 For example, the first prime gap of length 8 is between the primes 89 and 97, much smaller than formula_63 It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2; this is the twin prime conjecture.Polignac's conjecture states more generally that for every positive integer formula_64 there are infinitely many pairs of consecutive primes that differ by formula_65
Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, and Oppermann's conjecture all suggest that the largest gaps between primes from formula_66 to formula_1 should be at most approximately formula_68 a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér conjecture sets the largest gap size at formula_69 Prime gaps can be generalized to prime formula_70-tuples, patterns in the differences between more than two prime numbers.Their infinitude and density are the subject of the first Hardy–Littlewood conjecture, which can be motivated by the heuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.