Problem Statement
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 $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$?
Categories:
Number Theory Factorials
Progress
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.Sothanaphan, using ChatGPT, has found a disproof of the alternative statement with 'for all large $n$' instead: take $n=p_{r+1}^k-1$ for arbitrary $k$, whence the divisibility condition forces $a_1+a_2\leq n+2p_{r+1}$.
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]).
See also [400] and [729] (arguably this problem is a more precisely stated form of the latter).
Source: erdosproblems.com/401 | Last verified: January 14, 2026