I am trying to prove that $(m+1)^3+(m+2)^3+\cdots + (2m)^3$ can never be square. $m$ is a positive integer.
I have managed to show this but it was long and not very elegant. I am interested in seeing other solutions.
Outline of my solution: using the formula for sum of cubes we end up having to show $(5n+3)(3n+1)$ cannot be square. The two terms in brackets can have a gcd of 1,2 or 4. Then I considered the four cases of n modulo 4, and ended up with a square term multiplied by two terms in brackets which are coprime, hence each must be square, but this turns out to be impossible when considering the squares mod 3 and 5.