Why there are infinitely many positive integers n such that $1^5+2^5+...+n^5$ is a square?

Question

I have no clue how to show some kind of positive integers has infinitely many. I guess perhaps I can use proof by contradiction? Say there is only a finite many, which means it will eventually stop at number $x$, and then show we can always find some integers after $x$, say $y$, such that $1^5 + ...x^5 + ... y^5$ is a square?

I am not sure it's the correct approach since I haven't figure out how to prove such y exists.

Thanks.

Answer 1

Hint: $$\sum_{i=1}^n i^5 = (n^2+n)^2 \frac{(2n^2+2n-1)}{12}$$