FormalConjectures.ErdosProblems.«1101»

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import FormalConjecturesUtil

Erdős Problem 1101

Reference: erdosproblems.com/1101


open Nat Filter


namespace Erdos1101

The set of integers not divisible by any u_i.

def ASet (u : ℕ → ℕ) : Set ℕ :=
  { a | ∀ i, ¬ u i ∣ a }

The sequence of integers A_u which are not divisible by any u_i arranged in a monotonic sequence.

noncomputable def A (u : ℕ → ℕ) (n : ℕ) : ℕ :=
  Nat.nth (fun a => a ∈ ASet u) n

t_x such that u_0 ... u_{t_x-1} ≤ x < u_0 ... u_{t_x}.

noncomputable def t (u : ℕ → ℕ) (x : ℕ) : ℕ :=
  sSup { k | ∏ i ∈ Finset.range k, u i ≤ x }

A sequence is "good" if

it is strictly monotone

it is pairwise coprime

the sum of reciprocals converges

the gap between consecutive elements in A(u) is bounded relative to t_x.

def IsGood (u : ℕ → ℕ) : Prop :=
  StrictMono u ∧
  (∀ i j, i ≠ j → Coprime (u i) (u j)) ∧
  Summable (fun n => 1 / (u n : ℝ)) ∧
  ∀ ε > 0, ∀ᶠ x in atTop,
    ∀ k, A u k < x →
      (A u (k + 1) : ℝ) - A u k < (1 + ε) * (t u x : ℝ) * (∏' i : ℕ, (1 - 1 / (u i : ℝ)))⁻¹

There is NO good sequence with polynomial growth.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1101.parts.i :
    ¬ ∃ u, IsGood u ∧ ∃ k : ℕ, (fun n => (u n : ℝ)) =O[atTop] (fun n => (n : ℝ) ^ k) := by⊢ ¬∃ u, IsGood u ∧ ∃ k, (fun n => ↑(u n)) =O[atTop] fun n => ↑n ^ k
  sorryAll goals completed! 🐙

There is a good sequence with sub-exponential growth.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1101.parts.ii :
    ∃ u, IsGood u ∧ (fun n => Real.log (u n : ℝ)) =o[atTop] (fun n => (n : ℝ)) := by⊢ ∃ u, IsGood u ∧ (fun n => Real.log ↑(u n)) =o[atTop] fun n => ↑n
  sorryAll goals completed! 🐙


end Erdos1101