I've been thinking about such an interesting question these last few days. I wonder if anyone has studied it.
Question: Find all postive integers $n$ such that $$S(n)=1^n+2^n+3^n+4^n+\cdots+n^n$$ is a square number?
Here $n=1$ is clear and if $n=3$ then $S(3)=36=6^2$. Now I can't find any other $n\le 20$, so I conjecture these $n$ are the only two values.