I read today that a student in Michigan came up with the largest known prime number. Here’s the article.
Why is this important? I seem to remember from math classes I’ve taken discussion about the largest prime, but I’ve never understood why it would be significant. The article makes mention of number theory, but I haven’t the slightest clue what that is.
I’ve always liked math a lot, but it’s never liked me. Please help.
One of the main uses for large prime numbers is in encrytion systems. These systems work because it is easy to multiply two large primes to obtain a large number, but much harder to factor that number back into its prime factors. Beyond that, it’s nothing more than a “look what I did!” sort of thing.
Finding the largest known prime number used to be a measure of how good a particular computer was (the record holders a few years ago were usually found on Cray supercomputers). Most of the searches were based on Merssene numbers, using the Lucas-? algorithm, so there is no new math here.
Just to clarify, it’s not the largest prime number possible, it’s just the largest known so far. You said that in the OP, but not in the thread title. Just wanted to point that out. There are, of course, an infinite number of primes, so we’ll never find “the largest”.
Large primes also provide good tests for the best known primality algorithms, which aren’t always right.
Does anyone else find it amusing that the acronym for the project is G.I.M.P.S? 
I am a mathematician and I do not know a single thing that will follow or be clarified by this. It is interesting as a feat of distributed computing and that is all. It has nothing to do with cryptography (100–or at most 1000–digit primes will do fine for that) until quantum computers are built and then they will all fall and only quantum crytpgraphy, which is unbreakable in principle, will remain. It does not test primality algorithms, which are mathematically proven.
And I just heard a talk on mathematical cryptography that confirmed my suspicion that the RSA cryotosystems in common use don’t even bother with true primality tests but content themselves with numbers that are strong pseudoprimes to a hundred or so different bases. The odds on such a number not being prime are too remote to even talk about and the tests are extremely simple.
its also something useful for showing off your quantum computer if you can build one 
I about choked when I heard this announced on the radio late last night. The announcer (it was a national feed - CBS or ABC radio) ended his story by explaining that a prime number is “…a number which can be divided only by itself and zero.”
D’OH!
Here is what the master says about modern desktops vs. Crays. Kids, now you can crunch your own prime numbers at home!!
Mainly cos we can really. The same reason we search for digits of pi.
[Hillary]
Because it’s there.
[/Hillary]
Mathworld links on Mersenne Primes* and The Lucas-Lehmer Test. The Lucas-Lehmer Test is actually very nicely done on binary computers. (It’s also a lot simpler than the article implies.)
Note that there is the destination and there is the journey along the way there. Many people reporting Science don’t understand that the goal isn’t getting there, it’s finding interesting stuff along the way. The Lucas-Lehmer Test involves all sorts of neat properties of quadratic fields which are no doubt going to be really useful someday. We just don’t yet for what or when.
Having an interesting target to aim at means that there are people trying to figure out ways of speeding up Lucas-Lehmer, finding alternate methods, etc. Sometime in the next few decades there will be a big advance and finding billion* digit primes will be easy. What we will do with billion digit primes? Nothing. But the tools developed to generate that breakthrough will be very useful for other things.
Getting your name in the papers is a carrot on a stick used to advance Mathematics.
(*Note the postage meter stamp mear the bottom. I have a postage meter stamp from the '70s announcing a new largest known prime, which wasn’t a Mersenne, from IBM. So announcing record primes on stamps isn’t new.)
Thanks for all of the responses, and thanks for the links Adam and ftg. I think I get it now.