Field

Number — page 3

The whole numbers, and how much structure they turn out to have.

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.

$The tangent built from a continued fraction. The curve tan x on (−1.55, 1.55) with 4 of Lambert's convergents: a straight line, then rational curves that bend ever closer to the tangent and follow it towards its poles.$

The fraction Lambert built for the tangent

The first proof that π is not a fraction, from 1761, does not look at π at all. It writes the tangent as an endless continued fraction, shows that the fraction's value at any rational point other than zero cannot be rational — because its tails are trapped between nothing and one — and then notes that tan(π/4) = 1.

$The continued fraction of e. Bars for the first 30 continued-fraction terms of e: mostly ones, with every third bar rising in a straight staircase, 2, 1, 2, 1, 1, 4, 1, 1, 6, ….$

The pattern in e's continued fraction

Written as a continued fraction, e is 2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8 — two ones, then the next even number, for ever. Euler found the pattern and proved it with a differential equation. A proof from 2006 needs only three integrals, each of which turns out to be exactly the error of one of e's own convergents.

441 factored two ways among the numbers of the form 4k + 1. Two factor trees for 441 among the Hilbert numbers, those one more than a multiple of four: one splits it as 9 times 49, the other as 21 times 21, and every factor is irreducible there.

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.

Where discs of radius one cover the lattice, and where they leave holes. Six lattices of algebraic integers drawn as points in the plane with a unit disc around each; for five of them the discs cover the whole plane and for the sixth, the integers of the field of the square root of minus nineteen, uncovered holes remain.

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.

A witness that 97 is prime: 5 has order 96. A table of the checks Lucas's test makes on 97 with base 5: the power (n − 1)/q for each prime q dividing n − 1, none of which is 1.

An order that proves a prime

Fermat's little theorem is a test that primes pass and composites mostly fail, and it can be fooled. Run backwards, it cannot. If some number a has order exactly n − 1 modulo n, then n is prime — because only a prime has n − 1 numbers to cycle through. Checking that takes the prime factors of n − 1, which need proofs of their own, and the proofs nest into a tree that anyone can check: Pratt's certificate, which shows every prime has a short proof of being one.

The Fermat quotient of 2 at every prime up to 4000, and where it is 0. A scatter of the Fermat quotient of 2 modulo p, divided by p, against p for the primes up to 4000; the points spread evenly between zero and one and reach zero at the Wieferich primes.

Two primes where Fermat holds twice

Fermat's theorem says p divides 2^(p−1) − 1. Usually p² does not. It does at 1093 and at 3511 and at no other prime anyone has found, in searches reaching past 10^19. The leftover, (2^(p−1) − 1)/p taken mod p, behaves like a random number, so a prime has about a one-in-p chance of the extra divisibility — and a random count with that chance grows so slowly that two by now is unremarkable, while nobody can prove there are any more, or that there are infinitely many primes where it fails.

The solutions of x² − 2y² = N, class by class. Rows for several right-hand sides N, each marking the solutions of x² − 2y² = N at the logarithm of x + y√2, coloured by class, over alternately shaded windows one unit-step wide.

Two families of solutions, and a box that holds both

Replace the 1 in Pell's equation by 7 and x² − 2y² = 7 still has infinitely many solutions — but they fall into exactly two families, each one an orbit of the same multiplication, and every family has a member inside a box whose size is fixed in advance. How many families there are is then a count of factors, and 3 has none.

How many digits Pell's first solution has, for D up to 1000. A scatter of the digit count of the smallest solution of Pell's equation against D, with the record-setting values ringed and labelled.

Sixty needs two digits and sixty-one needs ten

The smallest solution of x² − 60y² = 1 is x = 31. For x² − 61y² = 1 it is x = 1,766,319,049. The jump is not an accident of 61: the solution is exactly the product of one period's complete quotients, so its size is set by how long the continued fraction takes to come home — and for the cattle problem Archimedes is said to have posed, that product has 103,273 digits.

The shape a random partition takes

There are about twenty-four thousand billion billion billion ways to write 1,000 as a sum of whole numbers. Pick one at random, draw its Ferrers diagram, shrink it by the square root of a thousand, and it is almost exactly the curve e^(−cx) + e^(−cy) = 1 with c = π/√6. So is the next one, and the next. A random partition of a large number has a shape, and the shape is known exactly.

Two ways of counting that agree at every number. For n up to 40, the counts of partitions with gaps of at least two against partitions into parts congruent to 1 or 4 mod 5, on a logarithmic scale, equal at every n, with the second identity's counts beside them.

Two counts that agree for no visible reason

Write 10 as a sum of whole numbers that differ from each other by at least two, and there are six ways. Write 10 as a sum of numbers that each leave 1 or 4 on division by 5, and there are six ways. The same happens for 20 (thirty-one each), for 40 (three hundred and seventy-four each), for every number anyone has checked and every number there is. The two lists look nothing alike, and no one has found a simple way to turn one into the other.

Numbers whose divisors add to two, three, four, five and six times themselves. A table of multiperfect numbers with the multiple their divisor sum makes of them, the number itself and its factorisation into prime powers.

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 next sum of two squares after 150. A quarter of the circle of radius √150 on the integer lattice, with the column x = 12 and the first lattice point above the circle in it, (12, 3), on the circle of radius √153.

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.

From a rational point on x² + y² = 17 to a whole one. The circle of radius √17 on the integer lattice, a rational point on it, and 4 reflections through nearest lattice points, the denominators 3757, 205, 25, 5, 1, ending at (1, 4).

A fraction on the circle forces a whole point

If a number is a sum of two squares of fractions, it is a sum of two squares of whole numbers. Draw the circle, mark the rational point, join it to the nearest lattice point and follow the line to where it meets the circle again: the new point is rational too, with a smaller denominator. Repeat, and the denominators fall until they reach one. The argument needs no primes at all — only the fact that every point of the plane is within distance one of the lattice.

Selberg's parity example: the same sieve data, the primes on one side and nothing on the other. Horizontal bars for the numbers up to 100000 with an odd and an even count of prime factors, before sifting (50144 and 49855) and after sifting by the primes below √100000 (9527 and 0).

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.

Prime pairs a fixed distance apart, counted, against Hardy and Littlewood's prediction. Bars for the number of prime pairs (p, p + d) up to 2000000 for even d from 2 to 60, jagged with peaks at multiples of 6 and 30, each with the Hardy–Littlewood prediction marked.

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.

$The 46 Farey fractions of order 12, and how far each strays from even spacing. Farey fractions of order 12 against 46 evenly spaced points, with the deviation of each drawn as a bar; largest deviation 0.0616.$

How evenly the fractions spread

List every fraction between nought and one with denominator at most n, in order. They spread across the interval almost evenly, and how fast the unevenness shrinks as n grows is — exactly, provably — the Riemann hypothesis. The link runs through a second fact: set the fractions round a circle and add them as arrows, and what is left is a whole number.

The 4 reduced forms of discriminant −84, each in its place. The reduced binary quadratic forms of discriminant −84 (x² + 21y²; 2x² + 2xy + 11y²; 3x² + 7y²; 5x² + 4xy + 5y²) plotted at their roots in the upper half-plane, all inside the modular fundamental region.

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.

The necklaces of 7 beads with 3 black. C(7, 3) = 35 arrangements in 5 rotation classes of sizes 7, 7, 7, 7, 7.

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 2 is a primitive root, measured against the prediction. base 2: 37.429% of primes up to 2000000, predicted 37.396%.

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.

Which D up to 400 make x² − Dy² = −1 solvable. D from 2 to 400: 69 solvable, 302 blocked by 4 or a prime 4k + 3, 9 unsolvable with no such obstruction, first at 34.

When minus one can be reached

x² − Dy² = 1 always has a solution. Put −1 on the right and it sometimes does and sometimes does not. A remainder rules out most D at a glance, every prime of the form 4k + 1 is guaranteed one, and between the two lies a set of D decided by arithmetic several layers deep — of which, a million numbers in, 78% are solvable, on the way to a limit of 58% that was proved only in 2022.

$The first 600 terms of the continued fraction of ∛2. Partial quotients of ∛2 computed exactly: 1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3 …; the largest in the first 600 is 7451.$

A cube root that looks like chance

The continued fraction of √61 repeats after eleven terms, and every square root's must. The cube root of 2 obeys no such law. Computed exactly, by a method of Lagrange's that never leaves the whole numbers, its terms run 1, 3, 1, 5, 1, 1, 4 … and throw up a 534 at the thirty-fifth place and a 7,451 at the 571st — and in every statistic anyone has measured, they look like the terms of a number picked at random.

Class numbers of x² + ny², with Euler's idoneal numbers marked. A scatter of class number against n up to 2000, on a logarithmic scale, with the 65 idoneal numbers marked on the power-of-two levels.

Euler's sixty-five convenient numbers

For some n, whether a prime can be written as x² + ny² is settled by its remainder on division by 4n alone, the way a prime's remainder on division by four settles whether it is a sum of two squares. Euler found sixty-five such n, from 1 to 1,848, called them convenient, and used the largest to prove that 18,518,809 is prime. Every one of them is a number whose class group has no element of order more than two — and whether the list is complete is still not known.

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