Ivars Peterson's MathTrek

September 16, 1996

# Mining Prime Terrain

Venturing again into largely unexplored digital territory, computer scientists at Cray Research have unearthed another gargantuan prime number, setting a new record for the largest known prime.

This number, 2^1,257,787 - 1, has 378,632 digits, putting it well ahead of the previous record holder, which came in at 258,716 digits when it was found in 1994. If written out in full, the new prime would cover about 120 typed pages.

It is also the 34th Mersenne prime to be discovered. Expressed in the form 2^p - 1, where the exponent p is itself a prime, Mersenne numbers hold a special place in the never-ending pursuit of larger and larger primes. These particular numbers have special characteristics that make it relatively easy to check whether a candidate is either a prime number or a composite number.

The smallest Mersenne prime is 3 (2^2 - 1). After that comes 7 (2^3 - 1), then 31 (2^5 - 1), and so on. With an exponent of 1,257,787, the new champion holds the distinction of being the largest Mersenne prime so far identified. However, because no one has yet checked all Mersenne numbers having smaller exponents, mathematicians can't be sure that no Mersenne primes lurk in the vast expanse between the record holder and the second-place Mersenne prime, or even between the third-place and second-place Mersenne primes.

The new prime was discovered last spring by David Slowinski and Paul Gage in the course of routine testing of a new Cray T94 supercomputer in preparation for delivery to a customer. The number surfaced during one particular 6-hour run. Slowinski and Gage then asked other researchers to double-check their work before making it public.

Over the years, Slowinski has been involved in the discovery of seven Mersenne primes. Gage has shared in the identification of the last three behemoths.

However, the hit-or-miss approach of Slowinski and Gage in merely going for the record has little value for computational number theorists interested in such questions as the distribution of Mersenne primes, or even primes in general. These mathematicians would prefer a more thorough search that checks every eligible exponent.

In fact, George Woltman, a computer programmer in Florida, is now leading such a hunt. He and more than 400 volunteers, using Pentium-chip-based desktop computers and special software, have started systematically checking all the numbers in the gaps between the known Mersenne primes, and beyond. "Our goal is to test every Mersenne number with an exponent less than 1,300,000 by the end of 1997 and to test every exponent below 2,630,000 by the end of 2000," Woltman declares.

Anyone can play the game and join "The Great Internet Mersenne Prime Search." Indeed, Woltman is looking for additional recruits. All you need is a Pentium computer with plenty of idle time on its chips. There's no shortage of numbers to test.

### References:

Geier, Thom. 1996. Prime stake. U.S. News & World Report (Sept. 16):18.

Gillmor, Dan. 1996. Researchers discover prime example of mathematicians' love. San Jose Mercury News (Sept. 3).

Holden, Constance. 1996. Grassroots search for primes.... Science 273(Aug. 9):743.

Peterson, Ivars. 1992. Striking pay dirt in prime-number terrain. Science News 141(April 4):213.

Ribenboim, Paulo. 1995. The New Book of Prime Number Records. New York: Springer-Verlag.

Schwartz, John. 1996. Computers: Number crunchers' 1 and only. Washington Post (Sept. 9).

Additional information about Mersenne primes and the new record is available at http://www.utm.edu/research/primes/mersenne.shtml.

George Woltman can be reached at 74473.2626@compuserve.com ( http://ourworld.compuserve.com/homepages/justforfun/prime.htm).

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.