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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number that can be written in the form for some integer ''n''. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The first four Mersenne primes are 3, 7, 31, and 127. If ''n'' is a composite number then so is . The definition is therefore unchanged when written where ''p'' is assumed prime. More generally, numbers of the form without the primality requirement are called Mersenne numbers. Mersenne numbers are sometimes defined to have the additional requirement that ''n'' be prime, equivalently that they be pernicious Mersenne numbers, namely those pernicious numbers whose binary representation contains no zeros. The smallest composite pernicious Mersenne number is . , 48 Mersenne primes are known. The largest known prime number is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search” (GIMPS), a distributed computing project on the Internet. ==About Mersenne primes== Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes congruent to 3 (mod 4), for these primes ''p'', 2''p'' + 1 (which is also prime) will divide ''Mp'', e.g., 23|''M''11, 47|''M''23, 167|''M''83, 263|''M''131, 359|''M''179, 383|''M''191, 479|''M''239, and 503|''M''251. The first four Mersenne primes are : , , and . A basic theorem about Mersenne numbers states that if ''M''''p'' is prime, then the exponent ''p'' must also be prime. This follows from the identity : This rules out primality for Mersenne numbers with composite exponent, such as . Though the above examples might suggest that ''Mp'' is prime for all primes ''p'', this is not the case, and the smallest counterexample is the Mersenne number : . The evidence at hand does suggest that a randomly selected Mersenne number is much more likely to be prime than an arbitrary randomly selected integer of similar size. Nonetheless, prime ''M''''p'' appear to grow increasingly sparse as ''p'' increases. In fact, of the 2,007,537 prime numbers ''p'' up to 32,582,657,〔(【引用サイトリンク】title=Number of primes <= 32582657 )〕 ''M''''p'' is prime for only 44 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, since Mersenne numbers grow very rapidly. The Lucas–Lehmer primality test (LLT) is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following. Consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing. Mersenne primes are used in pseudorandom number generators such as the Mersenne twister, Park–Miller random number generator, Generalized Shift Register and Fibonacci RNG. 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「Mersenne prime」の詳細全文を読む スポンサード リンク
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