Today I came through the following Diophantine equation in a maths contest: $$125(m^2+n^2)=(m+n)^3, m,n\in\mathbb Z_+$$ of which it asks for the sum of all possible values of $m$ in the solution set.
Obviously $m^2+n^2$ is a perfect cube, but also trivially the equation
have infinite many solutions, so it does not work out with only this.
Also we know that it is impossible that $m,n$ are both even, or else by counting the multiplicity of $2$ on both sides yields a contradiction. And by modulo $4$ the possibility for $m,n$ both odd is also sort out.
By a simple approximation one could also see $m,n<125$. The rest could actually verified by computer, but it is obviously not allowed in the contest. The answer to the question is $150$, but I cannot see how they got that, and for the question itself I can hardly proceed further, so I am asking for help in this community.
Thanks in advance.