A prime number is one that is evenly divisible only by itself and 1. The integers 1, 2, 3, 5, 7, 11, 13, 17, and 19 are prime numbers. Euclid (c. 335–270B.C.E.) proved that there is no “largest prime number,” because any attempt to define the largest results in a paradox. If there is a largest prime number (P), adding 1 to the product of all primes up to and including P, 1 1 (1 3 2 3 3 3 5 3 … 3 P), yields a number that is itself a prime number, because it cannot be divided evenly by any of the known primes. In 2003, Michael Shafer discovered the largest known (and the fortieth) prime number: 2^20996011 – 1. This is over six million digits long and would take more than three weeks to write out by hand. In July 2010, double-checking proved this was the fortieth Mersenne prime (named after Marin Mersenne, 1588–1648, a French monk who did the first work in this area). Mersenne primes occur where 2^n–1 is prime.

There is no apparent pattern to the sequence of primes. Mathematicians have been trying to find a formula since the days of Euclid, without success. The fortieth prime was discovered on a personal computer as part of the GIMPS effort (the Great Internet Mersenne Prime Search), which was formed in January 1996 to discover new world-record-size prime numbers. GIMPS relies on the computing efforts of thousands of small, personal computers around the world. Interested participants can become involved in the search for primes by going to: http://www.mersenne.org/default.php.