In my elementary number theory book (Rosen, 8th edition), there's a problem formulated as followed:
- Show that $1^{(p-1)} + 2^{(p-1)} + 3^{(p-1)} + \cdots + (p-1)^{(p-1)} \equiv -1 \pmod p$ whenever $p$ is prime. (It has been conjectured that the converse of this is also true.)
Does there exist a proof of the converse, and if not, why is it so much harder to prove than the original statement?