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Name: yendor
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995


Question:
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?


Replies:
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.

rcwinther



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