I recently found this problem saying:
Find all $n$ such that there exists a solution of $a^2 + b^2 = n!$ for positive integers $a, b, n$.
I first thought to show $n!$ will contain some prime of the form $4k+3$ odd number of times after some value $n_0$. Then I thought by Bertrand's postulate $n!$ (for $n >3$ ) contains at least one prime only once. If any of them is of the form $4k+3$ we would be done. But I failed to make any more progress.