What is Mersenne prime?
In [[mathematics]], a '''Mersenne number''' is a number that is one less than a ''power of two''. :''M<sub>n</sub>'' = 2<sup>''n''</sup> − 1. A '''Mersenne prime''' is a ''Mersenne number'' that is a [[prime number]]. This however, is not sufficient. Many mathematicians prefer the definition of a Mersenne number where exponent ''n'' to be a prime number. For example, ''31 = 2<sup>5</sup> − 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 = 2<sup>11</sup> − 1'' is not a prime because it is divisible by 89 and 23. And ''2<sup>4</sup> − 1 = 15'' can be shown to be composite because 4 is not prime. Throughout modern times, the largest known prime number has very often been a Mersenne prime. Most sources restrict the term ''Mersenne number'' to where ''n'' is prime, as all Mersenne primes must be of this form as seen below. Mersenne primes have a close connection to [[perfect number]]s, which are numbers 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. In the 18th century, [[Leonhard Euler]] proved that all [[even number|even]] perfect numbers have this form. No [[odd number|odd]] perfect numbers are known, and it is suspected that none exist (if one does, it would have to be greater than 10<sup>2200</sup>). 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 numeral system|binary]] representation of 2<sup>''n''</sup> - 1 is ''n'' repetitions of the digit 1. <!-- making it a base-2 [[repunit]]. --> For e