Section 13.6 A One-Sentence Proof
¶There is a completely different approach to this problem which has gained some notoriety. Often one wants multiple approaches in order to understand a problem more deeply; here, we have picked a geometric approach. It happens that D. Zagier provided the culmination of a series of proofs using only sets and functions, and that proof takes only one sentence to write down! This is reproduced from the famous article [C.7.2] with the following title:Proposition 13.6.1. A One-Sentence Proof that Every Prime p≡1 (mod 4) is a Sum of Two Squares.
Proof.
The involution on the finite set
\begin{equation*}
S=\{(x,y,z)\in \mathbb{N}^3 \mid x^2+4yz=p\}
\end{equation*}
defined by
\begin{equation*}
(x,y,z)\to \begin{cases}(x+2z,z,y-x-z)& \text{if }x<y-z\\(2y-x,y,x-y+z)& \text{if }y-z<x<2y\\(x-2y,x-y+z,y)& \text{if }x>2y\end{cases}
\end{equation*}
has exactly one fixed point, so \(|S|\) is odd and the involution defined by \((x,y,z)\to (x,z,y)\) also has a fixed point.