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.

Mersenne numbers 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.