OPEN
Show that the equation
\[n! = a_1!a_2!\cdots a_k!,\]
with $n-1>a_1\geq a_2\geq \cdots \geq a_k$, has only finitely many solutions.
This would follow if $P(n(n+1))/\log n\to \infty$, where $P(m)$ denotes the largest prime factor of $m$ (see Problem
[368]). Erdős
[Er76d] proved that this problem would also follow from showing that $P(n(n-1))>4\log n$.
Hickerson conjectured the largest solution is
\[16! = 14! 5!2!.\]
The condition $a_1<n-1$ is necessary to rule out the trivial solutions when $n=a_2!\cdots a_k!$.
Surányi was the first to conjecture that the only non-trivial solution to $a!b!=n!$ is $6!7!=10!$.
