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Mersenne Prime Number discovery - 21398269-1 is Prime!

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GIMPS Discovers 35th Mersenne Prime,
21,398,269-1 is now the Largest Known Prime.

ORLANDO, Florida, November 23, 1996 — On November 13, Joel Armengaud discovered the largest known prime number using a program written by George Woltman. Joel Armengaud, a 29-year-old programmer for Apsylog, is from Paris, France. George Woltman is a 39-year-old programmer living in Orlando, Florida.

Early this year, Woltman launched the Great Internet Mersenne Prime Search (GIMPS). This web site offers free software for ordinary personal computer owners to use in searching for big prime numbers. Large prime numbers were once the exclusive domain of supercomputer users. "By using a large number of small computers, we negate the supercomputer's speed advantage," said Woltman. Armengaud is one of more than 700 people searching for new primes. Even though Armengaud was the one lucky enough to find this new prime, credit must also go to all the other searchers. Without their efforts, this discovery would not have been possible.

The new prime number, 21,398,269-1 is the 35th known Mersenne prime. This prime number is 420,921 digits long. If printed, this prime would fill a 225-page paperback book. It took Joel 88 hours on a 90 MHz Pentium PC to prove this number prime.

Armengaud said of his discovery, "Finding this new Mersenne prime was quite a thrill! Mersenne primes are very rare, there was only one chance in 35,000 that this Mersenne number would turn out to be a prime."

An integer greater than one is called a prime number if its only 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 2p-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.

The previous largest known prime, also a Mersenne Prime, was discovered earlier this year using a Cray supercomputer. The new Mersenne prime was independently verified by David Slowinski, a co-discoverer of the last Mersenne prime.

Mersenne primes are useful in the field of cryptography. NeXT Chief Scientist Richard Crandall patented the Fast Elliptic Encryption system based on Mersenne primes. The program that found this new prime, uses an algorithm developed by Crandall to find Mersenne primes twice as fast as previous known algorithms. The current search has also proven useful in at least three other respects.

First, the search for Mersenne primes is a great way to get youngsters interested in mathematics. One Florida middle school teacher uses the program as a reward. After giving a lesson on prime numbers to her students, anyone who passes the "Prime Quest Test" gets to use a school computer to try and find a new largest known prime.

Second, the project has once again proven the power of distributed computing on the Internet. PCs in their spare time can tackle problems that would otherwise require multi-million dollar supercomputers. Said Woltman, "Many other important research projects could use this approach, especially if the funding isn't available for months of supercomputer time. It gives the average person a chance to participate in the scientific discoveries of tomorrow."

Finally, the program is a great stress test for PCs. In fact, the program Armengaud used has identified hardware problems in over 3% of the PCs that have run it. Says Woltman, "I'd run the program on any new PC. If it can't pass the self-test, I'd return it."

The search for more Mersenne primes is already under way. You can join the Great Internet Mersenne Prime Search at You do not need to be a mathematian or computer whiz to use the program. Said Armengaud, "You won't even know the program is running. It uses computer time that would otherwise go to waste."

Even though this is the 35th known Mersenne prime, there may be a smaller, as yet undiscovered Mersenne prime. Not all Mersenne numbers between the 31st and 35th have been checked.

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 35th known perfect number is 21,398,268 * (21,398,269-1). This number is 841,842 digits long!