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GIMPS Discovers 39th Mersenne Prime,

2^{13,466,917}-1 is now the Largest Known Prime.

### Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid.

**SAN DIEGO, California and ORLANDO, Florida, December 6, 2001 — **
Michael Cameron, a 20 year-old volunteer in a worldwide research project called the Great Internet Mersenne Prime Search (GIMPS),
has discovered the largest known prime number using his PC and software by George Woltman and Entropia, Inc. as part of an international
grid of more than 205,000 interconnected computers operated by the company.

The new number, expressed in shorthand as 2^{13,466,917}-1, contains 4,053,946 digits and was discovered November 14th.
It belongs to a special class of rare prime numbers called Mersenne primes. The discovery marks only the 39th known Mersenne
prime, named after Marin Mersenne , a 17th century French monk who studied the numbers.
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.

Cameron used a 800 MHz AMD T-Bird PC running part-time for 45 days to prove the number prime. He said, "A friend informed me that if I was going to leave my computer on all the time I should make use of that wasted CPU time. I put GIMPS on my PC because it does not interfere with my work on the computer. Finding the new prime was a wonderful surprise!"

"Finding this prime is by far our most impressive accomplishment to date, having taken two years of non-stop work. In addition to congratulating Michael Cameron, we wish to thank all 130,000 volunteer home users, students, schools, universities and businesses from around the world that contributed to GIMPS," said GIMPS founder George Woltman. "Joining GIMPS is a great way to learn about math through participation – plus you might find a new Mersenne prime, like Michael."

"Entropia is delighted to have a role in this discovery," said Entropia founder, Scott Kurowski. Kurowski developed the PrimeNet system for GIMPS to demonstrate Entropia's distributed computing scalability and power with a demanding parameter search application. He added, "It's great seeing such a repeatedly successful research effort having volunteers provide hundreds of thousand of PCs from virtually every time zone of the world. George runs an amazing and fun project."

PrimeNet registered discovery of the new prime the day after IBM (NYSE:IBM) announced a partnership with Entropia. The number-crunching power and searching performed by GIMPS is similar to pharmaceutical, bioinformatics, chemical, materials and financial applications that Entropia accelerates.

Acquiring and using computing capacity of this size would be much more costly without the distributed computing power harnessed by Entropia's PrimeNet system. PrimeNet performs 2 trillion calculations per second, or teraflops, around the clock. The GIMPS project spent 13,000 years of computer time to find this prime number, yet the research work was done using spare background computer time that would otherwise be wasted, even while the PCs were in use by their owners.

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. (www.perfsci.com).

Michael Cameron is from Owen Sound, Ontario and works for NuComm International, Inc. 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 development manager and entrepreneur in San Diego, California. He founded Entropia in 1997.

The new Mersenne prime was independently verified by Ernst Mayer and Paul Victor Novarese using three weeks of computer time on a 667 MHz Alpha workstation. The discovery is the fifth record prime found by the GIMPS project, and the third discovered using Entropia's distributed computing grid. In recognition of every GIMPS contributor's effort and PrimeNet, credit for this new discovery will go to "Cameron, Woltman, Kurowski, et. al."

In May 2000 a GIMPS participant received a $50,000 cooperative computing award from the Electronic Frontier Foundation for the discovery of the first million-digit prime number. A $100,000 award awaits discovery of a ten-million-digit prime number, a challenge GIMPS participants are already working upon.

"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 https://www.mersenne.org/. Please get permission to install the software on computers you do not own.

#### About The 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, and there are certainly larger Mersenne primes waiting to be discovered. Anyone with a reasonably powerful personal computer can join GIMPS and become a big prime hunter. All the necessary software can be downloaded for free at https://www.mersenne.org/. GIMPS is based in Orlando, Florida.

#### About Entropia, Inc.

Entropia's enterprise distributed and grid computing products harness the vast untapped processing power of PCs on corporate networks to build virtual computing clusters of a selected design and scope. The technology, scalable to millions of PCs, delivers computing capacities on par with that of supercomputers at a fraction of the cost. Companies deploy Entropia applications to accelerate their time to market using the PCs they already own, dramatically increasing the ROI in their desktop computing and network assets. Commercial applications include critical pharmaceutical, chemical and materials research, and financial services. Entropia application grids power mathematics research (www.mersenne.org), HIV drug research (www.FightAIDSatHome.org), and economic research (www.SaferMarkets.org). Founded in 1997, the company has offices in San Diego, California and Cambridge, England. For more information visit the Entropia Web site at www.entropia.com.

#### For More Information on Mersenne Primes

Previous GIMPS Mersenne prime discoveries were made by members in various countries. In June 1999, Nayan Hajratwala discovered the previous largest known prime number 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 39 known Mersenne primes.

An integer greater than one is called a prime number if its only positive divisors are one and itself. 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 study of Mersenne primes has 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 Cameron has identified hidden hardware problems in many PCs.