Unique factorisation
Named by 19 essays across 2 fields — each of them below, with the objects they name alongside it.
One way to factor, and no other
Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.
The shape of a number's divisors
Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.
Two squares, and a lattice
Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.
A tree that holds every triple
Three fixed matrices, applied to 3-4-5 over and over, produce every primitive Pythagorean triple there is — each of them once, none of them twice, and with no test for common factors anywhere in the procedure.
Always one before the double
A density says what happens on average and permits long empty stretches. This says something a density cannot — that the stretch from any number to twice it contains a prime, at every scale, without exception.
The sieve written as a product
Multiply out one geometric series for each prime and every whole number appears exactly once, as a single term. That identity turns a statement about factorisation into a statement about convergence, and it is where the analytic study of the primes begins.
Which roots refuse to be fractions
The square root of two is not a fraction, and neither is the square root of three, five, six or seven. The rule behind the list turns an infinite question into a search over the divisors of a single number — and the search finishes.
Which primes a form takes
A prime is the sum of two squares exactly when it is 1 modulo 4. Change the form slightly, to x² + 27y², and no congruence on p decides it at all — which is where the elementary subject ends and its successor begins.
The integers among the quaternions
The obvious integer quaternions are the ones with whole coordinates, and they are the wrong ones. Sixteen more, with every coordinate a half, have to be let in — and with them comes a division algorithm, twenty-four units instead of eight, and a regular solid that exists only in four dimensions.
Infinitely many of one kind
Euclid's argument produces a prime nobody had listed, and says nothing about what it looks like. Ask for infinitely many primes ending in 3, or leaving a remainder of 1 on division by 4, and the same construction has to be aimed — and for most targets nobody knows how to aim it.
On the circle and never home
Every root of unity lies on the unit circle, and so does the point (3 + 4i)/5 — yet no power of it ever returns to 1. A root of a whole-number polynomial of degree ten does the same. What forces a point home is a condition on the numbers its polynomial ties it to, and how far one of them may stray is a question open since 1933.
Almost no number is one
Sums of two squares look common — a sixth of all numbers up to a million are one. The fraction is falling to nothing, at a rate so slow that no computation will ever make it obvious, and the constant in front of it has been computed to fifty places and identified with nothing.
A factorisation that hides its primes
Keep only the whole numbers one more than a multiple of four. They multiply among themselves and nothing is lost — yet 441 is 9 × 49 and also 21 × 21, and every one of those factors is unbreakable there. Unique factorisation turns out not to be a fact about multiplication at all.
Factoring uniquely with no way to divide
Unique factorisation is proved by dividing with a small remainder, and in the Gaussian integers that works because discs of radius one cover the plane. In the integers of ℚ(√−19) the discs leave holes, no division algorithm of any kind can be made to work — and factorisation is unique anyway. The same field is why n² + n + 41 is prime forty times running.
Counting the classes that break factorisation
The class number measures how badly unique factorisation fails in a field, and defined through ideals it looks impossible to compute. Gauss computed it by hand, for every field he wanted, by counting quadratic forms — and every form can be squeezed, by changes of variable that keep its values, into exactly one small standard shape.
Real fields that factorise uniquely
Among the fields Q(√−d) only nine have unique factorisation, and Gauss guessed as much. Among the real fields Q(√p) he guessed the opposite — infinitely many — and the guess is still unproved. Counting cycles of reduced quadratic forms for the 16,900 primes p ≡ 1 mod 4 below 400,000 finds 79 per cent with class number one, drifting slowly down towards the 75.45 per cent that Cohen and Lenstra's heuristic predicts — and finds the reason the real fields behave so differently: their units are enormous.
The class number is a degree
The integers of ℚ(√−23) fall into three classes of ideals, and that 3 can be found by counting forms. It is also the degree of a single number: the modular function j, evaluated at the three forms' points, gives three values that are not whole but are the roots of one cubic with whole-number coefficients. When there is only one class, the value is a whole number outright — and that is why e^(π√163) misses being whole by less than a trillionth.
Most numbers are missing from the table
An N × N multiplication table has N² cells, and every product in it is less than or equal to N². Yet only 22.5% of the numbers that small turn up in the 10,000 × 10,000 table, and the share keeps falling. Erdős proved in 1955 that it falls to nought; Kevin Ford found in 2008 how slowly — like a tiny power of the logarithm, 0.0861 — because a product of two numbers has too many prime factors to be typical.
Sums with too few primes
Add 2 to 6,436,341 and the sum is 23⁵: three numbers with only four primes between them, and their product 15,042 is far smaller than the sum. The abc conjecture says this can happen only so far. Of 606 such triples up to 200,000, only seven reach quality 1.4, and if the conjecture holds, Fermat's last theorem and much else follows for large exponents in a line.
Named alongside it
The objects these essays reach for when they reach for this one.
PrimesGaussian integersLatticeModular arithmeticNormAlgebraic integerCounting two waysGreatest common divisorProof by contradictionQuadratic formSums of two squaresClass number