Generator

The arithmetic of GF(4)

A generator in the computation library, called 47 times across 12 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.

finite-field 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

The arithmetic of GF(4). Addition and multiplication tables of a finite field, optionally beside the table of a ring of the same kind of size.

The non-zero elements of GF(16) as the powers of one of them

The cubic curves over GF(43) with the most and the fewest points

The number of points of a cubic over GF(43), read as a walk

The range of point counts of cubic curves over GF(p), for p up to 61

How many cubic curves over GF(101) have each number of points

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.

y² = x³ + 0x + 1 over GF(43) has a count inside p + 1 ± 2√p ×1806
a curve over GF(101) has exactly 82 points ×41
in GF(7) the sum of the 0th powers is 0 ×33
every count the bound allows over GF(5) occurs ×16
every cubic over GF(5) is inside the bound ×16
the 9 zeros of a sum of 3 squares over GF(3) are divisible by p to at least the promised power ×12
over GF(7) no cubic misses p by more than 2√p ×11
there is a field with 4 elements only if 4 is a prime power ×7
in GF(7) the sum of the 6th powers is −1 ×5
A is a set of residues modulo 13 ×3
B is a set of residues modulo 13 ×3
the largest cap in dimension 1 ×3
the monomial count at n = 4 ×3
the prime is between 7 and 61 ×3
the size of the field is a whole number between 2 and 9 ×3
2 zeros and 2 ones contain no 3 with a sum divisible by 3 ×2
every 5 residues mod 3 contain 3 with a sum divisible by 3 ×2
the largest prime is a whole number between 13 and 97 ×2
1 − f² is 1 at a zero of f and 0 elsewhere ×1
A and B are read only by the sumset and cdgrid views ×1
a conic y² = x² + b misses p by exactly one ×1
a progression's own operation gives exactly 2k − 1 ×1
a ring of composite size has a pair of non-zero elements multiplying to zero ×1
addition and multiplication are commutative ×1
an irreducible polynomial of the right degree was found ×1
and it is exactly the composite ones that leave an element without a reciprocal ×1
and no curve y² = x⁵ + ax + b by more than 4√p ×1
and so is the mean fourth power, an eighth ×1
and stays below the limit ×1
and that variable's power sums to 0 over GF(3) ×1
and the fewest is the smallest ×1
and the parabola construction is the smallest there is ×1
between 3 and 8 variables ×1
dim is read only by the dvir view and the cap views that take a dimension ×1
each walk ends within 2√p of zero ×1
every monomial has a variable whose power is not a positive multiple of p − 1 ×1
every non-zero element has a reciprocal ×1
every other point is the third point of some pair of the cap ×1
every point of every chosen line is in the set ×1
mod is read only by the cdmin view ×1
modulo a composite number the bound fails somewhere ×1
modulo a prime the smallest sumset is exactly min(p, a + b − 1) ×1
multiplication distributes over addition ×1
multiplication is associative ×1
no Kakeya set beats the counting bound ×1
no line holds three points of the cap ×1
no random cap beats the largest possible ×1
no set is small under both ×1
no two non-zero elements multiply to zero ×1
no two points of the cap have their third point in it ×1
odd primes up to 11 ×1
one to three prime-power fields from 3 to 16 ×1
over a cap, x + y + z = 0 only on the diagonal ×1
over a set with a line, it happens off the diagonal too ×1
q is an odd prime from 3 to 13 ×1
q is read only by the tables, the log view and the kakeya view ×1
so the number of zeros is 0 mod 3 ×1
so, since (0, 0, 0) is one, there is another ×1
some part of every split is at most two thirds ×1
some row is divisible by exactly the higher power Ax and Katz promise, and no more ×1
the cap drawn is a largest one ×1
the cap has twenty points ×1
the cap's share of the space falls with every dimension ×1
the constant is 2.7551 ×1
the dimension is 2, 3 or 4 ×1
the drawn points and the count agree ×1
the expansion is drawn over GF(3), where it fits on a page ×1
the exponent of a product is the sum of the exponents ×1
the field is closed ×1
the largest cap blocks every point outside it ×1
the largest cap in the plane of nine points has four ×1
the largest cap in the space of twenty-seven points has nine ×1
the largest power summed is a whole number between 8 and 20 ×1
the lower bound never exceeds what a construction achieves ×1
the mean square of the normalised error is the semicircle's, a quarter ×1
the modulus is 5, 7, 8, 9, 11 or 12 ×1
the modulus is 7, 11 or 13 ×1
the modulus is 7, 11, 12, 13 or 17 ×1
the monomials at or below two thirds of the top degree are a minority ×1
the monomials of every degree number 3ⁿ ×1
the most points any curve has is the largest whole number the bound allows ×1
the number being factorised is a whole number between 1 and 1000000 ×1
the number of primitive elements is φ(q−1) ×1
the number of random caps is a whole number between 200 and 5000 ×1
the plane of nine points has twelve lines ×1
the powers of a primitive element close back onto one ×1
the prime is 101 or 1009 ×1
the rate climbs with n ×1
the set has q(q + 1)/2 + (q − 1)/2 points ×1
the set holds a line in each of the q + 1 directions ×1
the size of the ring drawn beside it is a whole number between 0 and 12 ×1
the sumset is at least as large as Cauchy and Davenport promise ×1
the theorem is checked for n = 3 and n = 5 ×1
the three-variable form is drawn over GF(3), GF(5) or GF(7) ×1
the tuples are all counted ×1
the two-variable form is drawn over a prime from 3 to 11 ×1
the view is one the family draws ×1
the walk ends at the count less p + 1 ×1
the zeros of x² + y² + z² over GF(5) number a multiple of 5 ×1
x² + y² over GF(7) has only the zero at the origin, since −1 is not a square ×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.

Computation

A field's worth of squares

Two orthogonal squares of order five are easy to stumble on. Four of them, every pair orthogonal, is not a stumble — it is one line of arithmetic over a field, and the field supplies as many as the order allows.

Computation

A memory of four bits

A register holding four bits, shifting them along and adding two of them back, runs through all fifteen nonzero states before it repeats. Which two are added back is a question about a polynomial, and getting it wrong costs fourteen of the fifteen.

Computation

A polynomial through the gaps

Write the message as the coefficients of a polynomial and send its values instead. Any k of them determine the polynomial, so it does not matter which ones are lost — and it does not matter how many, as long as k survive.

Computation

Every element is a power of one of them

Pick the right element of a finite field and its powers run through every other non-zero element exactly once before returning to one. Multiplication becomes addition of exponents, and a table of q − 1 entries replaces the whole multiplication table.

Computation

Give or take twice the square root

A cubic curve over the integers mod 43 should have about 44 points — one for each value of x, on average, and one at infinity. No curve misses by more than 13, the largest whole number below 2√43, and every count from 31 to 57 belongs to some curve. The first fact is Hasse's theorem, the second Deuring's, and the way the counts spread between the limits is a semicircle.

Computation

No set with a line in every direction is small

In the plane over the integers modulo 7 there are 49 points and lines in 8 directions. A set holding a whole line in every direction needs 31 of the points — more than half — and in any dimension such a set fills a fixed share of the space. In the real plane the same sets can have area zero. Over a finite field one polynomial of low degree shows they cannot be small.

Computation

Seven points, seven lines

A geometry with seven points, in which every two points lie on exactly one line and every two lines meet in exactly one point. There are no parallels, the whole thing is built out of the two-element field, and one of its lines has to be drawn as a circle.

Computation

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

The field with four elements

The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.

Computation

The polynomial that bounds the caps

For forty years the best bound on a set of SET cards with no set among them shrank only like one over the dimension. In 2016 a two-page argument made it shrink exponentially, and the whole proof is a count of monomials: a table that is diagonal on a cap, one polynomial that describes it, and the fact that three parts of a degree cannot all be large.

Computation

Twenty cards with no set among them

The card game SET is a four-dimensional space over the integers mod 3, and a set is a line in it. Twenty cards can avoid every line and twenty-one cannot — a fact that took a proof in 1970 — while laying cards down at random and stopping when nothing more fits reaches twenty about once in two thousand tries.

Computation

A sum of two sets modulo a prime cannot be small

Add every element of one set of residues to every element of another. Over the whole numbers the sums always number at least |A| + |B| − 1. Modulo a prime the sums can wrap round and collide, and still they never number fewer — the theorem Cauchy proved in 1813 and Davenport again in 1935. Modulo 12 they can. A polynomial of low degree explains the difference in a paragraph.

The whole library · What the figures prove