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.