Prime numbers are those positive integers whose only divisors are 1 and itself. (The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, ….) Recently, a computer discovered a new prime number. Illinois State’s Sunil Chebolu of the Department of Mathematics explains the race to find large prime numbers.
Chebolu:
Prime numbers are the fundamental building blocks of our number system and are also among the most mysterious and fascinating objects.
Although we know a lot about the structure, properties, and distribution of prime numbers, there is still a lot that is not known about them. Though there are many conjectures about prime numbers we can explain to grade school kids, resolving them has eluded mathematicians.
Euclid, around 300 BCE, showed that there are infinitely many primes, but we still don’t have a formula or an algorithm to generate them efficiently. Since mathematicians like challenges, they have engaged themselves in the race (possibly never ending) to find larger and larger prime numbers.
It is worth noting that the quest for large primes is also inspired by the important role large primes play in cryptography–the science of encryption which gives security to online banking, emails, and other communication channels. One can use prime numbers to encrypt sensitive information that is being transmitted through communication channels in such a manner that only the intended recipient can decipher it; in particular an eavesdropper will not be able to make any sense of the information that he or she intercepts.
Instead of searching for new primes within the list of all integers (a very difficult task), mathematicians narrow their search and try to find primes (using clever mix of theory and computers) of a particular type.
One such type is the Mersenne family—numbers that are one less than a power of 2. These primes are named after a French mathematician Marin Mersenne who studied them in the early 17th century.
Mersenne primes, like other primes, are rather sparse. As of last year only 48 Mersenne primes were known. Last month, Curtis Cooper at the University of Central Missouri discovered a new largest known prime number, which is again a Mersenne prime.
This prime is more than 22 million digits long–almost five million digits longer than the previous record. This was a result of the GIMPS (Great Internet Mersenne Prime Search)–a collaborative project in which volunteers use freely available software to search Mersenne primes. This project discovered 15 largest Mersenne primes in the 20 years it has been running. Although we don’t have a proof, it is believed that there are infinitely many Mersenne primes. So the quest for finding new primes has a long way to go.
