Mersenne prime

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In mathematics, a Mersenne number, named after Marin Mersenne, is a positive integer that is one less than a power of two:

Some definitions of Mersenne numbers require that the exponent p be prime.

A Mersenne prime is a Mersenne number that is prime. It is known[1] that if 2p − 1 is prime then p is prime, so it makes no difference which Mersenne number definition is used. As of October 2009, only 47 Mersenne primes are known. The largest known prime number (243,112,609 − 1) is a Mersenne prime.[2] 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.

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About Mersenne primes

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite. The Lenstra–Pomerance–Wagstaff conjecture asserts that, on the contrary, 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.

A basic theorem about Mersenne numbers states that in order for Mp to be a Mersenne prime, the exponent p itself must be a prime number. This rules out primality for numbers such as M4 = 24 − 1 = 15: since the exponent 4 = 2×2 is composite, the theorem predicts that 15 is also composite; indeed, 15 = 3×5. The three smallest Mersenne primes are

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