Ivars Peterson's MathTrek

December 16, 1996

Another Record Prime

The new record holder is (2^1,398,269) - 1, which has 420,921 digits and comfortably beats the previous champion, (2^1,257,787) - 1 (see Mining Prime Terrain).

Armengaud is a participant in a project known as GIMPS (Great Internet Prime Search). Started by George Woltman, a programmer in Orlando, Fla., it brings together more than 700 volunteers in a systematic effort to check so-called Mersenne numbers to see if they are primes. Mersenne numbers, written in the form 2^p - 1 (where the exponent p is itself a prime number) have special characteristics that make it relatively easy to check whether a candidate is a prime number or a composite number (the product of smaller primes).

The new champion is the 35th Mersenne prime to be discovered. Armengaud found it by using Woltman's software implementing the Lucas-Lehmer test for primes and working together with hundreds of volunteers via the Internet. In the final step, it took him 88 hours on a 90-megahertz Pentium personal computer to prove his number to be a prime. The find was double-checked by others on two different computers.

"We were incredibly lucky to find this prime," Woltman comments. "The [prime exponent] range from 1,000,000 to 2,000,000 was expected to contain 1.78 Mersenne primes. There are already two known [in this range], and probabilities suggest there will be one more."

Interestingly, running Woltman's program also serves as a stringent test of a personal computer. Users have identified hardware problems of various kinds in more than 3 percent of the personal computers that have run his prime-finding software.

Woltman is still looking for more volunteers to join in the GIMPS project. Not every Mersenne number between the 31st and 35th known numbers has yet been checked. So it's quite possible that another Mersenne prime lurks undiscovered in the gaps. Woltman expects the GIMPS team to test every prime exponent less than 1,345,000 by the end of 1997.

"The Great Internet Mersenne Prime Search is turning computer cycles most people waste on screen savers into new mathematical discoveries," says Chris Caldwell of the University of Tennessee in Martin. "Even some grade school kids are in on the act."

"By using a large number of small computers, we negate the supercomputer's speed advantage," Woltman adds. "Many other important research projects could use this approach, especially if funding isn't available for months of supercomputer time. It gives the average person a chance to participate in the scientific discoveries of tomorrow."

It would be pretty silly for anyone to try to memorize the 420,921 digits of the new prime record holder. But the digits of some other big primes are relatively easy to remember.

"What is the largest prime for which you can recite all digits?" Richard L. Francis of Southeast Missouri State University in Cape Girardeau, Mo., asks in a recent issue of Mathematics Magazine. "Is it the Mersenne prime 8191, or the last prime-numbered year of the twentieth century (1999), or the Fermat prime 65537?"

"The largest know repunit prime, which is the fifth of its kind, represents a case for easy memorization," he continues. "With virtually no effort, all 1031 of its digits (each of which is 1) can be recited."

You just have to keep careful count!

Copyright © 1996 by Ivars Peterson.

References:

Francis, Richard L. 1996. Math bite: Recitation of large primes. Mathematics Magazine 69(October):260.

Gillmor, Dan. 1996. Move over, supercomputers. San Jose Mercury News (Nov. 23). http://www.sjmercury.com/business/compute/prime1122.htm/

. Peterson, I. 1996. Mining Prime Terrain. MathLand/MAA Online (Sept. 16).