Congruence lattice proof of the prime sum of two squares ($\Lambda =\{(a,b)\in {\mathbb{Z}}^2:b\equiv ua(\mathrm{mod}p)\}$)

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.