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GIMPS Discovers 43rd Mersenne Prime,

2^{30,402,457}-1 is now the Largest Known Prime.

**ORLANDO, Florida, December 24, 2005 — **
A collaborative effort at Central Missouri State University (CMSU), led by professors Curtis Cooper
and Steven Boone, has discovered the largest known prime number as part of the volunteer
Great Internet Mersenne Prime Search (GIMPS) project. It is an achievement that also fuels researchers' hopes around the globe that a $100,000 prize
is within reach.

The CMSU faculty used idle time on 700 campus lab PCs and free software from www.mersenne.org as part of a world-wide computing grid of tens of thousands of computers working together to make this discovery. The software was developed by GIMPS founder, George Woltman, in Orlando, Florida, and grid computing pioneer Scott Kurowski, in San Diego, California.

The new prime number, known as M30402457, surfaced December 15th on one computer in the Department of Communication lab after running on and off for about 50 days. Dr. Cooper and Dr. Boone have joined together with 21,000 other researchers worldwide participating in GIMPS. In addition to pursuing new prime number discoveries, these individuals also have an opportunity to compete for a cash award offered by the Electronic Frontier Foundation for discovery of the first 10 million digit prime number. If GIMPS claims the $100,000 award, of which $25,000 will go to charity, a large portion will be given to the GIMPS participant that discovers the prime number.

CMSU's research team has come the closest to claiming the award with this discovery of M30402457, or 2 to the 30,402,457th power minus 1, which is a 9,152,052 digit number.
It is the largest known prime number, eclipsing GIMPS last discovery of a 7,816,230 digit prime in February 2005.
The new prime was independently verified in 5 days by Tony Reix of Bull S.A. in Grenoble, France using 16 Itanium2 1.5 GHz CPUs of a Bull NovaScale 6160 HPC
at Bull Grenoble Research Center, running the Glucas program by Guillermo Ballester Valor of Granada, Spain. A second verification was done by
Jeff Gilchrist of

The new prime is the 43rd discovery in a special class of rare prime numbers known as Mersenne primes, named for French monk Marin Mersenne, who studied these numbers more than 350 years ago.

George Woltman, who started GIMPS in 1996, said Mersenne primes today are important primarily to number theorists. He added, however, most GIMPS participants study them for the fun of having a role in pure mathematical research and the chance of finding a new Mersenne prime. "While we think we understand the frequency and distribution of Mersenne primes, it has not been proven. Finds such as this one give us 'another piece of the puzzle' in confirming our theories," Woltman said. He noted that in the past Mersenne prime searches have led to important advances in Fast Fourier Transforms (used in countless applications) as well as discovering computer hardware problems via rigorous stress testing. "The research project also promotes interest in math by capturing the imagination of younger participants," Woltman said.

The 700 campus computers are part of an international grid called PrimeNet, consisting of 70,000 networked computers in virtually every time zone of the world. PrimeNet organizes the parallel number crunching to create a virtual supercomputer running 24x7 at 18 trillion calculations per second, or "teraflops". This greatly accelerates the search. This prime, found in just 10 months, would have taken 4,500 years on a single PC. Kurowski said, "GIMPS is an amazing project and CMSU's team exemplifies the dedication to international cooperative computing demonstrated by all of GIMPS' participants."

"We've worked with Information Services to make sure we are not compromising the campus computing infrastructure," said Dr. Cooper, who got interested in this project over 7 years ago with colleague Vince Edmondson. Edmondson, professor of mathematics, was instrumental in the campus effort until he passed away in 2003. "We owe a lot to all of the people on campus who have helped with this project," Dr. Boone added.

The discovery is the ninth record prime found by the GIMPS project. In recognition of every GIMPS participant's contribution, credit for this prime will go to "C. Cooper, S. Boone, G. Woltman, S. 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 9.1 million digits displayed in an extremely small font — optional magnifying glass sold separately!

#### 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 found. 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 https://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 number of the form 2^{p}-1. The first Mersenne primes are 3, 7,
31, and 127 corresponding to P = 2, 3, 5, and 7 respectively. There are only 43 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 February 2005, Dr. Martin Nowak discovered the previous largest known prime number in Germany. In May 2004, Josh Findley discovered the 41st Mersenne prime 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 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. In August 1997, Gordon Spence discovered the 36th Mersenne prime in the United Kingdom. In November 1996, Joel Armengaud discovered the 35th Mersenne prime 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^{30,402,456} * (2^{30,402,457}-1). This number is 18,304,103 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 assembly 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 CMSU has identified hidden hardware problems in many PCs.