Erdős Problem #401

The problem

Is 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 $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$?

Statement quoted from erdosproblems.com/401, maintained by Thomas Bloom. Original source: [ErGr80,p.78]. 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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