A friend asked me this question:
If $p$ is a prime, prove that $(p - 1)! + 1$ is a power of $p$ if and only if $p = 2, 3$ or $5$.
Clearly one direction is obvious, namely that $p=2,3,5$ implies $(p - 1)! + 1$ is a power of $p$.
The other direction is not clear to me. Since by Wilson's theorem $p$ divides $(p - 1)! + 1$ so we need to show that if there are no other prime factors then $p=2,3,5$. Can someone give me a hint for establishing this?