#
GIMPS Sets Another Record!
2^{2976221}-1* is prime*

### By Chris Caldwell

There is no getting around it, the 895,932 digit number 2^{2976221}-1
is a giant--it would easily fill a 400 pagepaper
back book! Nevertheless, on August 24th, 1997 Gordon Spence
completed the proof of its primality on a 100 MHz Pentium. The complete
proof took fifteen days using a *highly optimized* program written by the
founder of the **G**reat **I**nternet **M**ersenne **P**rime **S**earch,
George Woltman. David Slowinskiof Cray Research finished verifying the primality on August 29th. It is now
the largest known prime, the 36thknown Mersenne Prime, gives rise
to the 36th known
perfect number (with 1,791,864 digits!) and smashes
all previous records for size!

#### How do you show a number that large number is prime?

A prime is an integer that has only one and itself for positive divisors (for example, the prime factors of 10 are 2 and 5). The first few primesare 2, 3, 5, 7 and 11. We can check small numbers by just dividing by the lesser integers. But to do that with a number this size would take far longer than the universe has been in existence. The key to slaying a giant this size is a theorem developed by Lucasin the late 1870's and simplified by Lehmer, now called the Lucas-Lehmer Test. Even as simple as this test is, it could not be done quickly without using very clever programs to multiply the numbers. In 1968 Strassen discovered how to multiply quickly using Fast Fourier Transforms (*). He and Schönhage refined and published the method in 1971. GIMPS now uses an improved version of their algorithm developed by the long time Mersenne searcher Richard Crandall[see CF94].#### Is that it, just get a fast program?

Even a fast program is not enough, alone a systematic search would take nearly
a millennium on Spence's computer. What GIMP's does is to coordinate the
efforts of over two thousand computer users--by
offering free software, a database of known results, and by
allowing individuals to check out regions to test. **Gordon Spence
and George Woltman share the credit and glory with all of the others
involved in this effort!** There are still infinitely manymore giants left
to slay, so why not surf over to Woltman's
GIMPS siteand join the search for the next record prime?

For more information click on one of the following:

- About
**this prime**- The official press release
- Spence's announcement
- Woltman's announcement
- Articles: Science News 9/13/97.
- The complete decimal expansion
- How long is this number when typed?

- About
**the GIMPS project**- Introduction(Home Page)
- Search Credits
- Search status(there is also a graphical view)
- June 96 San Jose Mercury News article

- About
**Mersenne primes**- Mersenne Primesdefinitions, theorems, lists...
- Marin Mersennereferences, images, sounds...
- Mersenne Prime Freeware
- Mersenne Primes Bibliography

- About
**primes in general**