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A number of the form 2p−1, where p is a prime number. Such a number which is itself prime is also called a Mersenne prime.
‘Primes of the form 2 n - 1 are known as Mersenne primes, and these numbers figure prominently in the search for the largest known prime.’
‘Written in the form 2 p - 1, where the exponent p is a prime number, Mersenne numbers hold a special place in the never-ending pursuit of larger and larger primes.’
‘‘Fermat had this idea that there were families of primes other than Mersenne primes, and there would be an infinite number of them,’ he adds.’
‘A few months later an engineer at Cray Research discovered a new record Mersenne prime number, and I wrote a 700-word piece describing the discovery.’
‘And the Great Internet Mersenne Prime Search continues to unearth new Mersenne primes.’
‘A computer search turned up the 40th Mersenne prime.’
‘The largest known prime is the 40th Mersenne prime: M 20996011 which has 6320430 decimal digits.’
‘After reading up on how known perfect numbers relate to Mersenne Prime Numbers, I wrote another program, this time to find Mersenne prime numbers and perfect numbers.’
‘Primes of the form are now known as Mersenne primes, and these numbers figure prominently in the search for the largest known prime.’
‘In 1876 he used his methods to prove that the Mersenne number 2 127-1 is prime.’
‘Usually you download a screensaver or something, and when your computer isn't busy it turns itself on and cranks away at calculating a Mersenne prime or a SETI signal or some other worthy semi-scientific endeavor.’
‘Expressed in the form 2 p - 1, where the exponent p is itself prime, Mersenne numbers have characteristics that make it relatively easy to determine whether a candidate is prime.’
‘The divisor itself is the fifth largest known prime number, and it is the largest that is not a Mersenne prime.’
‘Primes of the form 2 p - 1 are called Mersenne primes.’
‘How can you calculate a perfect number from a Mersenne prime?’
‘One must be the Lucas Lehmer primality test which uses the Fermat congruence, and in particular his application to testing whether a Mersenne number was prime.’
‘This project is seeking the largest Mersenne prime number, which is, at the moment, also the largest prime number of any form.’
Origin
Late 19th century: named after Marin Mersenne (1588–1648), French mathematician.