Mersenne prime

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. For example, 3 = 4 − 1 = 22 − 1 is a Mersenne prime; so is 7 = 8 − 1 = 23 − 1. On the other hand, 15 = 16 − 1 = 24 − 1, for example, is not a prime, because 15 is divisible by 3 and 5.

Related Topics:
Mathematics - Prime number - Power of two

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More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than a power of two; hence,

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:Mn = 2n − 1.

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Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century, Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exists.

Related Topics:
Perfect number - 4th century BC - Euclid - 18th century - Euler - Even - Odd

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It is currently unknown whether there is an infinite number of Mersenne primes.

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