#
GIMPS' Announcement of 2^{20996011}-1

### By Chris Caldwell

## 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

ORLANDO, Florida - December 2, 2003 -- Michael Shafer, a 26 year-old volunteer in the Mersenne.orgresearch project called the Great Internet Mersenne Prime Search (GIMPS), has discovered the largest known prime number. Shafer used a Michigan State University lab PC and free software by George Woltman and Scott Kurowski as part of an international grid of 211,000 networked computers in virtually every time zone of the world.

The new number, expressed as 2 to the 20,996,011th power minus 1, has 6,320,430 decimal digitsand was discovered November 17th. It is more than two 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 40th known Mersenne prime, named after Marin Mersenne, a 17th century French monk who first studied the 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. The new prime is 63% of the qualifying size for the $100,000 Electronic Frontier Foundationaward for the first 10-million-digit prime, also being sought by the project's volunteers. In May 2000, a previous participant won the foundation's $50,000 award for discovering the first million-digit prime.

Shafer, a chemical engineering graduate student at Michigan State University, described the find. "I had just finished a meeting with my advisor when I saw the computer had found the new prime. After a short victory dance, I called up my wife and friends involved with GIMPS to share the great news!" He used a 2 GHz Pentium 4 Dell Dimension PC running for 19 days to prove the number prime. "The software runs great without affecting the computer. I get my work done and contribute to the project at the same time."

Now in its eighth year, GIMPS has accomplished what no other distributed computing project has: Six consecutive successes. "Great teamwork has paid off for us again," said GIMPS founder George Woltman. "In addition to congratulating Michael Shafer, we wish to thank all 60,000 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 Michael."

"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 PrimeNetsystem that runs GIMPS using his company's technology to demonstrate its scalability and power with a demanding distributed search application. PrimeNet pulls together hundreds of thousands of computers in parallel to create a virtual supercomputer running at 9 trillion calculations per second, or 'teraflops'. This enabled GIMPS to find the prime in just two years instead of the 25,000 years a single PC would have required.

The new prime was independently verified by Guillermo Ballester Valor of Granada, Spain using twelve days of time on a 1.4GHz quad Itanium II server at the HP Test Drive center, and by Ernst Mayer of Cupertino, California using three weeks of time on a 1 GHz HP Alpha workstation. The discovery is the sixth record prime found by the GIMPS project, and the fourth discovered using distributed computing software from Entropia, Inc. In recognition of every GIMPS contributor's effort, credit for this new discovery will go to "Shafer, 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/prime.htm. 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.

Michael and his wife Liz Shafer live in Lansing, Michigan. He joined GIMPS in November 2000. 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/prime.htm. GIMPS is based in Orlando, Florida. Additional pubic 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 2001, Michael Cameron discovered the previous largest known prime numberin Canada. In June 1999, Nayan Hajratwala discovered the 38th Mersenne primein the U.S. In January 1998, Roland Clarkson discovered the 37th Mersenne primein the U.S. Gordon Spence discovered the 36th Mersenne primein August, 1997, in the U.K. Joel Armengaud discovered the 35th Mersenne primein 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 40 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. 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 Shafer has identified hidden hardware problems in many PCs.