Solinas, Generalized mersenne numbers
. Citeseer, 1999.
Our proposed moduli sets and algorithm are based on strict and looser views of Mersenne numbers
This provides the basis for a highly efficient test for the primality of Mersenne numbers
: start with 3 = [l.sub.2], apply the operation z [??] [z.sup.2] - 2, which doubles the suffix of [l.sub.2m], n - 2 times, and [2.sup.n] - 1 is prime if and only if it divides the resulting number (assuming always that [2.sup.n] - 1 ends in a 3 or 7).
The numbers [2.sup.[pi]] - 1 (with [pi] prime) are called Mersenne numbers
. There are many open problems about Mersenne numbers
Numbers of the form [M.sub.n] are called Mersenne numbers
; if this number is a prime, it is called a Mersenne prime.
have the form 2p-1, where p is a prime.
It is one of a set of numbers, known as Mersenne numbers
, that were originally conjectured to be prime by the French mathematician/priest Marin Mersenne in 1644.
To honour Mersenne these are called Mersenne numbers
. It is clear that all such numbers are odd but more importantly some of them are prime; as is the case with [2.sup.13466917] - 1.
By downloading software available at www.mersenne.org to their home or office computers, GIMPS volunteers can test Mersenne numbers
for primality whenever their machines are otherwise idle.
Participants can get into trouble, however, if they don't own the computer that they're using to run a climate model, check Mersenne numbers
, or crunch radio-telescope data.
Expressed in the form [2.sup.p] - 1, where the exponent p is itself a prime, Mersenne numbers
have characteristics that make it relatively easy to determine whether a candidate is prime.
Expressed in the form [2.sup.p]-1, where the exponent p is itself a prime number, Mersenne numbers
hold a special place in the never-ending pursuit of larger and larger primes.