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 identity $(a^2+b^2)(c^2+d^2)=(ac-bd{)}^2+(ad+bc{)}^2$ shows that a product of two sums of two squares is another sum of two squares. It follows by multiplying Gaussian integers and taking their squared absolute values.

If $u^2\equiv -1(\mathrm{mod}p)$, this lattice has basis $(1,u),(0,p)$ and covolume $p$. The Minkowski convex body theorem applied to a disk of squared radius strictly between $4p/\pi$ and $2p$ gives a nonzero lattice vector with $0<a^2+b^2<2p$. The congruence makes its squared length divisible by $p$, forcing $a^2+b^2=p$. The strict radius bounds avoid assuming the critical-volume form of Minkowski's theorem.