On the sparsity of integers $a$ in solutions to $a!b!=c!$
Joshua Cooper, Joseph Preuss
TL;DR
This work studies the sparsity of the first factor $a$ in solutions to $a!b! = c!$ with $a\le b$, building modular obstruction arguments for fixed class $k$ that force many $a$ out for a positive-density set of primes, and coupling these with irreducibility results in the falling-factorial basis. It then introduces an equidistribution framework for the falling factorial roots $a!^{1/k}$ modulo 1 and proves a conditional sparsity result: if these fractional parts are sufficiently distributed, the set of $a$ that occur in a solution has density zero, with an explicit upper bound under a weak equidistribution hypothesis. The results connect to Luca's density-zero finding for the $c$-values in solutions and suggest that, under plausible equidistribution assumptions, nontrivial solutions to $a!b! = c!$ are exceedingly rare, potentially aligning with the conjecture that only trivial or the sporadic (6,7,10) solution exists.
Abstract
We consider the Diophantine equation $$ a!b! = c! $$ due to Erdős, 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.