Factorial Products ❗

Problem Description

A very interesting property of $10!$ is that $$ 10!=6!\cdot 7! $$

Can we find other examples of factorials that are products of other factorials like this? Clearly there are “trivial” answer such as $$ 24!=4!\cdot 23! $$ where we say an example $n!=a!b!$ is “trivial” if $n=a!$ and $b=n-1$. As such we have the interesting problem of

Factorial Product¹: Does there exist any other non-trivial factorial products $n!=a!b!$ other than $n=10,a=6,b=7$?

It is known that there are no nontrivial identities for $n\leq 18,160$². It has also been shown that if the abc Conjecture is true, then there are only finitely many non-trivial examples³.