We have a prime $p$ and an integer $k > 0$. When does the following equation stand?
$(p-1)! + 1 = p^k $
I have obviously tried for some little numbers, and in some cases, it stands:
For $ p=2$ and $k=1, 1 + 1 = 2$.
For $ p=3$ and $k=1, 2 +1 = 3.$
For $ p=5$ and $k=2, 24 + 1 = 25.$
Any ideas, how to prove, if there are more, and if yes, what are the solutions?