FormalConjectures.ErdosProblems.«886»

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import FormalConjecturesUtil

Erdős Problem 886

References:

erdosproblems.com/886

[ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.


open Nat Filter


namespace Erdos886

The set of divisors of $n$ in the interval $(n^{1/2}, n^{1/2} + n^{1/2-\epsilon})$.

noncomputable def Erdos886Divisors (n : ℕ) (ε : ℝ) (C : ℝ) : Finset ℕ :=
  (divisors n).filter (fun d =>
    (n : ℝ) ^ (1/2 : ℝ) < d ∧ (d : ℝ) < (n : ℝ) ^ (1/2 : ℝ) + C * (n : ℝ) ^ (1/2 - ε))

Let $\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilon(1)$?

Erdős attributes this conjecture to Ruzsa.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_886 :
    answer(sorry) ↔ ∀ ε > 0, ∃ K : ℕ, ∀ᶠ n in atTop, (Erdos886Divisors n ε 1).card ≤ K := by⊢ True ↔ ∀ ε > 0, ∃ K, ∀ᶠ (n : ℕ) in atTop, (Erdos886Divisors n ε 1).card ≤ K
  sorryAll goals completed! 🐙

Erdős and Rosenfeld [ErRo97] proved that there are infinitely many $n$ such that there are four divisors of $n$ in $(n^{1/2},n^{1/2}+16n^{1/4})$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_886.variants.rosenfeld_infinite :
    Set.Infinite {n | 4 ≤ (Erdos886Divisors n (1/4) 16).card} := by⊢ {n | 4 ≤ (Erdos886Divisors n (1 / 4) 16).card}.Infinite
  sorryAll goals completed! 🐙

Erdős and Rosenfeld [ErRo97] proved that, for any constant $C>0$, all large $n$ have at most $1+C^2$ many divisors in $[n^{1/2}, n^{1/2}+Cn^{1/4}]$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_886.variants.rosenfeld_bound :
    ∀ C > 0, ∀ᶠ (n : ℕ) in atTop,
    ((divisors n).filter (fun (d : ℕ) =>
      (n : ℝ) ^ (1 / 2 : ℝ) ≤ (d : ℝ) ∧ (d : ℝ) ≤ (n : ℝ) ^ (1 / 2 : ℝ) + C * (n : ℝ) ^ (1 / 4 : ℝ))).card
      ≤ 1 + C ^ 2 := by⊢ ∀ C > 0,
  ∀ᶠ (n : ℕ) in atTop, ↑{d ∈ n.divisors | ↑n ^ (1 / 2) ≤ ↑d ∧ ↑d ≤ ↑n ^ (1 / 2) + C * ↑n ^ (1 / 4)}.card ≤ 1 + C ^ 2
  sorryAll goals completed! 🐙


end Erdos886