Proof that there is no positive integer n such that $n+2 | 1^k + 2^k + \dots + n^k$ where $k$ is an odd integer

Question

Proof that there is no positive integer $n$ such that $n + 2 | 1^k + 2^k + \dots + n^k$ where $k$ is an odd integer.

Answer 1

Using modular arithmetic ($\text{mod }n+2$); $$\begin{align} 1^k+2^k+\dots+(n-1)^k+n^k &\equiv1^k+2^k+\dots+(-3)^k+(-2)^k\\ &\equiv\begin{cases}1+\left(\frac{n}2+1\right)^k&n\text{ even}\\1&n\text{ odd}\end{cases}\\ \end{align}$$