Mersenne numbers


Numbers of the form

Mn=2n-1,(n1)

are called Mersenne numbers after Father Marin Mersenne (1588 - 1648), a French monk who studied which of these numbers are actually prime. It can be easily shown that if Mn is prime then n is prime. Indeed, 2ab-1 with a,b>1 factors:

2ab-1=(2a-1)(2a(b-1)+2a(b-2)++2a+1).

If Mn is prime then we call it a Mersenne prime. Mersenne primes have a strong connection with perfect numbers.

The currently known Mersenne primes correspond to n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13,466,917 and the newly discovered 40th number n=20996011, and even newer 41st number n=24036583. The latest Mersenne primes (as of 2/5/2006) are the 42nd Mersenne number which corresponds to n=25964951 (and which has more than 7.8 million digits) and the 43rd Mersenne prime for n=30402457 (the new prime is 9,152,052 digits long). For an updated list and a lot more information on how these numbers were discovered, you can check: http://www.mersenne.orgwww.mersenne.org.

It is conjectured that the density of Mersenne primes with exponent p<x is of order

eγlog2loglogx

where γ is Euler’s constant.

Title Mersenne numbers
Canonical name MersenneNumbers
Date of creation 2013-03-22 11:47:54
Last modified on 2013-03-22 11:47:54
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 19
Author alozano (2414)
Entry type Definition
Classification msc 11A41
Classification msc 11-02
Related topic TwoSmallResultsMersenneNumbers
Defines Mersenne prime