 Sum of two squares theorem

A positive integer is a sum of two integer squares exactly when every prime factor congruent to $3$ modulo $4$ has even exponent in its prime factorization.

The Sum of Two Squares Theorem states that a positive integer \( n \) can be expressed as a sum of two squares (i.e., \( n = a^2 + b^2 \) for some integers \( a \) and \( b \)) if and only if in its prime factorization, every prime of the form \( 4k + 3 \) appears with an even exponent.