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Concept

Primes

The whole numbers above one divisible by nothing but themselves and one, out of which every other whole number is built. How many lie below a bound is estimated by the logarithmic integral, and how they are arranged locally remains largely unknown.

Named by 18 essays across 4 fields — each of them below, with the objects they name alongside it.

Pascal's triangle mod 2, 32 rows. Only the odd entries are drawn; the pattern that appears is the Sierpiński triangle.

Pascal's triangle, in two colours

Shade the odd numbers in Pascal's triangle and a fractal appears. Nothing was designed to produce it, and the same shape arrives independently from a completely different construction.

discrete · Pascals triangle Ulam's spiral to 900. The integers up to 900 laid out in a square spiral, with the primes marked; they crowd onto diagonal lines.

The primes on a spiral, and a pattern nobody ordered

Wind the whole numbers outward in a square spiral, mark the primes, and they line up on diagonals. The observation is a hundred years old and there is still no proof it means anything.

discrete · Prime distribution Arithmetic on a dial of 12. A dial with 12 positions. Starting at 8 and stepping forward 9 places lands on 5, because the walk passes the top 1 time on the way.

Numbers that wrap

A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.

discrete · Modular arithmetic The sieve of Eratosthenes below 100. A grid of the whole numbers with the composites struck out by the prime that removes them.

The primes are what is left over

Eratosthenes' sieve does not find the primes. It removes everything else, one prime at a time, and whatever survives is prime by default — which is a strange way to reach the most studied objects in arithmetic.

number · Prime distribution Euclid's construction on 2, 3, 5, 7. The product of the listed primes plus one, divided by each of them in turn; every division leaves one over.

There is no last prime

Euclid's argument is often described as producing a new prime from any finite list. It does not, and the number it builds is frequently composite — which makes the proof more interesting rather than less.

number · Infinitude of primes Two factor trees of 360. The same number split two different ways, both ending in the same primes.

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.

number · Unique factorisation 28 is perfect, because its divisors form this rectangle. Two rows of divisors: the powers of two, and the same powers multiplied by the Mersenne prime.

Numbers that are their own parts

Six is one plus two plus three. Twenty-eight is one plus two plus four plus seven plus fourteen. Euclid explained where such numbers come from; Euler proved there are no others of that kind; and whether an odd one exists has been open for two thousand years.

number · Perfect numbers The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

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.

number · Sums of two squares Necklaces of 5 beads in 2 colours. Every string of beads, grouped by the rotations that carry one onto another.

Necklaces that prove a theorem

Thread five beads in two colours, thirty-two ways. Two of them are all one colour; the other thirty fall into rings of five. That count, and nothing else, is Fermat's little theorem.

number · Fermats little theorem Counting a 5 by 3 rectangle two ways. Lattice points in a rectangle cut by a diagonal of slope q over p, coloured by which side they fall.

Counting one rectangle, twice

Whether seven is a square modulo eleven, and whether eleven is a square modulo seven, are two unrelated-looking questions. Their answers are linked, and the link is a rectangle of dots counted along its rows and then along its columns.

number · Quadratic reciprocity π(x) against its two estimates, up to 20,000. The ratio of the prime counting function to x over the logarithm of x, and to the logarithmic integral, plotted against x. The first is above one and coming down slowly; the second is close to one throughout.

Counting what has no formula

There is no expression that gives the nth prime, and yet the number of primes below a bound is predictable to within a fraction of a per cent — by a function that is not a formula for the primes but an integral of the wrong-looking quantity.

number · Prime distribution Between every number and its double. The interval from n to twice n, drawn for n up to 26, with the primes inside each marked. Every interval contains at least one.

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.

number · Prime distribution The sieve as a product, and the sum over the primes. The whole numbers up to 60, with those built only from 2, 3, 5 marked — the numbers the product of three geometric series multiplies out to. Beside them, the sum of the reciprocals of the primes, which grows without bound.

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.

number · Prime distribution The two supplements, and the residue classes that decide them. A table of odd primes with the Legendre symbols of minus one and two beside the residue of p modulo four and modulo eight.

The two supplements, and where the eight comes from

The main law relates two odd primes to each other and says nothing about −1 or about 2. Those two are settled separately, by their own counts, and the answers arrive modulo four and modulo eight — which is a clue about where the whole subject is really taking place.

number · Quadratic reciprocity Where a congruence decides which primes a form represents, and where it does not. Rows of primes marked by whether each is represented by x squared plus n y squared, with the residue classes that decide it where such classes exist.

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.

number · Quadratic reciprocity The four-square identity, and how few squares a number needs. A product of two whole quaternions with both sides of the four-square identity evaluated, above a strip colouring every number by the fewest squares that add to it.

The identity that multiplies sums of squares

A sum of two squares times a sum of two squares is a sum of two squares, and the same is true of four and of eight. It is true of three of nothing: 3 and 5 are each a sum of three squares, 15 is not, and one small pair of numbers rules the case out forever.

algebra · Quaternions Every pattern, exactly as often. A table over 4 window lengths of a shift register's output stream: how many bit patterns are possible, how many actually occur, and the difference between the most and least frequent, which is one in every row.

Nineteen thousand bits of state

The generator most simulations actually use is not clever. It is a linear recurrence over the two-element field with an enormous state, and its virtues are a proved period, a proved equidistribution and speed — none of which is unpredictability, which it does not have and does not claim.

computation · Pseudorandomness 3 carries in base 2, and 2 divides it 3 times. The addition of 5 and 7 written in base 2, column by column, with the carries marked. There are 3, and 2 divides the binomial coefficient 792 exactly 3 times.

The carries decide the divisibility

How many times a prime divides a binomial coefficient is not a fact about the coefficient at all. It is a count of the carries that happen when two numbers are added in that prime's base, which is a question about column addition and has nothing to do with choosing anything.

discrete · Pascals triangle

Named alongside it

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

Counting two waysModular arithmeticParityUnique factorisationCompositeDensityDivisibilitySums of two squaresBinomial coefficientCounting argumentExistence proofGaussian integers

All concepts