'7 Things You Need to Know About Prime Numbers' - Dr Vicky Neale

hello everyone it's very nice to see you all here have I got the microphone working properly good excellent I'm slightly anxious about the technology it's great to see a roomful of keen enthusiastic mathematicians at least I hope you are so as Julia said I'm going to talk about prime numbers so my background my kind of mathematical interest are some extent in number theory so I'm just in properties of whole numbers and that sounds quite simple because whole numbers are just whole numbers but what I hope to show you over the course of this talk is that there are lots of really interesting quite deep questions that you can ask about whole numbers even seemingly simple very familiar whole numbers there's lots to kind of explore and investigate so maybe it's a good idea to start by being really clear about what a prime number is so I'm sure you've all come across prime numbers but let's be clear about the definitions definitions are important to mathematics so what is a prime number it's a whole number bigger than one that's divisible only by one and itself and the first thing I want to stress is that one is not a prime number not because of some interesting philosophical discussion about whether it has one factor or two factors or it's only factorize one and what none of that it's because part of our job as mathematicians is to make good definitions to make definitions that lead to interesting mathematics definitions don't arrive on stone tablets we make the definitions and it turns out to be a better definition to define one not to be a prime number so having got that straight let me tell you about the second thing I want to tell you about which is that two is a prime number mathematicians are good at looking for patterns so you've seen my first two facts you may have guessed how the rest of this talk is going to go one is not a prime number two is yeah the interesting thing about two being a prime number of course is that it's the only even prime number so if I take an even number that's bigger than two then it's divisible by one it's divisible by itself and it's divisible by two so it's not prime so there are all sorts of situations where somehow two is a slightly special case yes it is a prime number that's really important but also it's the the only even prime number so sometimes it's it's good to sort of separate deal with two separately let's have a look at the prime number so I've colored them in this nice shade of blue it doesn't quite match my shirt unfortunately so very familiar 10 by 10 grid I've just colored in the primes and the thing I like about this is that as a mathematician I look at this and I start seeing all sorts of patterns I start wondering I start asking questions because there are all sorts of interesting things to pick out there I don't care really about what happens in the numbers up to a hundred because I can see what happens with the primes up to a hundred there they are what I'm interested in is if I can see a pattern here will it continue if you like so beyond the bottom of the slide if I managed to fit more rows in here would those patterns that I'm seeing continue or is it just some feature of small prime numbers so what I'm trying to do is look for patterns but then seek to see can I can I see will that continue can I prove that it will continue can I show that it won't continue so there is an example of a kind of pattern so some of these columns seem to have no prime numbers in at all right so here's the before and the 6 and the 8 and the 10 columns they seem to have no prime numbers there are no blue numbers shaded there in the numbers up to 100 of course we just said the 2 is the only even prime number and all of the numbers in the 4 6 8 and 10 columns are even so we know they're not prime and in fact 2 is the only prime number in this column so that's not the most difficult example but it's an example of something that we can see here oh look these columns seem to be empty seem not to have any Prime's or we could be confident that feat that kind of feature pattern is going to continue beyond so there are other kind of similar things so for example 5 seems to be the only prime city in this column it's rather sad by itself at the top there and we might wonder does that continue you think about it for a moment to realize yes it does that 5 is the only prime in that column I can see a few nods which is great so all of the numbers in that column are multiples of 5 if you're a multiple of 5 you're bigger than 5 you're divisible by 1 and by yourself but also by 5 so you're not prime same kind of argument um what else with all sorts of things there are things about gaps between Prime's there are things about where they're distributed all sorts of questions to investigate um one of the questions this used to be really kind of important is if I continue this grid will I keep getting more and more prime numbers or at some point do I hit the biggest prime number there's two kind of very different worlds you could imagine there so in one world there is a largest prime number and after that point all of the numbers are not prime they're divisible by some smaller primes but they're not prime in the other world no matter how far down the grid I go I keep finding more and more Prime's so sort of tries to get a little bit of intuition about this I mean if we look at the grid the primes are becoming a bit more spread out right so there's this gap from 73 to 79 to 83 to 89 then this is really big gap to 97 looks like they might be becoming more spread out that sort of feels intuitively plausible right if you take some very large number like a number of this big it's going to be quite difficult for this number to be prime because there are lots and lots of smaller Prime's that might divide it it's quite easy for seven to be prime because there aren't many things that could divide into it number this big there are loads of things that might divide so it sort of feels plausible that the primes are going to become more sparse they're going to become more thinly distributed but it's sort of I mean that's just sort of an intuition that's not going to pin down is their biggest prime is there not a biggest prime I can't use a computer to help with this I can ask a computer please go and look for very large prime numbers I mean that's not a totally crazy thing to do it takes computer quite a long time to do it but but the trouble is that's still not going to answer the question right my computer is going to keep churning and okay it's find a very large prime number but I've got no idea whether that's the biggest prime number or whether if my computer keeps going it's going to find another one so so asking a computer go and look for big Prime's is never going to resolve the question of is there a biggest prime number so we need to do something more mathematical to get to the bottom of this and this is not a new question at all this question goes about more than two thousand years so this was known to the ancient Greeks a very long time ago this is a theorem this is a mathematical result we can prove to be true so it's no longer that kind of intuition we can prove that there are many Prime's there is no biggest prime number and proof is really important to me as a pure mathematician I really care about proof I care about this certainty of being able to prove something not beyond reasonable doubt but just completely utterly certain that something's true so I want to share this proof with you that goes back to Euclid so I guess some of you have come across Euclidian geometry may be named after you cleared who wrote about it in his books the elements more than two thousand years ago but he also wrote about other things like number theory properties of prime numbers for example and so he gave a proof and he's plan was was to use this kind of thought experiment so this is a mathematical technique called proof by contradiction my guess is that perhaps if you have come you'll come across it but lots of you haven't so let me try to describe this proof to you so he said very roughly speaking let's pretend for us so secretly we think that there are infinitely many prime numbers let's imagine that we're in a parallel universe where there are any finitely many prime numbers and let's see what would be the consequences of a world in which there are only finitely many Prime's and it turns out you reach an absurdity reach a thing that can't happen use your contradiction so that parallel universe can't exist that there must be infinitely many primes so he said ok so a bit more detail he said suppose there are only finitely many Prime's that means we can write them in a list it might it might be a very long list if they're a lot of them but they're only finitely many so I get a very large piece of paper I write all the primes in the world so my list starts to 3 5 7 all the way up to P the largest prime number whatever that largest prime number is and here's a Euclid's fantastic idea he said multiply all those numbers together and add 1 so I don't I don't know what this number is this is just a thought experiment but it's a product of all the primes in the world plus one other thing about this number is it has to have a prime factor right because every number bigger than 1 has to be a prime factor perhaps have a prime factor perhaps this number is itself prime perhaps it's divisible by a smaller prime but it has to be divisible by a prime so let's think about what that prime number is can it be - well no because this number is 2 times some stuff plus one right so it leaves remainder one when I divide by two it's not a multiple of two can this number be divisible by three well no because it's three times some stuff plus one so it leaves remainder one when we divide by three can it be divisible by five well the same thing right so for each of the primes on our list it can't divide this number because Euclid's built this number so that it leaves remainder one when you divide by any of them it's not exactly divisible by any Prime's and this is where we hit our contradiction right we've got this number that has a prime factor because every number has a prime factor but also it doesn't have a prime factor because it's not divisible by any of the primes in the world because we know what they are they were on our list none of them divide this number so that's the contradiction we sort of point we start feeling slightly ill you know this just doesn't work out so so I initial assumption our supposition that there are only finitely many Prime's must be wrong there must be infinitely many Prime's so don't worry if you didn't follow all the details of that but for me I love the fact that this is a really difficult problem right how could we possibly show that there are infinitely many Prime's we can't do it by writing a list of bigger and bigger prime so we need a clever idea and if you approach the problem in the right way if you say well what would happen if there only finitely many and so on then you end up with this really rather elegant proof so so for me this is quite exciting somehow I think the world would be a much less interesting place if there were only finitely many Prime's because in principle we could write a list of all of them there there would be let's move on with it this way we know there are infinitely many of them we're never going to run out of new prime numbers okay why am i why am i excited about prime numbers mostly I'm excited about prime numbers big they're really intriguing because they are difficult to understand because they seem simple I love the fact that it's quite easy to define a prime number we all sort of know what they are but then it you can ask really difficult questions about them but another reason that I care about prime numbers is that they're kind of the building blocks from which all numbers are made so this is another theorem this is a really important theorem this is so it's important that it's called the fundamental theorem of arithmetic so the clue to its importance is in that fundamental theorem and it says that every positive integer whole number bigger than one could be written as a product of prime numbers in an essentially unique way there's just one way of doing it what do I mean by essentially unique well if I give you the number 12 and ask you to write it as a product of prime factors perhaps somebody over here writes it as 2 times 2 times 3 perhaps somebody over here writes it's 2 times 3 times 2 I think we'd agree that was the same it's the same list of factors where it doesn't matter which order I mean write them in so that's what I mean by essentially unique so this is this is unbelievably important but somehow it's terribly easy to take it for granted I'm my guess is that when you've been at school sometimes your teacher has asked you to find the prime factorization of a number it probably never crossed your mind that the person sitting next to you might come up with a different answer but the fact that they won't is somehow really important to all sorts of more sophisticated mathematical things so the first property that you could just write every number as a product of prime factors is important but maybe not so subtle the subtlety comes from the uniqueness of prime factorization so there have been some work in generalizing number theory beyond the integers so we can do number theory with whole numbers we can ask you our properties of those it turns out there are some generalizations of that way you can try to do similar similar kinds of things this is something that mathematicians got really excited about in the 19th century tried to prove Fermat's Last Theorem so perhaps some of you have heard about farmer's Last Theorem this is a problem that goes back to the 17th century if you don't know about it you should look it up it's a fantastic story really old mathematical problems showing that there are no solutions to a particular equation in the 19th century mathematician started to get very excited because people start to say well look if we don't just work with the integers we work in these generalized settings we can prove Fermat's Last Theorem unfortunately somebody then later came along pointed out it just doesn't work because in some of these generalizations you don't have uniqueness of factorization which was a blow but on the other hand there's been a century of really exciting number theory coming out of oh this doesn't work instead of getting cross throwing in the towel it doesn't work going away let's try to understand why it doesn't work what's going on what's special about the integers that means this works so I'm not going to prove this result for you right now not that it's a massively difficult thing just there are other things that I want to tell you about but it is a really important thing that's worth being aware this is a thing that needs proving it is not completely obvious that if I take a number this big that I can write it as product of prime numbers in just one way I sort of clear with 12 because you can try it 12 as well if I pick an arbitrary very large number it's not clear without proving the theorem and it's deadly this is quite a good reason to define one not to be a prime number right because if one were prime then officially 12 could be written as 2 times 2 times 3 times 1 times 1 times 1 and officially that would be a different factorization and that would sort of be silly so so this is quite a good reason to say let's define one not to be prime okay so prime numbers are important because they're the building blocks from which we can make other things and that means we can use them to solve problems so sometimes I want to prove something about all whole numbers if I can prove it four prime numbers then I can use the fact that the prime numbers are the building blocks to sort of build up to proving it for other numbers as well but also they're just really intriguing of themselves and talking of intriguing I want to tell you about my mathematical pencil nobody gave this to me I wish I could remember who um few years ago they just said Oh Vikki have this pencil I think you'll like it and I said oh thank you very much it's a pencil you know and then I looked at it a bit more carefully and I discovered it had some numbers on you can't see the pencil from here so I took some photos of it which hopefully are here this is slightly grubby pencil because it's in my pencil case it comes everywhere with me just in case you never know when you'll need an emergency maths pencil so my pencil has a hexagonal cross-section right so there's six sides that's what these six photos are and the person who gave it to he didn't explain the significance of the pencil they just said Vikki have this pencil um so I was looking at it thinking what's going on okay so it's got some numbers on that's kind of interesting it's got some black numbers it's got some red numbers I realize the numbers kind of spiral round to the pen saw so I'm not sure I have that easy that is to see from the photos but it starts with the side with one and then two three four five six and then goes back to 7:00 on the same side as one okay so the numbers kind of spiral round some of them are black and some of them are red and I felt that's interesting what's going on and then I looked at the red numbers of the I recognize these numbers these are my old friends the prime numbers oh that's very nice somebody's giving me a pencil with a prime numbers one that'll be handy if I want to check is seventy-nine prime instead of having to work it out I'll check my pencil um and then I thought actually there's a bit more to it than that um these photos are not very clear because they're photos and it's a grubby pencil so I typed up the numbers for you but in order to fit them on the screen I had to tip them vertically okay so the the first horizontal photo is now at this vertical column so I hope sort of make sense of this so there's the six sides of my pencil in those six columns and the thing that really struck me was that there seemed to be some sides of the pencil with no prime numbers at all and then these sides just had two and three being poor sad lonely Prime's the only primes on that side and then there are a bunch of primes clustered on the others and of course this is like my grid from earlier I don't really care about the numbers on the pencil because I can check the pencil I care about what happens if I had a much longer pencil wouldn't fit in my pencil case but what would it tell me about the distribution of prime numbers so if I continued this if I go far enough will I find some Prime's on those sides will I not and I thought about this I thought actually this four and sixth column I've seen these numbers before these are even numbers I already know there are no even Prime's apart from two so two is the only prime on this side these are all even there are no Prime's here or here great fine what about three so I guess on this side of the pencil all the numbers are multiples of three so these numbers are all multiples of six these are all the odd multiples of three of course apart from three there are no multiples of three that are prime right if you're a multiple of three you're bigger than three divisible by one and by yourself and by three so you're not prime so three is indeed sad and lonely so then I realized what this pencil was telling me is that apart from two or three two and three every prime number is one less than a multiple of six or more than a multiple of six isn't that nice I thought that was quite good for a single pencil so apart from two and three every prime number in the world is one less than a multiple of six or one more than a multiple of six I thought that's good I feel well before I put this pencil away actually now I've got lots more questions I want to are asked I want to ask questions like I know there are infinitely many Prime's we saw Euclid's proof there are infinitely many Prime's now these two here that are special cases but apart from them I've got infinitely many Prime's split between these two sides of my pencil so are there infinitely many that are one less than a multiple six and infinitely many that are more than a multiple of six or does one of those sides only have finitely many you know how do the numbers in these two columns compare and what would happen if my pencil had a different number of sides what would happen if I had a pencil a seven side or hundred sides would I get this same kind of phenomenon of primes bunching up in certain columns would there be infinitely many Prime's in those columns and there's so many questions that you could go on and investigate and actually they all turn out to be really interesting questions there's there's lots to unpick there but let's not do that right now what I want to talk about right now is how many prime numbers there are and that's a stupid question because we already know the answer is infinitely many but here's a way to make it a sensible question I can ask how many Prime's there are up to a certain point how many Prime's there are other up to a million or billion or squillion so try to get a sense of the distributional prime so I know there are infinitely many I said before this sort of intuitive idea that they're becoming more spread out but how how fast are they becoming spread out you know if I go up to a squillion and then up to two squally and how many Prime's are there between a squillion and two squillion are they mostly up to a squillion or they're quite a lot between a squillion and to schooling schooling it's not a technical term I hasten to add so this turns out to be a really good question so mathematicians invented some notation for this so we write PI of X to mean the number of primes up to X so PI over 100 I showed you that grid earlier though 25 Prime's on that check don't believe me you should never believe anything you check it for yourself there are 25 Prime's up to 100 so pi 100 is 25 this is the Greek letter pi you are very familiar with this Greek letter pi from stuff about circles and things trigonometry there's lots of interesting things about pi this pi has nothing at all to do with that pi okay this is just a letter it could have been P of X or f of X or something but for reasons lost in the midst of time some mathematician somewhere decided it was a good idea to call it pi so now the convention is to call it pi so I have to call it PI this has nothing at all to do with circles it's just the name of a function that says how many Prime's are there up to X so back in the 18th and 19th century mathematicians got really interested in understanding the behavior of this function so how fast does X grow as a function of X so there are there are X numbers up to X how many of those X numbers are prime and the thing I like to remind myself when I'm thinking about this is the technology that the mathematicians in the 18th and 19th centuries had available to them if they wanted to know how many Prime's there were up to a million they did not Google it they did not hope that somebody had done this calculation before they did not write a little computer program they got out their piece of paper and they got out their pen or whatever and they journey well calculate it and you have to check yo which of these numbers are prime and I'm kind of humbled by the amount of calculation that these mathematicians were doing in order to start to make some predictions I mean it's very hard to have any guess about how does this function PI of X behave unless you gather some data and gathering data meant doing lots and lots of calculations and doing them right I mean you don't want to get that wrong and think this numbers prime well it isn't because then that's going to mess up your data so great mathematicians like Gauss Riemann kind of mathematical heroes of the past would working on this we're coming up with conjectures and there became this really famous problem to prove the PI of X is asymptotically equal to x over log X I will interpret this for you in a moment this became one of the kind of big problems people were very excited to try to prove this and then right at the end of the 19th century 1896 1897 about then to mathematicians Adam our Honda lavallee Poussin I managed to prove this and this is a kind of highlight of 19th century mathematics proving the prime number theorem let me try to make sense of this for you so my guess is that some of you have come across logarithms and some of you have them because it tends to come up in a level but it depends which order you do the topics in so don't worry if you haven't but if you have when I write log here I mean the natural logarithm to the base e which you probably write as learn but for reasons I don't understand people who work on this bit of number theory in fact lots of pure mathematicians in general will write log meaning learn where you write learn I don't know anyway it's the natural logarithm to the base e this twiddles thing this looks a little bit you know what does that twiddles mean it looks a little bit alarmingly is approximately or something that's very vague I'm a pure mathematician I like precise statements this is a precise statement officially this says PI of X is asymptotically x over log X which means PI of X divided by x over log X tends to 1 as X tends to infinity great what the way you should think about this is that PI of X grows like x over log X so it is a sort of is approximately equal to but there is a precise meaning so it's not just it's about this there is a technical kind of sense for what this thing means so there are X numbers up to X about x over log X of them will be prime and this is this is an asymptotic thing it it gets better for large values of X if you want to know how many Prime's there are up to 100 jolly well work the mountain count them if you want to know how many Prime's are there are up to a billion billion billion this is a reasonable estimate ok so it's a better estimate for larger values of X um there are so what an Mardela values are proved was that PI of X is x over log X plus an error term and then you show that this error term is somehow much smaller and less important so if we could prove the Riemann hypothesis which is one of the most famous kind of unsolved problems in mathematics we would get better information about this kind of approximation about that error term so we strongly suspect the better things are true this is conjecture sure a thing that mathematicians believe to be true but can't yet prove proving the Riemann hypothesis would be a really good way to understand that the Riemann hypothesis is great I'm not going to tell you about it right now because there are other things to understand the Riemann hypothesis properly you need to know some more kind of technical mathematics and I'm not going to try to do that in the next minute there are things that we know there are there are problems on which mathematicians who may progress like the prime number theorem there are problems that are still unsolved like the Riemann hypothesis like a better asymptotic estimate of how PI of X behaves let's go back to the primes because I want to talk about another famous problem so I showed you this before this just the primes on our familiar 1 to 100 grid and I said before the prime suspect - 9 89 89 to 97 it's a big gap this is a complete con right if I put another row on this slide you'd have seen 101 103 107 and 109 are all prime okay so the next row would have had lots of Prime's bunched up all together on average they are becoming more spread out the prime number theorem tells us is when you unpick the significance of this x over log X is telling us that they are becoming more spread out but they do seem to be these instances of primes that are still bunched up close together and and if we look at this grid there are some examples here we can see 17 and 19 71 and 73 59 and 61 107 109 these are pairs of primes that differ by just two so a very natural question somehow we know there are infinitely many Prime's are there infinitely many pairs of primes that differ by two I don't know how old this question is I mean the origins of it seem to be lost in the mists of time maybe Euclid maybe the Greeks ask themselves this question they might very well have done I don't know Mobius nobody knows the answer so this is called the twin primes conjecture conjecture we think it probably true we don't have a proof so the conjecture is that there are infinitely many pairs of these twin primes so there are some sort of plausible reasons to believe that it might be true like you get your computer to go and look for very large primes and there are some very large pairs of primes that differ by just two that we know about as I said before that doesn't prove anything either way but it's at least reassures us that we do seem to keep finding these there are some heuristic models of the primes so there are some ways of saying well the primes behave approximately like this and if you unpick the consequences of that you can predict that there are infinitely many two implants you can actually make predictions of how many pairs of twenty twin primes there should be up to a million or a billion or a squillion that's not a precise that that's a heuristic it's an idea it's it's based on the primes are a bit like this thing it's not a proof that that's how the primes actually do behave there are there applause herbal reasons for making this conjecture none of them are proof it would be possible for somebody to set up and say actually it's not true I've disproved it I'd be surprised but it's possible um but I don't want you to think that these problems get posed and then they never get answered right the prime number theorem did get addressed at the end of the 19th century and there's been some progress on the twin prime conjecture just in the last two or three years so I want to tell you a little bit about that because this is really exciting development within the world of mathematics mathematicians very excited about this so about just over two years ago a mathematician called me Tang Jiang based in the state not famously a world expert in the area you know he wasn't a hugely eminent mathematician published a paper in which he showed that there are infinitely many pairs of primes that differ by at most and by most 70 million the thing is we're aiming for infinitely many pairs of primes that differ by two and 17 million is quite a lot bigger than two right I mean this sort of okay there are infinitely many pairs Prime's that differ by most 70 million what of it but the thing is this was the first result of its type the mathematicians have been able to prove so 70 million is much bigger than two but it's a lot smaller than infinity right so so this development was a big news so mathematicians start writing to each other emailing say off you seen this paper this is really exciting and it became clear looking at his paper that he'd taken some recent work by other people he'd added his own clever ideas the kind of technical perseverance he needed to make this paper work was astonishing but there were little points in his argument where if you did this calculation in a different way you could get a bitter answer if you used a bit of computer help here you could improve on this bit if you applied this recent idea somebody else's had you could improve on the argument you've made that 70 million smaller so forty years ago what would have happened who had been mathematicians around the world would have got hold of Shang's paper and worked on making little improvements and they might work by themselves or they might have worked in little groups of two or three saying oh how can I improve on this little bit and then there'd be this kind of flurry of papers some say well I can get it down to I don't know 69 million or somebody else so I could do a little bit better every now and then there would be a big jump and somebody would miss out because they were sort of scooped by somebody doing more that they could do getting their first bit of a race and that's not what's happened this time thanks to the internet so there's this new way of working that some mathematicians have been looking into involving collaborating online so-called polymath so the first there's a joke in there somewhere the first polymath project was started as you'd like Tim Gauss who's based in maths here in Cambridge say can we have this kind of massive online collaboration can we by working together in public on blogs on wiki's solve a problem and I think sort of to a lot of people surprise possibly to everybody's surprise the first polymath project managed to solve a genuine research problem that mathematicians have been working on for some time in a surprisingly short period of time through his public collaboration and then there were several more polymath projects and following Zhang's announcement that he could prove there are infinitely many pairs of primes that differ by at most 17 million it became quite natural for somebody say well let's have a polymath project let's work on this together so there's no rule that says you can't sit in your office by yourself and work on it but there was an invitation to anybody who wanted you don't have to be a professional mathematician in a unit to Department it could be anybody to get involved in working on this problem of can we make this 70 million smaller can we get closer to two and so this started up a couple of years ago this month and over the summer happened it's all still there it's all online you can go and find it it's all on blogs it's all on wiki's it's in the public drain you can see the process of people trying out ideas there was a kind of leak table of you know Jang's got 70 million well somebody will come along say well I think I can get this by doing this bit of the argument and then somebody else will come along check say yeah this looks great tick or actually I think there's a problem with your argument here you want to take another look at that subscribe checking each other so there's a leak table of this number dropping and dropping and it's really exciting and you know progress is happening really fast in public rather than waiting a few months for it to be published in a journal whatever and by the end of the summer the polymath project who managed to show that there are infinitely many pairs of primes that differ by at most four thousand six hundred and eighty and the brilliant thing about this is that you're dead impressed by that because it's a lot smaller than 70 million if I'd say four thousand six hundred eighty the beginning you said that's a lot bigger than two but compared with 17 million four thousand six hundred eighty is fantastic right there are infinitely many pairs of primes that differ by most four thousand six hundred eighty and then progress kind of dried up somehow all of those little points where you could just improve a little bit here and a little bit there and so on it seemed like those had all been exploited so the people involved in the polymath collaboration wrote up the project wrote a paper for publication and sort of you need a new idea and what's exciting about research matters you don't know when that idea is going to come right it might be soon it might take a while and what happened was a couple of months later a British postdoc so a young mathematician who just finished his PhD was working in the University of module called James Maynard at he's one of my colleagues in Oxford had a new idea so he'd got his own work and he'd looked as Yang's work and he looks at the polymath work and he was able to show that there are infinitely many pairs of primes that differ by at most six hundred big drop it what gets very excited because you know can we get to two so the polymath project kind of wake Safin says well let's have another phase let's take James may not work and you tank Zhang's work and I'll polymath work and all these other things and see what we could do and next few months it drops and drops and drops and then it sort of dried up again just over a year ago so to the best of my knowledge the state of the art is that there are infinitely many pairs of primes that differ by at most 246 still not too but pretty good massive progress right in the space of a year and in the public domain so you can see what's been going on and so on which for me is a really exciting aspect to the project so what happens now well so here's a really good strategy if you're stuck on a math problem I remember I recommend this to you if you're stuck on a math problem assume that you can do some other difficult math problem that in fact you can't and then use that to help you so mathematicians do this all the time so I mentioned the Riemann hypothesis there are lots of papers that say if the Riemann hypothesis is true then dot dot so if somebody comes along approves the Riemann hypothesis then we'll know all of these other things you have quite a lot of papers that say if the Riemann hypothesis is false then dot dots so so in the case of the twin primes conjecture this there's this really useful conjecture called the Elliott Halberstam conjecture so the Elliott halberstam conjecture is not phrased in terms of pencils but should be so I said on my mathematical pencil that apart from two and three all the primes in the world are well more than a multiple of six and one less than a multiple of six we know there are infinitely many Prime's in the world it's not too difficult to show that there are infinitely many Prime's that are one less than a multiple of six and they're infinitely many Prime's that are more than a multiple of six so then there's this kind of race of you know if you're a prime would you rather be one less or one more if I look up to a million how many of the primes are one less than multiple six how many one more if I look to a billion or a squillion how many of the primes one less than a multiple of six how many at one more than a multiple of six how do those two options compare and there's no terribly clear reason why one of them should be more popular the other and you do some calculations and it it seems to be they're fairly evenly split I mean what you ain't expecting to be exactly evenly spit but fairly evenly split so I know that up to a million and there hi of X P of a million primes up to a million so I expect there are roughly Paiva million over two primes the one less than multiple six are roughly Paiva million Prime's over two Hiva million over two primes that one more than a multiple of six I expect to be roughly evenly split so the Elliott Halberstam conjecture does that for all possible number of sides on a pencil what happens if I have a hundred sided pencil which sides are the Prime's distributed between how many of them are then that's not too difficult to answer but then they should be roughly evenly split between those so there might be a few sides where there's just one prime but the cases where there are infinitely many they should be roughly evenly split if you assume the Elliott Halberstam conjecture if you assume a strong enough problem that nobody could do let's let's be clear about this then Maynard and polymath and Jiang and so on leads to better results like there are infinitely many pairs of primes that differ by at most 12 or even 6 if you assume enough so so one way of solving this problem is going to be to make some progress in the Elliott halberstam conjecture I'm watching that seems very difficult but you know mathematicians do difficult things all the time if you only worked on things that were easy nobody would ever make any progress so two is going to be really difficult it's sort of completely clear there are technical reasons it's sort of understood to some extent why two is difficult showing that infinitely many pairs of primes that differ by just two is a really hard problem but mathematicians are getting a lot closer and every time mathematicians get a bit closer we understand a bit more about how the primes behave we have some new techniques to apply you can take those techniques and apply them to other problems we can take techniques from other problems and apply this to this one so there's lots lots to keep working on there you can go home and have a look at the polymath blog the wiki of the league table see you can get involved right might need to learn a little bit more maths before you get involved but why not so I thought I would leave you with something to think about as a warm-up before you before you get to UM proving into a Prime's conjecture because I've got no idea when the twin prime conjecture is will be proved I sort of believe it will be I I'm inclined to believe that it's a true statement and I believe mathematicians will prove it I have no idea whether that will be next year whether it'll be in 30 years time whether it'll be 150 years time I've got no idea maybe it will be one of you who knows here's your warm-up problem so I talked about twin primes like 3 & 5 & 5 & 7 these pairs of primes that differ by just 2 so 3 5 & 7 it's like two pairs of twin primes glued together right there sort of three numbers with spacings of two each time so my question for you is are there any more triples like that and I think that's a good point for you to stop thank you very much what happens if you prove it sorry what do you yeah so nobody's going to come along say here's a big fat check or something so proving the twin prime conjecture would be a really exciting moment so the person who does it will go down in the history books as having proved this very famous old problem somehow mathematicians very excited when people solve problems but they're excited by seeing what the solution tells them about more mathematics so so what was exciting when Andrew Wiles proved Fermat's Last Theorem was not so much the Thurman's Last Theorem was true I mean that was great but yo what was exciting was that his proof led to all sorts more mathematics so the tools that he used to prove Fermat's Last Theorem have led to a whole kind of industry of mathematicians say oh I wonder whether I can take this idea that he had and apply it to this other problem what does this piece of work that he's done in the course of his proof tell me about this other thing so they're going to be all sorts of consequences of whoever comes up with a proof of the twin primes digits which might be one people person people one person it might be some massive online collaboration it's then going to be really exciting to see what comes out of that so knowing the twin prime conjecture is - is great would be great but it's also what can we do with that and there are prizes in the mathematical community and it seems possible that the person who proves the twin prime conjecture might get a prize if it's one person or a group or whatever but it's much more about what does this tell us about mathematics how can this tell us more about prime numbers I hope that's enough and incentive to work on it yeah how do you so put you're willing to assume that's a good question so somehow you want to assume as little as possible I mean ideally you only use things that are already known to be true best possible scenario is that you take lots of theorems that people have already proved and you put those together along with your own ideas and deduce something so you only start assuming other things if you really have to and I mean sometimes it's just the yeah yeah so people it becomes clear if only we knew the Riemann hypothesis was true we could prove this then somebody else might come along say well actually look you've had some really fantastical ideas here that's great I can see a way to get round that assumption so so it's not a disaster to publish a paper where you say suppose this thing is true and if you say suppose the Riemann hypothesis is true and then prove some stuff and somebody then comes on and proves the Riemann hypothesis is false that doesn't mean that what you've done is wrong it just means that you can't use the Riemann hypothesis to did you said so yeah I mean when you're doing your homework you have to be a little bit careful that you know you don't assume something your teacher wanted you to do but it's a well-established way to make progress see well what would be the consequences if we knew this what could we do with that so when people make conjectures they usually have some reason for thinking it's true so when people were counting Prime's up to a million or a billion or whatever and then making a guess for how that function PI of X behaves that was based on some kind of ideas but that wasn't a proof so you don't just say well suppose the following random thing is true then I could do some stuff you sort of want to write down something you've got some reasons to think might be true that's why you come up with a conjecture so I mentioned there were no reasons to believe why the twin prime conjecture is plausible but you don't have to have a proof when you say when people say suppose the Riemann hypothesis is true then saudade that's not because they have the outlines of a proof of the Riemann hypothesis it's because there's lots of other evidence from other places for why it might be true but that's not a proof as such yeah but it's a good question you know one to just kind of assume any old thing and then see what would be the consequences necessarily yeah that's a good question to the best of my knowledge the real life applications of proven to employ mr. Jetta are currently non-existent so I've completely unapologetic about this because for me it's really important the mathematicians some mathematicians are working on problems because of the intrinsic interest of the problem because within the context of pure mathematics making progress in these problems is really important of course there are mathematicians whose work does relate directly to real-world problems I have lots of colleagues in Oxford who are busy working on things that having the applications in engineering and biology and finance and so on but it's really important that we do these things as well because of the unforeseen consequences so the classic example is cryptography maybe you're not old enough to have a credit card I don't know so when I'm using my credit card on Amazon buying another exciting maths book I'm typing in my credit card number I want Amazon to read the credit card number I don't want the baddy along the way to read my credit card number because then they can use my credit card details to buy their own maths books right there would be a bad thing so I need some cryptography there that one of the standard techniques used for encrypting data with online shopping is based on mathematics that goes back to firma in 17th century an Euler a little bit later they weren't doing it because they were interested in online shopping they weren't even doing it because of the cryptographic applications they were doing it because they were really interested in properties of whole numbers in structural relationships about the whole numbers centuries later somebody came along and said look this is fantastic we can take these tools from pure mathematics and apply them it's very hard to predict what those applications might be or which things will turn out so maybe the twin prime conjecture in the future will have some application maybe it won't but maybe the understanding developed in the course of proving the term prime conjecture will lead onto something so for me it's really important that some people are doing that sort of blue skies pure mathematics just because it's really fascinating because we really want to know the answer as well as people looking for applications you you

Channel: Millennium Mathematics Project - maths.org

Views: 198,738

Rating: 4.7162471 out of 5

Keywords: Mathematics (Field Of Study), Prime Number (Literature Subject), Twin Prime Conjecture, Prime Numbers, Pure Mathematics (Field Of Study), Women in Maths, Math, Maths, Dr Vicky Neale, Number Theory

Id: _Nch1ho77gQ

Length: 44min 57sec (2697 seconds)

Published: Wed Jul 01 2015