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mathematics, in pictures
Ladder

Prime distribution — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. Ulam's spiral to 900. The integers up to 900 laid out in a square spiral, with the primes marked; they crowd onto diagonal lines.

    The primes on a spiral, and a pattern nobody ordered

    Wind the whole numbers outward in a square spiral, mark the primes, and they line up on diagonals. The observation is a hundred years old and there is still no proof it means anything.

    rung 1 · discrete
  2. The sieve of Eratosthenes below 100. A grid of the whole numbers with the composites struck out by the prime that removes them.

    The primes are what is left over

    Eratosthenes' sieve does not find the primes. It removes everything else, one prime at a time, and whatever survives is prime by default — which is a strange way to reach the most studied objects in arithmetic.

    rung 2 · number
  3. π(x) against its two estimates, up to 20,000. The ratio of the prime counting function to x over the logarithm of x, and to the logarithmic integral, plotted against x. The first is above one and coming down slowly; the second is close to one throughout.

    Counting what has no formula

    There is no expression that gives the nth prime, and yet the number of primes below a bound is predictable to within a fraction of a per cent — by a function that is not a formula for the primes but an integral of the wrong-looking quantity.

    rung 3 · number
  4. 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

    A density says what happens on average and permits long empty stretches. This says something a density cannot — that the stretch from any number to twice it contains a prime, at every scale, without exception.

    rung 4 · number
  5. 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

    Multiply out one geometric series for each prime and every whole number appears exactly once, as a single term. That identity turns a statement about factorisation into a statement about convergence, and it is where the analytic study of the primes begins.

    rung 5 · number

All ladders