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>> DR GRIME: In 2002 mathematicians published a new method to test for primes and finally they found a fast way that was 100%. This is now one of the latest things. So this is called that AKS test. And it's pretty much a very similar idea to what we did with the Fermat Little Theorem. It's very similar to that. And the test is, that if you've got a number p and we are going to do this. We're going to take (x - 1) to the power p and then we're going to subtract (x^p - 1). Alright. So it's a polynomial, I'll show you an example in a second. If all coefficients are divisible by p then p is prime. And if they're not it's composite. it's 100% this test. I'll show you an example. let's take (x - 1)³. Yeah, let's do that. (x - 1)³. >> BRADY: So 3 is the number we're testing? >> DR GRIME: Yeah, 3 is the number we're testing. We know it's a prime don't we? But we're going to test it. We're going to take (x - 1)³ and so that's, to make it explicit, (x - 1) by (x - 1) by (x - 1). OK. Multiply this out, you get a polynomial called x³ - 3x² + 3x - 1. That's what you get. Now you're subtracting the x³, I'm told- I'm told to that and you're, what's that, adding on the 1. So these, these dissapear. This says these numbers are divisible by 3. If it is divisible by 3, 3 is the number we're testing, then it is a prime. If it's not divisible by 3, it is not a prime. And that's our 100% test. So this was really important, this was huge news when they published this. 2002 they published it and it was huge news because it was a fast test as well. Or it could be made in to a fast test. There's a little thing they do just to make it a bit faster because it does get quite difficult, these polynomials get quite nasty. Just to make it a little bit faster what they did is (x - a)^p minus (x^p - a), so 'a' is a number like we had before. And they said this was equal to p lots of something plus, there you go, a little (x^r - 1) multiplied by something. So that's not quite the same, that's a little bit different, this was actually faster to work with, it makes a faster calculation. And for this version you need to test a few candidates for 'a' here. You test a few. And if it passes that test then you can determine that p, which is what we're trying to work out. Try and work that out, you can find it's prime or not. Mathematics is done all day, you know, every day. New mathematics is coming out all the time. And only, well I...