The existence of arbitrarily large prime gaps can be seen by noting that the sequence formula_63 consists of formula_64 composite numbers, for any natural number formula_65 However, large prime gaps occur much earlier than this argument shows. For example, the first prime gap of length 8 is between the primes 89 and 97, much smaller than formula_66 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_67 there are infinitely many pairs of consecutive primes that differ by formula_68