## 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 Product: Does there exist any other non-trivial factorial products $n!=a!b!$ other than $n=10,a=6,b=7$?^{1}

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 examples^{3}.

*Guy, R. (2004). Unsolved problems in number theory (Vol. 1). Springer Science & Business Media.*↩︎Information on Wolfram Math ↩︎

Luca, F. (2007, November). On factorials which are products of factorials. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 143, No. 3, pp. 533-542). Cambridge University Press. ↩︎