Erdős Problem #728 — Factorial divisibility near the diagonal

The question is attributed to Erdős, Graham, Ruzsa, and Straus, arising from their work on binomial prime factorizations. The intended formulation excludes trivial cases by requiring $a,b\leq(1-\varepsilon)n$; under that interpretation, the question is now settled.

Known results

• Erdős (1968): if $a!b!\mid n!$, then $a+b\leq n+O(\log n)$.

January 2026 formalized proof

An arXiv writeup proves that for every $0<C_1<C_2$ and $0<\varepsilon<1/2$, infinitely many triples satisfy $\varepsilon n\leq a,b\leq(1-\varepsilon)n$, $a!b!\mid n!(a+b-n)!$, and $C_1\log n<a+b-n<C_2\log n$. This implies the requested inequality for every fixed $C>0$. GPT-5.2 Pro supplied the argument and Aristotle by Harmonic formalized it in Lean; the proof reduces the claim to binomial divisibility and Kummer carry counts.

Current status (as of August 2026): The intended nontrivial formulation is resolved; the unrestricted wording has trivial solutions, and stronger quantitative bounds beyond the logarithmic window remain unoptimized.