# Mersenne prime

A Mersenne prime is a prime number of the form 2^{p} - 1, where *p* is prime. A prime exponent is necessary for a Mersenne
number to be prime but is not sufficient; for example, 2^{11} -1 = 2,047 = 23 × 89. In fact, after an early clustering of Mersenne
primes for fairly small values of *p*, further occurrences become increasingly
rare. At the time of writing there are 40 known Mersenne primes, corresponding
to values for *p* of 2; 3; 5; 7; 13; 17; 19; 31; 61; 89; 107; 127;
521; 607; 1,279; 2,203; 2,281; 3,217; 4,253; 4,423; 9,689; 9,941; 11,213;
19,937; 21,701; 23,209; 44,497; 86,243; 110,503; 132,049; 216,091; 756,839;
859,433; 1,257,787; 1,398,269; 2,976,221; 3,021,377; 6,972,593; 13,466,917;
and 20,996,011; however, it isn't known if the current largest Mersenne
prime is the fortieth in order of size because not all lower exponents have
been checked. Mersenne primes rank among the largest of all known primes
because they have a particularly simple test for primality, called the *Lucas-Lehmer
test*.

The search for Mersenne primes has been going on for centuries. They are
named after Marin Mersenne who, in 1644, helped
the search gain wide recognition by writing to many mathematicians of his
conjecture about which small exponents yield primes. Around the time that
Mersenne's conjecture was finally settled, in 1947, digital computers gave
a new impetus to the hunt for Mersenne primes. As time went on, larger and
larger computers found many more Mersennes and, for a while the search belonged
exclusively to those with the fastest computers. This changed in 1995 when
the American computer scientist George Woltman began the Great Internet
Mersenne Prime Search (GIMPS) by providing a database of what exponents
had been checked, an efficient program based on the Lucas-Lehmer test that
could check these numbers, and a way of reserving exponents to minimize
the duplication of effort. Today GIMPS pools the combined efforts of dozens
of experts and thousands of amateurs. This coordination has yielded several
important results, including the discovery of the Mersennes *M*_{3021377} and *M*_{2976221} and the proof that *M*_{756839}, *M*_{859433}, and *M*_{3021377} are the 32nd,
33rd, and 34th Mersennes.