This question is inspired by Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$.
Find all nonnegative integers $m$ and $n$ such that $m!+1=n^2$.
We know that $(m,n)=(4,5)$, $(m,n)=(5,11)$, and $(m,n)=(7,71)$ are solutions. Are there more solutions? Are there only finitely solutions?