TITLE: Prove $\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$
QUESTION [6 upvotes]: Let $T$ be the set of nonnegative integers, I need to prove that
$$\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$$
It's really easy to show that 
$$\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|\ge6$$
since for $n>2$
$$(n!)!=0!(n!)!=1!(n!)!=n!(n!-1)!=(n!-1)!n!=(n!)!1!=(n!)!0!$$
So I'm looking for a way to prove the apposite inequality.

REPLY [7 votes]: This equality was proved by Daniel M. Kane in this paper.