##
GIMPS Discovers 37th Mersenne Prime,

2^{3,021,377}-1 is now the Largest Known Prime.

**ORLANDO, Florida, February 2, 1998 — **
Roland Clarkson has discovered the world's largest known prime number using a program written by George Woltman and networking software written by Scott Kurowski.
The prime number, 2^^{3,021,377}-1, is one of a special class of prime numbers called Mersenne primes. This is only the 37th known Mersenne prime.
Roland Clarkson, a 19 year-old student at California State University Dominguez Hills, is from Norwalk, California. George Woltman is a retired programmer living
in Orlando, Florida. Scott Kurowski is a software development manager and entrepreneur living in San Jose, California.

The new prime number, discovered on January 27th, is 909,526 digits long! Roland used a 200 MHz Pentium computer part-time for 46 days to prove the number prime. Running uninterrupted it would take about a week to test the primality of this prime number.

Clarkson is one of over 4000 volunteers world-wide participating in the Great Internet Mersenne Prime Search (GIMPS). This prime number is the third record prime found by the GIMPS project. Gordon Spence discovered the previous largest known prime number last August. Joel Armengaud discovered the 35th Mersenne prime in November, 1996. The GIMPS project was started by Woltman in January of 1996.

This is the first Mersenne prime discovered using Scott's Internet software and server, which handles the huge number of GIMPS volunteers. His "PrimeNet" server distributes work to and collects results from thousands of copies of George's program, running all over the world, effectively operating as a single, massively-parallel supercomputer. In a typical day, it processes more than an entire year of desktop computing power, and this rate grows daily.

The PrimeNet server tracks the progress of each test program through ten different states, manages individual user accounts, and credits precise computer time for tested numbers. "Distributed Internet data processing is quickly maturing," says Scott. "GIMPS is now the world's foremost example of a new kind of computing service." To attract more would-be hunters, Scott has a sponsored cash prize for the PrimeNet discoverer of the 38th Mersenne prime. The prize pool is $1.00 for every 1000 digits in the new prime or US $1,000.00, whichever is larger. He adds, "If you have a desktop computer, we've got something for it to do between keystrokes and mouse clicks, and many more machines are needed for the next, even larger, prime."

Discovering prime numbers of this size would have been impossible just a few years ago. GIMPS is an example what can be accomplished when a large number of people, using spare computer time that would otherwise be wasted, combine forces over the Internet. In recognition of the entire group's effort, credit for this new discovery will go to "Clarkson, Woltman, Kurowski, et al."

Roland is the third youngest person to find a world-record prime number. Landon Curt Noll and Laura Nickel found the 25th Mersenne prime when they were high school students. Roland said of his discovery, "I am ecstatic and very, very lucky. I never expected this number to be prime because it was so close to the previous record prime."

The new Mersenne prime was independently verified on a Cray supercomputer by David Slowinski, discoverer of seven Mersenne primes between 1979 and 1996.

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^{3,021,376} * (2^^{3,021,377}-1). This number is 1,819,050 digits long!

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 http://www.mersenne.org/.

#### What are Mersenne Primes? Why are they useful?

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.

With undertakings such as the race to the moon in the 1960's, it is the byproducts that are most useful to society. The same is true in the search for large primes. This project has led to advances in distributed computing. That is, using the Internet to effectively harness the unused computing power of thousands of machines. Scott said, "Someday, many projects that require a supercomputer today will be solved using low-cost PCs."

When testing Mersenne numbers to see if they are prime one must repeatedly multiply very large integers. Recently Richard Crandall at Perfectly Scientific discovered ways to double the speed of some Fast Fourier Transforms which are used in numerous other scientific applications. Richard Crandall also patented the Fast Elliptic Encryption system which uses Mersenne primes to encrypt and decrypt messages.

School teachers in elementary through high-school grades have used GIMPS to get their students excited about doing 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 Roland has identified hardware problems in many PCs. Intel now uses part of the program to find manufacturing defects in Pentium II and Pentium Pro chips before they are shipped.