Prove $m-n$ and $2m+2n+1$ are coprime

Question

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.

My work:

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?

Answer 1

Assume for contradiction $p\mid \gcd(m-n,2m+2n+1)$ for some prime $p$.

But then $p\mid n^2\iff p\mid n$, and so $p\mid m-n\implies p\mid m$.

However, $p\mid 2m+2n+1$ is then impossible.