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