Mathematically, prime numbers are important because every natural number can be decomposed uniquely into a product of prime numbers:
Where are prime numbers. For example,
So primes are the “atoms” of the natural numbers.
They are real-life useful as well. The encryption used on webpages when you enter your credit card uses very, very large prime numbers. What it uses is the fact that it’s easy to multiply large primes together, but it’s hard to take a very large number and break it down into its prime factors.
Recently the Great Internet Mersenne Prime Seach (GIMPS) announced that computers have verified that
are both prime numbers. They are the first primes large enough to qualify for the $100,000 prize from the Electronic Frontier Foundation. The prize is for the first prime with more than 10,000,000 digits and these primes have 11,185,272 and 12,978,189 digits, respectively. If you would like to see the largest prime known to humankind, click here. Warning! That link takes a long time to load, so only do it if you have a fast internet connection and lots of time on your hands!
More on Mersenne Primes below:
A Mersenne Prime is a prime number of the form for some natural number n. They were consider way, way back by Euclid, but were first seriously considered by Marin Mersenne in the 17th century.
He compiled a list of all the which give prime numbers. No one is quite sure how he did such a massive calculation ( has 78 digits!). After 200 years mathematicians were able to verify his calculations and see that he was (mostly) correct!
Not all numbers of the form are prime. In fact, it is easy to see that if n can be factored, then so can . You just use the factorization:
However, just because n is a prime number doesn’t mean that is prime. For example, is not a prime because it can be divided by numbers other than 1 and itself (Which ones? That’s homework!).
Lots of things are still unknown about Mersenne primes. For example, nobody knows if there is finitely many or infinitely many! So far (counting the ones above) 45 Mersenne primes have been found so far.
If you’re interested in finding the next Mersenne prime, GIMPS is always looking for people to volunteer their computers to work on this problem in their free time. If you’re lucky enough to find the next one, you’ll be famous and wealthy (the EFF is offering $150,000 to whomever discovers a prime number with more than 100,000,000 digits and $250,000 for the first prime with more than 1,000,000,000 digits).
If you would like to buy a poster of showing one (only one can be squeezed onto a poster!) Mersenne prime, you can get them here. They are supposed to offer a poster with the newest primes in the near future.
Addendum: Today Terence Tao, Fields Medalist and UCLA faculty member, has a post on his blog where he talks about the math used by the GIMPS project to test if a Mersenne number is actually prime. You can check it out here. Note: UCLA computers were used to find one of the new primes.