Ivars Peterson's MathTrek

December 8, 2003

Megaprime Champion

The catalog of humongous prime numbers has a new entry—the champion prime (2^20996011 – 1), which has 6,320,430 decimal digits. It's now the largest known prime number and the 40th Mersenne prime ever found. A prime is a whole number (other than 1) that is evenly divisible by only itself and 1.

Written in the form 2^p – 1, where the exponent p is a prime number, Mersenne numbers hold a special place in the never-ending pursuit of larger and larger primes.

They are named for French cleric and mathematician Marin Mersenne (1588–1648). These particular numbers have characteristics that make it relatively easy to check whether a candidate is either a prime number or a composite number.

Written out in binary form, a Mersenne number consists of an unbroken string of 1s. The smallest Mersenne prime is (2² – 1) = 3, or in binary digits, 11. After that comes (2³ – 1) = 7, or 111, then (2⁵ – 1) = 31, or 11111.

Most Mersenne numbers are not themselves primes. Indeed, Mersenne primes are extremely rare. Finding the handful of Mersenne primes among all the possible candidates has proved a challenging exercise, once open only to those with access to the fastest computers available. Since 1996, however, participants in the Great Internet Mersenne Prime Search (GIMPS) have taken the honors for each new record-breaking prime.

Organized by George Woltman, a programmer in Florida, GIMPS brings together a host of volunteers throughout the world. By downloading software available at http://www.mersenne.org/ to their home or office computers, GIMPS participants can test Mersenne numbers for primality whenever their machines are otherwise idle. Each volunteer is responsible for checking Mersenne numbers within a specified range of exponents.

The GIMPS project relies on networking software developed by Scott Kurowski of Entropia, a computing-technology company in San Diego. His PrimeNet computer system distributes work to, and gathers results from, more than 200,000 computers scattered worldwide.

The new prime record holder was identified by GIMPS participant Michael Shafer, a chemical engineering graduate student at Michigan State University. It was found on Nov. 17, and the discovery was independently verified shortly afterward.

The new champion, (2^20996011 – 1), greatly surpasses the previous record holder, (2^13466917 – 1), which was discovered 2 years ago and has 4,053,946 decimal digits.

Volunteers haven't yet tested every Mersenne number smaller than the current champion, so another Mersenne prime may yet lurk among the untested numbers. So far, GIMPS participants have tested and double-checked all exponents less than 7,137,900 and tested all exponents less than 10,412,700 at least once.

There's still a lot more testing and checking to do! Who knows when the next champion will turn up?

References:

2003. Mersenne project discovers largest known prime number on world-wide volunteer computer grid: 2^20996011 – 1 is found with 25,000 years of computer time. GIMPS press release. Dec. 2. Available at http://www.mersenne.org/20996011.htm.

Peterson, I. 2001. Searchers capture a champion megaprime. Science News 160(Dec. 15):372.

______. 2000. Great computations. Science News 157(March 4):152-153. Available at http://www.sciencenews.org/20000304/bob1.asp.

______. 1999. Mersenne megaprime. MAA Online (July 26).

______. 1998. Calculating a record prime. Science News 153(Feb. 21):127.

______. 1997. Lucky choice turns up world-record prime. Science News 152(Sept. 13):164. Available at http://www.sciencenews.org/sn_arc97/9_13_97/fob1.htm

. ______. 1996. Another record prime. MAA Online (Dec. 16).

______. 1996. Mining prime terrain. MAA Online (Sept. 16).

The Great Internet Mersenne Prime Search (GIMPS) has a Web site at http://www.mersenne.org/.

Information about Entropia's Internet PrimeNet server can be found at http://mersenne.org/ips/.

Research, records, and resources devoted to prime numbers are available at http://www.utm.edu/research/primes/.

A poster showing all 6,320,430 decimal digits of the largest known prime (with an optional magnifier) can be purchased from Perfectly Scientific at http://www.perfsci.com/souvenirs.html.

Comments are welcome. Please send messages to Ivars Peterson at ip@sciserv.org.

A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.