The collection

Every essay — page 32

Page 32 of 50, 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.

$Every denominator has a fraction with small quotients. 2:1340 3:1624 4:32 5:3; need 5: 6,54,150$

A small quotient for every denominator

How close two sums of square roots come

The bases that still pass for 561. n=561: coprime 320, Fermat liars 320, strong liars 10 (Monier 10).

A quarter of the bases at most

A multiplication table and its repeats. n=30: 308 distinct products of 900.

Most numbers are missing from the table

Divisors that take a prime whole

How many fourth powers every number needs. Fewest fourth powers summing to n, n <= 10000000: 19 needed by 79, 159, 239, 319, 399, 479, 559; 18 by 24 numbers (last 1248); 17 by 65 (last 13792).

Nineteen fourth powers, and then sixteen

Amounts arranged by remainder. Payable and unpayable amounts for coins 5 and 7, in columns modulo 5: largest unpayable 23, 12 unpayable amounts.

The largest amount two coins cannot pay

Why 78,557 · 2ⁿ + 1 is never prime. For k=78557: 3 divides when n is 0 mod 2; 5 divides when n is 1 mod 4; 7 divides when n is 1 mod 3; 13 divides when n is 11 mod 12; 19 divides when n is 15 mod 18; 37 divides when n is 27 mod 36; 73 divides when n is 3 mod 9; every exponent covered with period 36.

Numbers that never make a prime

Sums whose primes are few. Four abc triples and their qualities: 1+8=9, radical 6, quality 1.2263; 5+27=32, radical 30, quality 1.0190; 1+63=64, radical 42, quality 1.1127; 2+6436341=6436343, radical 15042, quality 1.6299.

Sums with too few primes

Goldbach's comet. The number of ways to write each even number as a sum of two primes up to 12000, coloured by divisibility by 3 and 5; r(12000) = 302 and r(12002) = 115.

The comet Goldbach left behind

5 figures $Five powers of two that are seven more than a square. How close 2ⁿ − 7 comes to a square for n from 3 to 40, as a fraction of the gap between neighbouring squares: squares at n = 3, 4, 5, 7, 15 (1, 3, 5, 11, 181 squared), and at no other n up to 10000.$

Five powers of two, seven more than a square

6 figures · new

Every way to make change for two pounds. Number of ways to make each amount up to 200 pence from British coins, on a logarithmic scale: 451 for 50p, 4563 for £1, 73682 for £2; a dollar in United States coins has 292.

Two pounds in 73,682 ways

6 figures · new

Dynamics

One rule, applied over and over, and what the sequence does in the end.

the logistic map at 3.2, iterated from 0.2. A map drawn as a curve with the diagonal across it, and the staircase that iterating it produces.

The staircase that shows the whole orbit

A fixed point that attracts, and one that does not. The same map at two parameters, with the staircase walking towards the crossing in one and away in the other.

A point that pulls, and a point that pushes

The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points.

The road paved with doublings

Where the period doubles, and by how much the gaps shrink. The parameters at which the period doubles, with the ratio of consecutive gaps beside them.

A constant that does not care which map

Two orbits of the logistic map at 3.9, started 0.0001 apart. Two sequences from almost the same starting point, plotted together against the step number.

A difference too small to draw

The Lyapunov exponent, 2.8 to 4. The average rate at which nearby orbits separate, plotted against the parameter.

How fast two orbits part

The Mandelbrot set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region.

The shape in every picture of itself

A Julia set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region.

One c, one picture

The basins of Newton's method on z³ = 1. The complex plane coloured by which cube root of one Newton's method converges to from each starting point.

Where Newton's method goes instead

Elementary cellular automaton, rule 90. A row of cells evolving downward, each cell decided by the three above it.

Eight rules and a triangle

Elementary cellular automaton, rule 110. A row of cells evolving downward, each cell decided by the three above it.

The rule that computes

Rotating by φ − 1 of a turn, 21 times. Points on a circle produced by repeatedly turning through the same angle.

Three gaps and no more