Art and Craft of problem solving, 7.6.15
"Let $p$ be an odd prime and $P(x)$ a polynomial of degree at most $p-2$. Prove that if $P$ has integer coefficients, then $P(n)+P(n+1)+...+P(n+p-1)$ is an integer divisible by $p$ for ever integer $n$."
I've reduced the problem down to showing that $1^d + 2^d + ... + (p-1)^d$ is divisible by $p$ for all $d\leq p-2$. In the case where $d$ is odd this is straightforward but I can't seem to crack the even case. It may be that my approach is not the right one however, and there is another way. In particular, I have not been able to incorporate the primality of $p$.
Note there is a question on this website similar to mine, however, the other one is asking for a converse to what I am asking, and the two are not duplicates.