I have a small problem:
Let $m$ and $n$ be integers such that $2m^2+m = 3n^2+n$. Prove that $m-n$ and $2m+2n + 1$ are perfect square.
We have $$(m-n)(2m+2n+1) = 2(m^2-n^2) + m-n = n^2.$$
So, we need to prove that $m-n$ and $2m+2n+1$ are coprime. But I don't get further. Anyone can give me a hint?