Sum of two squares theorem

In number theory, the sum of two squares theorem relates the prime decomposition of any integer n > 1 to whether it can be written as a sum of two squares, such that n = a² + b² for some integers a, b.

An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no term p^k, where prime ${\displaystyle p\equiv 3{\pmod {4}}}$ and k is odd.[1]

If ${\displaystyle n}$ is a perfect square, it can be written with the trivial case applied, setting ${\displaystyle a}$ (or ${\displaystyle b}$) to zero: ${\displaystyle n^{2}=0^{2}+n^{2}}$ (Some perfect squares also have non-trivial cases such as ${\displaystyle 25=4^{2}+3^{2}}$ and ${\displaystyle 100=8^{2}+6^{2}}$, also known as Pythagorean triples).

This theorem supplements Fermat's theorem on sums of two squares which says when a prime number can be written as a sum of two squares, in that it also covers the case for composite numbers.