A question I found very interesting , which I found written on a blackboard while visiting a near by community science center is as follows.
Prove that there exist infinitely many $m,n,k$ for which $1^k+2^k+3^k \cdots +n^k$ is a perfect $m$ power.
I have been trying this from many days but I am unable to make out how to do it . I got one thing that $1^k+2^k+3^k \cdots +n^k | \; n+1$ but I don't thing this would help.
There are many solutions with $m=2$ -- as André noted, the case with $k=1$ gives you infinitely many solutions via Pell's equation. I can't find any solutions where $m$ is not a power of 2 (excepting the trivial cases $n\in\{0,1\}$).