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Divisibility Number Base Prime Matrix Even Number Mutual Information Strong Monomorphism General Power Connection Rectifiable Set Eulers Constant Perfect Number

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Mersenne Numbers

Numbers of the form

$\displaystyle M_n = 2^n - 1, (n \geq 1)$

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

is prime then

is prime. Indeed, $2^{a\cdot b}-1$ with $a,\ b >1$ factors:

$\displaystyle 2^{a\cdot b}-1=(2^a-1)(2^{a(b-1)}+2^{a(b-2)}+\ldots+2^a+1).$

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 $40^{\operatorname{th}}$ number , and even newer $41^{\operatorname{st}}$ 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: www.mersenne.org.

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

$\displaystyle \frac{e^{\gamma}}{\log 2} \log \log x$

where $\gamma$ is Euler's constant.

1	Matrix
1	Even Number
1	Number
1	Divisibility
1	Base
1	Prime
2	Mutual Information
3	Strong Monomorphism
3	Connection
3	General Power
4	Eulers Constant
4	Perfect Number
7	Rectifiable Set