Primes
Named by 49 essays across 7 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
One residue whose powers are all of them
Fermat's theorem says every order divides p − 1. It does not say that anything has order exactly p − 1, which is a separate and stronger claim — and what forces it is a count of how many numbers share each divisor with p − 1.
The exponent that is smaller than Euler's
Euler's theorem raises every unit to the count of the units and gets one. The smallest exponent that works for all of them at once is often much smaller — and a composite is invisible to Fermat's test exactly when that smaller number divides n − 1.
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.
Why a quarter of numbers overshoot
About one number in four has proper divisors adding to more than itself. That a proportion exists at all is not automatic — there are sets defined just as simply that have no proportion — and the reason this one does is that abundance is inherited by multiples, and the numbers it is first inherited from are sparse enough to add up.
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.
A quantifier is a shadow
'There is an x such that …' asks whether a column of a grid contains a mark — which is the same as asking whether a shape casts a shadow on the axis below it. Over the real numbers every such shadow can be described without the quantifier, by polynomial inequalities: 'x² + ax + 1 = 0 has a solution' is just a² ≥ 4. Over the whole numbers the same kind of shadow can carve out the primes, and any set a computer can list.
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.
Divisors that add to three times the number
The divisors of 6 add up to 12, twice 6: a perfect number. The divisors of 120 add up to 360, three times 120, and those of 30,240 to four times it. Numbers like these were a sport for Fermat and Descartes, and they are held together by one fact — the ratio σ(n)/n is a product over the primes, and each prime can add only a little.
A ratio nobody else has
Divide the sum of a number's divisors by the number and you get its abundancy: 2 for every perfect number, 12/5 for both 30 and 140. Numbers that share an abundancy are called friends. Some numbers provably have no friend at all, most have friends only far away — and for 10, whose abundancy is 9/5, nobody knows whether a friend exists.
The wait for the next sum of two squares
Sums of two squares thin out to a share of nought, and yet the gaps between them stay short. Take the largest square below any number; what is left over is small, and a small number is always close to a square. Two squarings in a row say the next sum of two squares is never more than about 2√2 times the fourth root of n away — a bound proved in 1947 that nobody has improved, sitting far above every gap anyone has found.
Fourteen fractions that list the primes
Change the Collatz rule so that the multiplier depends on the remainder modulo some other number than two, and the resulting maps can compute anything a computer can. John Conway proved it in 1972, which means no method can decide, for every such map, whether its orbits come down — and fourteen fractions, applied in order, turn out to be enough to print every prime.
Two halves a sieve cannot tell apart
Split the whole numbers into those with an even number of prime factors and those with an odd number. To a sieve — which sees only how many numbers each small divisor divides — the two halves look almost exactly alike. Yet sifting out the small primes leaves every prime in one half and nothing at all in the other. That is Atle Selberg's parity example, and it is the reason no sieve can prove there are infinitely many twin primes.
The patterns primes are allowed to make
Primes two apart are twin primes; primes six apart are about twice as common; three primes two apart happen exactly once, at 3, 5, 7. Which patterns of primes can recur for ever is decided by remainders alone — a pattern must leave some remainder free modulo every prime — and how often each one recurs is predicted, to within a few per cent, by a constant computed from nothing else.
The row that proves a prime
Fermat's little theorem can be fooled: 561 passes it for every base and is not prime. Thread necklaces with a fixed number of black beads instead of any colours at all, and the count becomes a statement about a whole row of Pascal's triangle — every middle entry of row n is a multiple of n exactly when n is prime — which no composite can fake. Written as polynomials it is (x + a)ⁿ = xⁿ + a, and cut down to size it is the first proof that primes can be recognised in polynomial time.
How often two generates every remainder
For some primes p, the powers of 2 run through every non-zero remainder before repeating; for others they get stuck in a smaller cycle. Among the primes up to two million the first kind make up 37.43% — and Artin predicted 37.40% in 1927 by treating each way of getting stuck as an independent accident. For the base 5 the same prediction is wrong, because two of the accidents are secretly the same one, and nobody can yet prove that 2 works for infinitely many primes at all.
Divisors that make every amount
Twelve's divisors — 1, 2, 3, 4, 6, 12 — can be chosen to add to every whole number from one to twenty-eight. Numbers like that are called practical, a test on their primes decides them, and they turn out to be counted like the primes: about 1.336 x / log x of them below x. For practical numbers Goldbach's conjecture and the twin conjecture are theorems.
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.
Which way a prime's two squares point
A prime one more than a multiple of four is a sum of two squares in exactly one way, and the two squares make a point on a circle. Draw that point for every such prime and ask which way it faces. It faces every way equally — the angles spread evenly over the sector, which Hecke proved by giving each angle a remainder and copying Dirichlet. Measured over seventy-four thousand primes, they are even more evenly spread than random angles would be.
A sum of factorials that converges at every prime
1 + 1 + 2 + 6 + 24 + 120 + … grows faster than any geometric series and has no sum in the real numbers. Measure size by divisibility instead and the same series converges at every prime at once, to a different number each time. Whether any of those numbers ends in zero is a question Đuro Kurepa asked in 1971, and a search through every prime below 100,000 says no while a coin-toss model says it should have said yes about twice.
When Euclid's number is itself prime
Multiply the primes up to p and add one. Euclid needed only that the result has a prime factor beyond p; it is itself prime for p = 2, 3, 5, 7 and 11, and then not again until 31, and then not until 379. Up to 1,100 there are nine such primes of each sign, against thirteen that a careful estimate expects. The estimate predicts infinitely many and grows by about 1.8 every time x is multiplied by e; nobody can prove there is even one more.
The numbers that fool every base
561 = 3 · 11 · 17 passes Fermat's test for every base it shares no factor with, and so do 1,105, 1,729 and 102 more numbers below ten million. They were proved infinite only in 1994. Counted to 10²¹ there are about twenty million, growing like x to a power that has crept from 0.21 to 0.35 — just past the bound the proofs guarantee and nowhere near the power of one that Erdős's argument predicts.
A walk on Gaussian primes stopped by a moat
Start at 1 + i and step from Gaussian prime to Gaussian prime, never more than 2 at a time: the walk reaches 720 primes and stops at 17 + 42i, surrounded by a moat. Allow steps of √10 and it reaches a quarter of a million primes before stopping at distance 1,024; steps of 4 carry it to 4,313. Whether some fixed step lets a walker reach infinity is a question Basil Gordon asked in 1962, and it is open.
The first number that is not a square
Modulo a prime, half the numbers are perfect squares. Count up from 1 and the first non-square is usually 2 or 3 — but modulo 366,791 the numbers 1 to 42 are all squares. How far the run of squares can last is one of the oldest open questions about primes: the truth is tiny, the proofs are huge, and the Riemann hypothesis would close most of the gap.
A constant that holds the primes
There is a number A = 1.3063778838… such that A cubed, A to the ninth, A to the twenty-seventh and every further power of three, rounded down, is prime: 2, 11, 1361, 2521008887 and on for ever. W. H. Mills proved in 1947 that such a number exists. It looks like a formula for primes and is the opposite of one. Its digits are read off the primes it produces, each prime fixes about as many digits as it has, and the whole argument stands on a theorem about gaps between primes — which is why the power is 3 and not 2.
A rule that counts lets the sum escape
Strike out every number containing a 9 and the harmonic series converges, to 22.92. Keep only the numbers with as many 7s as 3s and it diverges — slowly, like the square root of the number of digits. Ask for two such balances and it still diverges; ask for three and it converges. The dividing line is Pólya's theorem that a random walk comes home in one and two dimensions and not in three.
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.
Which powers of x are enough
Every continuous function on an interval is nearly a polynomial, and it seems that all the powers of x should be needed for that. They are not. The even powers will do, and so will the primes, though the primes run out at a rate of one number in ln n. The squares will not: however many are combined, they stay 0.0905 away from x itself. The dividing line, found by Müntz in 1914, is whether the reciprocals of the powers add up to infinity.
Six choices make a seventh
Bertrand Russell's millionaire needs the axiom of choice to pick one sock from each of infinitely many pairs. Does having that power for pairs give the same power for sets of four? It does, by a short trick that Tarski found: choosing from the six pairs inside a set of four makes a tournament, and all 64 tournaments single out something definite. For sets of three it fails, and whether a size follows from smaller ones is settled almost entirely by arithmetic: from pairs, 4 is the only other size that can follow.
Numbers that never make a prime
Multiply 78,557 by a power of two and add one. The result is never prime, for any power, though almost every other odd number gives a prime within a few doublings. The reason is that seven small primes take it in turn to divide the result, on a schedule repeating every 36 steps that leaves no step uncovered. Proving nothing smaller does this has taken sixty years of search, and five candidates are still running.
The comet Goldbach left behind
Count the ways each even number is a sum of two primes, and the points do not form a smear. They split into bands: 12,000 has 302 ways and 12,002 has 115. One small product over odd prime factors explains every band and brings them all to a single line, and it says the count grows without bound. Nobody can prove the count is never nought.
Named alongside it
The objects these essays reach for when they reach for this one.
Modular arithmeticExhaustive searchCounting two waysDensitySums of two squaresUnique factorisationParityDivisibilityMultiplicative functionBinomial coefficientCompositeConjecture