Zagier's Windmills: Primes as Sums of Two Squares
Why is every prime 4k + 1 a sum of two squares? Zagier’s one-sentence proof as windmills: flipping pairs them, only one stays alone, so their number is odd – and the remaining mirror windmill splits into p = x² + (2y)². With counterexamples 7, 11, 21 and 45.
From Erdős's BOOKTry it