= Congruence lattice proof of the prime sum of two squares
{title2=$\Lambda=\{(a,b)\in\mathbb Z^2:b\equiv ua\pmod p\}$}
If $u^2\equiv-1\pmod p$, 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.
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