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A Pierpont prime is a prime number of the form 2''u''3''v'' + 1 for some nonnegative integers ''u'' and ''v''. That is, they are the prime numbers ''p'' for which ''p'' − 1 is 3-smooth. They are named after the mathematician James Pierpont. It is possible to prove that if ''v'' = 0 and ''u'' > 0, then ''u'' must be a power of 2, making the prime a Fermat prime. If ''v'' is positive then ''u'' must also be positive, and the Pierpont prime is of the form 6''k'' + 1 (because if ''u'' = 0 and ''v'' > 0 then 2''u''3''v'' + 1 is an even number greater than 2 and therefore composite). The first few Pierpont primes are: :2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329. == Distribution of Pierpont primes == Andrew Gleason conjectured there are infinitely many Pierpont primes. They are not particularly rare and there are few restrictions from algebraic factorisations, so there are no requirements like the Mersenne prime condition that the exponent must be prime. There are 36 Pierpont primes less than 106, 59 less than 109, 151 less than 1020, and 789 less than 10100; conjecturally there are O(log ''N'') Pierpont primes smaller than ''N'', as opposed to the conjectured O(log log ''N'') Mersenne primes in that range. 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「Pierpont prime」の詳細全文を読む スポンサード リンク
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