Question 151
Let $m$ be a positive integer, and $p$ be a prime number such that $p \gt m$. Prove that the number of positive integers $n$, for which
$$
m^2 + n^2 + p^2 - 2mn - 2mp - 2np
$$
is a square of some positive integer, does not depend on $p$. (Bulgaria 2013)
$$
m^2 + n^2 + p^2 - 2mn - 2mp - 2np
$$
is a square of some positive integer, does not depend on $p$. (Bulgaria 2013)