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GIMPS Discovers 42nd Mersenne Prime,

2^{25,964,951}-1 is now the Largest Known Prime.

**ORLANDO, Florida, February 27, 2005 — **
Dr. Martin Nowak, an eye surgeon in Michelfeld, Germany, and a long-time volunteer in the Great Internet Mersenne Prime Search (GIMPS)
distributed computing project, has discovered the largest known prime number. Dr. Nowak used one of his business PCs and free software by George Woltman and Scott Kurowski.
His computer is a part of a world-wide array of tens of thousands of computers working together to make this discovery.

The formula for the new prime number is 2 to the 25,964,951st power minus 1. The number belongs to a special class of rare prime numbers called Mersenne primes. This is only the 42nd Mersenne prime found since Marin Mersenne, a 17th century French monk, first studied these numbers over 350 years ago.

Written out the number has 7,816,230 digits, over half a million digits larger than the previous largest known prime number. It was discovered February 18th after more than 50 days of calculations on a 2.4 GHz Pentium 4 computer. The new prime was independently verified in 5 days by Tony Reix of Grenoble, France using a 16 Itanium CPU Bull NovaScale 5000 HPC running the Glucas program by Guillermo Ballester Valor of Granada, Spain. The discovery is the eighth record prime found by the GIMPS project. In recognition of every GIMPS contributor's effort, credit for this new discovery will go to "Nowak, Woltman, Kurowski, et al". Dr. Richard Crandall, who discovered the advanced transform algorithm used by the GIMPS program, offers a framed or unframed poster with all 7.8 million digits displayed in an extremely small font — optional magnifying glass sold separately!

A second verification was completed by Jeff Gilchrist of Elytra Enterprises Inc. in Ottawa, Canada using fifteen days of time on 12 CPUs of a Compaq Alpha GS160 1.2 GHz CPU server at SHARCNET.

Dr. Nowak first read about GIMPS almost 6 years ago in his newspaper, the "Frankfurter Allgemeine Zeitung". As one of Dr. Nowak's hobbies is mathematics, the project seemed like a natural fit. He started with one PC in April 1999 and as his eye center has grown so has his GIMPS participation. He now has 24 computers calculating away.

Mersenne primes are most relevant only to number theorists. Most participants join GIMPS simply for the fun of advancing math knowledge – and the chance of finding a new Mersenne prime. Some also participate in hopes of claiming a portion of the $100,000 Electronic Frontier Foundation award for the first 10-million-digit prime. The new prime is 78% of the size needed. "An award-winning prime could be found next week or several years from now – that's the fun of math discoveries", said GIMPS founder George Woltman. The GIMPS participant who discovers a 10 million digit 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 GIMPS participant won the foundation's $50,000 award for discovering the first million-digit prime.

Now in its tenth year, GIMPS is one of the few distributed computing projects that has provided a string of success stories: eight record primes. "Persistence and teamwork are critical to our success." said Woltman. "Congratulations and thanks go to Dr. Nowak and all 75,000 volunteer home users, students, schools, universities and businesses from around the world that made this discovery possible." Woltman adds, "Joining GIMPS can be a great introduction to the math of prime numbers, as a bonus you have a chance to find a new Mersenne prime and claim a big prize."

"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 tens of thousands of computers in parallel to create a virtual supercomputer running at 17 trillion calculations per second, or "teraflops". This enabled GIMPS to find the prime in just nine months instead of the 1,500 years a single PC would have required.

"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 www.mersenne.org. The calculations are performed when the computer is idle. It is very important to get permission to install the software on computers you do not own.

#### 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 tens 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 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 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

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 42 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 May 2004, Josh Findley discovered the previous largest known prime number in the United States. In November 2003, Michael Shafer discovered the 40th Mersenne prime 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.

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 2^{25,964,950} * (2^{25,964,951}-1). This number is 15,632,458 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 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.

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 used by Dr. Nowak has identified hidden hardware problems in many PCs.