Mersenne numbers
Numbers of the form
Mn=2n-1,(n≥1) |
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, 2a⋅b-1 with a,b>1 factors:
2a⋅b-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 |