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31 (thirty-one) is the natural number following 30 and preceding 32. == In mathematics == Thirty-one is the third Mersenne prime ( 25 - 1 ) as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496, since 496 = 25 - 1 ( 25 - 1). 31 is the eighth Mersenne prime exponent. 31 is also the 4th lucky prime and the 11th supersingular prime. 31 is a centered triangular number, the lowest prime centered pentagonal number and a centered decagonal number. For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals. At 31, the Mertens function sets a new low of -4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31, making 31 a self number. 31 is a repdigit in base 5 (111), and base 2 (11111). The numbers 31, 331, 3331, , , , and are all prime. For a time it was thought that every number of the form 3w1 would be prime. However, the next nine numbers of the sequence are composite; their factorisations are: * = 17 × * = 673 × * = 307 × * = 19 × 83 × * = 523 × 3049 × * = 607 × 1511 × 1997 × * = 181 × * = 199 × and * = 31 × 1499 × . The recurrence of the factor 31 in the last number above can be used to prove that no sequence of the type RwE or ERw can consist only of primes because every prime in the sequence will periodically divide further numbers. Here, 31 divides every fifteenth number in 3w1 (and 331 every 110th). 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「31 (number)」の詳細全文を読む スポンサード リンク
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