Mersenne prime

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Mn = 2n − 1.

A Mersenne prime is a Mersenne number that is a prime number. The exponent n must be prime too. This however, is not sufficient. Many mathematicians prefer the definition of a Mersenne number where n has to be a prime number.

For example, 31 = 25 − 1, and 5 is a prime number, so 31 is a Mersenne number; and 31 is also a Mersenne prime because it is a prime number. But the Mersenne number 2047 = 211 − 1 is not a prime because it is divisible by 89 and 23. And 24 − 1 = 15 can be shown to be composite because 4 is not prime.



It is currently unknown if there is an infinite number of Mersenne primes. Most mathematicians who have studied the question think there are an infinite number of Mersenne primes. Carl Pomerance, Samuel S. Wagstaff Jr. and Hendrik Lenstra developed a formula which they think should say about how many Mersenne primes there are under a given number.

The binary representation of 2n - 1 is n repetitions of the digit 1. For example, 25 - 1 = 11111 in binary.

The first four Mersenne primes were known to the Ancient Greeks. They are 3, 7, 31 and 127. The next, 8191, was discovered in 1456. Some people like trying to find big primes. Since there are tricks to finding Mersenne primes quickly, the largest primes known are usually Mersenne primes.