I refer to this question.
Given a perfect square, can you prove that it is a sum of two perfect squares?
I recently saw this:
Let $p,q$ be primes. $p_i \equiv 1 \pmod 4$ and $q_i \equiv 3 \pmod 4$.
$N$ can be written as sum of 2 squares iff all $b$ are even.
I used an identity that can multiply two numbers that can be written as sum of two perfect squares into a number that can be written as two perfect squares. And I have proved that the $p$ part can be written as two squares. Hence it is left to prove that the $q$ part can be written as sum of two squares.
Any help is appreciated. Thanks.