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import FormalConjecturesUtilErdős Problem 401
References:
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
open Filternamespace Erdos401Is there some function $f(r)$ such that $f(r)\to \infty$ as $r\to\infty$, such that, for infinitely many $n$, there exist $a_1,a_2$ with $$a_1+a_2> n+f(r)\log n$$ such that $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$?
It is ambiguous in [ErGr80] what the intended quantifiers are on the variables (they write 'is it true that we can find $a_1+a_2>n+f(r)\log n$...'). Comparing to previous problems such as [728] and [729] it seems most likely that they intended to ask the formulation in the problem statement.
The answer is yes: Barreto and Leeham have used ChatGPT to provide a proof of the stated problem (in fact essentially the same construction as their solution to [729]).
@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos401.lean"]
theorem erdos_401 : answer(True) ↔
∃ f : ℕ → ℝ, Tendsto f atTop atTop ∧
∀ (r : ℕ) (hr : 1 ≤ r),
{n : ℕ | ∃ a₁ a₂ : ℕ, 0 < a₁ ∧ 0 < a₂ ∧
(a₁ : ℝ) + a₂ > n + f r * Real.log n ∧
a₁.factorial * a₂.factorial ∣
n.factorial * (∏ i ∈ Finset.range r, Nat.nth Nat.Prime i) ^ n}.Infinite := ⊢ True ↔
∃ f,
Tendsto f atTop atTop ∧
∀ (r : ℕ),
1 ≤ r →
{n |
∃ a₁ a₂,
0 < a₁ ∧
0 < a₂ ∧
↑a₁ + ↑a₂ > ↑n + f r * Real.log ↑n ∧
a₁.factorial * a₂.factorial ∣
n.factorial * (∏ i ∈ Finset.range r, Nat.nth Nat.Prime i) ^ n}.Infinite
All goals completed! 🐙end Erdos401