Every Prime $\equiv 1 \mod 4$ Is a Sum of Two Squares

Sun, 10 Sep 2023 15:35:44 +0000

I came across this “one sentence” proof that every prime congruent to 1 mod 4 can be written as a sum of two squares:

The proof involves constructing the following cryptic involution $$ (x, y, z) \mapsto \begin{cases}(x+2 z, z, y-x-z) & \text { if } x<y-z \ (2 y-x, y, x-y+z) & \text { if } y-z<x<2 y \ (x-2 y, x-y+z, y) & \text { if } x>2 y\end{cases} $$ on the set $S=\left{(x, y, z) \in \mathbb{N}^{3}: x^{2}+4 y z=p\right}$ and showing that the involution has exactly one fixed point $(1, 1, k)$ if and only if $p$ is a prime of the form $4k + 1$.

Elementary Number Theory on PostDoc Problems

Every Prime $\equiv 1 \mod 4$ Is a Sum of Two Squares