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Number Nabbers Bag Biggest Prime

On June 1, 1999, an ordinary Pentium PC in Plymouth, Michigan, found the largest known prime number,

a number having 2,098,960 digits (enough to fill a 1000-page paperback book). The newly discovered prime replaces the previous record-holder,

a number having 909,526 digits.

The champion computer belongs to Nayan Hajratwala, who works for Price Waterhouse Coopers in Plymouth, Michigan. The old record (set in January 1998) belonged to Roland Clarkson, a 19-year-old student at California State University. Mr. Hajratwala and Mr. Clarkson are participating in a unique international project to find new primes--and anyone who wishes to do so can join this effort.

An anonymous donor posted a $50,000 prize for the first million-digit prime; Mr. Hajratwala will be eligible for this award as soon as the discovery is published by a recognized mathematial journal. There's a $100,000 prize offered for the first ten-million-digit prime.

Prime numbers are whole numbers having no proper divisors, that is, a number p is prime if its only divisors are 1 and p itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Mathematicians can easily prove that there exist infinitely many primes, but it is very difficult to determine whether a given large number is prime or not.

Ancient mathematicians noticed an interesting possible connection between prime numbers and powers of 2. Their idea was to start with a prime number p and compute the number 2^p - 1. What they noticed was that in the first four cases this procedure yields a new prime:

Could it be that 2^p - 1 is always prime whenever p is prime? Sadly, no: 2¹¹ - 1 = 2047 is not prime, because 2047 is divisible by 23 and 89. However, Renaissance mathematicians proved that the next three cases do produce primes:

In 1644 the French monk Marin Mersenne (1588-1648) made a conjecture as to what the 8th through 11th primes on the list would be. He was wrong, but it took centuries for mathematicians to figure that out. As they worked, they came to call these primes of the form 2^p - 1 Mersenne primes.

As late as the 1940s only 12 Mersenne primes were known, the largest being 2¹²⁷ - 1, a 39-digit number. However, in the 1870s the French mathematician Edouard Lucas developed a theory allowing mathematicians to judge better whether large numbers are prime, and in the 1930s the American mathematician Derrick Lehmer used this theory to develop a reasonably efficient test for whether 2^p - 1 is prime. With the advent of modern computers, it became possible to use the Lucas-Lehmer Test to find many new Mersenne primes.

Clarkson's discovery is the 37th known Mersenne prime. We don't know yet if it is the 37th smallest Mersenne prime, though, because not all the smaller possibilities have been tested.

George Woltman, a programmer in Orlando, Florida, has developed PC software which allows any Pentium PC to search for new Mersenne primes "in the background." This means the program runs like a sceen saver does--when the computer is not being used for anything else. All over the world, PCs are patiently computing away, adding their bit to mathematical knowledge.

In addition to finding one new prime, this effort has also proved that there are no unknown Mersenne primes less than 2⁸⁵⁹⁴³³ - 1, which is the 33rd entry on the list.

If you have a Pentium PC, you can join the search and perhaps, like Roland Clarkson, have the thrill of breaking the record for the largest prime number. The software is free and downloads easily from the Internet.

So, what is your PC doing in its spare time?

Internet Sources

Great Internet Mersenne Prime Search

This site, maintained by Woltman, has the free software to search for Mersenne primes as well as news releases, historical information and charts showing the current status of the search.

Chris Caldwell's Mersenne Prime Page

An excellent, highly readable page with more complete mathematical information on Mersenne primes. Caldwell, a professor at the University of Tennessee at Martin, has posted all 909,526 digits of Clarkson's prime, if you have space to store them.

Luke Welsh's Marin Mersenne Page

Biographical information on Father Mersenne. Welsh was the co-discoverer of the 29th Mersenne prime, 2¹¹⁰⁵⁰³ - 1, in 1988.

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Posted July 2, 1997; updated September 8, 1997, July 10, 1998, and October 25, 1999. Features remain online as long as they remain current; they may be updated if new information becomes available.

Copyright © 1999, Center for Mathematics and Science Education. Teachers have permission to duplicate this page for use in teaching their own classes. All other rights reserved. You are welcome to link to this page, but do not copy its contents.

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