Thank you very much. I thought we could begin with some counting.
Of course, maths isn't only about counting, but I'm interested in number theory properties of whole numbers.
And I particularly like this way of visualising the numbers as it starts to give us an insight into Prime Factorisation.
And I want to talk all about prime numbers today, so maybe you can start to notice some patterns here.
And I want to talk about patterns. This isn't such a convenient way of of looking for some of the patterns that I want to do, though.
So I want to arrange the numbers on a slide like this.
So the idea here is we can start seeing patterns.
And as mathematicians, our job is to look for patterns, to look for structures, and then to understand what's going on behind the scenes.
So there's lots of patterns here that we might notice, and I'm not going to be able to talk about all of them today.
So an example might be if you look carefully.
So these columns, the four column, the six column,
the eight column and the ten column don't seem to have any prime numbers in from the basis of what we can see on the slide.
But I'm not going to be satisfied that I'm interested in what would happen if we kept going.
If we continued this slide further, what would happen?
And if you look carefully, these four, six, eight and ten columns, all the numbers in those columns are even, I think is a prime number.
It's only is a one in itself. So if you've got an even number bigger than two, it's divisible by one by itself, but also by two.
So it can't possibly be prime. So if two is the only even prime.
So in its column there, we're not going to find any more primes in that column, no matter how far we go.
And these four, six, eight and ten columns really are empty.
So that's not the most complicated example, but it gives you a flavour of noticing a pattern and then trying to understand what's really going on.
Why are we seeing that? And the kinds of patterns I want to focus on today.
It's to do with gaps between prime numbers.
So, in fact, that observation about even primes shows us that if we look at two and three, two or three consecutive numbers and that both prime,
but that the only example of two consecutive numbers that are both prime because if we pick any two consecutive numbers,
one of them is going to be even and we just said two is the only even prime number one, by the way, is not prime,
because it turns out that that leads to a better definition at least of better mathematics that way.
So we define one not to be prime.
So if the only pair of consecutive numbers that are prime primes two and three and up near the top, these primes of bunched up quite close together.
If we look further down, there's some quite big gap. So 83 to 80 9 to 1997, these primes are getting further apart and that makes intuitive sense.
If I give you a really big number, a number this big,
it's very hard for it to be prime because it's going to be divisible potentially by lots of smaller things.
There are lots of smaller numbers that might divide it that would stop it being prime.
It's very easy for seven to be prime because there aren't many smaller numbers that would stop it.
But this big, there are lots of potential factors, so it sort of makes sense that these primes are getting more spread out.
One question we might ask is whether we keep finding more and more primes.
If we continue down, if I could fit more and more on this slide,
would I keep finding new primes or at some point where we hit the biggest prime number?
Those are two very different scenarios. Right. And one world. There's a biggest prime number from that point on.
Nothing is prime. In the other scenario, no matter how far we go, we keep finding new prime numbers.
And it's sort of intriguing which of those situations it might be.
So I guess that intuition about if you're a very large number,
it's very hard for you to be prime is not quite enough to guess which way it goes because maybe they're getting further and further apart,
but they still keep happening. Or maybe they're getting so far apart and eventually they kind of stop.
But it's sort of I can't quite make that intuition help me figure out which it should be.
I could use a computer to help, but I see that's not going to get me very far.
I can ask my computer, please tell me the largest problem you can think of and it goes work and clunks
for a fortnight and comes back and says he got his very large prime number.
And that tells me nothing at all because maybe it's found the largest prime number in the world.
Or maybe if it thought for another month it would think of another prime number.
So computer is not going to help there, but happily, mathematics will help.
And this goes right back to the ancient Greek. So this is a theorem, this is a mathematical statement that we can prove to be true.
And the theorem is that there are infinitely many primes. There is no biggest prime number.
And for me, the certainty here is what's really important.
So when the radio comes on in the morning with the headlines, I don't have half a percent of my brain thinking, Oh,
I wonder whether today they'll find the largest prime number because I am absolutely, totally certain there is no largest prime number.
And the reason that I'm totally certain is because we have a proof. So for me, proof is really important part of what we do as mathematicians.
And I want to share this proof with you because proof is really important
because this is an especially beautiful proof and it's a very well known proof.
So lots of you will be familiar with this proof already, but I hope that like me, you will enjoy thinking about this again.
And if you haven't seen this before, it takes a little bit of getting your head around first time, but it's worth thinking about.
So this goes back to Euclid's sequence of books that he wrote.
The Elements is kind of textbooks that he wrote more than 2000 years ago, and he had this beautiful idea for proving this.
So he said, Let's do a thought experiment. So secretly we think there are infinitely many primes.
There is no biggest prime number. Let's imagine we're in a parallel universe where there is a biggest problem.
There are only finite, any many primes, and let's explore the consequences of that.
And it turns out you reach an impossibility. So if we're in this parallel universe where there are only finite,
how many points I can take very large pieces of paper, and on them I can write all the problems in the world.
So my list starts two, three, five, seven, 11, whatever. And I keep going, keep going till I get to the largest problem in the world.
And Euclid's beautiful idea was to take all the primes in the world.
Multiply them together. I'd add one.
So this is some enormous number, right? It doesn't matter what it is.
What matters is how we've built it from these primes. So this enormous number must have a prime factor.
Either it's prime itself or it's divisible by smaller prime every number.
It has a prime factor. Once you go beyond one. So conveniently, I've got this list on this piece of paper.
Of all the primes in the world, and I can just check which of them are factors of this large number.
So just to divide this number. Uh, no, because it's two times some stuff plus one.
So this. This we made one where we divide by two. Does three the next time on my list.
Buy this number? No, because it's three times and stuff plus one.
At least we made one. When we divide by three, does five divide by the same story.
What it's done is he's built this number so that it's not divisible by any of the primes on the list.
And that was supposed to feel slightly ill because somehow we've got this number that must have
a prime factor and yet isn't divisible by any of the primes in the world that were on our list.
So we call that a contradiction in that is this kind of impossibility,
this absurdity that arose from supposing that there were only finite, any many primes.
And that tells us that there must be infinitely many primes.
So this little square box symbol is a symbol that mathematicians often use to the end of a proof.
So, so more if you didn't take in all the details of that.
But I think it's extraordinary when you think about it, we're proving something really difficult, right?
We're proving that there are infinitely many prime numbers. And this argument is so elegant, so beautiful, that I can fit it on a whole slide.
So I'm very glad they're infinitely many prime numbers.
The world is a much more interesting place with infinitely many primes. There is no largest prime number.
So if we go back, we can then kind of keep thinking about patterns here.
I feel like I need to confess these. I've sort of slightly misled you.
You know what I was saying about the problems getting further apart towards the bottom?
Because if you're a big number, it's really hard to be prime. So the gaps get bigger.
Well, here's the thing about the primes. Every time you think you understand what's going on, it's a little bit more subtle than that,
because if I put another row on, I have to confess that 100, 100, 307 hundred and they're all prime.
So, yes, they were getting farther apart. But then all of a sudden, here's a cluster bunched up all together.
So what's going on? It is true that on average the primes are getting more spread out.
That's a consequence of a theorem that was proved right at the end of the 19th century called the Prime Number Theorem,
which gives us an insight into the distribution of the Prime. So on average, the primes are getting more spread out,
but on average is really important because then you get these little pockets of problems
that are very close and at least primes that are very close that I want to focus on today.
So Primes, there's these problems that differ by two, like three and five, like 29 or 30, 107, 109 pairs of primes that differ by two.
So we know there are infinitely many primes. So the question is, are there many pairs of primes that differ by two?
And this is called the twin primes conjecture. So the conjecture, the prediction is that there are infinitely many pairs of primes that differ by two,
and there are some sort of plausible reasons for thinking that this should be true.
So one plausible but not super good reason is that I can ask my computer to find examples of large pairs of twin primes.
Prime if it by two. My computer would go and think about it and they will come up with very large examples.
We know that there are very large pairs of primes that differ by two.
That's sort of not such a great reason, though, because, as I said, with there being infinitely many problems,
there's always this risk that we find the largest part of two in primes.
So a better reason is that we can kind of model the behaviour of prime numbers.
So if I pick a number like 61, either 61 is prime or it's not prime,
we we don't get to kind of toss a coin, but we can model the prime numbers by tossing a coin.
So the idea is you have a biased coin and the probability of getting a prime
is skewed a bit because the primes on average get more spread out later on.
We know that it should be harder for a large number to be prime.
So if you do this modelling very carefully, you can make a prediction for the number of pairs of twin primes,
up to a million or billion or whatever your favourite number is.
So we can make this kind of predicted estimate for the number of pairs of twin primes,
and then we can check that against the data that our computer has given.
That's where it's helpful to know that the computer has calculated these these large examples of twin primes.
And the prediction matches the data we have computationally beautifully.
But it's not a proof of anything because it's only this model. So we can't prove that this is exactly how the primes work.
We don't yet know how to do that, but it's sort of very compelling evidence because in particular,
it predicts that there should be infinitely many pairs of twin primes. So I thought it might be fun to have a go at trying to prove the twin primes
conjecture because we should be ambitious if we take the idea of Euclid's proof.
So Euclid said, Take all these primes of the piece of paper. Multiply them together.
Add one.
Nothing at all would change in Euclid's proof if instead we took all the points of a piece of paper, multiplied them together, and subtracted one.
So we could do this. What if I take two, three and five plus few primes multiplied together, subtract one, add one.
I get 29 and 31. That's a pair of twin primes. That's kind of nice that I could keep doing that, right?
So I could take two times, three times five times seven and add one and subtract one.
I get 209 211. The bad news is the 209 isn't prime.
So Euclid's argument doesn't say the number we come up with must be prime.
It says it's not divisible by any of the primes on the list we had.
So as has happened here with 209, it is divisible by smaller primes.
They were just too big to have been on the list. So.
So unfortunately, we can't prove the twin primes conjecture by adding and subtracting one from this argument,
which is a shame because that would be nice.
But that's how much guys, sometimes mathematicians spend a lot of time trying to prove theorems without necessarily succeeding.
So one of my mathematical heroes, a mathematician called Julia Robinson, who was a mathematician in the US in the 20th century and an administrator,
asked her one occasion to describe her typical week for some form or other.
So here's how Julia Robinson described her typical week Monday.
Try to prove theorem Tuesday.
Try to prove Theorem. Wed tried to prove theorem.
Thursday try to prove their. Friday Third Falls.
So we're sort of sounds very depressing, but it's not quite all like that.
But mathematicians do spend a lot of time trying out some ideas that don't necessarily get them.
So I like this because it gives us more insight into Euclid's argument, but it doesn't prove the twin primes conjecture.
In fact, the reason the twin primes conjecture is still called the Twin Primes conjecture is that nobody has proved it.
So we think it should be true. That's why it's a conjecture, but we don't yet have a proof.
I spend a fair bit of time going into schools and talking to young people about maths,
and I sometimes worry that I've given them the impression that maths is divided into things that we know how to do because
mathematicians did them a long time ago and those mathematicians have long since dead and things that we don't know how to do.
And that's not the case as all mathematicians are making progress on problems all the time.
So what I want to tell you about today is some of the recent progress on this particular problem, the twin primes conjecture,
because it gives us a bit of an insight into how progress is made and at least in some parts of mathematics.
And the excitement happened back in May 2013 when a mathematician, Cody Tanjung, announced this extraordinary breakthrough.
It got mathematicians super excited kind of progress on the twin primes conjecture.
Zang managed to show that there are infinitely many pairs of primes that differ by at most 70 million.
Now when you are aiming for the twin primes conjecture and you what pairs of points that differ by 270 million feels like a miss, right?
I mean, it's a long way off. I agree. But it was the first time anybody had managed to prove any theorem of that type before.
So 17 million is a lot bigger than two, but it's much better than infinity.
It was a finite number that was the important thing. So there are infinitely many pairs of primes where the gap between them is less.
They're equal to 70 million. Zhang was not a famous expert in this area of maths.
He's had hardly any publications since his PhD. He found it very difficult to get a job in a university.
He did, by this point, have a job as a lecturer at the university in the US.
But he'd spent some time working for the Subway restaurant chain, doing various bits of books, but was doing all of that.
He kept up with the research literature, so he really immersed himself in the area and that was what enabled him to do this.
So this wasn't some idea that everybody else had missed.
He'd taken what other people in the area were doing.
And so just the most extraordinary technical insights and perseverance to kind of push through
an argument that sort of everybody else had ruled out because they knew it wouldn't work.
It's just extraordinary. And everybody is super excited. And the experts in the area checked the proof and yep, it was all good.
And then people started to think, well, can we improve on that? In order to tell you a bit more about that,
I need to tell you a little bit more about finding other kinds of patterns in the primes, not just pairs of twin primes.
So I made myself a number line.
Here's my number line and I call it the primes a different colour.
So I hope you can just about see that the primes are in red there and the non primes are in black.
And then I make myself a twin prime detector.
So it's a piece of card and I've just cut out holes. It's a little bit like an old fashioned punch card.
And the idea is that you can slide it along the number line and detect pairs of twin primes.
So one of the three is no good, is not prime two or four not so good.
Three or five twin primes. Great. Four, six, no, five and seven.
So so I can take my piece of cards, slide it to the number line.
And the twin primes conjecture predicts that there are infinitely many positions of the card where both of the visible numbers are prime.
Right.
So the twin primes conjecture says as you slide the cards along, there are infinitely many pairs of positions where both of these numbers are prime.
And when you start looking at it that way, you start thinking, Well, what if I use other kinds of punch card instead?
Here's one I made earlier. So this one's got three holes and I've got a slightly larger gap.
And again, I can slide this along and see what happens. So no.
Two, four, seven. No good. Three, five, eight. No.
Four, six, nine. No. Five, seven, ten, no.
So again, the goal is to size it alongside infinitely many positions where all of the visible numbers are prime.
So far I have found no positions of the card where all of the visible numbers are prime.
And actually, if you look carefully, we can sort of see what the problem is here.
So if you look at the two numbers on the right. One of them's out, and one of them's even always.
So they differ by three. So if I start with order, if I get the other one afterwards.
So the problem is we know two is the uneven prime.
So no matter how I slide this along, I'm always getting one even one odd, I'm not getting primes.
So this was a very bad choice of card,
but I could try a more sensible choice of card where I fix it so that all of the visible numbers can be odd simultaneously.
Like three, five, seven. In fact, here all of the visible numbers are prime.
So this is great. This is a good triple from our point of view, and then I can slide it along.
And there's really no point in focusing on even numbers because we are not interested in even numbers bigger than two.
So five, seven, nine close. We're not quite.
Seven, nine, 11. Okay, nine is a problem here, so let's keep going.
11, 13, 15. No, now 15 is a problem.
Okay, so I found one triple so far where all of the visible numbers are prime.
But I'm sliding it along and I'm not having much joy.
So what's going on is not the problem of having all even because all of the visible numbers are odd.
Actually, the problem turns out to be multiples of three.
So the problem, if you look at the problematic numbers, they were nine and 15.
And it turns out if we keep going, that that keeps being the problem. So why do we always get a multiple of three that's visible?
I like to think about this in terms of bars of chocolate. So every number in the world is a multiple of three or one.
More than a multiple of three or two more than a multiple of three as illustrated by bars of chocolate.
So every number is one of these. And if we look back at those triples.
One of the visible numbers is of each type. So looking at the bottom 15 is a multiple of three, 13 is one more than a multiple of three,
11 is two more than multiple of 35795 is two more than multiple 371 more nine is a multiple of three.
We always seem to get one of each type and if you think about it, you can think about it with a box of chocolate.
That's sort of inevitable. If you start with a multiple of three, that's bad. If you start this a number, there's one more than a multiple of three.
When you add two one, you get a multiple of three and so on.
So, so what we can see by thinking about bars of chocolate or by some other way is that we always have one of each type of number,
and that means that we somehow deemed always to have a multiple of three that we can see.
And the problem is the only prime multiple of three is three. So in fact it was the same kind of thing with the odds.
And even it's just that we talk about ordered even relative to being a multiple of two or one more than multiple of two,
but it's the same kind of thing.
So this card we can see we're definitely not going to have infinitely many positions of the card where all of the visible numbers are prime.
In fact, the only one that works is three, five and seven. You could kind of convince yourselves of that.
And you can imagine if I had a code with more holes that I might find that the problem was the prime 11 So there are 11 types of number,
like 11 bars of chocolate.
If you think about divisible by 11, you could be a multiple of 11 or one more than a multiple of 11 or two more than a multiple of 11.
And so of the Big Ten, more than a multiple of 11,
if your punch card somehow means that the visible numbers between them always include one of each type,
you are always going to have a multiple of 11 and the only prime multiple of 11 is 11.
So that's going to be no good.
So if our game is think of punch cards where there are infinitely many positions of the card for which all of the visible numbers are prime,
we need to somehow rule those ones out. And mathematicians have a definition for this.
So we talk about a punch card being admissible if there's no problematic prime.
So we've seen a couple of examples of punch cards that are not admissible.
So the first one was this 11 1316 here where we always had an order and even the problem was the prime to the nine,
11, 13, the one I'm holding here, the problematic prime was three.
So not admissible punch cards we can rule out immediately from our game because somehow they are not going to work.
We can see straightaway they're not going to work. The admissible punch cards.
We can't rule them out, but we don't know for certain that they do work.
So the first punch card I've got here is just the twin primes detector.
So we think there are infinitely many positions of the card for which both these visible numbers are prime.
We can't yet prove that. I've got another example of a visible card here.
So you can you can check, but there aren't any primes that are going to cause this card problems.
All that means is that we can't rule it out.
We don't yet know that there are infinitely many positions of the card for which all of the visible numbers are prime.
So that was one of those things where I thought more if you didn't take in the details.
What we need to know is that admissible punch cards all good is still a work in progress.
So Zhang's work used this idea of admissible, admissible punch cards.
They're not called punch cards in the mathematical literature they could set so admissible sets.
And what Zhang proved was that if you've got an admissible punch card with enough holes applied that way, that means loads.
So you've got a punch card with loads of holes.
There are infinitely many positions of that punch card where at least two of the visible numbers are prime.
So we'd like to have all of the visible numbers, but that's way too much to hope for.
But at least two of the visible numbers will be prime.
And then, crucially, there is an invisible punch card that has enough holes where the length of the card is 70 million.
So as I take this card of length, 70 million is sliding along the number line.
Zhang's theorem says there are infinitely many positions where at least two of those visible numbers are both prime.
It might be lot more than that. We think it is. There's one that we know at least two.
So it might be these two over here and these two when we've moved the card further down.
So that's why we don't know exactly what the gap is. All we know is that some two of these visible numbers would be prime.
What do you think about the theorem in that way?
Just start having opportunities to start improving on this, because when people started reading Zhang's paper,
it became clear 70 million was not the best answer that his argument would give.
And you can totally see why that is from his point of view, because from Zhang's point of view,
it doesn't matter whether his paper says 70 million or 65 million.
Right. It really doesn't matter. What matters is being the first to get your paper out and writing a paper that is
clear enough and comprehensible enough that other people are going to read it.
So he focussed on writing this very clear paper rather than spending extra time trying to see, well, what's the best possible number.
But then other people will to go, what, what? What can I do? Because if I could use a narrower card, they still had enough holes.
So we get a better balance straightaway.
If we could somehow show that we could get away with having fewer holes in our punch cards, that we could have a narrower punch card and we'd improve.
So people started trying to improve with this. So this really started in public.
So a couple of people started writing papers, but it really took off when Scott Morrison wrote a blog post called I Can't Just Can't Resist.
There are infinitely many pairs of primes at most. 59,470,640 apart.
And he explained how he'd slightly tweaked Zhang's arguments to come up with this number.
Of course, everybody else then wanted to have their moments of having the world record for this problem.
Right. This is one of the most famous also of problems of the whole of mathematics.
If you have an opportunity to hold the world record for 5 minutes, wouldn't you?
So people started commenting on Scott Morrison's blog post and say, Well, what if we do this?
What if we do that? Maybe we should get this other thing. Somebody which paper a while ago. I think that might help.
And this was sort of going to get out of hand because how do you have that kind of mathematical conversation in public?
You sort of have to be a little bit organised about it. Fortunately, there was a way to organise that conversation.
So a few years earlier, a mathematician called Tim Gowers, who has a blog that's very,
very widely read amongst mathematicians and those beyond the academic maths community.
So he wrote a blog post called Is Massively Collaborative Mathematics Possible?
And he had this kind of vision for a collaborative project on a much larger scale than is usual in maths, in maths.
It's kind of unusual to have a paper with five authors on it, even in pure masses sort of.
There's lots of small groups, but not these big groups.
Tim Gowers wanted to know could we have a much bigger group where somehow everybody works on their little bits of the project,
but no one person has to think too hard. So sort of a little bit like an A.A. Everybody's doing their own thing.
The collective effort is great.
No one person is organising it, and he puts out this post and describes some rules for how this would work, how you should behave in the conversation.
If this led to a publication, who would get the credit? All of those kinds of things.
So he thought through that very carefully and he proposed a project for Polymath to work on.
So there's a joke in that name somewhere.
So the idea was Polymath is going to work on this particular research form, which was a genuine research problem.
People in the area had wanted to solve this problem for some time, have thought hard about it.
And I think to everybody's surprise, within a few weeks, Polymath had proved this result.
They found a new way of proving this conjecture. They'd solved the problem, which was kind of unexpected.
And it seems that that collaboration proved very effective for doing that.
So that led on to other Polymath projects trying out different kinds of problem.
And one of the things here is this very new way of working. If this is all happening in public, on blogs, on wikis, anybody can join in.
It's all in public. So what kind of problems lend themselves to that way of working?
So there's a lot of trying this out. Over the years, some projects were more successful than others,
but it meant it was very natural when this came along to say, Well, let's have a Polymath project to work on this.
So Terry Tao, who is another mathematician with a very widely read blog who'd been involved in the earlier Polymath project,
said, Let's have Polymath eight projects on bounded gaps between primes.
And I've included this quote This is his description, his proposal for what the project should do, because I think it's quite interesting.
Yes, the first one is improving those numerical bounds.
We've got 70 million. We've got Scott Morrison's bounce. How far can we go?
But the second aspect to the problems of the project is understanding what sounded did, really kind of getting to grips with the details.
What were his new insights that other people hadn't had before? Could those be simplified?
Could they be streamlined? How did it fit with what other people in the area had done?
Mathematics doesn't happen in isolation. Yes, John worked by himself on this, but he built on what other people had done.
It was going to feed into other things. So what were those relationships?
So this Polymath project started and of course lots of people were quite keen to join in because their opportunity to be part of the whole thing,
the world record on this very famous problem and it's all in public, it's all in blogs, it's all online, you can still find it.
I quote you if you're interested to have a look. So what I've done here is taken some screenshots from the wiki that was set up for the project.
This is like a league table for the best known things, so I know it's hard to see on the screen.
I'm not expecting you to kind of see all the details. The crucial call and this is comments.
The crucial column is the next one across, which is it's called H at the top.
That's the crucial number that we're interested in.
So the second row here is John with H 70 million, infinitely many pairs of primes that differ by at most 70 million.
And then this number tumbled.
So Scott Morrison is quite high up on that list, so fairly early and then this number just tumbles through the summer of 2013.
So we've got 2nd June 2013 already. The number of digits is decreasing significantly.
Even if you call me the number, they're getting much smaller. And it kept going.
Actually, this was quite an interesting moment for the league table.
I don't know whether you can see lots of these numbers have been crossed out because what would happen when people come along and say,
well, I think I could get this number.
Here's a link to the blog or wherever that I've explained the argument, and then other people will come along and check and go, Yep, that seems great.
Or Hang on, I don't understand. This bit seems to be a problem there. There was a whole sequence of results here that turned out to be problematic.
And they were all based on a preprint, like a draft paper by one of the experts in the area, Jonas Prince, who put up this draft paper.
And people said, Oh, that's great using these ideas. Here's how we can improve the bounds.
And it turned out there was a mistake in the draft. So these bans had to be retracted.
I see that the paper went on to be interesting and useful and have consequences, but not quite for these numerical bounds.
And I think one of the really interesting things about this polymath way of working, so we get to see all of this.
Normally when mathematicians work on a problem, they write up their nicely polished solution at the end for publication.
What appears in the journal, what appears in the paper online is the This is how I did it.
This is what works. This is all cleaned up. You don't see all the rest of the What if we try this?
What if we try that? Oops, that thing didn't work out. I got that bit wrong.
All of those other kinds of things. They get wiped off the boards at the end of the project.
They get put in the recycling bin or put in a pile in the office to look at later.
Gauss, the great number theorists of the past,
commented that the architects don't leave the scaffolding behind and this is somehow what mathematicians do.
But I see this quite frustrating because it means if you've got some famous problem that everybody's working well,
maybe there are three obvious things to try to solve this problem,
and everybody who works in this problem tries those three obvious things before they get on to maybe having some new ideas,
having something like this. There's a record of everything that you've tried so other people can see, Oh, I see.
That seems to be a bit of a dead end or that looks promising, but they didn't pursue that.
Maybe I can use that. The good news is that after those patterns were retracted,
there was then a continued improvement and it just dropped and dropped through the summer 2013.
I'm sure that that progress would have happened without Polymath, but Polymath was a very good way of streamlining it and making it more efficient.
Otherwise you'd have had maybe individuals, maybe some groups of collaborators putting out papers saying,
Well, I can get this band, I can get this band, I can get this band, this kind of race.
So doing it collectively was much more efficient way of doing it.
And through the course of doing it, it became clear that the different ways to improve on the band drew on different types of expertise.
So some people were very good at getting their computers to do a particular kind of calculation to find narrow punch cards with lots of holes.
Other people were more interested in the theoretical aspects of How could we decrease the number of holes?
How can we really understand this argument? So in order to go forward with the story, I want to go back a bit to before then.
So back in 2005, when there was the previous big breakthrough on this problem, which came from Goldstone Pinson,
Yildirim In fact the same points I just mentioned and Goldstone Pitts and Yildirim proved that if the Halberstam conjecture is true,
then there are infinitely many pairs of crimes that differ by, at most, 16.
That's like 16. The trouble is, we can't prove the Halberstam conjecture.
So this is a feel slightly disappointing. At the time. This was super exciting because it was progress on a problem that had seemed intractable.
They'd found a way to do something, make some progress, and actually assuming a conjecture,
you don't have to prove it's a really good way of making progress in maths.
Sometimes if you can isolate that the problem is here and you say, Well, if I could, if I'll allow myself to assume that and deduce this from it,
that could be a really helpful way of focusing attention on what's the important thing here.
What are the key questions we need to do? So in number theory, often the the assumption you want to make is called the Riemann hypothesis.
This is another of these famous unsolved problems. So there are lots of papers in mathematics and.
If the Riemann hypothesis is true, then whatever.
In the early part of the 20th century, a group of mathematicians proved if the Riemann hypothesis is true, then something or other is true.
And then a few years later, another group came along and proof. If the Riemann hypothesis is false, that same statement is true.
Isn't that beautiful? We know it's true. We just know. Oh, it's proof.
The go to conjecture in this particular area is called the Elliott Halberstam conjecture,
which is a conjecture about the distribution of prime numbers.
So let me try to give you a little bit of a flavour.
I don't want to use the technicalities, but let me try to give you a flavour of the Halberstam conjecture.
So what I've got here is the prime numbers, but instead of ten columns like I had earlier this time, I've got six columns.
I think this is quite visually striking. Patterns start to leap out.
So the four column and the six column. Appear not to have any primes.
Does that continue if I go below the slide? Yes. Because of the numbers in the fourth column.
In the sixth column. Even when I choose the uneven prime. So if two is very lonely in its column.
Two is the only even prime, no matter how far down we go. Don't get any more there.
Three. He's looking pretty lonely, too. But we know that the only multiple of three that's prime is three.
And if you look carefully at this column, these numbers are all multiples of three.
So these are the multiples of six, the even multiples of three.
These are the odd multiples of three. So what this slide and that little bit of reasoning shows is that apart from two and three,
every prime in the world is one list, less than a multiple of six.
Or one more than a multiple six. Isn't that nice?
It's like, good. I'm going to say it again. Apart from two and three.
Every problem in the world is one. Less than a multiple of six.
Over more than a multiple of six. And once you know that, then you can start asking more questions about the distribution of the primes.
Like we know they're infinitely many primes. We saw Euclid's proof of that earlier on,
other infinitely many that are one less than a multiple six other infinitely many that are one more than a multiple of six.
There must be infinitely many. At least one of those two. But is it just one or is it both?
So it turns out there are infinitely many primes in both of those columns.
So this one turns out you can prove quite nicely by adapting Euclid's argument.
This one takes a little bit more work, and you can ask other questions, like if you were a prime number.
Would you rather be one less than a multiple of six or one more than a multiple of six?
So I guess the more mathematical way to say that is, if we look at the numbers, up to a million or a billion or whatever.
Are there more problems that are one less than multiple six or more primes that are one more than a multiple of six?
And it's not really clear that the primes should favour one of those options over the other.
So we kind of expect the to be pretty evenly distributed.
So I think up to 90 here, they are pretty evenly divided between those two.
So don't expect to be exactly even, but pretty close. So the Halberstam conjecture is a prediction that, yes,
they should be they should be about evenly divided between those two columns
and a whole bunch of other similar predictions for other numbers of columns.
So if I've got seven columns or 101 columns, whatever, I can predict which columns should have infinitely many primes.
In fact, we can prove that that turns out to be a harder theorem to prove pretty interesting theorem which columns have infinitely many primes.
And the Halberstam conjecture says that the Prime should be pretty evenly distributed between the columns, whether infinitely many primes.
So it's a for all of these tables at once. So that turns out to be hard.
By which I mean we don't currently know how to prove it. But if you take that data, that kind of prediction about the distribution of the primes,
you can feed it in to this argument using sieve theory, which is what Goldstein yields from did.
And they were able to say, well, if this is true, then we can prove this result.
So there's kind of a whole family of these conjectures, the different, different parameters,
and they sort of needed a particular parameter where they couldn't prove it.
What Zang was able to do was to slightly weaken the assumption you needed to make so something slightly weaker than the hope of some conjecture.
Increasingly, then, he could prove that required facts about the distribution of the Prime.
So he took the goals of Pennsyl Grimm kind of idea and then adapted it in a really ingenious way to make it work out.
So back to Polymath.
So by August 2013, the Polymath Project had managed to prove that there are infinitely many paths of primes that differ by at most.
4680. I love the fact that some of you are looking at press.
If I say 4680 half an hour, you'd have said That's rubbish.
She wants it to be. But it is a big improvement on 70 million.
Right. It's fantastic. Very efficient. The trouble is, by August 2013, so totally said that the low and medium hanging fruit is being picked.
All those little tweaks you could do to Zhang's argument or those getting a computer to help us a bit,
that sort of apparently will be exhausted and it seemed like it was going to need to be a new idea.
The nature of research, of course, is you never know when the next idea is going to come or where it might come from.
In this case, it didn't take very long. In fact, it was November 2013,
and a mathematician called James Maynard was able to show that there are infinitely many pairs of primes that differ by, at most, 600.
Big improvement, James said recently finished his Ph.D. He's now one of our colleagues in Oxford.
He'd gone to Montreal as a postdoc at the time, and through his Ph.D. work, he'd become an expert in this kind of area.
And he found that he was able to go back to the work of Goldstein Points Yildirim, and find a new way through.
So not only did his argument give a better number than the work of Zhang, a polymath, but it also gave a slightly simpler argument.
And that's the kind of dream scenario, right, where a slightly easier argument gets you a better bound sort of on the right track with understanding.
One of the nice things about this pound is that it's small enough that I can show you an invisible set that works.
I didn't expect you to take in these numbers. This is like the the recipe for the punch card, the admissible punch card.
So there are infinitely many positions of this punch card where at least two of the visible numbers apply.
And I'm only putting this up to make the point that it's a small enough set so I can fit it on the screen.
So James Menon could say that infinitely many pairs of poems that differ by at most, 600.
Of course, a polymath gets very excited and said, Well, that's great. What if we look at James Maynard's?
Well, I can understand that. Can we do even better? So this league table resumes.
The pound kept dropping and dropping by April 2014.
So a few months later. Polymath can show that infinitely many pairs of primes that differ by, at most, 246.
And to the best of my knowledge, that's the state of the art.
You'll notice that April 2014 was a little while ago. So.
So Polymaths have written up their paper types may not just written up his paper, I should say about the same time as James Maynard had his ideas,
Terry Tao had some very similar ideas, that same kind of approach for adopting the theoretic argument.
But Toto's argument escapes the slightly less good numerical bounds.
What James Manos and Terry Tao both managed to do was find an argument that would deal with other clusters of primes as well.
So not just pairs of primes, which the differ by two, but if you want three primes,
very close together for primes close together, they started to be able to make progress on those kinds of questions.
So Polymaths have been working on improving those spans too.
And if you go and look at the league table, you kind of track down those bands.
There's still the game of if we're allowed to assume the Halberstam conjecture, what's the best band we can get?
So Goldstein Pitts deals to him back in 2005.
So if you see in the Halberstam conjecture, then there are infinitely many pairs of primes differed by, at most, 16.
So James Maynard's argument gets you down to 12.
So if you see the Halberstam conjecture, there are infinitely many pairs of pipes differ by at most, 12.
If you assume a stronger and more controversial form, it's the conjecture.
Maybe you could even get all the way down to six. That's like tantalisingly close to two, right?
So near and yet so far. And the trouble is that within Sift theory, there's a well known kind of problem.
That means that people don't think that the arguments that get down to six, even assuming this conjecture will get down to two.
It's called the parity problem.
It's because the theory is a very good at dealing with only looking for numbers with an even number, an odd number of prime factors.
This is kind of technical difficulty. That means the six is like a brick wall.
So they're going to need to be some new ideas. So where next?
Well, there's all this work on the ten primes conjecture to look at.
So, of course, people are trying to improve on the bounds.
People are taking the arguments that have been developed and saying, well, what else could we do with those?
What other problems can be solved with these kinds of ideas? Surprisingly, sometimes the answer is problems about large gaps between primes.
So how large a gap can you have between primes where you can have as large a gap as you like?
If you want a gap of 100 between two consecutive primes, you can arrange that.
That's not that's not too difficult.
That's the kind of problem that might appear on an undergraduate problem sheets is a nice go way to think about this kind of thing.
The interesting question is when those gaps are larger than the average.
So on average, the primes get more spread out and the gaps can be as large as we like.
But amongst primes around a million, we sort of know what the average gap looks like, and sometimes the gap can be much smaller, like two.
Sometimes the gap could be much larger. So you sort of say, well, four primes around this size.
Can the gap be much larger than the average? And this was a longstanding problem.
That was a problem posed by Addis, who who was a prolific poser of problems.
And he likes this problem and thought it was so difficult that he attached quite a large monetary value to this problem.
So I'd like to offer cash prizes in return for problems.
And the amount of cash he was willing to stump up depended on how hard he thought the problem would be.
And this died some years ago now, but a friend colleague of his has been honouring these promises subsequently and simultaneously.
Two groups made progress on this. How large can the gap be?
When can it be larger than the average gap problem?
So one of those groups was a group of four mathematicians involving in fact Ben Green here from Oxford,
Terry Tao, couple plus couple of others come forward and say again,
the other group was James Maynard, who they came up with these completely different approaches,
each based on things that those groups were experts in before and what's extraordinary.
So James Maynard took his work on understanding small gaps between primes and used it to understand the large gaps between primes,
which is very unexpected. So there's lots of potential for seeing where these ideas can take us,
even if they're not proof of the two in primes conjecture, they're all building our understanding of prime numbers.
I do believe that the twin primes conjecture is true. I do believe we managed to prove it.
I've got no idea when it might be next week.
It might be next year, it might be in 20 years time. I'm one of the young people in the audience today has solved it.
I'm looking forward to finding out. Thank you very much.