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A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g''''n'' or ''g''(''p''''n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. : We have ''g''1 = 1, ''g''2 = ''g''3 = 2, and ''g''4 = 4. The sequence (''g''''n'') of prime gaps has been extensively studied, however many questions and conjectures remain unanswered. The first 60 prime gaps are: :1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... . By the definition of ''g''n the following sum can be stated as : . ==Simple observations== The first, smallest, and only odd prime gap is 1 between the only even prime number, 2, and the first odd prime, 3. All other prime gaps are even. There is only one pair of gaps between three consecutive odd natural numbers for which all are prime. These gaps are ''g''2 and ''g''3 between the primes 3, 5, and 7. For any prime number ''P'', we write ''P''# for ''P primorial'', that is, the product of all prime numbers up to and including ''P''. If ''Q'' is the prime number following ''P'', then the sequence : is a sequence of ''Q'' − 2 consecutive composite integers, so here there is a prime gap of at least length ''Q'' − 1. Therefore, there exist gaps between primes which are arbitrarily large, i.e., for any prime number ''P'', there is an integer ''n'' with ''g''''n'' ≥ ''P''. (This is seen by choosing ''n'' so that ''p''''n'' is the greatest prime number less than ''P''# + 2.) Another way to see that arbitrarily large prime gaps must exist is the fact that the density of primes approaches zero, according to the prime number theorem. In fact, by this theorem, ''P''# is very roughly a number the size of exp(''P''), and near exp(''P'') the ''average'' distance between consecutive primes is ''P''. In reality, prime gaps of ''P'' numbers can occur at numbers much smaller than ''P''#. For instance, the smallest sequence of 71 consecutive composite numbers occurs between 31398 and 31468, whereas 71# has ''twenty-seven digits'' – its full decimal expansion being 557940830126698960967415390. Although the average gap between primes increases as the natural logarithm of the integer, the ratio of the maximum prime gap to the integers involved also increases as larger and larger numbers and gaps are encountered. In the opposite direction, the twin prime conjecture asserts that for infinitely many integers ''n''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Prime gap」の詳細全文を読む スポンサード リンク
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