Question 129
Let $p \geq 5$ be a prime number and $S_n = 1^n + 2^n + \dots + (p - 1)^n$. Show that there exists infinitely many $n \in \mathbb{N}$ such that $p^3 \mid S_n$, $p \mid S_{n-1}$ and $p \mid S_{n-2}$.
(proposed by N. Davii-Od)
(proposed by N. Davii-Od)