Concept

Factorisation

A way of writing an object as a product of smaller objects of the same kind. Whether the decomposition is unique is a property of the surrounding system rather than of the object, and uniqueness fails in arithmetics only slightly larger than the whole numbers.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

3 roots, and the two numbers the coefficients already knew. The roots of a degree-3 polynomial, found numerically, with the point they average to. That average, and their product, are readable straight off the coefficients without finding the roots at all.

What the coefficients already know

Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.

algebra · Polynomial roots

The 7 7th roots of unity. 7 points spaced evenly around the unit circle, at the vertices of a regular 7-sided polygon.

The polygon an equation forces

The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.

algebra · Roots of unity

The solutions of x² − 2y² = N, class by class. Rows for several right-hand sides N, each marking the solutions of x² − 2y² = N at the logarithm of x + y√2, coloured by class, over alternately shaded windows one unit-step wide.

Two families of solutions, and a box that holds both

Replace the 1 in Pell's equation by 7 and x² − 2y² = 7 still has infinitely many solutions — but they fall into exactly two families, each one an orbit of the same multiplication, and every family has a member inside a box whose size is fixed in advance. How many families there are is then a count of factors, and 3 has none.

number · Pell

Perfect but for one factor

In 1638 Descartes wrote to Mersenne with an odd number whose divisors add up to exactly twice the number — provided one of its factors, 22021, is counted as a prime. It is not; it is 19² × 61. Nearly four centuries later that number is still the only odd one of its kind known, and a search of every odd number below ten million finds no other. It is the closest anyone has come to an odd perfect number, and what it shows is how an odd perfect number would have to be built.

number · Perfect numbers

Euclid's argument run twenty times. The first 20 terms of the Euclid–Mullin sequence: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957.

Euclid's proof run as a machine

Euclid proved there is no last prime by multiplying the primes on any list, adding one, and noting that the result has a prime factor not on the list. Run the proof as a machine — start from 2, and each time take the smallest prime factor of one more than the product so far — and it produces 2, 3, 7, 43, 13, 53, 5, 6221671, … a sequence that never repeats, that reaches small primes late and large ones early, and that nobody can prove reaches every prime.

number · Infinitude of primes

Named alongside it

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

Complex numbers Exhaustive search Abundancy Coefficient Counterexample Cyclic group Cyclotomic polynomial Degree Discriminant Divisor sum Equivalence class Fundamental solution

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