$1^{k}+2^{k}+\cdots+n^{k}$ is not divisible by $n+2$ $k$ is odd number.
show that for arbitrary $n\in N$ , $1^{k}+2^{k}+\cdots+n^{k}$ is not divisible by $n+2$    

This is just a suggestion, nowhere near a complete answer. For the special case $k=3$, $1^3+2^3+3^3+4^3+ \cdot \cdot \cdot=(1+2+3+4+ \cdot \cdot \cdot)^2$. Maybe you could prove it for this case and then use induction.
Every perfect cube is the difference of two perfect squares?
