When is a prime the sum of two squares? — The Ross Program
Channel: Jim Fowler
Date: 31/07/2020

For primes p ≡ 1 mod 4, there are integers x and y so that x² + y² = p.  There are many proofs of this fact, but this videos rushes through Zagier's proof (itself inspired by Heath-Brown and Liouville) and then surveys the "windmill" argument from https://mathoverflow.net/questions/31113/zagiers-one-sentence-proof-of-a-theorem-of-fermat/299696 before finishing with some musings on the goals  of mathematical proof.