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GIMPS Discovers 41st Mersenne Prime,

2^{24,036,583}-1 is now the Largest Known Prime.

### Project Leaders Believe $100,000 Award Within Reach

**ORLANDO, Florida, May 28, 2004 — **
Josh Findley, a volunteer in the Mersenne.org research project called the Great Internet Mersenne Prime Search (GIMPS),
has discovered the largest known prime number. Findley used his home PC and free software by George Woltman and Scott Kurowski as part of an international grid of 240,000 networked
computers in virtually every time zone of the world.

The new number, expressed as 2 to the 24,036,583th power minus 1, has 7,235,733 decimal digits and was discovered May 15th. It is nearly a million digits larger than the previous largest known prime number, and belongs to a special class of rare prime numbers called Mersenne primes. The discovery marks only the 41st known Mersenne prime, named after Marin Mersenne, a 17th century French monk who first studied the rare numbers 300 years ago. Mersenne primes are most relevant to number theory, but most participants join GIMPS simply for the fun of having a role in real research – and the chance of finding a new Mersenne prime.

GIMPS is closing in on the $100,000 Electronic Frontier Foundation award for the first 10-million-digit prime. The new prime is 72% of the size needed, however "an award-winning prime could be mere weeks or as much as few years away – that's the fun of math discoveries", said GIMPS founder George Woltman. The GIMPS participant who discovers the prime will receive $50,000. Charity will get $25,000. The rest will be used primarily to fund more prime discoveries. In May 2000, a previous participant won the foundation's $50,000 award for discovering the first million-digit prime.

Findley, a consultant to the National Oceanic and Atmospheric Administration (NOAA) in La Jolla, California, described the find. "I'm still surprised at the discovery. Even after five years running GIMPS on my computers, I didn't expect to find a new Mersenne prime!" He used a 2.4 GHz Pentium 4 Windows XP PC running for 14 days to prove the number prime. "I joined GIMPS because it seemed the logical choice for using my spare CPU cycles."

Now in its eighth year, GIMPS has accomplished what no other distributed computing project has: seven consecutive successes. "Great teamwork has paid off for us again," said Woltman. "In addition to congratulating Josh Findley, we wish to thank all 75,000 volunteer home users, students, schools, universities and businesses from around the world that contributed to this discovery." Woltman adds, "Joining GIMPS is a great way to learn about math through participation – and you might find a new Mersenne prime, like Josh."

"PrimeNet organizes a vast computing resource for GIMPS. It's humbling to see so many people of varied lands, ages and vocations volunteering for this fun and amazing project," said Entropia founder, Scott Kurowski. Kurowski developed the PrimeNet system that runs GIMPS. PrimeNet pulls together hundreds of thousands of computers in parallel to create a virtual supercomputer running at 14 trillion calculations per second, or "teraflops". This enabled GIMPS to find the prime in just six months instead of the 2,500 years a single PC would have required.

The new prime was independently verified by Tony Reix of Grenoble, France using half of a Bull NovaScale 5000 HPC running Linux on 16 Itanium II 1.3 GHz CPUs for five days using the Glucas program by Guillermo Ballester Valor of Granada, Spain, and by Jeff Gilchrist of Elytra Enterprises Inc. in Ottawa, Canada using eleven days of time on a HP rx5670 quad Itanium II 1.5 GHz CPU server at SHARCNET. The discovery is the seventh record prime found by the GIMPS project. In recognition of every GIMPS contributor's effort, credit for this new discovery will go to "Findley, Woltman, Kurowski, et al".

"There are more primes out there," invites Woltman, "and anyone with an Internet-connected computer can participate." All the necessary software can be downloaded for free at http://www.mersenne.org. The calculations work by using spare background time that would otherwise be wasted. Please get permission to install the software on computers you do not own.

The mathematical algorithm Woltman uses for GIMPS, called the IBDWT (irrational-base discrete weighted transform), was discovered by Apple Distinguished Scientist Dr. Richard Crandall, director of the Center for Advanced Computation at Reed College, Portland, Oregon. A framed or unframed poster (and optional magnifying glass) displaying the multi-million-digit prime number is available from Perfectly Scientific, Inc. (http://www.perfsci.com/). The IBDWT and related algorithms are available in the book, "Prime Numbers: A Computational Perspective," by R. Crandall and C. Pomerance.

Josh Findley is an independent IT consultant living in Issaquah, Washington. He joined GIMPS in June 1999. George Woltman is a retired computer programmer living in Orlando, Florida. A life-long number theory enthusiast, he founded the Great Internet Mersenne Prime Search in 1996. Scott Kurowski is a software technologist and entrepreneur in San Diego, California. A GIMPS member since 1996, he built PrimeNet and founded Entropia, Inc. in 1997.

#### 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. GIMPS harnesses the power of hundreds of thousands of small computers like yours 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, but there are certainly larger Mersenne primes waiting to be discovered. Anyone with a reasonably powerful PC can join GIMPS and become a big prime hunter. All the necessary software can be downloaded for free at http://www.mersenne.org. GIMPS is based in Orlando, Florida. Additional information may be found at http://www.mersenneforum.org/.

#### For More Information on Mersenne Primes

Previous GIMPS Mersenne prime discoveries were made by members in various countries. In November 2003, Michael Shafer discovered the previous largest known prime number in the United States. In November 2001, Michael Cameron discovered the the 39th Mersenne prime in Canada. In June 1999, Nayan Hajratwala discovered the 38th Mersenne prime in the United States. In January 1998, Roland Clarkson discovered the 37th Mersenne prime in the United States. Gordon Spence discovered the 36th Mersenne prime in August, 1997, in the United Kingdom. Joel Armengaud discovered the 35th Mersenne prime in November, 1996, in France.

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, etc. For example, the number 10 is not prime because it is divisible by 2 and 5. A Mersenne prime is a prime of the form 2^{p}-1. The first Mersenne primes
are 3, 7, 31, 127, etc. There are only 41 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.

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 1990's, 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 machine language, thereby producing a prime-search program of unprecedented efficiency, and that work led to the successful GIMPS project.

Several hundred school teachers 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 used by Findley has identified hidden hardware problems in many PCs.