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. 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 $n=20996011$. 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. |
