FormalConjectures.ErdosProblems.«390»

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import FormalConjecturesUtil

Erdős Problem 390

References:

erdosproblems.com/390

[EGS82] Erdős, P., R. K. Guy, and J. L. Selfridge. "Another Property of 239 and some related questions." Congr. Numer. 34 (1982): 243-257.


open scoped Nat

open Filter Asymptotics Real


namespace Erdos390

Let f n be the smallest integer for which n! can be represented as the product of distinct integers greater than n, the largest of which is f n.

noncomputable def f (n : ℕ) : ℕ := sInf {m : ℕ | ∃ k, ∃ f : ℕ → ℕ, StrictMono f ∧
  n < f 0 ∧ f (k - 1) = m ∧ ∏ i < k, f i = n !}

f n - 2 * n = θ (n / log n). This is proved in [EGS82].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_390.variants.theta :
    (fun n => f n - 2 * n : ℕ → ℝ) =Θ[atTop] (fun n => n / log (n : ℝ)) := by⊢ (fun n => ↑(f n) - 2 * ↑n) =Θ[atTop] fun n => ↑n / log ↑n
  sorryAll goals completed! 🐙

Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_390 :
    answer(sorry) ↔ ∃ c,
      (fun n => f n - 2 * n : ℕ → ℝ) ~[atTop] (fun n => c * n / log (n : ℝ)) := by⊢ True ↔ ∃ c, (fun n => ↑(f n) - 2 * ↑n) ~[atTop] fun n => c * ↑n / log ↑n
  sorryAll goals completed! 🐙


end Erdos390