧ G.Adj a b ∧ G.Adj b c ∧ G.Adj c d ∧ G.Adj d a;\n    have cC4 := if hasC4 then 0 else 1;\n    ↑rad + ↑⌊G.l⌋ ^ cC4 ≤ ↑G.path","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture133.conjecture133"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 31](http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj31)\n(Chung 1988):\n\nFor every simple connected graph $G$,\n$\\mathrm{path}(G) \\ge 2 \\cdot \\mathrm{rad}(G) - 1$,\nwhere $\\mathrm{path}(G)$ is the floor of the average distance and\n$\\mathrm{rad}(G)$ is the graph radius.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture31","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],\n  G.Connected → 2 * ↑G.radius.toNat - 1 ≤ ↑G.path","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture31.conjecture31"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 32](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$,\n$\\operatorname{path}(G) \\ge \\operatorname{dist}\\_{\\operatorname{avg}}(A) + 0.5 \\cdot \\operatorname{ecc}\\_{\\operatorname{avg}}(M)$,\nwhere $\\operatorname{path}(G)$ is the floor of the average distance of $G$, $A$ is the\nset of minimum-degree vertices, $M$ is the set of maximum-degree vertices,\n$\\operatorname{dist}\\_{\\operatorname{avg}}(A)$ is the average distance from all vertices\nto $A$, and $\\operatorname{ecc}\\_{\\operatorname{avg}}(M)$ is the average eccentricity\nof the vertices in $M$.\n\nThe conjecture is false, the authors present a counterexample: \"The path on 5 vertices\nis a counterexample, path = 5, distavg(A) = 4 and the average of eccentricity of maximum\ndegree vertices is 8/3.\"\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture32","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α)\n    [inst_3 : DecidableRel G.Adj],\n    G.Connected →\n      have A := {v | G.degree v = G.minDegree};\n      have M := {v | G.degree v = G.maxDegree};\n      have eccavg := fun S => ↑(∑ v ∈ S, (G.eccent v).toNat) / ↑S.card;\n      G.distavg ↑A + 1 / 2 * eccavg M ≤ ↑G.path","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture32.conjecture32"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.annihilationNumber = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_annihilation"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.radius = 2","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"WrittenOnTheWallII.Test.house_radius"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.radius = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_radius"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.matchingNumber = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_matching"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.wienerIndex = 6","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_wiener"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.residue = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_residue"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.edgeFinset.card = 6","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_size"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.cvetkovic = 4","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_cvetkovic"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.annihilationNumber = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_annihilation"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.indepNum = 4","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_indep"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.indepNum = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_indep"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.szegedIndex = 25","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_szeged"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.matchingNumber = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_matching"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"Fintype.card ↑⊤.verts = 4","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_order"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.averageDistance = 5 / 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_avg_dist"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.ediam = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_diameter"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.matchingNumber = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_matching"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.dominationNumber = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_dom"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.wienerIndex = 75","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_wiener"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.averageDistance = 9 / 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_avg_dist"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.szegedIndex = 54","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_szeged"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.egirth = ⊤","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_girth"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.minDegree = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_min_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.girth = 6","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_girth"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.annihilationNumber = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_annihilation"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"Fintype.card ↑⊤.verts = 6","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_order"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.averageDegree = 12 / 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_avg_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.matchingNumber = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_matching"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.indepNum = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_indep"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.edgeFinset.card = 15","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"WrittenOnTheWallII.Test.petersen_size"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.averageDistance = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_avg_dist"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.ediam = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_diameter"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.radius = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_radius"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.cvetkovic = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_cvetkovic"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.wienerIndex = 27","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_wiener"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.radius = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_radius"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.ediam = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_diameter"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.szegedIndex = 6","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_szeged"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"Fintype.card ↑⊤.verts = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_order"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.cvetkovic = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_cvetkovic"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.girth = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_girth"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.cvetkovic = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_cvetkovic"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"Fintype.card ↑⊤.verts = 10","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_order"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.minDegree = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_min_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.girth = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_girth"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.maxDegree = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_max_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.averageDistance = 7 / 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_avg_dist"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.averageDegree = 5 / 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_avg_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.ediam = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_diameter"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"Fintype.card ↑⊤.verts = 6","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_order"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.maxDegree = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_max_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.averageDegree = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_avg_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.edgeFinset.card = 6","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_size"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.edgeFinset.card = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_size"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.szegedIndex = 24","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_szeged"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.wienerIndex = 14","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_wiener"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.dominationNumber = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_dom"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.residue = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_residue"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.minDegree = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_min_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.maxDegree = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_max_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.indepNum = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_indep"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.indepNum = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_indep"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.residue = 3","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"WrittenOnTheWallII.Test.petersen_residue"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.dominationNumber = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_dom"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.residue = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_residue"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.radius = 2","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"WrittenOnTheWallII.Test.petersen_radius"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.averageDistance = 5 / 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_avg_dist"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.maxDegree = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_max_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.edgeFinset.card = 6","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"WrittenOnTheWallII.Test.C6_size"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.girth = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_girth"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.szegedIndex = 135","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"WrittenOnTheWallII.Test.petersen_szeged"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.averageDegree = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_avg_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.dominationNumber = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_dom"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.maxDegree = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_max_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.wienerIndex = 25","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_wiener"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.residue = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_residue"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.PetersenGraph.minDegree = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.petersen_min_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.annihilationNumber = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_annihilation"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.dominationNumber = 1","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_dom"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.annihilationNumber = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_annihilation"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.minDegree = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_min_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.C6.matchingNumber = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.C6_matching"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.HouseGraph.ediam = 2","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.house_diameter"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":true,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.K4.averageDegree = 3","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.K4_avg_deg"},{"answerKinds":[],"category":"test","docstring":null,"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.Test","statement":"WrittenOnTheWallII.Test.Star5.cvetkovic = 5","subjects":["5"],"theorem":"WrittenOnTheWallII.Test.Star5_cvetkovic"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 58](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a connected graph `G`, the size `f(G)` of a largest induced forest satisfies\n`f(G) ≥ ceil( b(G) / average l(v) )` where `b(G)` is the largest induced\nbipartite subgraph and `l(v)` is the independence number of `G.neighborSet v`.\n\nThis conjecture is false. A counterexample is the graph described in the module docstring\nabove: a $K_{3,3}$ joined to a $K_{73}$ via vertex $0$, giving\n$\\lceil b/l_{\\mathrm{avg}} \\rceil \\ge 7 > 6 \\ge f(G)$.\n\nThe counterexample has been found by Moritz Firsching and Goran Žužić using an\nexperimental pipeline.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/4bd72a06842a10e1b8d7bb0fd6b1ef5e6bd20210/FormalConjectures/WrittenOnTheWallII/GraphConjecture58.lean#L772"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture58","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],\n    G.Connected → ⌈G.b / G.l_avg⌉₊ ≤ G.largestInducedForestSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture58.conjecture58"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 217](http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj217):\n\nIf $G$ is a finite simple connected graph on $n > 1$ vertices and\n$L_s(G) \\le 4 \\cdot \\chi_{\\mathrm{residue}=2}(G) + 2$,\nthen $G$ has a Hamiltonian path. Here $L_s(G)$ is the maximum number of\nleaves over all spanning trees and $\\chi_{\\mathrm{residue}=2}(G)$ is the indicator\nof $\\mathrm{residue}(G) = 2$.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/wowii-graph-conjecture-217-lean/blob/6a2fb82fcd17aa15ec734736740794bb8bd194c0/lean/GraphConjecture217Audit.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture217","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α)\n  [inst_3 : DecidableRel G.Adj],\n  G.Connected →\n    G.Ls ≤ 4 * ↑(WrittenOnTheWallII.GraphConjecture217.residueEqTwoIndicator G) + 2 → ∃ a b p, p.IsHamiltonian","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture217.conjecture217"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 200](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, if `tree(G) = ⌈1 + l_avg(G)⌉`, then `G` has a Hamiltonian path.\nHere `tree(G)` is the number of vertices of a largest induced tree subgraph, and\n`l_avg(G) = averageIndepNeighbors G` is the average over all vertices of the independence number\nof the neighbourhood.\nA Hamiltonian path is a walk visiting every vertex exactly once.\n\nThis conjecture is false. The counterexample family in the module docstring\nsatisfies the equality hypothesis and has no Hamiltonian path.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/infinityscroll/formal-conjectures/blob/9dd290db402c49922fa42793e4a7cfb802daf5c1/FormalConjectures/WrittenOnTheWallII/GraphConjecture200Counterexample.lean#L24-L195"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture200","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α),\n    G.Connected → ↑G.largestInducedTreeSize = ↑⌈1 + G.averageIndepNeighbors⌉ → ∃ a b p, p.IsHamiltonian","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture200.conjecture200"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 13](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, the size `b(G)` of a largest induced bipartite subgraph\nsatisfies `b(G) ≥ diam(G) + max_{v ∈ V} l(v) - 1`, where `diam(G)` is the diameter\nof `G` and `l(v) = indepNeighborsCard G v` is the independence number of the\nneighbourhood of `v`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture13","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α),\n  G.Connected → ↑G.diam + ↑((Finset.image (fun v => G.indepNeighborsCard v) Finset.univ).max' ⋯) - 1 ≤ G.b","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"WrittenOnTheWallII.GraphConjecture13.conjecture13"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 101](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$,\n$\\alpha(G) \\le \\lfloor (n + |\\mathrm{alphaCore}(G)|) / 2 \\rfloor$\nwhere $\\alpha(G) = G.\\mathrm{indepNum}$ is the independence number, $n$ is the\nnumber of vertices, and $\\mathrm{alphaCore}(G)$ is the set of vertices whose\nremoval decreases the independence number.\n\nThis is a theorem known to follow from inclusion-exclusion principles.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture101","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],\n  G.Connected → G.indepNum ≤ (Fintype.card α + (WrittenOnTheWallII.GraphConjecture101.alphaCore G).card) / 2","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture101.conjecture101"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 7](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$,\n$L_s(G) \\ge \\max_v \\lambda(v) - 1 + n - 2 \\alpha(G)$,\nwhere $L_s(G)$ is the maximum number of leaves over all spanning trees of $G$,\n$n = |V(G)|$, $\\alpha(G) = G.\\mathrm{indepNum}$ is the independence number, and\n$\\lambda(v) = \\mathrm{indepNeighborsCard}\\, G\\, v$ is the independence number of the\nneighbourhood of $v$.\n\nProved by DeLaVina, Fajtlowicz, Waller (2002).\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture7","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α)\n  [inst_3 : DecidableRel G.Adj],\n  G.Connected →\n    have maxL := (Finset.image (fun v => G.indepNeighborsCard v) Finset.univ).max' ⋯;\n    ↑↑maxL - 1 + ↑↑(Fintype.card α) - 2 * ↑↑G.indepNum ≤ G.Ls","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture7.conjecture7"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 36](http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj36):\n\nFor every finite simple connected graph $G$,\n$\\operatorname{path}(G) \\ge 2 \\cdot \\operatorname{rad}(G) / \\operatorname{dp}(G)$,\nwhere $\\operatorname{path}(G)$ is the floor of the average distance of $G$,\n$\\operatorname{rad}(G)$ is the radius of $G$, and $\\operatorname{dp}(G)$ is the number\nof *diametrical pairs* of $G$ — that is, the number of pairs of vertices at distance\n$\\operatorname{diam}(G)$.\n\nDisproved by Waller in Oct 2003 (counterexample: path number 5, radius 3, dp 1).\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture36","statement":"False ↔\n  ∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],\n    G.Connected →\n      0 < WrittenOnTheWallII.GraphConjecture36.dp G →\n        2 * ↑G.radius.toNat / ↑(WrittenOnTheWallII.GraphConjecture36.dp G) ≤ ↑G.path","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture36.conjecture36"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 20](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, `b(G) ≥ n(G) / ⌊deg_avg(G)⌋`, where `b(G)` is\nthe size of a largest induced bipartite subgraph, `n(G)` is the number of vertices,\nand `deg_avg(G) = (∑ v, deg(v)) / n(G)` is the average degree.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture20","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],\n  G.Connected →\n    have deg_avg := (∑ v, ↑(G.degree v)) / ↑(Fintype.card α);\n    ↑(Fintype.card α) / ↑⌊deg_avg⌋ ≤ G.b","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture20.conjecture20"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 145](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$,\n$\\mathrm{tree}(G) \\ge 2 \\cdot \\mathrm{ecc}(B) / \\lambda_{\\min}(\\overline{G})$\nwhere $\\mathrm{tree}(G)$ is the number of vertices in a largest induced subtree,\n$\\mathrm{ecc}(B)$ is the eccentricity of the boundary vertices (`eccSet` and\n`boundaryVertices`), and $\\lambda_{\\min}(\\overline{G})$ is the minimum local\nindependence number of the complement graph.\n\nWe state the inequality in the form\n$\\mathrm{tree}(G) \\cdot \\mathrm{lMin}(\\overline{G}) \\ge 2 \\cdot \\mathrm{ecc}(B)$\nto avoid division.\n\n## Provenance\n\nSolved by Dominic Dabish.\n\nProofOrchestrator, using OpenAI GPT-5.6 Thinking, assisted with the mathematical\nargument and Lean formalization; all formal claims were checked by the pinned\nLean compiler.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/DomTheDeveloper/crl/blob/2ee448baa80c98f0c8b9a0c1c3d9421200f99aa5/math/wowii145/WOW145/145.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture145","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [inst_2 : Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],\n  G.Connected →\n    0 < WrittenOnTheWallII.GraphConjecture145.localIndependenceMin Gᶜ →\n      2 * G.eccSet G.maxEccentricityVertices ≤\n        G.largestInducedTreeSize * WrittenOnTheWallII.GraphConjecture145.localIndependenceMin Gᶜ","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture145.conjecture145"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 3](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a connected simple graph `G`, the number of leaves in a maximum spanning\ntree satisfies `Ls(G) ≥ G.indepDominationNumber * MaxTemp(G)`, where `G.indepDominationNumber` is the independent\ndomination number and `MaxTemp(G)` is `max_v deg(v)/(n(G) - deg(v))`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture3","statement":"∀ {α : Type u} [inst : Fintype α] [DecidableEq α] {G : SimpleGraph α} [inst_2 : DecidableRel G.Adj]\n  [inst_3 : Nontrivial α], G.Connected → ↑G.indepDominationNumber * G.MaxTemp ≤ G.Ls","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture3.conjecture3"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 59](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$, the size $f(G)$ of a largest induced forest\nsatisfies $f(G) \\ge \\lceil \\sqrt{\\mathrm{residue}(G) \\cdot b(G)} \\rceil$, where\n$\\mathrm{residue}(G)$ is the Havel-Hakimi residue (the number of zeros remaining\nafter applying the Havel-Hakimi algorithm to the degree sequence until termination)\nand $b(G)$ is the size of a largest induced bipartite subgraph.\n\nSee: Favaron, Mahéo, Saclé (1991) for the residue; DeLaVina's Graffiti.pc for the conjecture.\n\nThis conjecture is false. There is a connected counterexample on 123 vertices\nwith `residue G = 101`, `b G = 122`, and `G.largestInducedForestSize = 111`.\nIndeed, `101 * 122 = 12322 = 111 ^ 2 + 1`, so the conjectured lower bound is 112.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/QDKStorm/wowii59-counterexample/blob/main/Counterexample59.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture59","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],\n    G.Connected → ↑⌈√(↑G.residue * G.b)⌉ ≤ ↑G.largestInducedForestSize","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture59.conjecture59"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 315](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nLet $G$ be a simple connected graph and let $P$ denote the set of pendant vertices\n(vertices of degree $1$). If $\\alpha(G) = |P|$, then $G$ is well totally dominated.\n\nA formal proof of this conjectures has been obtained by Goran Žužić and Moritz Firsching using an\nexperimental pipeline.\n","formalProofs":[{"conditions":[],"kind":"formal_conjectures","link":"https://github.com/mo271/formal-conjectures/blob/9ef80e1a3709ed3eda43d9ed6ff1087681621041/FormalConjectures/WrittenOnTheWallII/GraphConjecture315.lean#L43"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture315","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] (G : SimpleGraph α) [inst_2 : DecidableRel G.Adj],\n  G.Connected → G.indepNum = G.pendantVertices.card → G.IsWellTotallyDominated","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture315.conjecture315"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 5](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, `Ls(G)` is bounded below by the maximal size\nof a sphere of radius `radius(G)` around the centres of `G`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture5","statement":"∀ {V : Type u_1} [inst : Fintype V] [DecidableEq V] [Nontrivial V] (G : SimpleGraph V),\n  G.Connected →\n    sSup ((fun v => ↑{w | w ∈ (fun v => {w | G.dist v w = G.radius.toNat}) v}.card) '' {v | G.eccent v = G.radius}) ≤\n      G.Ls","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture5.conjecture5"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 316](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nLet `G` be a simple connected graph and let `P` denote the set of pendant vertices\n(vertices of degree 1). If `|P| ≥ deg_avg(Gᶜ)`, then `G` is well totally dominated,\nwhere `deg_avg(Gᶜ)` is the average degree of the complement of `G`.\n\n**Proof sketch.** In the trivial cases (`P = ∅`, or at most `2` vertices) `G` is complete,\nand complete graphs are well totally dominated. Otherwise the set `C` of non-pendant\nvertices satisfies `|C| ≤ 3` and is a clique of `G`, and a case split on the set `Q ⊆ C`\nof neighbours of pendant vertices shows that `G` is well totally dominated.\n","formalProofs":[{"conditions":[],"kind":"lean4","link":"https://github.com/KitaKen1/wowii-graph-conjecture-316-lean/blob/3335e07151bc43e86d5c104dd30fee3596f06410/GraphConjecture316.lean"}],"hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture316","statement":"∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] (G : SimpleGraph α) [inst_2 : DecidableRel G.Adj],\n  G.Connected → Gᶜ.averageDegree ≤ ↑G.pendantVertices.card → G.IsWellTotallyDominated","subjects":["5"],"subsets":["FC100OpenSet1"],"theorem":"WrittenOnTheWallII.GraphConjecture316.conjecture316"},{"answerKinds":["Prop"],"category":"research solved","docstring":"WOWII [Conjecture 33](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph `G`, `path(G) ≥ ⌈2 · dist_avg(M, V)⌉`, where `path(G)`\nis the floor of the average distance of `G`, `M` is the set of maximum-degree vertices,\nand `dist_avg(M, V)` is the average distance from all vertices to `M`.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture33","statement":"False ↔\n  ∀ (α : Type) [inst : Fintype α] [inst_1 : DecidableEq α] [Nontrivial α] (G : SimpleGraph α)\n    [inst_3 : DecidableRel G.Adj],\n    G.Connected →\n      have M := {v | G.degree v = G.maxDegree};\n      ⌈2 * G.distavg M⌉ ≤ ↑G.path","subjects":["5"],"subsets":["FC100SolvedSet1"],"theorem":"WrittenOnTheWallII.GraphConjecture33.conjecture33"},{"answerKinds":[],"category":"research solved","docstring":"WOWII [Conjecture 18](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)\n\nFor a simple connected graph $G$, the size $b(G)$ of a largest induced bipartite\nsubgraph satisfies\n$b(G) \\ge \\alpha(G) + \\lceil \\sqrt{\\mathrm{dist}_{\\max}(M)} \\rceil$,\nwhere $\\alpha(G)$ is the independence number, $M$ is the set of maximum-degree\nvertices, and $\\mathrm{dist}_{\\max}(M) = \\max\\{\\mathrm{dist}_G(u,v) \\mid u, v \\in M\\}$\nis the maximum distance between two maximum-degree vertices (DeLaVina's\n`dist_max(M)`). Proven by Benny John (Feb. 2006), generalizing Schindl's proof\nof conjecture 17.\n","hasSorryFreeProof":false,"module":"FormalConjectures.WrittenOnTheWallII.GraphConjecture18","statement":"∀ {α : Type u_1} [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],\n  G.Connected →\n    have M := {v | G.degree v = G.maxDegree};\n    ↑G.indepNum + ↑⌈√↑(G.distMaxSet M)⌉ ≤ G.b","subjects":["5"],"theorem":"WrittenOnTheWallII.GraphConjecture18.conjecture18"}],"provenance":{"answer_mode":"postpone","dependencies_sha256":"c8ffc0d59f492007162fe54c68396e4b9f2f305fcf69210a7fbaf0b18fcd8cd9","extractor":{"commit":"eee8f2ec741ac7772ba075fd8023e5716beb78c6","path":"scripts/extract_names.lean","repository":"williamjblair/formal-conjectures"},"lean_toolchain":"leanprover/lean4:v4.33.1","scope":"FormalConjectures","source":{"commit":"eee8f2ec741ac7772ba075fd8023e5716beb78c6","repository":"williamjblair/formal-conjectures"}},"schemaVersion":2}