Unit — where it appears
Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
![The integers of ℚ(√5), with ℤ[√5] inside them. Points a + bφ plotted against their conjugates for small whole a and b, with the index-two sublattice ℤ[√5] filled and the basic cells of both lattices shaded, of areas √5 and 2√5.](/f/the-integers-of-q-5-with-z-5-inside-them-a4f067e7.svg)
The integers a field contains
Inside the field of numbers a + b√5, the obvious integers are those with whole a and b. They are not all of them: the golden ratio has a one-half in it and satisfies x² = x + 1, a monic equation with whole coefficients, exactly as an integer should. The right integers form a lattice twice as dense as the obvious one — and a whole-number matrix proves they are closed under addition.
Units that form a lattice
In the whole numbers only 1 and −1 have whole-number reciprocals. In the integers of a bigger field there can be infinitely many such units, and they are not scattered: take logarithms of their sizes under each way of placing the field in the real or complex numbers, and the units land exactly on a lattice. How many dimensions that lattice has is a count of those placements, and the area of its cell is a number no formula gives.
![Three roots that one map rotates, for every t. Shanks cubic roots for t in [-9,6]; at t=2: 3.507019, -0.221876, -1.285142; map permutation 1,2,0.](/f/three-roots-that-one-map-rotates-for-every-t-bf803443.svg)
A line of cubics that are all cyclic
In a box of cubics the full group of six symmetries is almost universal, and the three rotations are a rare exception. Daniel Shanks's x³ − tx² − (t + 3)x − 1 has the rotations at every whole t, and at every rational one. Its discriminant is the square of t² + 3t + 9, its roots are turned into one another by x ↦ −1/(1 + x), and on a suitable circle every member is an equilateral triangle.
Named alongside it
The objects these essays reach for when they reach for this one.
Algebraic integerContinued fractionsConvergentDeterminantField extensionFundamental solutionLatticePell equationAlgorithmCharacteristic polynomialCyclic groupDiscriminant