The product of two numbers that can be written as the sum of two squares

Question

Prove that if n is the product of two numbers that can be written as the sum of two squares then n can be written as the sum of two squares.

Answer 1

See the Brahmagupta–Fibonacci identity.

Or consider the formula for the product of two complex numbers, and take the moduli.

Answer 2

A number can be written as the sum of two squares iff each of its prime divisors of the form $p = 4k + 3$ occurs to an even power. So if $n = ab$ and $a$ and $b$ have this property, so will their product.

See Integer as sum of Two Squares