Diophantine equation $^2$

Question

Let $x,y,n \in \mathbb{N} $. The $n$ is given and then we would like to solve: $x^2 + y^2 = n^2$ Is it possible? If yes, how to do it? Thanks in advance.

Answer 1

$x = k(a^2-b^2), y = 2kab, n = k(a^2+b^2), a > b$. So if $n$ is not of the form $k(a^2+b^2)$, it has no solution.