Concept

Convergent

A fraction obtained by cutting a continued fraction off after some number of terms. The convergents of a number approach it from alternate sides, and each is closer to it than any fraction with a smaller denominator.

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

Also named here as fundamental solution, unit — the same set of essays touches all of them, so they are one junction rather than several.

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 Fundamental solution Pell equation Unit Algorithm Hyperbola Lattice point Modular arithmetic Quadratic irrational

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