Which primes are sums of two squares?

Show that an odd prime $p$ can be written as $p=x^2+y^2$ with $x,y$ integers if and only if $p\equiv 1(\mathrm{mod}4)$.

Fermat announced it and Euler first proved it. Since then it has been proved by descent, by lattices, by Gaussian integers, and in a single sentence by an involution on triples of integers. Proofs this different about one theorem are exactly what this board is for.