Concept

Pell equation

The equation x² − Dy² = 1 in whole numbers, for a D that is not a perfect square. It always has infinitely many solutions, and every one of them is a power of the smallest under a multiplication that is really multiplication of numbers of the form a + b√D.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked.

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.

number · irrationality

Whole-number points on x² − 2y² = 1. The branch of the hyperbola x² − 2y² = 1 in the first quadrant, with the whole-number points on it marked and labelled, and the lattice drawn faintly behind.

One solution that makes all the others

The equation x² − 2y² = 1 has infinitely many whole-number solutions, and every one of them is a power of the smallest. The multiplication that produces them is what multiplying two numbers of the form a + b√2 comes to when the √2 terms are collected — so an equation about a hyperbola turns out to carry a group.

number · pell

Named alongside it

The objects these essays reach for when they reach for this one.

Continued fractions Convergent Counting two ways Descent Fundamental solution Hyperbola Incommensurability Infinite descent Irrationality Lattice point Parity Proof by contradiction

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