On the sparsity of integers $a$ in solutions to $a!b!=c!$

Abstract

We consider the Diophantine equation $$ a!b! = c! $$ due to Erd\H{o}s, where we assume $a \leq b$. It is widely believed that there are only finitely many nontrivial solutions, and considerable work has been dedicated to showing this. In one direction, Luca (2007) showed that the set of $c$'s which can appear in solutions has density zero. Here we show that the set of $a$'s appearing in solutions is also sparse. In particular, $a$ cannot be one less than a large fraction of primes, and, under the assumption that $\sqrt[k]{a!} \mod 1$ is equidistributed in an appropriate sense, we show that the set of such $a$ has asymptotic density zero.