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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 [E.7.2] with the following title:

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.

In Exercise Group 13.7.19–23, you will be asked to verify the various statements that this proof depends on. Although perhaps it is not the easiest single sentence after all, it is still fun – fun enough that you can watch a couple videos about it from Numberphile!