Why Do People Look For These Big Primes?

1. Tradition!

Euclid may have been the first to define primality in his Elements approximately 300 BC. His goal was to characterize the even perfect numbers.

Large primes have since been studied by Cataldi, Descartes, Fermat, Mersenne, Frenicle, Leibniz, Euler, Landry, Lucas, Catalan, Sylvester, Cunningham, Pepin, Putnam and Lehmer. We need to continue this tradition.

2. For the By-products of the Quest.

While searching for large primes, some of the greatest theorems of elementary number theory, such as Fermat's Little Theorem and the Law of Quadratic Reciprocity, were discovered.

More recently, the search has demanded new and faster ways of multiplying large integers. 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.

3. People Collect Rare and Beautiful Items.

Mersenne primes are both rare and beautiful. Just 38 have been found in all of human history so they are indeed rare.

But they are also beautiful. Mersenne primes have one of the simplest possible forms for primes, 2^p-1. The proof of their primality has an elegant simplicity.

4. For the Glory!

The Mersenne prime 2^6972593-1 has 2,098,960 digits. If you join GIMPS, you could be the next one (along with George Woltman and Scott Kurowski) to discover a million digit prime.

In addition, mountain climbers wish to climb Mt. Everest because it is there. Several number theorists want to find large Mersenne primes out of curiosity and the desire to discover new things.

5. To Test the Hardware.

This reason has historically been used as an argument to get the computer time, so it is often a motivation for the company rather than the individual).

Software routines from the GIMPS project were used by Intel to test Pentium II and Pentium Pro chips before they were shipped.

Slowinski, who has helped find more Mersenne primes than anyone else works for Cray Research and they use his program as a hardware test. The infamous Pentium bug was found in a related effort as Nicely was calculating the twin prime constant

The GIMPS programs are intensely CPU and bus bound. They are relatively short, give an easily checked answer (when run on a known prime they should output true after their billions of calculations). They can easily be run in the background while other "more important" tasks run, and they are usually easy to stop and restart.

6. To Learn More About Their Distribution.

Though mathematics is not an experimental science, we often look for examples to test conjectures (which we hope to then prove). As the number of examples increase, so does (in a sense) our understanding of the distribution. The prime number theorem was discovered by looking at tables of primes.