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GIMPS Discovers 36th Mersenne Prime,

2^{2,976,221}-1 is now the Largest Known Prime.

**ORLANDO, Florida, September 1, 1997 — **
Gordon Spence, using a program written by George Woltman, has discovered what is now the largest known prime number. The prime number, 2^{2,976,221}-1,
is one of a special class of prime numbers called Mersenne primes. This is only the 36th known Mersenne prime. Gordon Spence, a 38-year-old I.T. Manager for
Thorn Microwave Devices Ltd, is from Hampshire, England. George Woltman is a 39 year-old programmer living in Orlando, Florida.

Peter Butcher, Managing Director at TMD said "We congratulate Gordon & George on their discovery. As a world leader in Microwave R&D we appreciate the value of research and were glad to donate thousands of hours of computer time to the search."

The new prime number, discovered on August 24th, is 895,932 digits long – more than twice the length of the previous record prime! If printed, the number would fill a 450-page paperback book. It took Spence's 100 MHz Pentium computer 15 days to prove the number prime. Alan White, Managing Director at Technology Business Solutions, who provided the historic PC, said "We were delighted to donate the computer that has made this exciting discovery."

The new Mersenne prime was independently verified on a Cray T90 supercomputer by David Slowinski, discoverer of seven Mersenne primes between 1979 and 1996.

Spence is one of over 2000 volunteers world-wide participating in the Great Internet Mersenne Prime Search (GIMPS). This prime number is the second record prime found by the GIMPS project. Joel Armengaud discovered the previous largest known prime number last November. The GIMPS project was started by Woltman in early 1996.

Discovering prime numbers of this size would have been impossible just a few short years ago. GIMPS is an example what can be accomplished when people, using spare computer time that would otherwise be wasted, combine forces over the Internet. Working alone, it would have taken Spence's computer 940 years to find this prime number. Woltman said, "All 2000 volunteers share in the credit of this discovery — Gordon would not have suceeded without their help."

Gordon Spence said of his discovery, "I was just lucky to get the right range of numbers to check, but it is a great feeling to become a part of history and join a very exclusive club."

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^^{2,976,220} * (2^{2,976,221}-1). This number is 1,791,864 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 https://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 monkMarin 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. 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 Spence has identified dozens of hardware problems in PCs. Intel now uses the program to test every Pentium II and Pentium Pro chip before it ships.