BG-MO 2013 Problem 6

Given $m\in\mathbb{N}$ and a prime number $p$ , $p>m$ , let $$M=\{n\in\mathbb{N}\mid m^2+n^2+p^2-2mn-2mp-2np \,\, \text{is a perfect square} \}$$ Prove that $|M|$ does not depend on $p$ .

Proposed by Aleksandar Ivanov