Why participate in GIMPS?
Euclid may have been the first to define primality in his 'Elements' approximately 300 BC. His goal was to characterize the even perfect numbers (numbers like 6 and 28 who are equal to the sum of their aliquot divisors: 6 = 1+2+3, 28=1+2+4+7+14). He realized that the even perfect numbers (no odd perfect numbers are known) are all closely related to the primes of the form 2p-1, for some prime p (now called Mersennes). So the quest for these jewels began near 300 BC.
Large primes (especially of this form) were then studied (in chronological order) by Cataldi, Descartes, Fermat, Mersenne, Frenicle, Leibniz, Euler, Landry, Lucas, Catalan, Sylvester, Cunningham, Pepin, Putnam and Lehmer (to name a few).
Much of elementary number theory was developed while deciding how to handle large numbers, how to characterize their factors and discover those which are prime. In short, the tradition of seeking large primes (especially the Mersennes) has been long and fruitful. It is a tradition well worth continuing.
For the by-products of the quest
Being the first to put a man on the moon had great political value for the United States of America, but what was perhaps of the most lasting value to the society was the by-products of the race. By-products such as the new technologies and materials that were developed for the race, are now common everyday items.
The same is true for the quest for record primes. In the tradition section above are listed some of the giants who were in the search (such as Euclid, Euler and Fermat). They left in their wake some of the greatest theorems of elementary number theory (such as Fermat's little theorem and quadratic reciprocity).
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 transform. Strassen 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.
The Mersenne search is also used by school teachers to involve their students in mathematical research, and perhaps to excite them into careers in science or engineering.
People collect rare and beautiful items
Mersenne primes, which are usually the largest known primes, are both rare and beautiful. Since Euclid initiated the search for and study of Mersennes approximately 300 BC, very few have been found. Just 51 in all of human history - that is rare!
But they are also beautiful. Mathematics, like all fields of study, has a its own notion of beauty. What qualities are perceived as beautiful in mathematics? Mathematicians look for proofs that are short, concise, clear, and if possible that combine previous disparate concepts or teach you something new. Mersennes have one of the simplest possible forms for primes, [math]2^p-1[/math]. The proof of their primality has an elegant simplicity (to a mathematician). Mersennes are beautiful and have some surprising applications.
For the glory
Why do athletes try to run faster than anyone else, jump higher, throw a javelin further? Is it because they use the skills of javelin throwing in their jobs? Not likely. More probably it is the desire to compete (and to win!).
This desire to compete is not always directed against other humans. Rock climbers may see a cliff as a challenge. Mountain climbers can not resist certain mountains.
Look at the incredible size of these giant primes!, like the currently Largest Known Prime. Those who found them are like the athletes in that they outran their competition. They are like the mountain climbers in that they have scaled to new heights. Their greatest contribution to mankind is not merely pragmatic, it is to the curiosity and spirit of man. If we lose the desire to do better, will we still be complete?
Athletes do not only compete as individuals, but also as teams or groups of individuals, with the same desire to be first and best. In the same manner, so to speak, can prime searchers compete in teams, like ARS Team Prime Rib. As with other competitions we closely follow our standing against other teams and rejoice over any gain we make.
To test the hardware
This one 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.
Since the dawn of electronic computing, programs for finding primes have been used as a test of the hardware. For example, software routines from the GIMPS project were used by Intel to test Pentium II and Pentium Pro chips before they were shipped. So a great many of the readers of this page have directly benefited from the search for Mersennes.
David Slowinski, who found more Mersennes than any other (from 1994-2004), works for Cray Research and they use his program as a hardware test. The infamous Pentium Bug was found in a related effort as Thomas Nicely was calculating the Twin prime constant.
Why are prime programs used this way? They 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.
Intel uses just a few iterations of Prime95 to test the FPU. Obviously, they incorporated part of the code in their test program rather than running Prime95 directly. This is only one of many tests - but it's a good one.
OverClockers often use programs like Prime95 and SuperPi to test the stability of systems. If the system can run the Prime95 and SuperPi programs for any length of time without failing, they are regarded as stable.
To learn more about their distribution
Though mathematics is not an experimental science, mathematicians often look for examples to test conjectures (which they then hope to prove). As the number of examples increase, so does (in a sense) their understanding of the distribution. The prime number theorem was discovered by looking at tables of primes.
For the challenge
Why do anything, why climb that rock, why read that book, why learn a new language, why build a machine? On the whole there are a lot of why do questions. And they can all, although not always, be answered with: For the challenge of doing it and because it can be done. Mankind seems to have, this need to overcome challenges built into them from the start. We are always pushing borders and limits. I believe the answer to the question raised above, "If we lose the desire to do better, will we still be complete?", is no. We need the desire to do it, to overcome the challenge.
For the money
There are a few who seek primes just for the money. There are prizes for the first prover ten-million digit prime, the first hundred-million digit prime, and the first billion digit prime.