The collection

Every essay — page 28

Page 28 of 49, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Number

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

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.

Counting one rectangle, twice

Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked.

The square that cannot shrink

The partition 5 + 4 + 2 + 1 and its conjugate. A row of dots for each part, and the same dots read down the columns instead.

A diagram turned on its side

The tree of Pythagorean triples. A tree rooted at 3-4-5. Each triple has three children, obtained by three fixed integer matrices, and every primitive triple appears exactly once somewhere in it.

A tree that holds every triple

Counting what has no formula

Between every number and its double. The interval from n to twice n, drawn for n up to 26, with the primes inside each marked. Every interval contains at least one.

Always one before the double

The sieve as a product, and the sum over the primes. The whole numbers up to 60, with those built only from 2, 3, 5 marked — the numbers the product of three geometric series multiplies out to. Beside them, the sum of the reciprocals of the primes, which grows without bound.

The sieve written as a product

Whole-number points on x² − 2y² = 1. The branch of the hyperbola x² − 2y² = 1 in the first quadrant, with the whole-number points on it marked and labelled, and the lattice drawn faintly behind.

One solution that makes all the others

Which roots refuse to be fractions

The tail that would have to be a whole number. For each denominator, the value of q! times the tail of the series for e, plotted against the band between zero and one where no whole number lies, with the bound 1/q above it.

A tail too small to be a whole number

The polynomial that squeezes π. On the left, xⁿ(π − x)ⁿ/n! drawn at several degrees, its largest value falling toward nothing; on the right, the derivatives of the same polynomial at zero, every one a whole number.

An integral that cannot be a whole number

6 figures $How closely a fraction can come, and the barrier that says no closer. Two panels at very different scales: the approximations to √2, which stay above the barrier a degree-two number obeys, and the truncations of a constructed number, which fall below every barrier drawn.$

Approached too fast to be algebraic

The partition product's coefficients to q¹². A row of series coefficients computed by expanding a product, beside the same numbers obtained another way.

Every partition, hidden in a product

The product of (1 − qᵏ), and what survives at 12. The coefficients of the pentagonal product drawn as signed bars, with the partitions into distinct parts that Franklin's move leaves unpaired.

The terms that cancel almost everything

p(n) to 60, against the Hardy–Ramanujan estimate. The number of partitions of each number up to sixty on a logarithmic scale, with the asymptotic estimate drawn over it and the ratio of the two tabulated.

The size of a number with no formula

Every fifth partition count divides, and the rank that says why. A row of partition counts with the ones in a congruence class marked, and a histogram of partitions sorted by rank.

Every fifth one divides

The two supplements, and the residue classes that decide them. A table of odd primes with the Legendre symbols of minus one and two beside the residue of p modulo four and modulo eight.

The two supplements, and where the eight comes from

Multiplication by 3 modulo 11, and the sign of the shuffle. Residues in two rows joined by strings showing where multiplication sends each one, with a strip beneath comparing the sign of the shuffle to the Legendre symbol for every multiplier.

The symbol is the sign of a shuffle

The Gauss sum for 13, added one root at a time. Partial sums of the p-th roots of unity signed by the Legendre symbol, drawn as a walk closing on a point at distance root p from the origin.

One sum, squared two ways

Where a congruence decides which primes a form represents, and where it does not. Rows of primes marked by whether each is represented by x squared plus n y squared, with the residue classes that decide it where such classes exist.

Which primes a form takes

The primes below 100,000, by remainder mod 4. A bar for each remainder on division by 4, showing how many primes below 100000 leave it. The 2 classes sharing no factor with 4 hold near-equal counts; the rest are empty or hold one prime.

Infinitely many of one kind

Five families of primes, counted below 100,000. Five counting curves on logarithmic axes: every prime, the primes one more than a multiple of four, the twin pairs, the primes one more than a square, and the Mersenne primes. Two of the five families are known to be infinite and three are open questions.

Which infinitudes are proved

The primes 3 mod 4 against the primes 1 mod 4, out to 30,000. A plot of the difference between the counts of primes in two residue classes, against the bound, showing a persistent lead for one class and where it is lost.

Every class, and in equal shares

Two sums of reciprocals: 2.89 and climbing against 1.71 and level. Two curves against the logarithm of the bound: the sum of reciprocals of all primes, rising steadily, and the sum over the twin primes, flattening towards a limit.

The sieve that cannot finish