Generator

Counting a 5 by 3 rectangle two ways

A generator in the number library, called 27 times across 6 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

reciprocity is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

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.

The multiples of 5, modulo 13

The two supplements, and the residue classes that decide them

The Gauss sum for 13, added one root at a time

Forty-two squares in a row

Half the primes stop at two

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

two primes agree modulo 3 and disagree about x² + 27y² ×158
3 is represented by x² + 1y² exactly when 3 mod 4 is one of the classes ×134
every step is a whole unit, because 1 is not a multiple of 13 ×70
the sign of multiplication by 3 is (3 | 11) ×22
(−1 | 3) is decided by 3 mod 4 ×17
(2 | 3) is decided by 3 mod 8 ×17
Gauss's lemma counts 1 folds for −1, and its parity is the symbol ×17
Gauss's lemma counts 1 folds for 2, and its parity is the symbol ×14
the modulus deciding x² + 1y² is at most 4n ×3
the modulus is a whole number between 5 and 31 ×3
3 folds gives the same answer as Euler's criterion ×2
(p | q) comes out of the count above it ×1
(q | p) comes out of the count below the line ×1
a prime with both symbols positive is 1 modulo 8, which is the two conditions met at once ×1
and every one of them lands in the bottom half ×1
and exactly half the multipliers are squares ×1
and it has no imaginary part ×1
and points up, which is the part Gauss took four years over ×1
and positive, which is the part Gauss took four years over ×1
and their product is minus one to the power of the whole rectangle ×1
between two and five forms are drawn ×1
both are odd primes and they are different ×1
each form's coefficient is a whole number between 1 and 27 ×1
every moved cycle has the same length, which is the order of the multiplier ×1
every number from 1 to 42 is a square mod 366,791 ×1
every point is counted exactly once, on one side or the other ×1
every record lies below 2(ln p)², the bound the Riemann hypothesis would give ×1
for a prime 1 modulo 4 the sum is real ×1
for a prime 3 modulo 4 the sum is purely imaginary ×1
no lattice point lies on the diagonal, because the primes are coprime ×1
the average approaches Erdős's constant from below ×1
the bound the congruence check runs to is a whole number between 2000 and 40000 ×1
the cycle count and the inversion count give the same sign ×1
the drawing holds both kinds of form, which is what it is for ×1
the first prime is a whole number between 3 and 31 ×1
the first prime whose least non-residue is 43 is 366,791 ×1
the folded values are all different ×1
the folds for 2 are the k above p/4 ×1
the largest modulus tried is a whole number between 20 and 400 ×1
the largest prime drawn is a whole number between 60 and 400 ×1
the largest prime in the table is a whole number between 11 and 97 ×1
the least non-residue is always a prime ×1
the length of the sum, squared, is the modulus ×1
the modulus is an odd prime ×1
the multiplier is a whole number between 2 and 30 ×1
the multiplier is not a multiple of the modulus ×1
the points below the line are the sum of the floors of kq/p ×1
the remainder mod 24 decides whether n(p) is 2, 3 or larger ×1
the second prime is a whole number between 3 and 31 ×1
the share with least non-residue the k-th prime is close to 2⁻ᵏ ×1
the square of the sum is plus or minus the modulus ×1
the view is one the family draws ×1
the walk never strays more than √p ln p from nought ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Number

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

One sum, squared two ways

Add the p-th roots of unity, each taken with a plus or a minus according to whether its index is a square. The walk that results closes on a point at distance √p from the origin — and squaring that one number, evaluated two different ways, is the reciprocity law.

Number

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

The symbol is the sign of a shuffle

Multiplying every residue modulo p by a fixed number rearranges them. That rearrangement is a permutation, permutations have a sign, and the sign is exactly the Legendre symbol — so a question about squares becomes a question about crossings.

Number

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

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 whole library · What the figures prove