Concept

Prime gaps

The distances between consecutive primes. Near a large number x they average about the logarithm of x, and the longest gap below x appears to grow like the square of that logarithm, though far less than that is proved.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

number · Prime distribution

Record gaps between primes against the square of the logarithm. 24 record prime gaps below thirty million as a staircase of points against log p, under the curve (log p)² and a dashed curve 1.12 (log p)²; the last record is 210 after 20831323.

The longest wait for a prime

Near a number x the primes are on average log x apart, and the longest stretch without one below x is much longer: about the square of log x, if the primes behave like random numbers with the right density. Sieving to thirty million finds twenty-four record gaps, all climbing in the shape of that square and all below it. A random model of the primes gets the scale right and the details wrong, and the correction for what it gets wrong predicts gaps larger still — larger than any anyone has found.

discrete · Prime distribution

Each prime fixes the next digits of A. A = 1.306377883863080690468614492602605712916…; digits fixed by the primes from 11 to the 254-digit one: 1, 4, 11, 30, 87, 257.

A constant that holds the primes

There is a number A = 1.3063778838… such that A cubed, A to the ninth, A to the twenty-seventh and every further power of three, rounded down, is prime: 2, 11, 1361, 2521008887 and on for ever. W. H. Mills proved in 1947 that such a number exists. It looks like a formula for primes and is the opposite of one. Its digits are read off the primes it produces, each prime fixes about as many digits as it has, and the whole argument stands on a theorem about gaps between primes — which is why the power is 3 and not 2.

number · Infinitude of primes

Named alongside it

The objects these essays reach for when they reach for this one.

Prime number theorem Primes Composite Counting two ways Cramér's model Density Exhaustive search Existence proof Miller rabin Nested intervals Prime Prime counting function

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