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Concept

Divisibility

The relation between two whole numbers when one is a whole multiple of the other. It is the relation the whole of number theory is built on, and it is what a factorisation records.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

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 A schedule on 9 points where every pair meets exactly once. Points around a circle with the triples of a Steiner system drawn between them, beside the list of triples.

A schedule where every pair meets once

Sort n people into groups of three so that every two of them share a group exactly once. Two divisions have to come out whole, that rules out most sizes — and at every size the divisions permit, a schedule exists.

computation · Finite geometry 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 256 consecutive pairs from xₙ₊₁ = 137xₙ + 187 mod 256. Consecutive outputs of a linear congruential generator plotted as points of a square, falling on a small family of evenly spaced parallel lines.

The planes a recurrence cannot leave

One multiplication and one addition, taken modulo a fixed number, produce a sequence that passes for random one value at a time. Taken two or three at a time it does not, and the reason is a whole-number relation that pins every point onto one of a small family of parallel lines.

computation · Pseudorandomness Every fifth partition count divides, and the rank that says why. A row of partition counts with the ones in a congruence class marked, and a histogram of partitions sorted by rank.

Every fifth one divides

p(4) is 5, p(9) is 30, p(14) is 135, and every partition count at a number leaving four on division by five is divisible by five. Ramanujan read it off a table; the explanation is a way of splitting those partitions into five equal heaps.

number · Partitions 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 Pascal's triangle modulo 4, where one digit at a time is not enough. 32 rows of Pascal's triangle coloured by remainder modulo 4 — hue for the last base-2 digit, depth for the second. The digit-by-digit product that gives every remainder modulo 2 gets the remainder modulo 4 wrong at 100 of the 243 entries 2 does not divide.

A remainder read two digits at a time

Lucas' theorem reads a binomial coefficient's remainder on division by a prime off its digits one at a time. On division by the prime's square the same reading is wrong at four odd entries in ten. What replaces it still reads digits — in overlapping pairs, with the prime taken out first and a sign that the carries decide.

discrete · Pascals triangle The zeros of x² + y² + z² over GF(5), and of x² + y² over GF(7). Grids of every point over a small prime field with the solutions of a quadratic equation filled in: the three-variable equation drawn as one slice per value of z, beside a two-variable equation with far fewer solutions.

Solutions that come in multiples of p

Count the solutions of x² + y² + z² = 0 in the field with five elements and there are 25; with seven, there are 49. Whenever a system of equations has more unknowns than its total degree, its number of solutions is a multiple of the characteristic — which forces a solution besides zero, and the reason is a sum over the field that vanishes because its non-zero elements form one cycle.

computation · Finite fields The powers of 2 in 1 to 12: one number, 8, stands alone. The whole numbers from 1 to 12, each with a bar whose height is the power of 2 dividing it. A single number has the tallest bar, which is why the harmonic number H(12) has an even denominator and an odd numerator.

The sum that steps over every whole number

The harmonic sum 1 + 1/2 + 1/3 + … passes 2 at the fourth term, 3 at the eleventh, 4 at the thirty-first, and eventually every whole number there is. It never lands on one. The proof is a single number in the list 1, 2, …, n that carries more factors of two than any other — and the same arithmetic makes the numerators divisible by squares of primes they have no business knowing about.

analysis · Harmonic series

Named alongside it

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

Counting argumentPrimesExistence proofModular arithmeticProof by contradictionBinomial coefficientFermats little theoremHarmonic seriesKummer's theoremLucas' theoremPlace valueUnique factorisation

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