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 2 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

Named alongside it

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

Prime number theorem Composite Counting two ways Cramér's model Density Prime Prime counting function Primes Random model Sieve Sieve of eratosthenes Smallest prime factor

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