so this is

Asking about consecutive prime numbers p 1 as the first prime which be equal to [2] and there'll be the second time

P 2 which would be equal to 3

And you can look at the gaps between the primes, so the gap between three and two...

is one, this is the only time you can possibly have two prime numbers which differ by exactly one, so the next smallest possible gap

Would be a gap of size, [two]

It's only not the case that all gaps become size 2

But the question is how many of these gaps of size two? So for example,

101 and 103 are two rather large prime numbers which differ by exactly 2. And then the twin prime conjecture

says that this having two primes which differ by exactly 2 happens infinitely often

If the twin prime conjecture is false,

At some point you go along the number line, and then all prime gaps would always be bigger than two

But we don't believe that's the case. We believe that it should be the case the twin prime conjecture is correct

So no matter how far you go along

There's always going to be some bigger primes and in fact infinitely many bigger primes that differ by

Just exactly two so you can test this on a computer

And we found pairs of primes that differ by exactly [2] with over

300,000 decimal digits and in fact they're surprisingly common that it's certainly not all the time

but if you look at the first

how far you have to go after a million to find a pair of twin primes

It's not very far. If you look after a billion, It's not very far if you look after a thousand billion

It's actually not very far until you start finding twin pines

Most primes aren't going to have the one after it been just two apart

But we still believe that you should be relatively plentiful and so far if you check on a computer this seems to be true

but whether it's really true all the way as far as you go is something that you can't check on the computer and

That's where you need to try and prove the real mathematical result

So we don't actually know how old the twin prime conjecture is

that some people have claimed that it should go all the way back to euclid and it's certainly the kind of question that the

Ancient Greeks could easily have thought about the most recent time where we've seen someone really write down the conjecture

officially was just over 100 years ago by de Polignac

But he actually conducted a rather stronger more general form of the twin prime conjecture that's a bit more complicated

It's not just gaps of size 2 but also gaps of size 4 and gaps of size six

And so there's a suspicion that the twin prime conjecture itself is rather older than that

But we know it's at least a hundred years old possibly thousands of years old. If we're still finding these things

Way way down the number line, and we know they're an infinite number of primes

This is a known thing... if they did stop

What would stop them occurring like what is... what beast out there could stop twin primes occurring?

It doesn't seem as any roadblocks of them happening forever.

This is one of the interesting and frustrating things about prime numbers,

that often it's clear what the right answer should be if you give me a simple problem

Like the twin prime conjecture, you can think about it heuristically a bit, you can check it on the computer

And you can quite quickly convince yourself that this should clearly be the right answer

And you can do it in a fairly confident way so lots of problems about prime numbers

we believe we know what the correct answer should be it's just that we don't know how to prove it and

The game is always trying to rule out there being some very bizarre conspiracy amongst prime numbers

That would mean that they would behave in a rather different way to how we believe that they should behave

So you're completely right that if the twin prime conjecture was false there'll be some

strange switch that happens at some strange point far down the number line

There's an absolutely huge breakthrough by Yitang Zhang just relatively recently in 2013

He showed that although he couldn't prove the twin prime conjecture (that two is a gap infinitely often)

And he similarly couldn't conclude that four is a gap infinitely often, he showed that either two or four or six or eight or some number

Less than 70 million has to be a gap infinitely often. So the gap wasn't 70 million

It was just somewhere between two and 17 million, but he didn't know what it was?

Right, he couldn't say which one of these came as a gap,

but he was able to show that at least one of these occurs as a gap infinitely after and 70 million might sound like a

Huge number to most people it's certainly a lot bigger than two for example

But before that we had no idea of, how we could possibly prove anything like this

That he showed that no matter how far you go down the number line

There's always these primes that get clumped bizarrely close together

Because when you start looking at really really really big prime numbers, prime numbers become quite rare

so if you're looking at numbers with billions and billions of decimal digits for example

Then suddenly the average gap will be well over a billion and if you look at even bigger ones you get even bigger gaps

So typically prime numbers will have these very big gaps

And he showed that no matter how far you go down the number line you will get these pairs that come really close together

At least compared to how they typically look I guess.

Earlier on around 2005,

there was this earlier work of Goldston Pintz and Yildirim, who came up with a new method for thinking about the twin prime conjecture

and problems closely related to it. People worked really hard to try and push

Goldston Pintz and Yildirim's ideas a little bit further

but there seemed to be this big obstacle that we kept on not being able to make progress on and

It was this problem which Yitang Zhang managed to break through, came as a complete shock to everyone

I think precisely because people had worked so hard on this

particular approach and all the experts had been completely stumped on this. so 70 million is a

Slightly arbitrary number that just comes out of the method and in fact his method really just showed some number

And he didn't try and optimize the 70 million too much. So if you take exactly his argument

But you're ever so slightly more careful with the numerical calculations

You can bring 70 million down to 30 million

And then if you tweak his arguments and rearrange things ever so slightly it's still fundamentally the same argument as what he had

But you can bring 30 million down to 20 million and so as soon as Yitang Zhang's result came out

there was this huge excitement at the result in trying to understand the result, but they also became a

Small online competition amongst

Mathematicians as to

How low can you make the gap that we knew 70 million was an upper bound, this is the first time we had a

Finite bound, but we also knew that by tweaking the arguments

You should be able to bring 70 million down to some smaller number

How low could you go just using what he had put out there and putting in the extra work?

So there was a polymath project

Which was a big online collaborative project that set up precisely to try and optimize the arguments in Zhang's work,

Now, they came up actually with quite a lot of neat ideas to make Jang's argument as efficient as possible

They were able to bring Zhang's bound of 70 million down to I think

4680.

so I'd actually been thinking about a

Completely separate approach to this bound of gaps problem before Zhang's work came out

so I was super excited that Zhang's work came out 'Cause it's fascinating to me because I was

still a phd student at the time. Excited or disappointed? Like did you feel like "Oh no I've been beaten" or were you really excited?

So I never had any belief that my ideas would lead to bound of gaps between primes in any way,

So I was just excited. I think you always super excited when you

Suddenly one of these big results comes out and you know that collectively we understand something new about prime numbers

So I was just desperate to get into the details of Zhang's work. There's always this

Intense frustrating game that you play when you do mathematics that you sort of feel that you have an idea

but you don't quite know

Exactly how it fits in and it takes a long time of fumbling around in the darkness before you really get intuition for how

What you're thinking about works... It's a huge motivation to me that this is part of the fun

It's the chase of the problem and trying to understand what's going on

So I was still completely addicted to this thing that I've been working on

And I viewed it as a side benefit that it might actually be able to help this

big collaborative project if I could sort my ideas out, but then the huge surprise to me was that actually once I managed to

Grasp a hold of this and look at things in from the right perspective

A variation on the ideas that I was thinking about

Were able to prove bound of gaps in a completely different way to Zhang's work.

This gave a completely new way of proving bound of gaps between times

so both my work and Zhang's work is very much based on the earlier work of Goldston Pintz and Yildirim

and Goldston Pintz and Yildirim used an argument that's fundamentally using a technique known as Sieve methods

They proved a great result about primes which come close together

And they said if we could prove this other technical result that isn't to do with Sieve methods

Then we'd be able to bound of gaps between primes, and so what Zhang managed to do is

Essentially proved something like what they wanted to prove and therefore use their argument to get bound of gaps between primes.

Whereas

my approach was based on their original ideas rethinking some of the way their original argument worked which meant that you didn't need such a

Strong technical input and so I could use weaker results that be known for the past 50 years or so instead

On the one hand I tend to sort of feel my way to a proof and know it's right before I

Actually write down all the details so I did have this feeling that yes, this has the feeling of being right

But at the same time I remember

Being pretty scared when I first came up with the ideas

Because you know I was going to be very much putting myself out there

And it's very important for mathematicians to really be strict with themselves to check that they've got all the details right

but you often have this internal feeling about these things and it felt like it to me.

I was really surprised, so yeah there was suddenly lots and lots of interest

I was a postdoc at the time in Montréal when this came out and suddenly I was getting invitations to fly all over

to give talks in different places in the US, so it was really exciting for me being able to

Meet all these famous mathematicians, give talks of my work suddenly everyone is very excited. I hadn't really expected this at all

I thought you know I was pleased that I'd proven a nice result

But I was more pleased because I'd finally managed to get an understanding of these things and

I had the satisfaction of the proof. I hadn't really expected there to be any

Media interest or all these people asking me to give talks and things.

So some people have called it the Maynard and Tao method

So Terry Tao was also thinking about similar ideas

He was very actively involved in leading this polymath project

And so he came up with essentially the same method at the same time

It was another bizarre coincidence that there were

There was this one proof by Yitang Zhang that came out and then both terry and I came up with

This new alternative method. So some people been calling it the Maynard Tao method. So the current state of the art, trying to optimize

All these techniques including with these new method and these new ideas is

Is that they're infinitely many pairs of primes that differ by no more than two hundred and Forty-six. We expect that every even number

between 2 and 246 occurs, it's no different, in particular we expect two

But we know there's at least one between 2 and 246

Can your method get us to two?

Unfortunately there's a

Fundamental barrier to our method get my sequel to to you so in some later work

we said that if you [a] really optimistic about how we could maybe get all the technical stages of the argument to fit together and

If you able to prove these other very strong results that were sort of strong versions of what yu tang jang managed to prove then

Maybe you can just about push our methods to get gaps of size at most six infinitely often

But we also showed that there's not going to be any small variation on

These sorts of ideas that can really get down to two so you can get close to you

But not quite all the way

there's a new idea that's needed to get all the way to the twin type of injection for sure the reason for it is that

We have these mathematical devices that are trying to detect time numbers

But they don't work perfectly there are only approximately detected prime numbers

So maybe they get it like one in ten of the time or one in

six of the time or something like that, and if you're getting like one in ten at the time

Then maybe if you're looking for primes that come close together you miss a whole load of being between primes

But you still find to that a pretty close together

So you may be missed out 10 of them

But on the tenth one you find one and there may be 10 times further apart in you love hopes

But they're still pretty close to get there no variation on these techniques even if you're really optimistic

Can ever improve your way of finding primes to one into?

Without using some extra automatic information and really a big new idea the fact that your proof has this

Trait to it that it involves approximation of things that aren't always right

Doesn't sound like what I mentioned a method proof