Mersenne numbers
Numbers of the form
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 is prime then is prime. Indeed, with factors:
If 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 = 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 number , and even newer number . The latest Mersenne primes (as of ) are the nd Mersenne number which corresponds to (and which has more than million digits) and the rd Mersenne prime for (the new prime is 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 is of order
where is Euler’s constant.
Title | Mersenne numbers |
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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 |