Prime gaps

A prime gap is the difference between consecutive prime numbers. The sequence of gaps records how prime numbers are spaced along the integers.

Consecutive prime numbers: Two prime numbers with no prime number between them. Each gap is measured between a consecutive pair.
Prime number theorem: A theorem describing the asymptotic density of primes near large numbers. It predicts the average gap near a large number is approximately its natural logarithm.
Euclid: An ancient Greek mathematician whose surviving work includes a proof that there are infinitely many primes. His proof establishes an unending sequence in which prime gaps can be studied.
Bounded gaps between primes: The theorem that some fixed finite bound contains the difference of infinitely many pairs of primes. It proves recurring bounded gaps without showing that any particular gap recurs infinitely.
Polignac's conjecture: The conjecture that every positive even integer occurs infinitely often as a difference between consecutive primes. It asks whether every permitted gap size recurs without end.
Prime number: A natural number greater than one whose only positive divisors are one and itself. Prime gaps describe the spacing within this sequence of numbers.
Cramér's conjecture: A conjecture that the largest prime gap below a number x is asymptotically at most proportional to the square of log x. It proposes a scale for how large unusually long gaps can become.
Christian Goldbach: An eighteenth-century mathematician known for correspondence that produced Goldbach's conjecture. His conjecture concerns sums of primes and has inspired questions about their spacing.
Green–Tao theorem: The theorem that the primes contain arithmetic progressions of every finite length. It concerns additive patterns among primes rather than the differences of consecutive primes.
Prime k-tuples conjecture: A conjecture predicting which finite patterns of integer offsets contain infinitely many simultaneous primes. Its predictions include the frequency of many specific consecutive-prime gap patterns.

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