The Badly Behaved Prime - with James Maynard

James Maynard is one of the young guns of modern mathematics. Specializing in prime numbers, he made his name for pioneering work on one of math's most famous problems, the twin prime conjecture. He's since made breakthroughs on a number of other problems, and he's accumulating an impressive collection of prizes along the way. In fact, since we recorded this episode, the American Mathematical Society announced he's won another.

They're giving him the 2020 Cole Prize in number theory. Not bad for a 32-year-old who, by the way, has already been a full research professor for two years. I met James in his office.

at oxford university let's start at the beginning little boy james were you like were you going to be a mathematician were you always like mathematical Were you a prodigy? Okay, I definitely don't think I would have counted as a prodigy. It's very interesting looking back, because at the time, I was always quite good at maths. Maths was...

probably my favourite subject at school most of the time in school. So it's easy to look back and say, oh, clearly I was going to be a mathematician the whole time. Somehow I was written in stone. But it certainly never felt like that when I was a kid. It was definitely never clear to me.

at a young age that i wanted to be a mathematician or anything like that or even that being a mathematician was a job that yeah sort of when i was choosing school subjects and things at each stage okay yeah maths is one of my favorite subjects so i guess i'll continue doing maths but it wasn't that

yes, I definitely want to be a mathematician or I'm clearly going to be a mathematician or, you know, maths is the one and only thing for me. Was a favourite subject of yours just because you were good at it and found it easy or because you got pleasure from it, you liked it? It's very difficult to say when you're... kid sort of you definitely like things that you're good at I think yeah I think it was a bit of both that I definitely liked it a bit more because I found it

more straightforward and i thought i was good at it and i'll get high marks and things in maths without too much effort so that's one thing that i liked about it but i definitely also genuinely liked just thinking about the ideas so i remember there was one time when is maybe when I was in primary school, I think, that I had music lessons, I was learning the violin.

I was pretty pathetic at learning the violin, but I was struggling along either way. And for one year, I was particularly pathetic at learning the violin because my violin lessons were always directly after my maths class. And I was... Constantly just thinking about the maths that I've been doing in the maths class when I was supposed to be playing the violin with my violin.

tutor. And yeah, this meant that I was just completely pathetic that year learning anything on the violin. So you did have this curiosity, though, to know more than what just was in the textbook and what you did in the lesson. You were like, oh, I like these ideas. I want to go a bit deeper.

Yeah, and certainly the more it went on, the more I got more and more into trying to think about these things more independently and go a little bit beyond the standard syllabus and things. So, yeah, when I was in primary school, it was more just...

thinking about what we've done and thinking about the ideas and trying to understand them a bit better and things like that. As I grew a bit older, it was being a bit more adventurous and trying to think out, hey, what's the extent of these things? What are other questions you can ask?

Did you have those other interests? Did you want to play cricket for England or be an astronaut and things like that? Oh, yeah, you bet. There were lots of different... aims for my life and mathematician was never one of them when i was a kid give us a few what were a couple of them um if i went if i went and met the little boy and said what do you want to be when you grow up what would the likely answers have been uh so

There was one phase when I definitely wanted to drive a tractor. I wanted to be an astronaut for a bit. I remember I had a big dinosaur phase and I wanted to be a paleontologist. So I think that would have maybe... already signalling me out as a slightly odd kid if you go and ask them what they want to be and i say oh i want to be a paleontologist i don't know i reckon there's a lot of dinosaur obsession these days among kids but yeah so yeah i think particularly maybe it was when i was a

younger kid i really liked dinosaurs and then i had another dinosaur phase around when jurassic park came out i think of course of course obviously you are like mr prime numbers these days do you remember Your early encounters with prime numbers? I mean, they're something you encounter quite early at school, aren't they? Did you have a fascination with prime numbers when you were young like you do now or was it just part of the whole math conglomerate back then?

again it's very difficult for me looking back to be honest that my sort of initial answer would be that I didn't think of primes as particularly different or particularly special but having said that I remember For my university applications, I explicitly put down that I was interested in number theory for my university application, despite the fact we weren't taught number theory.

at school and I just sort of learned it independently and so there was clearly something going on that was sort of fostering a fascination with number theory and prime numbers from an earlier age even though again I think I wasn't so aware of it at the time that i was maybe fascinated by these things without

really being aware of it because for me it was just all sort of fun different logical ideas were you surrounded by mathematical people was anyone in your family mathematical or like was it something you just picked up yourself or uh yeah not at all really so Both my parents were linguists. They were both essentially French and German teachers. So we'd go on holiday quite a lot abroad, but...

There was no one in my family who was explicitly mathematical or even scientific. Yeah, if you look at my family, I'm definitely the black sheep of the family. The rest of them are all... linguists and historians and things and i'm the one scientist so i stick out like a sore thumb in that sense at family gatherings then like is it hard for you to talk about your work and interest like do they are they extra interested because you do the different thing or are they a bit less interested

because it's sort of not their wheelhouse. There's a sort of niche interest where there's a very brief joke about me testing them to see if they can tell someone else. the sorts of things that i work on um but normally once i've made them sweat a little bit after that then it's accepted amongst everyone that we're not really

hear a family gathering to talk about maths, there's other things that we talk about then. So let's role play for just a second. If you were like your mum or your dad and I said, what does James do for a living? Tell me what you think their answer would be. Well, he works on prime numbers. So it's something to do with the pairs of the primes that are close together once or maybe infinitely close together.

Something like that, yeah. Prime numbers close together. They'd know what a prime number was, though. Okay, I'd hope so. Maybe that'll be the next question that found me gathering. Can you define a prime number for me?

All right, then. And just kind of in a condensed way before we get to you, to kind of where you're at at the moment. I'm assuming you've got good... marks in high school for mathematics do you perform well uh yeah i think i was good i was I was not one of these child prodigies who was miles beyond everyone else, but I was definitely getting good marks in maths, I'd say. And then where did you sort of study and go to university after school? So I...

Studied at Cambridge University. I was at Queen's College there. And yeah, that was when I sort of got more into sort of proper maths in a certain sense that there's a... fairly big difference between the sort of maths you learn in high school and the sort of maths you learn at university. And so I think the rigour of the logical ideas and things is when I sort of really was quite keen on.

Yeah, maths is the sort of subject for me. That really increased your passion. It wasn't like, oh, this is different to what I expected. Well, it was, but in a good way. Yeah, it was different to what I expected, but in a good way, exactly. And then when you finished at Cambridge, you did like a PhD? Yeah, that's right. So I then...

Came over to Oxford to do a PhD, did a PhD here in Oxford. And then after that, I had a few postdoc positions. I spent a year in Montreal. I spent basically a year in the US. partly in Berkeley, partly in Princeton, but otherwise I was also back in Oxford for quite a lot of the time.

Anyone who kind of follows what's been happening with twin prime conjecture and gaps between primes in this research that's been going on, it's kind of portrayed in the news or on the internet since Zhang made his original. breakthrough. It's been portrayed as this team of really amazing mathematicians, Terry Tao and a few others.

you know, really doing great stuff. And you're sort of portrayed as this one guy on his own on the other side, like sort of working in parallel and you're quite friendly with them, but it's always sort of portrayed as this. Manchester United superstars and this one guy on his own who's also having all these amazing breakthroughs. Is that the reality of it? Does it feel like that? Yeah, not really at all. So I guess...

They were working on optimising lots of the really nice ideas that Zhang had come up with, and I was thinking about something that was a slightly different approach that I'd been thinking about from... before Shang's work and things. And it turned out my thing worked out in a way that it was way better than I'd possibly expected and was therefore very relevant for the stuff that they'd been thinking about.

But then I joined their team. And so it was more as if I was recruited into Manchester United than I was the League One side that was beating Manchester United. All right. It wasn't David and Goliath then. It was more like...

It was me teaming up with Goliath rather than anything else. Okay, so these days it's all kind of one big group now, is it? It's not like they're doing what they're doing and you're still working on your alternative way. Yeah, so maybe the... big polymath project that we had as this collaborative project to try and reduce the gap down as small as possible has ended now so i don't think there's a team that's actively working on the problem because we seem to have reached

the natural limit of most of the ideas. And to really make big progress, we'd need to have a big new idea. But I think this is also very representative of how mathematics works, that in some ways it's very collaborative. You have lots of... colleagues aboard and things and you're trying to come together to mesh ideas together and then understand new ideas when they come out but equally there's lots of individual discoveries that happen and that changes

slightly how you think about things and you adjust to the new ideas to try and incorporate them and try and bring as much together as possible to always get the best results so just so i'm completely across where

the twin prime conjecture is now. This is this elusive proof that there are an infinite number of primes separated by just two. Where's that now? You're jammed at a higher number, aren't you? There's an infinite number of... primes but they're separated by yeah no more than 246 so we know there's infinitely many players of primes that differ by at most 246 so if we could bring 246 down to two

we'd approve the twin prime conjecture. But unfortunately, we're pretty stuck now at 246. Is this something that people, and in particular you, are still working on? Is this like your number one... thing you think about when you go to sleep at night or you're moving on to other things like where where's this at now is it going to get to two in my lifetime uh okay so it's a fool's game to ever guess how long a conjecture should be open

for when things are going to get proven I remember one person telling me that if someone ever asks you how long will it be until we prove a certain conjecture then You should always say, well, the length of time that a conjecture has been opened for, because if you're just randomly disputed in time, you're most likely to be around the middle of when a conjecture is. But that doesn't even work for the twin prime conjecture because there's...

people don't really know how long it's been open for. It's certainly over 100 years old, but some people have speculated it could be thousands of years old. So even if I was trying to use this... way of suggesting how long it should be open for um it's not clear what the answer should be it could be thousands of years it could be thousands of years for all you know that

Again, this is another feature of mathematics that somehow we make big breakthroughs, but then we get stuck and we're stuck on patos for quite a long time. It's clear that for the twin prime conjecture, you definitely need some big new idea from somewhere. In my day-to-day research, I'm thinking much less directly about twin primes or small gaps between primes now, and thinking more about developing other techniques for understanding the primes in different ways.

maybe ultimately would lead to some new insight that could be combined with the previous ideas to get Results about capturing primes. Is twin prime then like a touchstone for you? Every time a new technique is developed, do you go back and think, does this help with twin primes? Or is it not like that? It's like, you know...

I'm trying to think, is this like a white whale for you? Is this like what Fermat's last theorem was to Andrew Wiles, or is it not a big deal to you? Okay, so, yeah, the twin prime conjecture has a sort of special place in my heart, and so, yeah, it would be... Amazing for me if someone proved it in our lifetime, that it's sort of my favourite problem in mathematics, I guess. However, I guess, unlike Andrew Wiles...

there doesn't seem to be any clear route to prove from the twin prime conjecture. It really needs some radically new perspective from somewhere. And so because of this, I... tend not to directly think about the twin prime conjecture so much but more a circle of different easier problems where hopefully we can build up enough of a toolkit to make progress but whenever someone does come up with a new idea

suddenly I guess we have a new hammer basically and everything looks like a nail so it's very natural for me to think through a whole list of different problems that I have in my head and to say hey we've got this new idea can this idea help with any of these different problems and so the twin prime conjecture is always on that list even if maybe it's a bit ambitious to ever hope that

one smallest new development could really make big progress on twin primes it's certainly one of the things that i mentally test against can i ask you about the age of mathematicians because you seem quite young to be an oxford professor i don't know if that's true or not but i mean obviously the success you've had has resulted in you having like quite a nice office here and a good job so congratulations to you you seem quite young but then we also hear that

I don't know if it's a cliche or true that the best mathematics is done by people when they're young, perhaps perpetuated by the fact they give the Fields Medal to people who are under 40. What do you think about that belief, that you're probably at your absolute peak now? Okay, I...

I'd like to think for myself that it's not just downhill from here and I don't really buy this idea that mathematicians do their best work when they're young that there's always a slight advantage of young people in the sense that their minds aren't set into ruts in the same way that you naturally get when you've thought about a problem for quite a long time and so sometimes new people can come up with

very different perspectives on things so often when you have a new mathematical breakthrough it's very much someone from who's not super familiar with the field who's coming in from slightly outside and that could be because they work in a different field or it could be because they're relatively junior in the field and so they haven't got

the same perceived wisdom and so they think about things in a slightly different way. So there's that one advantage but I definitely wouldn't buy this idea that mathematicians are at their peak when they're less than 40. On a gradual decline after that, I think there's lots of really top-level mathematicians who work well into their 60s and beyond, and much of their best work happens later on in life.

That makes me feel a bit better because I always feel really guilty when I ask you to do an interview because I feel like maybe I'm eating into a few hours of the few hours you've got left. I feel a bit less guilty. I want to get some idea how you do research. I always hear, like, because as a professor, you have teaching time. I understand what that means. You're in a lecture theatre teaching or you're with a group of students talking to them. But then I know you also have research time.

which is the most valuable time to you, I know. But what does that look like? How do you do research? If you said to your friends or someone, okay, I'm spending a day doing research today. It's a research day. What does that look like? Does that involve having a bath and thinking in the bath full of hot water, going for a walk around Oxford, sitting at your desk with a piece of paper and a pencil? Like, what does research look like when you're trying to have a breakthrough? Yeah, so...

Most of the time it'll be me working at my desk with a pen and paper and there'll be some mathematical idea that I'm working on and I have a project on at the moment that I basically don't understand at all. And I'm just... trying out different ideas to test what on earth is going on that somehow... So like calculate, are you sitting there...

calculating or manipulating equations or... Yeah, so I'll be playing around with equations and things on pen and paper and sort of writing down rough ideas and maybe trying to visualise things. So it's often a... very good tool if you can somehow draw a picture somehow that seems to me at least to help me try and understand what's going on but i've certainly heard mathematics described as you sort of spend

six months blind in a dark room fumbling around trying to look for a light switch and you're just spending the entire time tripping over chairs and things and then eventually you find the light switch and then everything is straightforward and you go into the next stream and then it's another six months of bumbling around in a dark room and so I'm very much doing this chaotically trying to find

understand what on earth is going on with these things that I don't understand by manipulating equations and looking at toy examples of the problem. And most of that is pen and paper in my office. Is there anything you do to... create a better environment like do you do you take the phone off the hook do you switch off your computer do you play music do you have to have a glass of coca-cola like what what are the things that you do that will that help you um i guess i have

a few different sort of little ticks if you like that um i find that going for a walk often helps clear my mind a bit that um it gets pretty depressing if you've spent three hours straight sort of intellectually banging your head against a brick wall and so going for just a wander around the maths department just the building yeah just the building it doesn't have to be a long walk but i find it helps sort of clear my head a bit i

drink water semi-obsessively as a way of sort of trying to clear my head again in some way um so lots of it is things like that that i yeah walk in and trying to distract myself a bit because There's some very strange process that I never quite understand where I think the subconscious is doing a lot of the thinking for you. So I'm consciously trying out a few ideas, but the real sort of...

Building up intuition, which is the most important thing, is somehow done on a slightly lower cognitive level. And I feel like I need to give enough space for that somehow. by going for a walk and drinking water and clearing my head slightly to try and process some of the things that I've encountered during the day. You talk about this moment when you flick the switch and everything's in place.

That must be an amazing feeling. And you've had quite a few of them early on. What's that like? Yeah, it's a big mixture of emotions. So one thing that's always slightly depressing with maths is that you spend... 99 of your time really not understanding what's going on at all and then suddenly you have the light bulb moment

But then you completely understand what's going on, so it doesn't seem complicated at all. So I look at other people's work and I think, oh my goodness, they're so clever. They're doing all these really complicated things that I don't understand at all. And all of my work is super trivial. It's really easy because I... understand exactly what's going on and it's really clear and this is obviously just the sort of i hope it's just the different perspectives that you naturally have but

Yeah, that's one funny thing with maths, that somehow everything that you understand is easy and everything that you don't understand is impossibly complicated and there's no in-between grounds at all. Similarly, if you have proven a result that...

sort of you know other people have tried hard to prove and things and is a cute result in itself it's really rewarding when you feel like you have this you could break through moment and so I tend to get this big adrenaline rush and I get very excited but I also get pretty scared. So after this wave of adrenaline and euphoria, I also have this big fear that I'm about to claim a big mathematical proof.

which has some really obvious flaw in it, and I'm going to destroy my reputation in the mathematical community. And so I start getting very worried that maybe I've made some really idiotic mistake. How does one avoid making the idiotic mistake? You just go through it again, go through it with some collaborators and check your work. Yeah, so it's a very important mathematical skill that you gradually develop.

about being sceptical about your own work and having an intuitive idea of what feels right and what doesn't feel right when you try and do proofs. So when you're very new to a problem, often your intuition isn't... too well developed and so it's much easier to make small mistakes and things but this mathematical intuition I find is a very good guiding process as to whether i'm on the right track or whether i'm not on the right track and again this is just formed through trial and error

But there's also a general mathematical skill about being sceptical of your own work and being able to look at it critically and be like, hang on, have I really proven this? Or am I missing something here?

And that's something that's often quite difficult to develop, but is a really important mathematical skill to develop. Do you ever worry that the tap will switch off at a time where it seems like you're going through such a fertile period of ideas and proofs and that? Do you worry about it switching off or does it feel like it's just like it will continue forever? In the back of my mind, there is a small fear about this that, yeah, I guess at the moment I feel like I have...

many more problems to work on than I have time. And I feel that there's so many different, really interesting mathematical problems out there. And you really don't need to be a genius to make some decent... progress on lots of mathematical problems that if you put in the time and the effort and you learn the state-of-the-art techniques then there's almost always some like small variations that can gradually

inch forward the techniques so i feel that even if you lose a sort of creative streak or something then there should always be a decent supply of problems out there like grunt work sort of well But I think this grunt work is really important for the whole of mathematics. So grunt work sounds maybe a little bit negative about it, but it's easy to notice individual breakthroughs or a few amazing people who come up with singular discoveries.

but the context for which those discoveries always made are when the field as a whole has gradually developed techniques and has understood very subtly the limitations and of those techniques and how far those techniques can go and so Lots of the slightly less glamorous work is still hugely important for mathematics and you wouldn't possibly be able to have the breakthroughs without this grunt work, as you say, being done. So I feel that there's always going to be lots of...

fertile problems where you can make good progress and understand techniques and push things forward. But at the back of my mind, there's always this fear of what if suddenly the tap goes off?

That at the moment, as I say, I feel like I've got more things to work on than I have time to do. But often if I'm working on a challenging problem, I'll go three or four months with... making absolutely zero focus it won't be three or four months where i feel like oh i haven't proven the thing but i've understood the problem a lot better um i can go three or four months and feel like i've made

no progress whatsoever on this and so this can be a little bit disheartening and if you're certainly if you're not used to this certainly after a while there's a little bit of is it that this is just the natural process of research or is it that i've lost it And so there's always a small fear in the back of my mind. But as I said, there's so many interesting problems and so many different ways you can make progress and contribute to mathematics that I don't spend too long worrying about it.

Can I come back to prime numbers for just a bit? Yeah, sure. You must always get asked what your favourite number is. What's your favourite number? Two. Really? Yeah, so obviously I have to choose a prime number. Yeah. And... But two is somehow special in all kinds of ways. The reason we don't choose one as a prime number is because... if you did define one lots of people ask me why is one not a prime number you know it divide it can't be divided by anything other than itself but

Then when you have lots of results about prime numbers, you'd have to say prime numbers apart from one. But it is the case that we nonetheless have all these results about prime numbers where it says prime numbers apart from two, because two somehow... special and different to them so two is also a spanner in the works surprise yeah so one would be a huge spanner in the works um and most of the time things work okay with

two but often twos are spanning the works and so you say well we have to ignore two because somehow that's a badly behaved prime and so because two's special in this way and it stands out i think two would be my favorite number a badly behaved prime yeah That's a category I haven't heard before. Are there any other primes besides one and two that behave badly? So in some ways, they get better behaved the bigger they are. Yeah. So in many ways...

Two's the worst. I think the phrase that other people have used is two is the oddest prime. Right. But then three would maybe be the next... worse behaved and then five and seven and so on and so two just by virtue of being the smallest prime number is often the worst behaved prime number i've heard you give this answer before but i think i think i should ask you about it

Why is it important to research prime numbers? Why has it got a little bit of extra importance about it for us to understand these numbers so well? So primes are like the atoms of arithmetic. key point about prime numbers is that every whole number can be broken up into prime numbers and so if you can understand the prime numbers well then you can understand the whole numbers well so somehow to me

The whole numbers are the most natural possible things in the universe. The numbers 1, 2, 3 that we use to count are... the most fundamental objects in mathematics and somehow there's all these questions about whole numbers when you're looking at them being multiplied together, that boil down to just questions about prime numbers. And even though whole numbers maybe sound like the simplest questions in the world, they boil down to these questions about prime numbers.

that we somehow don't understand at all. So we have the most natural things in the world, but they're built out of these objects that somehow we don't understand. So there's lots of questions about whole numbers that... turn into questions about prime numbers so for example maybe one of the most famous questions ever about whole numbers was Fermat's last theorem which is asking are there solutions of x

to the power n plus y to the power n equals z to the power n. And this is now a theorem due to Andrew Wiles, but one of the, like... very basic first steps in the theorem is to say that because prime numbers generate all whole numbers If you want to show you that there's no solutions to x to the n plus y to the n equals z to the n, it's sufficient to consider only the case when n is a prime number. And it turns out that because...

prime numbers are special in many different ways. It's much easier to understand the situation when n is a prime number than when n is not a prime number. And so this is maybe... the very first step in the very long and complicated proof of Andrew Wiles on how to prove that there's no solutions to x to the n plus y to the n equals z to the n when n is bigger than or equal to 3. I didn't realise that, so I know it's an epic 200-page proof, but...

One of the key foundations of it at the start is actually he's only doing it for powers that are prime. Yeah. And again, you have this slight issue with... two coming up that uh when badly behaved too yeah when the exponent is two then you do get solutions to x squared plus y squared equals z squared because of course that's just pythagesis equation yeah

So of the various problems that are out there with prime numbers, twin prime conjecture we've already spoken about, like that seems like it might be top of... top of the heap, or Riemann hypothesis as well. Can you run me through the three or four biggest problems in prime numbers that you would most love to see chalked off? Okay, so I think the Riemann hypothesis is clearly the most important.

problem on prime numbers and that's the problem that I would really love to see chalked off. Are you able to describe what the Riemann hypothesis is with just your voice? Because I've seen you do it I think before quite... simply, way more simply than I've seen where people are drawing strips and graphs and that. How would you describe what the Riemann hypothesis is? To your linguist parents, maybe. Yeah, so I think of it as just a problem about counting prime numbers.

how many primes are there up to a million? There's a very natural guess as to approximately how many primes there should be. And the question is, how good is this guess? And the Riemann hypothesis... just says that this guess is actually very accurate indeed. And so if the Riemann hypothesis is true, regardless of what number you chose, whether it was a million or a billion or a gazillion, then we would have...

we would know that the number of primes up to a gazillion is this natural guess plus some very small error term, which is basically as small as you could possibly hope it to be. Okay, so Riemann hypothesis... Top of the list. What else have we got that you'd like to see knocked off? Okay, so as I said, the twin crime conjecture is the one that sort of sits specially with me. And yeah, I really like that because...

If you're interested in the distribution of primes, somehow the Riemann hypothesis is talking about the large-scale distribution of primes. How many primes are there up to a gazillion? Whereas the twin prime conjecture is talking about the small-scale distribution of primes. How many primes are there that... really close together so in some ways they're like the two extremes of the most natural questions you'd ask about the distribution so for me those two are both the most two

famous problems but the most important problems as well. Then for other problems it turns out that Golbat's conjecture is very closely related to the twin prime conjecture. So Golbat's conjecture is probably one of the most famous problems and crimes. But for me...

It's very likely that if you can solve the Twin Fine conjecture, you might well be able to solve Goldbach's conjecture. And so you kind of get two birds with one stone in this sense. They have two famous birds. If someone does that, they're going to get a prize or something. Yeah, very much so.

But it sort of means that because I'm in love with the twin prime conjecture, Goldbach's conjecture isn't so important to me as it might be to some other people and as famous as it is. Maybe the next most important problem for me would be... prime values are polynomials so are there infinitely many primes of the form n squared plus one this is another one of these very simple to state conjectures and

The answer, as with lots of these things, is, oh, it clearly should be the case that there's infinitely many times of the form n squared plus one. How could it possibly not be? But if we had... techniques that could handle this then there's all kinds of interesting questions again linked to solutions of integer equations and there's applications in groups here as well that would follow from questions about primes of the form n squared plus one or in general like

polynomials taking prime values so that maybe be my number three question the twin prime conjecture always seems to me like why is this so important like the reaman hypothesis and how many primes are there seems like, okay, I can see why that's a really important question. But the twin prime conjecture always just feels a bit like, oh, how many little freakish moments are there where two of them were near each other? Like, just like...

How many coincidences are there? How many times will I walk down the street and be lucky and find a pound coin? Like twin primes are nice. They tickle my brain as nice things, but they don't seem important. They just seem like... happy coincidences, and yet there's such importance is placed on this problem. Can you help me understand why it's important to understand these freaky moments where you have the consecutive, basically, primes?

Okay, so there's two reasons in my head. The first reason is that the primes are an interesting sequence, and if you can't get really good specific mathematical control over the sequence, you may be asked statistical questions, like how many primes are there? the Riemann hypothesis. But also, in general, if you want to understand a sequence, as well as, as I said, looking at these large scale statistics, you also often get a lot of information about small scale statistics.

how well they're distributed in short intervals and how often they come close together and things like that. And so if you can prove the twin prime conjecture, then it's likely that you'd know roughly how many primes there are that divide.

two but also how many differ by four how many differ by six and the question although the twin prime conjecture is stated as just are there infinitely many times you get these lucky coincidences it's quite likely that sort of the real twin prime conjecture is about how often do you have gaps of a certain size and it's all about a statistical understanding of when primes come close to each other so it's just

The same as the Riemann hypothesis, but looking at small-scale statistics rather than big-scale statistics. And that's what statisticians might do for any other sequence that comes out of physics or nature or something.

The second reason that I... So that's why I view it as a natural problem in and of itself, if you accept that primes are interesting. The second thing is that it mixes multiplication and addition, and... primes are somehow naturally multiplicative objects because they're defined by every number can be multiplied by primes put together, but then you have this plus two feature that...

means that there's this slightly mathematically jarring interaction between addition and multiplication. And lots of our... ideas behind it are trying to understand how multiplication and addition interact with one another and so for mathematicians the sort of technical side of things i think the twin prime conjecture is fascinating because it's really trying to understand on a

It's a sort of toy case of trying to understand a much bigger problem of how multiplication and addition interact with one another. And there's lots of very different subtle questions about how... this comes about and if you could prove the twin prime conjecture you've likely made a big advance on how you can understand this interaction between multiplication and addition which are the two most basic

techniques in mathematics, operations. When I think of prime numbers, I think of them as like individuals, seven and 13. Or if I see a big number, I'm like, oh, is that prime? You know, I'm... I often will get a big number and just put it into an internet search to find out if it's prime because I'm weird like that. But it seems like you're not particularly interested in prime numbers anymore as individuals. To you, they're just always P, you know.

P1, P8. Do you feel like you've moved on from prime numbers as personalities? It's almost like you watch football and you don't see, like, Wayne Rooney and the players. You just see, like, footballers. Yeah, this is certainly a bit like it. We know that primes are these really complicated numbers, and so...

trying to understand them on an individual level is just a hopelessly difficult task. And so we have to take this sort of more zoned out, blurred statistical... way of trying to understand them because this is the only way we have hope of getting any sensible answers and so the sort of overall patterns of prime numbers are definitely the sorts of things that interest me and not any individual

number itself where you write it down explicitly so when they announce a new prime numbers new massive prime numbers being found you're a bit like meh yeah it's it has some interest and it shows sort of the computation how far computational techniques and things can go and sometimes the ways the algorithms that they use to find the prime numbers are

saying something interesting about the prime numbers themselves, but I definitely view prime numbers as some huge family and I'm only interested in properties of the whole family. rather than properties of any one individual in the family. So if you write down someone's phone number or you see a number somewhere, you don't, in the back of your head, think, hmm, I wonder if that's Prime.

Yeah, I don't really do this. I definitely somehow notice numbers a lot more than I think a normal member of the public. So I remember there was a funny experience that my girlfriend was showing me her... old textbook from undergraduate studies where i think it's maybe a biology textbook i can't remember exactly and she was just flicked to a random page and was sort of showing me some of the things that she'd learned during her undergraduate studies and i looked at one of the diagrams

and the first thing I did was I zoned in on the numbers because I was trying to work out I think it was some picture of a cell or something and so it was very small and I noticed that two of the numbers were inconsistent with one another and so I pointed this out and

She found it completely bizarre that the first thing she did was open a random page of her textbook to explain some of the things we were doing. And because I just zoned in on the numbers, I noticed a typo on this textbook that's famous and is on its fifth edition or something. just because the numbers didn't quite match up. So I definitely have something slightly odd about noticing numbers, but I don't tend to test an individual number.

for whether it's a prime number or not. I'm normally much more interested in the sort of vague properties of numbers rather than the individual properties of individual numbers. Because you've had a few headline-grabbing... proofs and breakthroughs do you find everyone asks you what you're working on at the moment what's next actually they don't do that as much as

you think so I certainly get asked a little bit um oh what are you working on at the moment but I think this is normal for mathematicians that it's um sort of maybe British people talk about the standard opening line is what's the weather maybe mathematicians are like oh what are you working on at the moment

because you get a good feeling for what other people are working on so beyond that I find that people are always interested in what everyone else is working on and it's always good I find because I'm still relatively new to the subject and junior overall it's good to see how other people pick their problems and how they choose what to focus on and where they put their focuses but i don't think i get asked what am i working on much more than any other mathematician really

What are you working on at the moment? So at the moment I'm working on a result about primes and arithmetic progressions. So I've got a paper that I've been... writing up slowly over the past month or so so that's almost finished now so that's almost ready to go oh well i know what our next video is going to be about

That's all for today. I'll have links to some of James's work and his videos on Numberphile in the show notes. The Numberphile podcast is made possible by the Mathematical Sciences Research Institute. And this episode was supported by the audio engineering company, Maya Sound. Again, links in the show notes. You can support the Numberphile project on Patreon. Go to patreon.com slash numberphile. I'm Brady Harron. We'll be back again soon with another mathematician and another episode.