Proof by contradiction — where it appears
There is no last prime
Euclid's argument is often described as producing a new prime from any finite list. It does not, and the number it builds is frequently composite — which makes the proof more interesting rather than less.
One way to factor, and no other
Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.
The square that cannot shrink
The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting two waysDivisibilityPrimesCompositeContinued fractionsCounting argumentEuclidEuclid lemmaExistence proofGreatest common divisorHarmonic seriesIncommensurability