Does there exist any even integers $m,n$ such that $m$ divides $1^n + 2^n + \cdots +(8m+2)^n$ ?
After my initial attempt below that i'm not so sure of, i feel that Bernoulli polynomials might be the best approach, but not sure how ?
My attempt: Suppose $m$ divides $1^n + 2^n + \cdots +(8m+2)^n$. Applying the factor theorem, we arrive at $1^n + 2^n = 0$, which is absurd ?
EDIT: For $m$ even, the question was settled by @Vrugthagel below, what about when $m$ is odd ?