The problem from Burton: show that the equation $n^2 + (n+1)^2 = m^3$ has no solution in the positive integers.
So far, I can see that gcd($n$,$n+1$)$=1$ and $m \equiv_4 1$ and $m=a^2 + b^2$ for some integers a,b. I'm guessing I need to reach a contradiction.
At this point, I am stuck. Any hints?