Theorem - Sum of Squares in the Integers

theoremnumber-theory

Intuition

A prime integer is equal to the sum of two squares if and only if it is congruent to $1$ mod $4$.

Formal Statement

For odd prime $p$,

$$-1\equiv \square \mathrm{mod}p⟺p=x^2+y^2\text{ in }\mathbb{Z}.$$

Equivalently, we have,

$$p\equiv 1\mathrm{mod}4⟺p=x^2+y^2\text{ in }\mathbb{Z}.$$

First seen in Álvaro’s book Number Theory and Geometry. Subsequently mentioned in class by Keith Conrad.