Ask A Scientist Mathematics Archive

Author:      yendor

Could someone please explain mersenne primes to me, examples, like the
smallest Mersenne prime which is not a prime number, also is there any
interesting problems associated with Mersenne primes?

Response #:  1 of 1
Author:      rcwinther

A number of the form (2^p)-1, where p is a prime, is called a Mersenne
number.  If a Mersenne number is prime, it is called a Mersenne prime.  It
is not known if there are infinitely many Mersenne primes.  The smallest
non-prime Mersenne number is (2^11) - 1 = 2047 = 23*89; the largest Mersenne
prime known as of 1992 is (2^756,839) - 1, which has 227,832 digits!  It is
also the largest prime known as of 1992, and the 32nd known Mersenne prime. 
Mersenne numbers are named for the French mathematician Marin Mersenne
(1588-1648), who studied them.  Actually, interest in these numbers goes
back at least as far as Euclid, who considered sums of powers of 2, that is,
sums of the form 1 + 2 + 4 + . . . + 2^N  [we recognize this as a geometric
series, whose sum is 2^(N + 1) - 1].  Euclid observed that sometimes this
sum is a prime number, and he proved that if the sum 
     (1 + 2 + 4 + . . . + 2^N) is prime, then the number 
     (2^N) * (1 + 2 + 4 + . . . + 2^N) is a perfect number (a number equal
to the sum of its proper divisors).  In carrying this over to the discussion
of Mersenne primes, we must now make the identification N = p - 1.  Mersenne
showed that if (2^p) - 1 is prime, then p must be prime (the converse is not

true, e.g. for p = 11), and Euler showed that ALL of the even perfect
numbers are given by [2^(p - 1)] * [(2^p) - 1] where (2^p) - 1 is prime. 
Nowadays, most of the interest in Mersenne numbers is connected with the
search for larger and larger prime numbers.  This is because Euler also
developed an algorithm that greatly facilitates checking for primeness of a
Mersenne number.