OK, great. Thank you very much for the kind introduction and thanks to all of you guys for coming to listen to me talk today.
So as you've probably got by now, I'm here to talk about prime numbers.
And in particular, prime numbers seem to have this space in the public consciousness of being the ultimate symbol of obscure mathematical geekery.
And what I like to try and persuade you today is that actually prime numbers are some of
the most simple and fundamental objects you can possibly imagine in the whole universe.
But more than just them being important in this way,
they're also completely mysterious and fascinating because they're very poorly understood and difficult,
despite at the same time, it's been so simple.
So the basic question that I'd like to talk about is why on earth should you care about numbers that maybe you've heard about prime numbers,
but why on earth should a mathematician or someone in the general public have even heard of prime numbers,
let alone spend any time thinking about them? And.
If you're of my millennial generation, then maybe you are told that you have a very short attention span.
So the one minute tagline is that primes are these very fundamental basic building blocks of the universe.
But despite on the one hand being the most natural, simple objects you can possibly imagine,
they're also somehow unbelievably complicated, and we really don't understand them at all.
Even though mathematicians and some of the smartest mathematicians in the history of
mathematics have been trying to understand them for thousands and thousands of years.
So if all you wanted was one minute summary, that's fine, you can wander out now.
If you'd like to hear a tiny little bit more,
this is roughly how I envisage trying to persuade you that times are both important but also beautiful in their own right.
So first, I'd like to explain why the times are important, particularly to mathematicians as fundamental building blocks of nature in the universe.
And then I'd like to illustrate this. Point about claims being both important on the one hand,
but exceptionally complicated and poorly understood, on the other hand, through two different examples.
So my first example is going to be problem, a problem coming out in the real world.
Both of the examples are slightly artificial, but the header to illustrate the point.
This is going to be a real world problem.
The both sees the fact that times of these fundamental building blocks but are somehow, at the same time, very poorly understood.
And then secondly, I'd like to give an example in pure mathematics, which is completely on the other end of the spectrum,
where again, times as fundamental building blocks play a super important role.
But they quite quickly lead to questions that we've been thinking about for hundreds and hundreds of years, and we don't understand it to.
Now, at this point, you might be slightly depressed, because maybe if I'd been successful in convincing you,
you will accept that there are these important objects,
but you might also feel that all these super smart people, these old white guys from thousands of years ago who've spent their time thinking about it.
And if they can't do anything and we've been trying for hundreds of years to do anything, how on earth can we possibly say anything at all?
So then like to try and end on a slightly positive note that despite there are been all
these very basic questions about farm numbers that we really don't understand at all,
we have nonetheless been able to make some progress on time numbers and we
are able to say something new about it and progress is really happening now.
OK, so let's move on. And the first stage is famous is building blocks.
So this is really why I think of times as important objects, because they're somehow very much the building blocks for mathematicians.
So but before we get ahead of ourselves, let's remind ourselves what a prime number is.
So if you nicon a textbook or maybe if you go on Wikipedia,
you'll find the following or something like this is a rather dry definition of what prime numbers are.
So a prime number is a whole number, which is bigger than one, and it can't be written as two smaller numbers.
Two smaller numbers multiplied together. So 15 is not a time number, because 15 is three times five, which is black and it's two smaller numbers,
but certainly is a prime number because you can't write seven as two smaller numbers multiplied together, but.
A fairly natural question, if you see this definition for the first time is, so what?
OK, fine, you can call upon no, this. But this doesn't mean anything.
It seems completely artificial. We've got this somewhat arbitrary point that we send that time number has to be big and one wasn't one to find.
No. And why on earth should you care about writing numbers is too small, the whole numbers.
And this is a pretty reasonable question, I think. But it is the case at times important.
And that's because this definition is actually precisely the right definition, and it's incredibly useful for mathematicians.
So what do I mean by useful? Well, scientists and I think in my experience, mathematicians in particular are lazy?
OK. We often spend our time thinking about very complicated big systems of interacting things, and it's really difficult to think about these things.
It really hurts. You had to think about them. So rather than often directly attacking this big, complicated problem, we like to,
we found this trick of just breaking problem and unit has lots of smaller problems, all multiplied together.
And if we can understand the little elements that make up the big problem, then maybe we can understand something about the big problem.
And this is a principle in mathematics only. It's a principle that cuts all the way across science and wider.
If you're studying for your GCSE, then but I remember when I was studying for my Jesuses,
those GCSE Bite-Size, which was about breaking up your vision into smaller blocks.
So for example, if you're a chemist, then maybe you want to study large chemical compounds, very complicated objects of lots and lots of things.
But we know that chemicals are made up of lots of atoms join together.
And if you know what atoms make up a chemical and how they join together,
you can understand some things, not necessarily everything about the big, complicated chemical.
Similarly, if you're a geneticist, maybe you're interested in some genetic disease.
And you can begin to understand something about the genetic disease by looking up,
looking at the base pairs and sequencing parts of the genome to understand different values as part of a gene which determine the genetic phenotypes.
And so both of these are big, complicated problems that you can understand, at least in part by looking at the small constituent parts.
And there's just a relatively small number of choices of possible atomic elements.
This is the famous periodic table.
Similarly, there's only a very small number of possible choices of base pairs and so relatively few relevant values for your genes.
OK, so it's all very well talking in this abstract language about nice principles of taking things into small blocks.
How would this possibly work in maths?
Well, I'm a no theorist and so as a no theorist, I want to investigate and study properties of the whole numbers.
And, OK, you might think that the whole numbers are very simple, but to me, at least the whole numbers are very complicated.
If you questions that I might be interested in is if you write down some equation.
Does this have a solution in the whole numbers or not?
And in fact, this seems to be a very, very difficult question, which we know in general doesn't have a computer to answer.
Even it's undesirable. But in slightly simpler cases, we might hope to understand something about questions involving the whole numbers.
If we can break that question down into simpler building blocks. But phone numbers come with very, very natural operations that you can do to them.
In fact, there's two natural operations. You can add whole numbers together, and if you had to hold numbers together,
you get another whole number or you can multiply whole numbers together to get another whole number.
Subtraction and division are essentially the inverse of addition and multiplication.
And so you can study the whole numbers from the point of view of addition or the whole numbers from the point of view of multiplication.
And from the point of view of addition, all the numbers are generated just by the number one,
you get any number by, just keep on adding one to itself.
And this means that certain questions about how numbers, particularly a bit of an additive flavour,
very quickly become exceptionally trivial because they just boil down to a question about the number one.
So what sort of gaps between how numbers? Well, all gaps between hole numbers are just one.
So this is a weak example of this concept of very complicated things down into simpler things that works exceptionally effectively.
Basically, because the problems are just generated by this one, no one from the times, the whole numbers are generated by just this one, no one.
But from the point of view of multiplication, things suddenly become much, much more complicated.
So whole numbers are generated by prime numbers, every whole number is can be broken down into prime numbers,
but then they can't be broken down any further.
And this means that if you have a problem about the whole numbers that involves multiplication in some way,
then you might hope to break it down into a simple question about prime numbers.
But the sudden first difficulties, because you need to understand the simpler problem about crime numbers,
and so you need to understand something about times.
So my life would be a lot easier if no terrorists had our own periodic table where it had a list of every pride number.
And then maybe you would have some problem involving multiplication and we could reduce this to a problem about farm numbers.
And then we could go through each poem in the periodic table and solve the problem for each time.
And then we'd be done. However, it's a famous torso from the ancient Greeks that there's infinitely many prime numbers.
And so this isn't a very good approach to doing things since we have an infant sized periodic table.
And so we naturally need a more theoretical understanding of how things can work so we can do this with all times at the same time.
But this concept of times been the building blocks of numbers and from the point of view of multiplication,
have you no been able to be broken down into times is an absolutely fundamental one in mathematics.
It's so fundamental that we give it the funny title, the fundamental theorem of arithmetic.
So this essentially goes back to the ancient Greeks, and this is that every hole, no,
which is bigger than one can be written down uniquely, apart from the order in which you multiply the numbers as a product of times.
So 15 can be broken down as three times five. And there's no other prime numbers that can possibly come up.
You can't have another way of writing. 15 that would be two times seven. It always has to involve one three and one five and nothing else.
And this is also why we say that one is not a prime number, because if one was a prime number,
you could write 15 as three times five or three times five times one or three times five times one times one and so on.
And there'll be lots and lots of different ways of doing it, which then mean that one would somehow be this annoying exception.
And every result about time numbers would say let PBA Prime.
That's not one, because once it's funny time, that keeps them messing everything else up.
So in this way, you can think of if you want to visualise things,
you can think of times as being the atoms that make up the chemicals of integers with respect to multiplication.
So one hundred and thirty five is five times, three times three times three.
And so here's my visual representation of five times, three times, three times three.
And if you somehow understand questions about three and five very well, then you might hope to understand questions about 135 very well.
And I at least claim that somehow questions about three and five by virtue of these atoms that can't be broken down
anymore are fundamentally simpler than questions about these more complicated numbers that have lots of different factors.
And this is really the reason that I think times important that these are these fundamental building blocks of whole numbers,
so concur was somewhat unusual for a mathematician in the sense that philosophically he didn't
really believe that maths was out there and mathematicians just discovered truths of the universe.
He felt that lots of lots of the work of mathematicians was somehow artificial constructs of man.
But even to concur, he said, God made the whole numbers.
Everything else is the work of man.
So even to Conacher, he decided that the whole numbers are somehow so fundamental to the universe that they were made by gods.
And similarly, if I try to imagine an alternative universe,
I could potentially use go a bit crazy and think of something where gravity didn't work in the same way we had different fundamental forces.
Maybe life didn't exist.
But I can't possibly imagine a world where there isn't some concept of the whole numbers of just counting one, two, three, four, five and so on.
And if I could imagine a world without whole numbers,
then I certainly can't imagine a world without the building blocks of whole numbers, which are the final.
And so it's this fact that the prime numbers of these really fundamental building blocks of the whole numbers,
which are maybe the most natural objects in the entire universe. That is really why I think of times is important.
But that's not necessarily why I find time interesting. And.
The really the real reason that I find primes so fascinating is this contrast between, on the one hand,
you having these exceptionally simple numbers, just the whole number is almost the simplest thing you can possibly imagine is an abstract concept.
And very naturally from them multiplying numbers together,
you break them down to their building blocks and then you ask some exceptionally elementary question about prime numbers,
and we don't know how to solve it.
So it's the fact that on the one hand, Prime seem to be exceptionally simple, but on the other hand, exceptionally complicated.
That is really what fascinates me about times. So Donziger uses it much better than I possibly can.
So this is a quote from him. Despite the simple definition and goal is the building blocks of the natural numbers,
the prime numbers grow like weeds amongst the natural numbers seeming to be no other lower than that of chance,
and nobody can predict where the next one will sprout.
And somehow, this very much captures the fact that even though primes are completely vigorously pinned down,
you can ask your computer, Is this large of a problem or not?
And it will say yes or no tool extents and purposes from the point of view of mathematicians,
they look as if they just appear randomly, and we have no understanding at all.
So just to give one example of this, you might have heard of the winning hypothesis, it's dubbed sometimes the most important problem in mathematics.
And if you had a good answer to how many times are there, which are less than some large number X?
Then essentially, you would have solved the women hypothesis,
and this would make someone like me exceptionally happy, we'd have a fantastic and it's only a permanent base.
If you couldn't care less about making me happy than maybe you'd be happy because the Climate Institute,
based in Oxford, have offered $1 million for the solution.
And so you'd be a little bit richer. But this is one of the most basic questions you can possibly ask about how often department members come.
And the answer is, well, we don't know. We haven't known for hundreds of years,
and we really we're so stuck on this that we view it as one of the most important problems in the whole of mathematics.
OK, so that's given a slight feel of why it's important, because there are these fundamental building blocks of the whole numbers,
which are the most natural objects in my mind in the entire world.
And also why they're fascinating because even though they have this very fundamental definition,
they're really so poorly understood and they look as if they could just become random.
And I'd like to I'd like to explain these two properties see two examples.
And so the first one is real world example. So maybe some of you at the back here a little bit bored and you playing on your phone.
And if you're planning on your phone,
maybe you decide that a good thing to do would be to make the world's richest man a little bit richer by buying something on Amazon.
So, OK, I'm a no since, say the sort of things that I find exciting to buy on Amazon books about No3.
Maybe you find something else exciting.
But certainly, if you were going to buy something on Amazon, it would be pretty important that your credit card details could remain secure.
So depending on your politics, you can view this nefarious figure as some teenage anarchist or some shady government organisation.
But one downside of the internet is that whenever you send a message to someone else over the internet in principle,
it's very easy for someone else to eavesdrop and to hear what you're saying.
And so to get around this, you need to have some code so that you can communicate with someone else.
And even if someone can literally hear what you're saying over computer cables,
they can't work out what that means is this if you were talking in the secret code language
and say the way that this is done is by turning your secret information that you'd like to
pass along into some number and then doing some mathematical operations to encrypt the number
in a way that it's very difficult for someone who knows nothing secret at all to decrypt.
But someone on the other side can decrypt without too much difficulty. So this first clever ideas as to how to do this.
But the key point is that all of these commonly used crypto systems for this sort of setup
involve multiplying numbers together and doing things that look like multiplication.
So this is a problem that's involving whole numbers being multiplied together.
And so because part numbers are the building blocks of whole numbers, you might naturally think that you can try and understand the situation a bit
better if you look at it from the point of view is either if you want to yourself,
be the hacker who's hacking someone else's details or if you want to be the person who's
reasonably confident that someone else isn't going to be able to hack your details. So there's a few different ways of encrypting things,
but one of the common ways roughly you can analyse from the point of view point numbers and the
whole question comes boils down to is it easy to factor some large number into its final bits?
So we've seen that every number can be written uniquely as part of the phone numbers.
But that doesn't mean it's necessarily easy to do this if I give you some large number.
Can you like it as its prime factors? So for example?
You could take the number four hundred and forty nine thousand six hundred twenty three.
Can you write it as foreign factors?
OK, I can see that everyone can tell you that this is obviously five hundred and twenty one times eight hundred and sixty three.
And this isn't too difficult, at least for your computer, because it can find these factors very quickly.
However, when the numbers are larger, we believe that this very quickly becomes very difficult.
And this means that it's very difficult to hack any internet communications.
So, for example, you could take the slightly larger number,
which I'm not going to eat out for you and those of you at the back who are bored and buying things on Amazon.
Well, you can go on a bit more of a spending spree because you get seventy five thousand dollars if you even know how to factor this number.
And one thing I'd like to point out is, although this number is certainly quite a lot larger than the other number,
it has maybe 100 times as many digits. But your laptop can do this factorisation in less than one hundredth of a second very easily,
whereas it would take hundred supercomputers more than a billion years.
The fact this number, as far as we know. So it really becomes much, much harder to factor numbers as you get to bigger and bigger numbers.
We don't have any theoretical proof of this, but this is at least what we believe.
And so we therefore believe that when you buy things on Amazon, you card details won't get hacked.
But this but the example is less about the precise cryptographic protocol, but more.
This is a very real world example that boil down to a problem about multiplying numbers together,
and because it involves multiplication of whole numbers, it could be understood via prime numbers.
And firm numbers allowed you to get a feeling of whether this was easy or difficult.
But you can actually analyse this a little bit more. If you think a bit more from the point of view of prime numbers.
So it's a fact, and it's not super complicated fact that you can factor no easily if it has factors.
Prime factors say could one of the time factors P where P minus one itself only has small boat taxes?
So this seems like a fairly odd thing.
You take a number, you take one of its factors, you take one away and then you factor that and have all of those factors very small.
Then there's a quick way of factoring the original number.
So this is another example of prime numbers being used to say something about the regional problem.
So for example, you could take the time to time three times, four times up to three hundred twenty then and one that is in fact the phone number.
And this when you take one away from it, it has.
Foreign partners, which are smaller than 320. And then if you gave me some large number, which is this prime times,
some other big thing, but you didn't tell me that it was actually most of that time.
But you did tell me that it has a has a prime factor P. Where people on this one in his small town factors that would be very,
very quick for me to factor it, even though this number is much larger than the $75000 number that I put up on the previous slide.
And so this is another example of. Question of times understanding the situation.
But then the sorts of questions that aren't really interested in are how many times are there with certain properties?
And so an actual question you might have based on this is how often is it the P minus one in his small prime factors?
If this happened virtually all the time, then you should be pretty worried about buying things on Amazon.
Now, fortunately, we know a little bit about this problem with a not a huge amount about this problem.
And we know that it's not very common for people one to only have lots and lots of small factors,
and therefore you don't need to worry too much about your internet purchases. But despite this.
It has been suggested by some people who work on this, but because it's really bad at P minus one has lots of small form factors,
you should do the opposite thing and have it as having as few small time factors as possible now all times bigger than to your aunt.
So p minus one is always going to be an even number. So maybe the furthest thing you could possibly have away from p minus one only having
lots of small fun factors would be if P minus one was two times some other large fine.
This way, this way of trying to hack your credit card details will completely fail.
And you could be a little bit secure that this is a safe time to use.
And so these are indeed called safe times.
And actually, it is an option in some of the very common internet protocols to use safe firearms, even if it's not the typical default option.
But then. There's another question, how many of these safe forms are there?
Imagine you to only use any subprime pee where p minus one or two was a prime number.
Well, if there aren't any of these times, then you're not going to be able to communicate with Amazon at all.
You're not just going to sit there thinking and thinking, looking to try and find one of these things that doesn't exist,
which means you certainly won't be able to buy your number textbook or whatever it is that you are wanting to buy in the first place.
However, it almost be even worse if there were a few of them, but only a small number,
imagine there's only 10 of them out there of the right sort of size that your computer's looking for.
Then maybe your computer will find it or communicate,
but maybe the hacker can also find one of each of these 10 numbers can try them each one by one, and then can actual details very, very easily.
So a natural question is, well, what do we know about these safe times where payments one for two is prime and.
We believe there should be lots and lots of these.
But this is an example of a famous unsolved problem on prime numbers, which has been open for hundreds of years.
So you become very famous if anyone solves this during the process of my talk.
Just other infinitely many times such that P minus one I for two is itself a prime number.
This is very much an open problem, and we really don't know how to solve it, it's all.
But this is a question that naturally arises if you're using these cryptographic schemes to buy something on Amazon.
So that's just basically what I said before, it's one of the most notorious problems.
OK, so that's one example of a real world situation, just buying something on Amazon.
The very quickly gets related to encryption and multiplying numbers together because it involves whole numbers multiplied together.
You can break it down into problem, but times you can start to understand the situation by primes.
But quite quickly you start getting into questions about how many problems are there with this sort of property.
And unfortunately, you quite quickly run into famous 100 year old problems that people don't know how to solve.
OK, so now let's see. Change gears completely and go to the complete opposite end of the spectrum and
give an example as to why a pure mathematician might care about prime numbers.
So you are no serious, but you don't work on time numbers per se.
And so my example is going to be may be the most famous theorem in pure mathematics, fermat's last thing.
So if you went to where this was originally suggested by the farmer that there are no
solutions to eight then plus between 16 and whole numbers whenever and is bigger than two.
So we're not in the Pythagoras system case, apart from the obvious ones where one of the numbers is equal to zero.
And so you can have people who see an accuser. So this was.
Conjectured a very long time ago by a farmer and.
Driven by Andrew Wiles, and of course, you're in the Andrew Wilds building, so I felt obliged to use this as an example.
And OK, so this is a problem to do with how numbers you just taken on a you multiply it by itself and times you add it to number B,
which is multiplied itself, and times and distance equals C for some numbers, c multiplied by two itself and times.
And this involves whole numbers, and it involves multiplication,
so you might naturally think that you can start to analyse this by thinking about prime numbers.
And it's the result that rather than having to worry about all possible home numbers and you can just concentrate on when and is a prime number.
So you're only multiplying your numbers together a prime number of times.
And this is actually it might seem that this doesn't make the problem much easier,
but there's lots of special properties when and it's fine, which does actually make this problem technically quite a lot easier.
So let me roughly sketch the idea of this claim. So imagine.
Let's think about the case when he's 15. So we're thinking about how you multiply.
So 15 times plus be marked by 15 times. Does that ever frequency multiplied 15 times?
And I imagine that format was wrong. This is the sort of fake news thinking that mathematicians like still the time we like to
imagine all of these crazy worlds where true things are false and false things are true.
So imagine there was some example of some numbers A, B and C such that A15 plus B-15 equals 50.
Well. 1815 is a most fight together 15 times.
But one thing doing that, I could take a multiply together five times and then multiply that number to itself three times.
So mathematical language shorthand 1815 is 85 to the power three.
It's just two different ways of collecting 15 lots of money together and multiplying them all together.
But that means that if I did have numbers A, B and C, where if the 15 plus B 15 was equal to CS 15,
I would have that eight to five to the three plus beats.
The five to the three would equal C to the five to the three.
And so this means that I would have a counterexample for Fermat's last year when I was three.
Because I've now found three numbers where when I raised them to the third palace, I get a counter example to Fermat's not to.
So if I had a counter example for 15, then I will get a counterexample for three.
And very similarly, the 15 is eight to three to the five.
So I'd also get a example for five.
And if you think about it a little bit, this example works in general that if I have a counter example for any integer and that I get a,
for example, for any prime factor of an. And so it's essentially enough to Fermat's last June,
when it is time I've skipped over the case when and it's just got lots of practise, which are all equal to two.
But this was a case that was already nine Typekit firmer. So we can just assume that.
And the point of this is, again, we've taken a problem involving whole numbers and multiplication,
and we've reduced it down to a problem involving five members. So.
Wales's result is famous, and it's really a pinnacle of modern mathematics, it's exceptionally complicated.
So unfortunately, there's not enough space in the slide for me to give the foolproof.
But before wowsers work, there was nonetheless lots of other important partial progress to Fermat's Last there.
One reason the Fermat's Last became such a celebrated problem in mathematics is because
it led to the development of so many different beautiful ideas in mathematics.
And so one. Very. Old result, but maybe one of the first sort of fundamental it's trying to attack famous last name in general, was weaker.
Sophie Germain back in the 1820s.
So she was thinking about from its last term, and she knew that I she only needed to consider the problem when Ed was a prime member.
And she realised that if he was applying no such, that 2+1 was also a prime number, then there was a good way of attacking Fermat's last year.
So if he was prime of two plus one was prime, then essentially this new solutions to Fermat's Last Theorem with the exponent P.
Now there is a technical condition about ABC not being the rupee, which is like ABC not being equals zero.
But this allows you to show lots and lots of cases of Fermat's Last, in which one and at all.
Back in the 1820s. So this was really very clever ideas.
Apart from there's one natural question that arises, when does the assumptions of Sophie Germain stadium actually occur?
How many programmes are there such that two plus one is also a prime number?
Does this even ever occur? OK, so it works for three.
Because two people, one when three is seven.
So we get a few examples concretely. But now if we think about.
I've given you two examples of questions which boil down to probing about prime numbers.
And just like the other example, we don't know how to solve this.
It's a famous problem that's been open for a hundred years and we don't know how to solve this.
Like the other example, because they're the same problem. But before I was asking, is he minus one over to apply?
No, but if p minus one I for two was a prime number. If you call that phone number Q, then two plus one is a number.
So this is just exactly the same question rewritten slightly.
And so here's two examples of problems that occur in the complete opposite ends of the spectrum.
One is button something on Amazon. One is trying to solve some cases of the most famous problem in pure mathematics.
Both of them involve maybe not obviously, but involve integers multiplied together because they involve integers multiplied together.
You can start to study the problems using prime numbers, and you can certainly start making some progress.
If you knew that there were certain times with certain doctors and these questions arise so naturally.
And it turns out that for these two cases, exactly the same problem arises either from many times such that 2+1 one subprime.
And this is a problem that despite having been studied for several hundred years, mathematicians still really don't have any clue of how to prove it.
We all believe it is the case, but we have no idea of how to prove it.
OK. So finally, I.
For the time I have left in this talk, I'd like to talk a little bit about some positive results.
So as I mentioned, maybe at this stage in the talk,
you're not really convinced that at least with my examples, there's some problems involving numbers.
They can be broken down into similar problems involving firearms,
but you might feel that every time I have a problem involving phone numbers and break it down into problem involving primes,
I run straight into a brick wall because I get into some problem involving prime numbers,
which all these super smart guys from antiquity have thought about and failed to solve.
And so how on earth we possibly can make any progress? So, um, and these are really the sorts of problems that I like to think about day to day.
I do not work on encryption because maybe people in this audience who are much, much more expert on this to me simply,
I don't work on trying to solve explicit equations like from its them, although I do like things like that.
But I really like to think about is how many just very basic problems of the times, how many times out there with a certain property.
So you've seen a couple of examples. Today, the question really is how many primes are there such the P minus one in his small prime factors?
Now the question was how many times are the such that p minus one i produce prime?
So I guess I mentioned the Lehman hypothesis. How many firms are there, which are less than some larger number X?
So this is a case of something where we did make some partial progress.
So we know the prime number theorem that says approximately how many numbers there are.
Even a hypothesis is really about what does approximate mean and how accurately can we describe this number?
And then one other question that's again, a big, famous opium problem,
but is particularly close to my heart is how close confinement be to each other, other infinitely many passing times, which differ by exactly two.
So I mentioned right at the beginning of the talk that we can understand gaps
between industries very easily because industries are just generated by one.
So somehow all gaps in the number line on a science one. And I also mentioned the old times bigger than two.
Even so, apart from the Times two and three or gaps between times have to be of size at least two.
And then one very natural question on the basic distribution of times.
Almost one of the easiest questions you can possibly ask is do these gaps occur often or is it the case that the gaps gradually get bigger?
If you think about the square numbers, for example,
the gaps between square numbers always get bigger and bigger and bigger as you look at bigger and bigger numbers.
And we know that typically for prime numbers, the gaps get bigger.
But actually, we believe they should be. And for me, often these rare films that come clumped very close together.
So these are about as simple as questions you can possibly ask if you're just interested in basic distributional questions about the prime numbers.
And so these are the sorts of questions that I think about,
I would be exceptionally happy if I could really answer any of these questions in a very strong way.
But my aim is to develop flexible tools to try and understand something to do the distribution of the firearms.
And to give some hints that there has been progress made on these problems for the question of downstream times,
we now know that there is some even no less than two hundred and forty six two hundred forty six is
just in slightly artificial number that comes out from some computations that they're infinite.
Many pairs of times the difference by exactly age. So if I then send some number eight thousand two hundred forty six,
I could take equals to our proof, the term conjecture, and I'll be very happy indeed.
Unfortunately, we don't know how to prove the time conjecture, but up until six years ago,
we didn't know at all that it couldn't be the case that the gap between the parties got bigger and bigger and bigger
in the same way that the gap between the squares could gradually get because you look at bigger and bigger numbers.
But we now know, at least, that they do come together and fairly often, even if we don't know precisely how it happens.
Similarly, I mentioned several times the two examples in my talk boil down to a question about other than filling me primes.
Such the P minus one divided by two is a prime number. And again, unfortunately, I don't know how to solve this.
I would love to be able to solve that. But in the same spirit as the first term, it is the case that relatively recently.
We do know that there are two fixed numbers. Are you one of to such as those infinitely many times PE with p minus three one divided by two?
Also a prime number. So if I could take a one to be one in 80 to be two out of some of the original problem, I can't quite do that.
I need a little bit more flexibility. But this is exciting progress that we've made on some of these famous 100 year old problems.
So I'm nearing the end of my time now,
so I just want to recap some of the things that I've hopefully talked about today, and I hope you found interesting.
So there's lots of interesting and important questions that arise both in the real world and in pure mathematics that involve whole
numbers and whole numbers of somehow some of the most basic fundamental objects that you can possibly imagine in the universe.
But whenever these problems involve multiplication, which is empirically quite often.
Then often these problems can be broken down into simpler problems involving prime numbers.
And so if you understand the problems very well, then you could solve the original problem and you'd be really happy.
Now. So therefore,
people like me try and understand the distribution of the firearms in general as
a flexible toolbox so it could work for any problem that someone else comes,
mathematicians tend to get quite specialised as they get older. So I welcome prime numbers, but I have lots of colleagues who work on other things.
So the sorts of interactions are often someone will come to my office and say, Hey, I could do this really great thing in my field.
If only I could understand this other problem to do prime numbers. Do you know if this is solvable or not?
And if it's very if we're very lucky, I say yes, I know how to solve that.
We can write a paper together more often. I say, unfortunately, this is a famous 100 year old problem, so we can't do anything.
But such is life. So unfortunately, often you can break it down and you can see a path to solving the original problem.
But you run into these big hard walls of these famous fundamental problems about the distribution of primes because realistically,
we hardly know anything about the distribution of primes.
Slightly embarrassing for me to say, but very slowly we are making progress and we are gradually understanding the times, slowly but surely.
And so it's actually a very exciting time in the field and we feel that it's current
and progress is being made and we are able to understand these problems much,
much better. So I think I'm going to stop there. Thanks a lot for listening.
I hope you enjoyed the talk.