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

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

https://doi.org/10.2307/2323918

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 \(4k + 1\).

This implies that \(S\) contains an odd number of elements and hence every involution of \(S\) will have a fixed point. One involution of \(S\) is \((x, y, z) \mapsto (x, z, y)\) and a fixed point of this map gives a way of writing \(p\) as a sum of two squares.

The details do not seem too hard to check (I haven’t checked them though). The real question is - how did the authors come up with this mad involution?