What's a Mersenne number? It's a number of the form 2p-1, where p is a prime number. (Some consider numbers of the form 2n-1, where n is any positive integer to be a Mersenne number.)
What's a Mersenne prime? It's a prime number of the form 2p-1, where p is a prime nunber.
What's a prime number? It's a positive integer, greater than one, that is wholly divisible only by the integers one and itself.
Prime numbers are quite common. With the exception of 2, all primes are odd. The first ten prime number are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Up to 1,000,000,000, there are 50,847,534 prime numbers.
Mersenne primes by contrast are exceedingly rare. The first five Mersenne primes are 22-1 (= 3), 23-1 (= 7), 25-1 (= 31), 27-1 (= 127) and 213-1 (= 8191). At the time of writing only 46 Mersenne primes are known: the largest (which may or may not be the 46th largest Mersenne prime) is 243112609-1, the decimal representation of which has 12978189 digits.
While I don't want to repeat all the information that is available elsewhere on the subject, I will just note that the values of the exponents that are offered for primality testing are themselves prime (e.g., 29630527 is a prime number). That is a consequence of the fact that Mersenne primes (2p-1) can occur only when p is itself prime. But not all (or even most) prime values of p produce Mersenne primes, even though an examination of the first few Mersenne primes might suggest otherwise (seven of the eight smallest prime values of p in the expression 2p-1 produce Mersenne primes). Copious information on prime number practice and theory is also available.
The Great Internet Mersenne Prime search is a distributed computing project aiming to find more Mersenne primes. The link above will allow anyone interested in finding out more to locate everything they will need. Do note that even with a fast CPU, testing a single exponent for primality can take three or four weeks and that the chance that an exponent you are given for testing is a Mersenne prime is about 1 in 300000. Still far, far better odds than the big prize in state lotteries...
My latest PrimeNet Individual Account Report shows the following statistics (see up to date position):
To see the list of exponents I have worked, you can select a listing of the 100-most recent exponents processed or the full list in reverse assignment order or in decreasing numerical order. The columns are as follows: Order indicates the order in which the exponents were tested; Exponent gives the exponent (p) in the expression 2p-1; T indicates whether the exponent was Lucas-Lehmer tested (L), Lucas-Lehmer double-checked (D), P-1 factored (P) or trial-factored (F), the number n indicating that it is the n-th number to be so tested [except that the count of P and F are combined]; Comp indicates which of the four computers (coded as Pr2 [3.0 GHz Core 2 Extreme], Pr1 [2.66 GHz Core 2 Duo], Cy [3 GHz Pentium 4] or CyL [2 GHz Pentium M laptop]; which core of a multi-core was used is indicated by _1, _2, _3 or _4) was used, the number n indicating that it is the n-th number processed on that processor/core; Date Completed is the date on which the primality test or trial factoring completed; and Status indicates whether or not the listed exponent leads to a Mersenne prime or a factor and which test was used.
In addition, I've been running my own searches for factors of Mersenne numbers, doing what is known as Lone Mersenne Hunting (or LMH). This searching is proving to be very successful: a summary page is available. Two different types of searches are being undertaken: for exponents without any known factors, I try to extend the range of possible factors that has been searched and hopefully find a factor (this is successful in about 5% of cases); for exponents with at least one known factor, I try to find additional factors by searching to higher limits (this is successful in about 29% of cases, with an average of 1.20 new factors for each successful case and including two cases where five new factors were found!). The number of "1st factors" claimed on the summary page is based on the very useful datafiles provided by Will Edgington. This work is not credited in the V4 PrimeNet statistics:
Recently, I have moved the reporting of my LMH results over to the V5 PrimeNet server. These results are included in the summary page. The V5 server does give credit for the work, albeit not very accurately for the type of searches I undertake when a factor is found, and when the whole GIMPS project moves over to the V5 server the LMH results will be merged into my GIMPS results and will not be available separately:
Upon hearing that my first assigned exponent, 230219223-1, was not prime, a certain, unnamed, male relative (by marriage) opined that I (or more specifically, my computer) had just wasted three weeks. This is patently not so. The knowledge that 230219223-1 is not prime has increased (just slightly) the total sum of human knowledge.
Document last updated 2010 Feb. 3.045 UTC.