FormalConjectures.ErdosProblems.«951»

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import FormalConjecturesUtil

Erdős Problem 951

References:

erdosproblems.com/951

[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.

open scoped Finsupp Nat.Prime Topologyopen Filternamespace Erdos951

A sequence a : ℕ → ℝ is said to have property Erdos951Prop if for any pair of distinct finitely supported sequences k l : ℕ →₀ ℕ their corresponding Beurling integers are of distance at least one apart.

def Erdos951Prop (a : ℕ → ℝ) : Prop :=
  ∀ (k ℓ : ℕ →₀ ℕ), k ≠ ℓ → |beurlingInteger a k - beurlingInteger a ℓ| ≥ 1

If a has property Erdos951Prop and 1 < a 0, then a is a set of Beurling prime numbers.


@[category API, AMS 11]
theorem erdos_951.variants.isBeurlingPrimes {a : ℕ → ℝ} (ha : 1 < a 0)
    (hm : StrictMono a) (he : Erdos951Prop a) :
    IsBeurlingPrimes a := bya:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop a⊢ IsBeurlingPrimes a
  refine ⟨ha, hm, tendsto_atTop_atTop.2 fun x => ?_⟩a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝ⊢ ∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1
  by_contra h_contraa:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1⊢ False
  obtain ⟨L, hL⟩ : ∃ L, Filter.Tendsto a Filter.atTop (𝓝 L) :=
    ⟨_, tendsto_atTop_isLUB hm.monotone (isLUB_ciSup ⟨x, Set.forall_mem_range.2 fun n =>
    le_of_not_ge fun hn => h_contra ⟨n, fun m hm' => hn.trans (hm.monotone hm')⟩⟩)⟩a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)⊢ False
  obtain ⟨N, hN⟩ := Metric.tendsto_atTop.mp hL (1 / 2) (bya:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)⊢ 1 / 2 > 0a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2⊢ False norm_numAll goals completed! 🐙a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2⊢ False)a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2⊢ False
  have := hm (bya:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2⊢ N < N + 1a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)⊢ False linarithAll goals completed! 🐙a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)⊢ False : N < N + 1)a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)⊢ False
  have h_diff : a (N + 1) - a N ≥ 1 := bya:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop a⊢ IsBeurlingPrimes aa:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ False
    rw [← abs_of_nonneg (bya:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)⊢ 0 ≤ a (N + 1) - a Na:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)⊢ |a (N + 1) - a N| ≥ 1a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ False linarithAll goals completed! 🐙a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)⊢ |a (N + 1) - a N| ≥ 1a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ False : 0 ≤ a _ - _)]a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)⊢ |a (N + 1) - a N| ≥ 1a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ False
    simpa using he (.single (N + 1) 1) (.single N 1) (bya:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)⊢ (fun₀ | N + 1 => 1) ≠ fun₀ | N => 1a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ False simpa [Finsupp.ext_iff] using ⟨N, bya:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)⊢ ¬(fun₀ | N + 1 => 1) N = (fun₀ | N => 1) Na:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ False simpAll goals completed! 🐙a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ False⟩)a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ False
  linarith [abs_lt.1 (hN N le_rfl), abs_lt.1 (hN (N + 1) (bya:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ N + 1 ≥ N grindAll goals completed! 🐙))]

If 1 < a 0 < ... has property Erdos951Prop, is it true that #{a i ≤ x} ≤ π x?

@[category research open, AMS 11]
theorem erdos_951 : answer(sorry) ↔
    ∀ a : ℕ → ℝ, 1 < a 0 → StrictMono a → Erdos951Prop a →
      ∀ᶠ (x : ℝ) in Filter.atTop, {i : ℕ | a i ≤ x}.ncard ≤ π ⌊x⌋₊ := by⊢ True ↔ ∀ (a : ℕ → ℝ), 1 < a 0 → StrictMono a → Erdos951Prop a → ∀ᶠ (x : ℝ) in atTop, {i | a i ≤ x}.ncard ≤ π ⌊x⌋₊
  sorryAll goals completed! 🐙

Beurling conjectured that if the number of Beurling integer in [1, x] is x + o(log x), then a must be the sequence of primes.

@[category research solved, AMS 11]
theorem erdos_951.variants.beurling :
    ∀ a : ℕ → ℝ, IsBeurlingPrimes a →
    ((fun x => (BeurlingIntegers a ∩ .Iic x).ncard - x) =o[atTop] Real.log) →
    a = Nat.cast ∘ Nat.nth Nat.Prime := by⊢ ∀ (a : ℕ → ℝ),
  IsBeurlingPrimes a →
    (fun x ↦ ↑(BeurlingIntegers a ∩ Set.Iic x).ncard - x) =o[atTop] Real.log → a = Nat.cast ∘ Nat.nth Nat.Prime
  sorryAll goals completed! 🐙

end Erdos951