Concept

Lattice point

A point of the plane or of space whose coordinates are all whole numbers. Counting the lattice points inside a region estimates its area or volume, and the error in the count is where much number theory lives.

Named by 2 essays across 2 fields — 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

A ball counted slice by slice. Ball of radius 7.5: slices 0:177, 1:177, 2:169, 3:145, 4:129, 5:97, 6:69, 7:21; total 1791, volume 1767.146.

The dots a ball catches

Count the lattice points inside a ball and compare with its volume. In five dimensions the error is about the size of one spherical shell, and that is a theorem; in four a logarithm sneaks in; in three and two the true size of the error is unknown. The difficulty runs backwards because the arithmetic of sums of squares is most erratic when there are fewest squares to add.

discrete · Pick theorem

Named alongside it

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

Asymptotics Class number Continued fractions Convergent Dimension Divisor function Error term Fundamental solution Hyperbola Pell equation Quadratic irrational Sum of squares

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