English Script Request

So P1 is 2, 2 - 2 is 3 and so forth, so we can look at prime gaps which is just the difference between one prime and the next. Ok. Given one prime Pn how long do you have to advance before you see the next prime? OK, so the first prime gap is 1. After that they are all even numbers because all primes are odd, after 2. So, the prime num, prime gap starts at 1 and then you have 2 , 2 , er 4 and so forth, urm and so you have the sequence of even numbers and you can , er, you can ask two basic questions about this prime gap, so, one is how small can this prime gap be in terms of ‘n for large’ and how large? K er almost the most basic questions you could ask. Ok, um, in both of these questions we have a conjecture which looks impossible to solve by our current methods er but we also have progress and in both cases I can –[sounds like brur - see?] progress in the last two years and there is a nice symmetry, er and ultimately as it turns out we actually use the same method to attack both questions which was not, which was not obvious actually er we start with this. Ok, so er start with first question. Ok so the er, the basic conjecture here is one that most of you probably know, is the twin prime conjecture. ??[unintelligible! Could it be ‘Infinity means prime twins and the distance is two a pod????’] so another way of saying that is that the prime gap is equal to 2 infinity of 10. Ok, as the primes, as you go up into the primes, we know the primes do get sparser and sparser so on average the prime gap actually increases, er, roughly like the logarithm of the primes actually er but we still expect every so often just by pure chance basically the prime gap should the prime gap should every now and then return back to be as small as it can be, which is 2, remember it has to be even. Ok, er, so this is still open, this was first explicitly posed by de Polignac in 1860 something, er, and its still open erm and for a long time it was basically just seen as completely out of reach, urm, but we had this amazing breakthrough by Yitang Zhang last year. And the precise date is actually May 14 2013, er, who for the first time managed to get a bound which looks like this, er , what he showed was that, er, the prime gap is bounded infinity of ten and the bound, he even got an explicit bound, pretty large but explicit .. ok, er, that there, that there are infinitely many pairs of adjacent primes whose distance between each other is not 2, itcould be 2, but certainly bounded by some large and fixed constant, er so this is the first bound of this type . So previously the best bound like this , erm just for comparison…...

Just for comparison. One could get a bound which is basically a square root of a logarithm infinitely often.

This is (name) about 10 years ago. This was the first actual bound. Um, so this was an amazing result. It's now published in the Annals of Mathematics. It's amazing for several reasons probably because of Yitang's life story. He was an assistant professor in the US, New Hampshire, who actually hadn't published for a while. He even left academia for a while, but still was working on this problem. And he actually solved it and wrote a completely correct proof. So there's nothing too special about this number seventy million. It's what came out of Yitang's argument. And he wasn't completely careful. His goal was not to get the sharpest (?) possible number. Just to get sort of a nice round number that was large enough for everything that he wanted to do. So very quickly after his preprint was made available people started noticing, oh if I optimize this line I can change seventy million down to sixty-three million. Somebody else said I can shave it down to fifty-eight million by this extra trick. And so just spontaneously people started going through the papers and finding, very briefly holding the world record for the boundary gaps, the boundary gaps between primes.
And so eventually myself and several other people organized this into an online project we called Polymath Project. It's a play on words. Polymath means someone who is good at many different things. But here we use it to mean many mathematicians, poly/mathematicians working together. Each sort of focussing on one aspect of Yitang's argument and doing some numerics. Other people were adding some theoretical advances. So we worked online for several months.

And we managed to improve this bound (by July 27th) -- managed to get the bound down to about four thousand. Uh, and then at which point we began to get stuck. And we started writing up the results, and we were all very happy with that. Actually, before we had finished writing up, uh, actually, turns out what was almost a hundred sixty pages worth of stuff, of calculations that we actually had to write up. It's since been trimmed down a bit. But, before we finished, we actually found, we were surprised, that independently of our work, but probably inspired by what Yitang did, a postdoc at Montreal, his name is James Maynard--and the time here was November 19th--found a shortcut that cut out a lot of the difficulty in Zhang's argument. Zhang used quite a bit of firepower. Just give you one example, there's a very famous result of Deligne, proving the Riemann Hypothesis over finite fields.

And it won him the Fields medal and the Abel prize and so forth. Uh, and Zhang used that to, uh, to get this. And we also did, as well. But Maynard found a shortcut that cut around a lot of this, and uh, he managed to find a shorter argument, which, of the same, okay, but he found a much better bound. Of, uh, 600, actually. Ok, so after that came out... my group got very excited and we started talking to Maynard and so he decided to join forces with us. And so we worked a little more to combine our two methods and so we worked on that for quite a while and we eventually got, uh, so the final thing that we uh, that we declared victory at was like in April. Ok, so our bound is now 246. And it's possible, with lots of computer power, to maybe shave down 4 or 6 out of this.

But uh, but it got really complicated and we thought this was a good place to stop. Now this was a, I mean- For the purposes of an online project it was really good to have this number as a sort of a score, right? Because you could- it was a very tangible measure of progress which is something that you don't often get in a mathematics project. I should say though that this there's nothing special about these numbers. You know I mean it's- What's more interesting ultimately is the techniques that go to proving these things and the various advances in technology that you need to get from one to the next. But still, it's nice to have this very clean progression here. If- but unfortunately you know I mean this looks like it's approaching two but slowly. But we- Unfortunately we do know that these methods cannot attain the twin prime conjecture. So the most- What we can prove now is that if you assume- so all these results are proven using the methods of sieve theory which I'll talk about later. And there's a certain conjecture in sieve theory which is kind of like the Riemann hypothesis. There's something called the Elliot-Halberstam conjecture. I'll explain what it is a little bit later- well roughly it's sort of like a generalized Riemann hypothesis for sieve theorists and if it is true we can improve this bound all the way to six. That we can get infinitely many pairs of primes six apart. So pairs of primes that are two apart are called twin primes, pairs of primes that are four apart are called cousin primes, and pairs of primes that are six apart are called sexy primes. This would mean that there's either infinite twin primes, infinite cousin primes, or infinite sexy primes. This is a cute way of phrasing it. But unfortunately we know also that this is the limit of the method. That there is no purely sieve theoretic method that can lower six to anything smaller. So if you want to actually get all the way to two you have to use something different from what anything that we do here. So this is- These are some of the results we have. We have other results as well. So one way to think about these results—for example this result here—is saying that you can find infinitely many intervals of length six hundred which contain two primes. It turns out that some of the later methods starting with Maynard and going forward can also find intervals which contain many primes. Um so- not just- so which means- yeah so rather than talk about prime gaps you can also talk about you know: P N plus two minus P N and there's something bound for this and there's something bound for P plus 3 minus P N. You can find intervals of bounded length that contain three primes, four primes, and so forth.