A Confusing Question By Mohammed Imran

$m^2 - n^2 = 2mn \Rightarrow (m - n)^2 = 2n^2 \Rightarrow m = ( \sqrt 2 + 1)n$

Not possible.

The given equation can be rewritten as $m=(√2+1)n$ . If $n$ be an integer, then $m$ will obviously never be an integer.

The answer is 0. If there exists a solution in positive integers to $m^2-n^2=2mn$ then there should exist a Pythagorean triple of the form: (a,a,b) where a,b are positive integers.But this leads us to the equation $2a^2=b^2$ which further gives $(a/b)^2=2$ .But this implies that $2^(1/2)$ is rational,a contradiction.Hence the number of solutions is 0.