Erdős Problem #728

The problem

Let $C>0$ and $\epsilon>0$ be sufficiently small. Are there infinitely many integers $a,b,n$ with $a\geq \epsilon n$ and $b\geq \epsilon n$ such that

$$a! b! \mid n!(a+b-n)!$$

and $a+b>n+C\log n$?

Statement quoted from erdosproblems.com/728, maintained by Thomas Bloom. Original source: [EGRS75,p.91]. Current status there: proved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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