Matpic
mathematics, in pictures
Concept

Riemann hypothesis

The conjecture, from 1859, that every non-trivial zero of the Riemann zeta function has real part one half. It is equivalent to the primes being as evenly spread as randomness allows, and it is the most famous open problem in mathematics.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

number · Infinitude of primes 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.

number · Stern brocot The fifty-four sums of four weighted squares that reach every number. The 54 universal diagonal quaternary forms, as coefficient lists: 1,1,1,1; 1,1,1,2; 1,1,1,3; 1,1,1,4; 1,1,1,5; 1,1,1,6; 1,1,1,7; 1,1,2,2; 1,1,2,3; 1,1,2,4; 1,1,2,5; 1,1,2,6; 1,1,2,7; 1,1,2,8; 1,1,2,9; 1,1,2,10; 1,1,2,11; 1,1,2,12; 1,1,2,13; 1,1,2,14; 1,1,3,3; 1,1,3,4; 1,1,3,5; 1,1,3,6; 1,2,2,2; 1,2,2,3; 1,2,2,4; 1,2,2,5; 1,2,2,6; 1,2,2,7; 1,2,3,3; 1,2,3,4; 1,2,3,5; 1,2,3,6; 1,2,3,7; 1,2,3,8; 1,2,3,9; 1,2,3,10; 1,2,4,4; 1,2,4,5; 1,2,4,6; 1,2,4,7; 1,2,4,8; 1,2,4,9; 1,2,4,10; 1,2,4,11; 1,2,4,12; 1,2,4,13; 1,2,4,14; 1,2,5,6; 1,2,5,7; 1,2,5,8; 1,2,5,9; 1,2,5,10.

Fifteen numbers decide every number

Lagrange proved that x² + y² + z² + w² takes every whole value. Ramanujan asked which other sums of four weighted squares do, and in 1917 listed fifty-five. One of them is wrong: x² + 2y² + 5z² + 5w² misses 15. The mistake points at a theorem found eighty years later — to know whether such a form reaches every number, it is enough to check that it reaches 1, 2, 3, 5, 6, 7, 10, 14 and 15. Nine checks decide infinitely many cases, and the nine numbers come out of an escalation any one can run.

geometry · Figurate numbers Forty-two squares in a row. p = 366791: partial sums of (n/p) for n ≤ 200: 0, 10, 20, 30, 40, 46, 54, 60, 64, 72, 78, 78, 88, 92, 98, 102, 108, 116, 122, 126, 132; least non-residue 43.

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.

number · Quadratic reciprocity Each prime fixes the next digits of A. A = 1.306377883863080690468614492602605712916…; digits fixed by the primes from 11 to the 254-digit one: 1, 4, 11, 30, 87, 257.

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.

number · Infinitude of primes

Named alongside it

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

PrimesConjectureExhaustive searchAlmost primeDiscrepancyEquidistributionExistence proofFarey sequenceFinite checkGauss sumLegendre symbolLiouville function

All concepts