Prime numbers cause me pain. It's a tingly, warm pain just on the inside of my skull, signaling that said skull's contents are within thinking proximity of ugly math--math, in other words, that resists patterns or analytical tools or even a good graph. It is inelegant. Math involving a high proportion of guessing and checking gives me this soft pain, as does the limbo between not knowing if a solution to something exists (because it might not) or me just being an idiot.
Prime numbers, numbers that don't have any components besides themselves and "1," are ugly math of the most epic proportions. In the case of the world's new largest prime number, announced this week by the GIMPS distributed computing project, it is epic to the tune of 17,425,170 digits. This is up about five million digits from the previous discovery, in 2008 by computers at UCLA. And the new prime is hardly the largest prime: they just keep going up, forever.
But, at the same time, primes also resist infinity, unlike more pretty number series that can be described forever by some formula. For example, the series beginning with "1 + 2 + 4 + 8 + ..." can be described forever without actually stating all of its numerical components, so long as we have a place to start it. However, primes don't have a neat formula that describes all of them; their overall distribution seems to be by chance. The distribution of primes can be modeled imperfectly, and there are some good algorithms that can test for the primality of a number (testing the new prime took several week of computing), but, yeah, super ugly.
The search for big prime numbers focuses on a subset of numbers known as Mersenne primes, named after the 17th century theologian who made a bunch of wrong predictions about them. The Mersenne primes take a certain form, 2^P-1, where P is an already known prime number. The GIMPS project (the Great Internet Mersenne Prime Search) takes Mersenne prime prospects of that form, and kicks them out to different computers belonging to volunteers taking part in the project (so, kind of like SETI). They're tested for primality and, should something come up "yea," the number is checked and double-checked and well beyond. Getting an answer from a home PC can take months for a single number, and it sure doesn't involve WolframAlpha.
Interest in the largest primes is kind of a "because it's there" deal. In an NPR interview a few years back, mathematician Chris Caldwell put it like this: "Nobody there looking at the Hope Diamond ever asks, 'Why did they bother to dig it up?' or 'What is it good for?' — even though it really isn't good for much other than to just hang there and people to look at. And in many ways the Mersennes play that same role — that they really are the jewels of number theory." If they could be described to infinity, were elegant, the primes wouldn't be nearly as valuable.
It's worth noting that primes, and their randomness, do play a key role in modern computing. In fact, it is (smaller but still big enough) primes that allow for the encryption that we rely on to protect credit card numbers online and other top secret stuff. Basically, it's incredibly difficult to find the prime number factors (e.g. components of) large numbers. It's so difficult, computationally, that it might as well be impossible. If you were able to find those factors, you'd be able to decrypt whatever it is being kept secret, at least if you knew some number theory.
This is one of the many reason quantum computers are so interesting. One of these bad boys, once created, should be able to chew through a prime factorization quick enough to make the entire method of encryption moot. Which is pretty ugly.
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