What are the prime numbers $p$ such that $(p-1)^{p}+1$ is a power of $p$?
$p=2$ works. Otherwise, we apply LTE to $p$ odd to get $v_{p}\left((p-1)^{p}+1\right)=2$, so $(p-1)^{p}+1=p^{2}$, thus $(p-1)^{p-1}=p+1$. It is easily verified that the left-hand side is too large for $p \geq 5$. But $p=3$ works, so the problem has two solutions: 2 and 3.