
New record prime (GIMPS): 2^{82,589,933}1 with 24,862,048 digits by P. Laroche, G. Woltman, A. Blosser, et al. (7 Dec 2018). 

"Why?" we are often asked, "why would anyone want to find a prime that big?"" I often now answer with "did you ever collect anything?"" or "did you ever try to win a competition?"" Much of the answer for why we collect large primes is the same as why we might collect other rare items. Below I will present a more complete answer divided into several parts.
Perhaps these arguments will not convince you. If not, just recall that the eye may not see what the ear hears, but that does not reduce the value of sound. There are always melodies beyond our grasp. (*) 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 (numbers like 6 and 28 which 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 2^{p}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). How can we resist joining such an illustrious group? 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. (Look, for example, at the concepts required to develop simple proofs such as [1] or [2].) In short, the tradition of seeking large primes (especially the Mersennes) has been long and fruitful It is a tradition well worth continuing. 2. For the byproducts of the questBeing 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 byproducts of the race. Byproducts such as the new technologies and materials that were developed for the race that are now common everyday items, and the improvements to education's infrastructure that led many man and women into productive lives as scientists and engineers.The same is true for the quest for record primes. In the tradition section above I 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 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]. 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. And these are just a few of the byproducts of the search. 3. People collect rare and beautiful itemsMersenne 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. Less than fifty in all of human historythat is rare!But they are also beautiful. Mathematics, like all fields of study, has a definite notion of beauty. What qualities are perceived as beautiful in mathematics? We 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, 2^{n}1. The proof of their primality has an elegant simplicity. Mersennes are beautiful and have some surprising applications. 4. 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! 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? 5. 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. Slowinski, who has help find more Mersennes than any other, 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. 6. To learn more about their distributionThough 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. Simple calculations have found patterns, such as the prime number races, which have led to significant amounts of research. 7. For the money?There are a few who seek primes just for the money. There are prizes for the first prover tenmillion digit prime ($100000), the first hundredmillion digit prime ($150000), and the first billion digit prime ($250000). 
Another prime page by Chris K. Caldwell <caldwell@utm.edu> 