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'Mersenne numbers'
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| Title of object: |
Mersenne numbers |
| Canonical Name: |
MersenneNumbers |
| Type: |
Definition |
| Created on: |
2001-10-18 09:10:39 |
| Modified on: |
2006-02-05 15:31:30 |
| Classification: |
msc:11A41 |
| Keywords: |
number theory |
| Synonyms: |
Mersenne numbers=Mersenne prime |
Revision comment (for changes between this and next version):
| Changes for correction #9263 ('emphasis on defined terms'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Numbers of the form
\[
M_n = 2^n - 1, (n \geq 1)
\]
are called Mersenne numbers after Father Marin Mersenne, a French monk who wanted to discover which such numbers are actually prime. It can be easily shown that if $M_n$ is prime then $n$ is prime. Indeed, $2^{a\cdot b}-1$ with $a,\ b >1$ factors:
$$2^{a\cdot b}-1=(2^a-1)(2^{a(b-1)}+2^{a(b-2)}+\ldots+2^a+1).$$
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 $40^{\operatorname{th}}$ number $n=20996011$, and even newer $41^{\operatorname{st}}$ number $n=24036583$. The latest Mersenne primes (as of $2/5/2006$) are the $42$nd Mersenne number which corresponds to $n=25964951$ (and which has more than $7.8$ million digits) and the $43$rd 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: \PMlinkexternal{www.mersenne.org}{http://www.mersenne.org}.
It is conjectured that the density of Mersenne primes with exponent $p<x$ is of order
$$ \frac{e^{\gamma}}{\log 2} \log \log x $$
where $\gamma$ is Euler's constant. |
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