All Random Solved Random 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]). 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 $n=a_2!\cdots a_k!$.