Concept

Fundamental solution

The smallest solution in positive whole numbers of Pell's equation x² − Dy² = 1, leaving aside x = 1, y = 0. Every other solution comes from its powers, multiplying x + y√D by itself, so this one pair determines the whole infinite family.

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

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

Bhāskara's cyclic method on x² − 61y² = 1, step by step. A table of the cyclic method's rows: the helper m chosen at each step and the near miss a² − Db² = k it produces, ending at k = 1 with the fundamental solution.

A method that is allowed to miss

Bhāskara's cyclic method solves x² − Dy² = 1 by aiming at the wrong target. It keeps a pair a, b with a² − Db² = k for some small k, combines it with a helper chosen so that k can be divided out, and repeats until k is 1. For D = 61 it reaches the ten-digit fundamental solution in 13 steps, where walking the convergents of √61 takes 22 — and for every D up to 100 it is faster.

number · Pell

Named alongside it

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

Continued fractions Convergent Pell equation Unit Algorithm Hyperbola Lattice point Modular arithmetic Quadratic irrational

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