Largest Known Prime Number with 17 Million Digits Discovered

On January 25, 2013, the largest known prime number was discovered on a Great Internet Mersenne Prime Search (GIMPS) volunteer’s computer. The new prime number, 2 multiplied by itself 57,885,161 times, less one, has 17,425,170 digits. With 360,000 CPUs peaking at 150 trillion calculations per second, 17th-year GIMPS is the longest continuously-running global grassroots supercomputing project in Internet history.

The volunteer, Curtis Cooper, is a professor at the University of Central Missouri. This is the third record prime for Cooper and his University. Their first record prime was discovered in 2005, eclipsed by their second record in 2006. Computers at UCLA broke that record in 2008 with a 12,978,189 digit prime number. UCLA held the record until University of Central Missouri reclaimed the world record with this discovery. The new primality proof took 39 days of non-stop computing on one of the university's PCs. Cooper and the University of Central Missouri are the largest individual contributors to the project. The discovery is eligible for a $3,000 GIMPS research discovery award.

The new prime number is a member of a special class of extremely rare prime numbers known as Mersenne primes. It is only the 48th known Mersenne prime ever discovered, each increasingly difficult to find. Mersenne primes were named for the French monk Marin Mersenne, who studied these numbers more than 350 years ago. GIMPS, founded in 1996, has discovered all 14 of the largest known Mersenne primes. Volunteers download a free program to search for these primes with a cash award offered to anyone lucky enough to compute a new prime. Chris Caldwell maintains an authoritative Web site on the largest known primes as well as the history of Mersenne primes.

To prove there were no errors in the prime discovery process, the new prime was independently verified using different programs running on different hardware. Serge Batalov ran Ernst Mayer's MLucas software on a 32-core server in 6 days (resource donated by Novartis IT group) to verify the new prime. Jerry Hallett verified the prime using the CUDALucas software running on a NVidia GPU in 3.6 days. Finally, Dr. Jeff Gilchrist verified the find using the GIMPS software on an Intel i7 CPU in 4.5 days and the CUDALucas program on a NVidia GTX 560 Ti in 7.7 days.

GIMPS software was developed by founder, George Woltman, in Orlando, FL. Scott Kurowski, in San Diego, California, wrote and maintains the PrimeNet system that coordinates all the GIMPS clients. Volunteers have a chance to earn research discovery awards of $3,000 or $50,000 if their computer discovers a new Mersenne prime. GIMPS' next major goal is to win the $150,000 award administered by the Electronic Frontier Foundation offered for finding a 100 million digit prime number.

Credit for GIMPS' prime discoveries goes not only to Cooper for running the software on his University's computers, Woltman and Kurowski for authoring the software and running the project, but also the thousands of GIMPS volunteers who sifted through millions of non-prime candidates. Therefore, official credit for this discovery shall go to "C. Cooper, G. Woltman, S. Kurowski, et al."

**About Mersenne.org's Great Internet Mersenne Prime Search**

The Great Internet Mersenne Prime Search (GIMPS) was formed in January 1996 by George Woltman to discover new world-record-size Mersenne primes. In 1997 Scott Kurowski enabled GIMPS to automatically harness the power of hundreds of thousands of ordinary computers to search for these "needles in a haystack." Most GIMPS members join the search for the thrill of possibly discovering a record-setting, rare, and historic new Mersenne prime. The search for more Mersenne primes is already under way. There may be smaller, as yet undiscovered Mersenne primes, and there certainly are larger Mersenne primes waiting to be found. Anyone with a reasonably powerful PC can join GIMPS and become a big prime hunter, and possibly earn a cash research discovery award. All the necessary software can be downloaded for free at www.mersenne.org/freesoft.htm. GIMPS is organized as Mersenne Research, a 501(c)(3) science research charity. Additional information may be found at www.mersenneforum.org and www.mersenne.org; donations are welcome.

**Further information on Mersenne Primes**

Prime numbers have long fascinated amateur and professional mathematicians. An integer greater than one is called a prime number if its only divisors are one and itself. The first prime numbers are 2, 3, 5, 7, 11, etcetera. For example, the number 10 is not prime because it is divisible by 2 and 5. A Mersenne prime is a prime number of the form 2P-1. The first Mersenne primes are 3, 7, 31 and 127 corresponding to P = 2, 3, 5 and 7 respectively. There are only 48 known Mersenne primes.

Mersenne primes have been central to number theory since they were first discussed by Euclid in 350 BC. The man whose name they now bear, the French monk Marin Mersenne (1588-1648), made a famous conjecture on which values of P would yield a prime. It took 300 years and several important discoveries in mathematics to settle his conjecture.

Previous GIMPS Mersenne prime discoveries were made by members in various countries.

• In April 2009, Odd Magnar Strindmo et al. discovered the 47th known Mersenne prime in Norway.

• In September 2008, Hans-Michael Elvenich et al. discovered the 46th known Mersenne prime in Germany.

• In August 2008, Edson Smith et al. discovered the 45th known Mersenne prime in the U.S.

• In September 2006, Curtis Cooper and Steven Boone et al. discovered the 44th known Mersenne prime in the U.S.

• In December 2005, Curtis Cooper and Steven Boone et al. discovered the 43rd known Mersenne prime in the U.S.

• In February 2005, Dr. Martin Nowak et al. discovered the 42nd known Mersenne prime in Germany.

• In May 2004, Josh Findley et al. discovered the 41st known Mersenne prime in the U.S.

• In November 2003, Michael Shafer et al. discovered the 40th known Mersenne prime in the U.S.

• In November 2001, Michael Cameron et al. discovered the 39th Mersenne prime in Canada.

• In June 1999, Nayan Hajratwala et al. discovered the 38th Mersenne prime in the U.S.

• In January 1998, Roland Clarkson et al. discovered the 37th Mersenne prime in the U.S.

• In August 1997, Gordon Spence et al. discovered the 36th Mersenne prime in the U.K.

• In November 1996, Joel Armengaud et al. discovered the 35th Mersenne prime in France.

There is a well-known formula that generates a "perfect" number from a Mersenne prime. A perfect number is one whose factors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3. The newly discovered perfect number is 257,885,160 x (257,885,161-1). This number is over 34 million digits long!

There is a unique history to the arithmetic algorithms underlying the GIMPS project. The programs that found the recent big Mersenne finds are based on a special algorithm. In the early 1990s, the late Richard Crandall, Apple Distinguished Scientist, discovered ways to double the speed of what are called convolutions — essentially big multiplication operations. The method is applicable not only to prime searching but other aspects of computation. During that work he also patented the Fast Elliptic Encryption system, now owned by Apple Computer, which uses Mersenne primes to quickly encrypt and decrypt messages. George Woltman implemented Crandall's algorithm in assembly language, thereby producing a prime-search program of unprecedented efficiency, and that work led to the successful GIMPS project.

School teachers from elementary through high-school grades have used GIMPS to get their students excited about mathematics. Students who run the free software are contributing to mathematical research. Historically, searching for Mersenne primes has been used as a test for computer hardware. The free GIMPS program has identified hidden hardware problems in many PCs.

**Citation:** *Science* (American Association for the Advancement of Science), May 6, 2005 p810.