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

We consider the Diophantine equation $$a!b!=c!$$ due to Erdős, where we assume $a\le b$. 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 $\sqrt[k]{a!}\mathrm{m}\mathrm{o}\mathrm{d}1$ is equidistributed in an appropriate sense, we show that the set of such $a$ has asymptotic density zero.