The Music of the Primes - Marcus du Sautoy

good evening everyone welcome to the second of the clay public lectures for this academic year my name is Jim Carlson I'm president of the clay mathematics Institute the clay mathematics Institute was founded in 1999 with the purpose of increasing and disseminating mathematical knowledge primarily what it does is to support mathematical research it has a large postdoctoral program every year it organizes a Summer School in a topic of interest in mathematics it supports two very important programs for high school students the Ross and the promise program and of course as part of its dissemination efforts it organizes by annually or twice a year these public lectures so this evening it's my great pleasure to introduce Marcos Dusautoir despite the French Sahni name he is a professor of mathematics at Oxford his specialty is theory of numbers and groups and symmetries he's a very talented gifted expositor of mathematics I has been involved in the direction of films is the author of two well-known books one is the music of the primes the other is has just appeared it is called finding moonshine a mathematicians journey through symmetry and I believe there will be a book signing after the event before introducing Marcus formerly I would also like to publicly thank Candice Bott Candice would you like to stand and be recognized Candice Candace does a superb job of organizing and publicizing these public lectures and we're very grateful forever so without further ado I would like to turn it over to Marcus thank you when David Beckham moved to Real Madrid a couple of years ago there was a lot of speculation in the British media about why he chosen this number-23 shirt to play in every newspaper had their own different theory most of the newspapers went for the Michael Jordan theory this one goes it's a real Madrid wanted to sell a lot of more football shirts or soccer shirts you call them here out in America I mean in America I understand that soccer is not a very big game and so Real Madrid wanted to try and somehow break the American market and sell more shirts soccer shirts over here so that you know in America you like things like baseball and basketball and so one of the most famous basketball players in the world of course is Michael Jordan they used to wear the 23 shirts so one theory was that the Real Madrid were just choosing 23 so you'd all go and buy it thinking it was a baseball shirt basketball shirt and and not actually a football shirt Real Madrid will become even richer than they already are so so that was one theory but the other said that's far too cynical actually maybe much more sinister reason why Beckham chose the number 23 if any of you know your history then you might remember that Julius Caesar was in fact assassinated by being stabbed 23 times in the back so some newspaper said it's a very bad idea for Beckham to be putting 23 on his back but being a math nerd as soon as I saw this number I said oh no it's a really interesting number because it's a prime number one of these numbers which is only divisible by itself and one and I'm particularly sensitive to prime numbers kusa about the same time as beckham's moved to Real Madrid I wrote this book called the music of the primes or all about prime numbers and my British publisher this is the British hard back cover and the British publisher let me choose my favorite primes to put on the front cover of this book so I live at number 53 my local bus in London where I live is a number 73 bus I only go on prime number buses never tunes anything else and my local football team that I support is Arsenal Football Club and we have a rivalry I'm sure you have it between teams here I don't know who to the Red Sox who's the big rival with the Red Sox here there you go okay so they four up for Arsenal its Tottenham Hotspur and Spurs we just stolen one of their players called Sol Campbell who plays in defense and we given him the number 23 shirt so to rub it all in to all my friends who are Spurs supporters I put a 23 football shirt on the front cover of my book now a lot of public media saw this is it aha here's the guy you can really explain why Beckham chose the number 23 shirt so I got asked onto various radio stations to explain my theory about the 23 football shirt and we have a you probably have it here in America we have a radio station called talkSPORT radio which just talks about sport all the time and I regard it as a real coup to get higher mathematics talked on this radio station it's one of my big achievements in life I think anyway on the way to the radio station I was trying to think okay well I've got to come up with a theory about this so what's my theory about why Beckham might have chosen a prime number shirt so I started to think well what's important about primes for me as a mathematician well Prime's are literally the building blocks of my subject if I take a number like 105 most of you I hope know that that's not a prime number divisible by five then I go down to twenty one times five twenty one still isn't prime I can divide that into three times seven times five but now I can't divide any further because then I got to these indivisible numbers the primes which built that number a hundred and five so so for me the primes I like to call them the atoms of arithmetic because there any the building blocks of all numbers for me they're a little bit like the periodic table I see we've got a periodic table on the wall here one of the most fundamental things in chemistry and the atoms which build all molecules starting with hydrogen helium and lithium well for me the primes are like my hydrogen helium and lithium there they're the building blocks of the whole of mathematics I started to look at real madrid's football team and it's clear somebody on the bed chat Real Madrid knew that primes are building blocks because all the key players in Real Madrid at the time they were all playing in prime number shares you had Carlos the building block of the defense he was in number three Zidane he was in number five Raul number seven Ronaldo at the time was playing in the eleven shirt so clearly this might purse on the bench said okay becomes a key player he's a building block of our team we have to give him a prime number shirt so the game in the number 23 shirt so that's my theory at least and and I can actually talk about prime number soccer shirts from some experience because I also play for a soccer team out in East London we called recreativo Hackney we chose a Spanish named kind of frighten the on position but actually as soon as they see is playing they know that anyway our team isn't big enough to have 23 players so I play in the number 17 shirt very nice prime of Fermat's prime I'm fortunate we play in the Super Sunday lead Division two I'm pushing our primes didn't do too well for us because you look for recreativo we're somewhere yeah right down here at the bottom so in there but don't worry this is the lowest division in London so the only way is up from here so on but my prime number sure did help me if you look at the goals for we scored 25 goals and one of those is my goal which I'm very proud of that Sam boy if you look at the goals against we had 64 goals scored against us and one of those is also my goal I'm trying not so proud about Sam anyway I realized that we had to do something about this and maybe Beckham was on to something so I persuaded our team for the next season to change our kids so we now all play in prime numbers 2 3 5 all the way up to 43 and it transformed our season we became second in the league we got promoted to the Super Sunday lead division to one where I'm first unfortunately we learnt that primes they only last for one season because we mean relegated back down again so I'm looking for a new theory now oops in fact Sam this is a t-shirt I got a company in London to design which is the quadratic equation that every footballer soccer player has to solve when they're trying to work out where to stand to knock the ball into the back of the net also baseball players when they they're trying to catch a ball they're actually solving quadratic equations to work out exactly where to stand you didn't think these sports people were clever but in fact they can solve quadratic equations in their head okay so that's my latest theory it's working quite well but we're still not doing fantastically this season now every summer Real Madrid get bigger and bigger and they want more prime number shirts so they come to me the mathematician and say can we buy a formula off you for finding these new prime numbers and then they'll be very surprised to learn that we don't actually have some magic formula to try and find these prime numbers in fact trying to find where the next prime number is represents one of the biggest mysteries in the whole of mathematics and it goes really to the heart of what it means for me to be a mathematician and when I'm at a party and somebody asks so what do you do I kind of dread this question and look forward to it slightly and I say I'm a mathematician and you can see their faces drop and they sort of back away very slowly and their glass gets very empty and they dash to the other side of the party but I'm very persistence I run after the machine oh no way we're a very misunderstood breed mathematicians them and I try to explain to them what I do is a mathematician all day is that I try to look for patterns I'm a pattern search I try and look for logic and structure in the kind of messy world we have around us and the challenge of the primes is somehow the ultimate puzzle in the whole of mathematics now I'm going to start by showing a little movie clip from one of my favorite movies called PI has anyone seen the film PI here yeah a few math movie buffs they're excellent um in this movie there's a mathematician called max Cohen who is obsessed with looking for patterns his particular obsession is looking for patterns in the decimal expansion of pi so pi is this number which starts 3.14159 then goes on and on and he's convinced that he's found the secrets to the stock exchange hidden inside this decimal expansion as the movie goes on he kind of gets madder and madder it's kind of typical stereotype of the mathematician in the movies it's that we always go mad by the end of the movie this is no exception I start seeing some cabbalistic messages from God start appearing inside this decimal expansion but I love his passion for looking for patterns and he starts every morning with his mantra of what it means for him to be a mathematician so here's a max cone from the film Pi it's got a great soundtrack as you say so here is the decimal expansion of Pi if you spot any patterns you can shout out I spotted a pattern there you see as you and I say I'm a mathematician at parties um I think that's what they think I do all day that I'm sitting in my office kind of doing long division to lots of decimal places and I had to show you I suit do something slightly more interesting three if you graph the numbers of any system therefore there are patterns everywhere in nature there are patterns everywhere in nature and I think that's a belief of the mathematician if that something is significant then there is some structure and pattern there to understand but sometimes that pattern can be a little bit hidden a little bit mysterious and with the primes I'm going to show you some patterns that are hidden inside these numbers and what we're going to do is to take max cons advice we're going to graph these numbers and we will see some rather strange patterns emerging in the primes I think I search for patterns is perfectly encapsulated in the kind of problems you probably all had at school when you're given a sequence of numbers and you have to try and spot the structure the pattern the logic behind that sequence to get the next number in the sequence so I brought a few challenges along see how good you are at pattern searching now obviously if you've got a maths degree or if you're studying for a math degree you're not allowed to play on the first two okay you can look at the third one but if you're not studying for a math degree don't have a math degree what's the next number in the sequence 1 3 6 10 15 21 28 36 very good and you've got the ease by adding you add to two number three and four and five and six and these are called the triangular numbers because you can view them in a very geometric way as the number of stones you need to build a triangle with an extra layer added on each time now mathematicians understand these numbers very well for example we have a formula which will help you to calculate for example the hundredth triangular number without having to do the work of adding up all the numbers from 1 to 100 mathematicians love looking for patterns but we're also incredibly lazy at heart so we like these kind of shortcuts these formulas to help us find these numbers without having to do hard work and you can build up the formula by putting together two triangles building a rectangle and then it's very easy to count things in a rectangle so these numbers we understand very well ok the next sequence um if you're not studying from math degree and you also have not read The Da Vinci Code then you're also allowed to play on this one okay so if you haven't read The DaVinci Code what's the next number in this sequence 34 very good you've got 34 by adding the two previous numbers together so 13 plus 21 gives you 34 21 plus 34 gives you 55 and these are very famous numbers called the Fibonacci numbers some and they're really nature's favorite numbers you find them all over the natural world so for example if you take a flower and you count the number of petals on that flower invariably it's a number in the Fibonacci sequence or sometimes double you get two layers of a flower on it and if it isn't the number in the Fibonacci sequence and then that means a petal has fallen off your flower which is how mathematicians get round exceptions so now we understand these numbers very well as well and we also have a formula which will help to calculate the 100th Fibonacci number without having to add up all the pairs all the way up to 100 it's a little bit more complicated than the one for the triangular numbers and involves taking powers of this magic number called the golden ratio which expresses the perfect proportions in art and nature okay so I can see you're all fantastic pattern searches and so here's a little bit more of a challenging sequence for you what's the next number in the sequence 2 9 10 11 13 16 we all thought it was so easy didn't you Fibonacci up top okay the mathematicians are allowed to join in I know so there are some professors of mathematics here they're allowed to play as well a little bit more difficult this one well if you could get that 26 to the next number in this sequence I recommend you buy a lottery ticket next Saturday because these were in fact the lottery ticket number winnings in September last year they in England so uh so none of you fell for finding it yeah I don't have some magic formula not as any other mathematician for finding the lottery ticket numbers if I did have a formula I wouldn't be standing here talking to you now I'd be sitting on some tropical island enjoying myself so I'm bushi so you have to be careful you have to choose your battles carefully because not everything does have patterns the point about the lottery is that there is no structure there for you to help to you to find the next number in the sequence so of course the last sequence here are the primes the primes the numbers at the heart of my talk which go from 19 to 23 then you get another big gap a big gap the biggest so far scene 23 jumps to 29 then a very another close prime 31 and I would say of all the numbers in mathematics these ones are the most important because as I said they're the building blocks of the whole of mathematics I compared them to the chemists periodic table now the chemists have done incredibly well with their atoms here we have this lovely table telling you all the stable atoms from which you can build molecules now they've also done another fantastic thing which is to build a machine a spectrometer which if you give it a molecule can tell you what the atoms are that built that molecule so how well are the mathematicians doing compared to the chemists well let's take that second problem first so what about some a machine that's if you give it a number will tell you the primes which built that number well this is in crap in fact a very difficult problem and to show you how difficult it is I'm going to give you a little challenge for the course for this lecture here's a number nine million nine hundred and ninety nine thousand nine hundred eleven this is not a prime number it's a bit like salts it's made out of a sodium and chlorine but you have to find the sodium and chlorine which built that number and as an incentive I've got a bottle of champagne which I will give to the first person who finds the two primes which built that number now I notice some people came in with their laptops so they me I was hoping to go away and enjoy myself tonight but anyway there's your challenge I only have one bottle it's the first person to shout out and gets the bottle cooling in my minibar in so it's warming up in the light so if you want it you solve it quickly and you'll be cold okay I'll come back a little later to this problem and show you why if you can win that bottle of champagne there might be more prizes out there for you to win it you can really solve this problem okay what about the first problem about perhaps producing a periodic table of the primes well if you write down the primes in some sort of table the trouble is that the primes they don't seem to have any pattern to them if you look at here the primes from 1 to 100 and I've written it as a kind of heart beat so the the prime or the the heart beats every time it goes over a prime number and I can't really feel like the primes as are the heartbeat of mathematics they run our subject but you can see this is a this is a heart which probably needs to visit the cardiac Department because look at it jumps at 23 and has a big pause as the subject died no double beat there and then suddenly a beat there so the beat seems to be incredibly random and in fact mathematicians believe there's far more in common between the prime numbers and the lottery ticket numbers than between the primes and the Fibonacci and triangular numbers it seems to us hard to predict where the next prime is going to be as to predict the lottery the way the primes are laid out seem to be as random as the lottery this is deeply frustrating for me as a mathematician a pattern searcher trying to understand these numbers now in fact it wasn't mathematicians who were the first to discover these numbers at all but a curious little insect which lives here in North America this insect I can see somebody cheering over there it's that because you like this insect or oh great excellent good so we got a fan for this insect up the back this insect is a Secada and it has a very curious lifecycle this Sakura hides underground doing absolutely nothing for 17 years then after 17 years the cicadas all seem to know to emerge on mass simultaneously into the forest and they sing away here's the sound of one Sakura the sound of the cicadas is so unbearable you got to multiply this by about 100,000 of these things the residents move out of the area because it's so loud the cicadas party way they eat the leaves they may say lay eggs and then after six weeks of partying they all die and the forest goes quiet again for another 17 years before the next generation appears now 17 a prime number is it just a coincidence that they've chosen the prime number and well it seems not there's also another species which hides underground for 13 years and another species that strides underground for seven years 7 13 17 all prime numbers there must be something about the primes which is helping this Secada but what is it well we're not really too sure but the best hypothesis we have so far is that maybe there was a predator that also used to appear in the forest periodically and the predator would try and time its arrival to coincide with the secada now the secada that had a prime number line cycle found that it could keep out of sync much better than the predator um the predator then those are the non prime number life cycle for example if I've got a predator that appears every 6 years then the Secada that appears every 7 years won't meet it until year 42 but it's a condor that appears every eight years or every nine years is going to meet that predator much earlier and get wiped out so sins like the cicadas that had a prime number lifecycle were much better at avoiding the predator and seems me in a real competition that went on in this forest where the Predators perhaps found of the cicadas prime the Sakana had to push its lifecycle up and it's a real a good example of you know your mass then you survive in this world because this clever sacada found the number prime 17 the predator couldn't find it died out and now we seem to be left with these cicadas with this prime number life cycle and now it isn't only cicadas that depend on prime numbers for their survival one of my favorite math movies of all time is a movie called the cube has anyone seen the cube yeah great to form a few more movie buffs some math movie bus and this in this movie six characters wake up inside this cube shaped room and they don't know how they've got there but they start exploring the the cube shaped room has four doors or four four doors in the walls one in the ceiling one in the floor next or through these these these rooms and they find another room on the other side another cube-shaped room it's a very low-budget movie it's the same set just lit differently but Sam and but after a while they discover that some of the rooms are booby-trapped and oh yeah you've got it already have you gone in yeah I think that's right yes very good well there's my bottle gone Wow see ya ha okay very fast he's got one of those pesky electronic gadgets the other tier well I'll ask you later how you did that okay what you've got your computer to do but um okay so in it so these characters have to work out before they go into a room whether the room is booby-trapped and in this sequence they finally discover the key to working out whether the room is booby-trapped or not these on you anything's random why they here what are you doing school 1:39 time numbers I can't believe I didn't see it before see what it seems like if any of these numbers are prime then the room is trapped 6:45 you won't be surprised to find that they don't survive very long in this maze so 11 times 59 it's not over so that really safe you make their assumption based on one prime number track I'm not the incinerator thing was prime zero eight three the molecular chemical Feeney had 137 the acid room had 149 you remember all that in your head I have a facility for you beautiful three okay out of the way and if you all go away as excited about prime numbers as she is I'll know I've done a good job and if any of you have kids who are having trouble with their multiplication tables or prime numbers any teachers here and this is what happens to you if you don't know your Prime's okay so um if you do get nightmares I suggest you look away now but if you want some tricks to kind of terrify your kids then here here's what happens to you if you don't know your primes there are some younger people here who might not like this they're probably the ones who will like it the adults learn your primes children learn your primes I recommended our government in Britain get every school in England one of these cheese graders and we have numeracy sorted out in the UK so maybe you're your next president might take a trick from this one anyway so um the cicadas and the Hollywood seem to know about the primes but it's really the ancient Greeks that we credit with the first great discoveries about the primes and in particular this Greek the mathematician Euclid who proved what I believe is the first great theorem of mathematics um which is some the chemist you see they've made this table which lists all of their atoms maybe mathematicians could just produce a table of all the primes whenever I have a problem about the primes I go to this table and look things up and I'm finished well Euclid proved that anyone who tried to write down the primes in some great big table would be writing forever because he proved that there are infinitely many Prime's the primes never run out there are infinitely many of these indivisible numbers so I'm going to get my football team here in fact to explain this prime number proof to you this a little movie that I made which is on the web it's a five-minute movie but I'm just going to show you a little clip um how do we know how can I really prove my football team has not shirts from two up to forty three how can I actually be sure there's another prime number shirt out there for my football team if we manage to sign some new players this summer um how do I know we'll be able to always find another prime number shirt for him to play in I want to show that the numbers in my football team ahem to 243 suppose those are all the prime numbers there are perhaps can build all other numbers by multiplying that Prime's in our football team together eucla came up with this clever way to show why there must be a number which can't be built out of those primes room 223 what he did was to take home for up foot butchers multiply them together so you did 2 times 3 times 5 times 7 all the way up to 43 then here was his act of cheapness what he did was to add 1 to this number now can this new number the Euclid bills be built out of any of the primes in our 40 well no because if you divided that number of Euclid's by any of the numbers on our football team you always get remainder 1 and so Youkilis found a number which isn't built out of any of the primes in our football team so there must be another football shoe with a different prime number which is helping you to build that number that Euclid's built it's such a beautiful argument if any of you came to me and said look I've got a table with all the prime numbers in I can show you that you've missed some from your list I multiply all the prime numbers together in your table then I add 1 to that number and then this new number is not divisible by any of your Prime's because always leaves remainder 1 so you must be missing some primes sometimes that's a new prime number but very often it is not a new prime number it's just built by primes that are missing from your list even if you add those new prime so your list I can play the same trick again multiply the new list of Prime's together add 1 and you still miss some Prime's now the real trouble you could have proved that their infant many primes but he couldn't tell you how to find them and this was the great challenge for 2,000 years been trying to find a way to find prime numbers maybe we can find a formula to help us to find the primes and every formula never really worked now what makes a really great mathematician in my mind is somebody who can do lateral thinking somebody who can look at a problem in a new way and what we needed with the prize was somebody to take a new perspective I like this quote from Enrico Bombonera who's an expert in prime numbers here in American Princeton he said when things get too complicated it sometimes makes sense to stop and wonder have I asked the right question and maybe with the primes we're just asking the wrong question trying to predict when the next prime is going to occur where whether there's a formula for the primes and the person who changed the question when it the primes and found a way into finding patterns in these numbers was the great mathematician carl friedrich gauss now gauss was clearly going to be a great mathematician from the word go already at the age of three he was apparently correcting his father's arithmetic when he was handing out wages at the end of the week so perhaps not surprising that when he got to fifteen the thing he asked for his birthday that year was a book of mathematical tables kind of thing we'd like to get for our birthdays book of mathematical tables now this about a book of tables it's a book of logarithm tables now I just need to find out how many put your hand up if you know what a logarithm is please okay good excellent how many of you have actually used a set of logarithm tables yeah it's all the old people in the room now putting also put mine and the point about a logarithm and if you don't know what logarithm this is all you need to know what about it it's a logarithms with these kind of ancient form of calculator because a logarithm does something very clever it turns multiplication into addition multiplying two large numbers is quite difficult to get to do but adding two numbers together is very easy so these logarithm tables facilitated doing arithmetic and so merchants navigators engineers would use these tables in order to do calculations now it wasn't actually the logarithms which interested gals at the back of this book of logarithm tables was a book of prime numbers was a table of prime numbers and Gauss began to get obsessed with these numbers he couldn't understand any pattern to them at all for the rest of his life he would add more and more Prime's to this list instead of going out on the town in Goethe Kani to say no I'm sorry guys I've got to stay and do some more prime numbers and we'd had more and more Prime's for these tables and by asking a new question changing the perspective cows amazingly found a connection between the primes at the back of the book and the logarithms at the front of the book almost spooky that he got given this book with the two things together and he found a connection between them so what was a new question that gals ask that made him make this connection well girls said okay let's not get obsessed with trying to find the next prime trying to get a formula let's try and count how many Prime's there are okay you might say that's a very stupid thing to do because you cleared in my football team we've already shown you there are infinitely many primes so how can you count them well gal said no let's try and count how Prime's are there in the first ten numbers for example so you've got two three five and seven so there are four Prime's up to ten up to a hundred well that part beat that I showed you it beat twenty five times all the way up to 100 so there are 25 Prime's less than a hundred so what gauss wondered is is there any way we can predict how the number of primes grows as you count higher and higher now gauss like to view things graphically actually a time in mathematics when pictures were viewed with a lot of suspicion so here's actually a graph of this function so the height of the graph say above at hundred the height is 25 because it tells you there are 25 Prime's less than 100 and I'd like to call this the staircase of the primes because every time you go over a new prime number the graph takes a step upwards so for example we have a new prime number at 101 so that the graph would take a step up at 101 okay well what gal said was let's not get obsessed with them I knew Sharri of this graph graph let's take a step backwards and see if there's any overall pattern to the way this staircase is growing and the pattern that Gauss found is in this last column here so let me tell you what this last column records and so for example the number of primes less than 100 there are 25 Prime's less than 100 so this last column records the proportion of Prime's amongst all numbers up to that point so there it this means that one in for 25 out of 100 one in four numbers is a prime number up to 100 so that's what this last column here record so for example around 10 million here are the number of primes s and ten million well now one in fifteen numbers so this is the important column one in fifteen numbers is a prime number so I live in London my London telephone number has about that many digits so the probability that my London telephone number is a prime number it's a one in fifteen chance that my London telephone number is a prime number now of course being a math node as soon as I get a telephone number I always check to see whether it really is a prime number I know I moved house last a couple of summers ago and I had to change my telephone number so I phoned the woman up to get a new telephone number and she gave me this number and I put it into my computer and tested it and it wasn't the prime number so I said I'm never going to remember that ones can you give me another one so okay all right so she gave me another one I quickly tapped it in and still wasn't prime so I don't know where remember that either so she gave me about five numbers by which time she got so annoyed with me that she just gave me the next number which came out so I now have an even telephone number of all things hopeless but anyway I probably would have had to have waited for about fifteen numbers and one in 15 of those would have a chance of being Prime and what gal spotted was a pattern in this last column here and you can think of this in a way what Gauss was looking for was a good model to help him to predict the number of primes and in a way you can think of this as some say that it's the probability that a number is prime so for example around a thousand there's a one in six chance that a number is prime so what Gauss produced was a kind of model to help him predict the way the primes behaved and in some sense this is a slightly modern interpretation of this model but in a way this last column here sort of says well the primes look very random some maybe nature chose them with a set of prime number dice and there's a prime on one side and five sides blank and around a thousand she'd say okay with it's a thousand a prime toss the dice lands on the p-side circle that is a prime and now clearly a thousand isn't a prime the point is that Gauss thought this might be a good model to predict the way the primes might behave so another way to interpret this last column is it's the number of sides on the prime number dice as you climb higher and higher and work out where the Prime's are so the pattern starts to emerge here so around around 10,000 there are eight point one sides on the prime number dice okay eight point one sighs I'm happy with mathematical dice with eight point one sides on okay what happens when I multiply by ten I go from 10,000 to 100,000 the number of sides on the prime number dice goes up from eight point one to ten point four I've added two point three to the number of sides on the dice now multiply by ten again go from a hundred thousand to a million the number of sides and the prime number dice goes up again by adding to point 3 multiplied by ten here add to point 3 multiplied by ten okay I've got two point four but what happen is there's a bit of rounding up going on but essentially there's a very strong pattern emerging in the way the primes are thinning out I always seem to be adding two point three every time I multiply this first column by ten and here's the connection with the logarithms at the front of that book logarithms turn multiplication into addition so the logarithm is the key to working out when the probability that a number is prime you take the logarithm of the number and that will tell you whether the number what the probability is that the number is prime so the logarithm function is actually telling us the number of sides on the prime number dice the probability that a number is prime and that was Gauss's that was Gauss's prediction this is a guess that he made as a fifteen year old child absolutely astounding that he found this pattern this regularity in the way the primes thin out at the age of fifteen so the logarithm function tells us how many Prime's there are on the prime number how many sides are on the prime number dice and once you know what the probability that a number is prime you can start to make predictions about how many Prime's are going to get if I toss this dice thirty times it will tell me there are roughly five Prime's that I'm expecting to get so between a thousand and a thousand and twenty nine I can make a guess that there should be about five Prime's inside there I won't exactly know where they are but I can predict they're about five um so this is the number of primes that Gauss would guess there are less than any number in and this is a slight refinement that he made in later life something called the logarithmic integral you can see it honks the UM the shape the way the staircase of the primes grows quite well but that's a theoretical analysis of this dice and of course dice don't necessarily land exactly five times on the prime side you actually look how many Prime's there are between a thousand and a thousand and twenty nine you find there only four so there's a little bit of error cropping in so Gauss has got a good first guess at helmet on the number of primes emerges but it's only an approximation now that's good enough for an engineer but for a mathematician we like things to be precise we like to have exact formulas so how can we correct this formula that Gauss has and get an exact prediction of the number of primes that this dice is predicting well it wasn't gals but Gauss's student Riemann who found way into this problem and actually understood what's making this prime number dice tick how its distributing the primes now the best way to describe what Riemann did is to explain a little bit about the theory of music and this is why I've called the talk the music of the primes so I'm going to start to really push you now so brace yourselves for perhaps not understanding everything that I'm going to say now but I really just want to show you some of this mathematics because it's my favorite bit of mathematics in the whole of my subject and okay so scientists around Riemann's time discovered in particular a guy called Fourier discovered that just as Prime's of the building blocks of numbers the building blocks of sound and of graphs is a sound that a tuning fork makes so here's a tuning fork here whoops well I've got it in here let some air through here now a tuning fork when you record the sound of a tuning fork on an oscilloscope it produces a perfect sine wave and what charm scientists discovered is that these sine waves are the building blocks of all sounds so anyone who has an mp3 player what say what that mp3 player is doing is saving the sound of a band or an orchestra and breaking it down into these sine waves the building blocks which build up the sound of that Orchestra or band so for example a violin the sound of a violin when it's praying an a it sounds very different from the tuning fork very sort of sharp sound to it and the oscilloscope records a completely different shape it's almost like the saw the teeth on a sore the point is the violin isn't just playing the a of the tuning fork it's also playing a lot of other harmonics as well a lot of other sine waves so in fact the violin is playing a lot of tuning forks all together so the first tuning fort you're hearing is the the this sine wave here which is safe essentially the sine wave which fits from the bridge of the violin to the top of the violin here there's another sine wave which fits here which is vibrating twice as fast it has half the wavelength and that produces a note an octave higher which is also contributing to the sound in fact you're hearing all the sine waves which fit perfectly in the length of the violin I don't know whether I can do something clever oh I have to get closer to it there but essentially that's the the sine waves which fit the length of the violin are all the notes you're hearing and if you combine those sine waves together so I'm going to add the graphs of these sine waves together you see the the shape of the tuning fork gradually turning into the shape of the sawtooth of the violin so what I'm going to do is here's the the tuning fork the first approximation to the violin then I'm going to add another sine wave which is vibrating twice as fast so you see it's going to push this bit of the graph up because going to add this then it's negative here you're going to pull this bit of the graph down so I sad I add on all of the harmonics so the stir of the violin the graph gets pushed and pulled until the the tuning fork turns into the sawtooth shape of the violin and sudden your hearing if you play all those sine waves together you'll get tricked into hearing a violin in fact of course that's what my computer is doing it's vibrating the loudspeakers but all of those sine waves together and you think you're hearing a violin now this is very important for example for a trumpeter a trumpeter especially before barks Sara a trumpeter didn't have bowels and couldn't change the length of the string with length of the pipe and so he would have to depend on the harmonics to get more notes so for example I'm not going to change the length of the pipe but by pushing more energy through the trumpet you're going to hear the different sine waves which essentially fit the length of the trumpet so those are essentially the sine waves which make up the sound of the trumpet and trumpeter has to use these sine waves a lot whenever he's playing some music so just to give you a little light entertainment here it's a little bit of music which depends on using these harmonics in order to be actually able to play something two one two three four afraid the band's had a chance to warm up so they were a bit better than even well each instruments has its own different harmonics so for example the clarinet which has a very close sound much not a sharp sound like the violin and you can see this on the oscilloscope it has a much square shape and this is because the harmonics which fit inside the clarinet they have to be open at one end and closed at the other one so we get sine waves of different frequencies which build up the sound of the clarinet so actually you have the first wavelength here then you have one which is a third the wavelength of the original frequency so clarinets actually like odd numbers and if we combine the different frequencies of the clarinet together the harmonics and we get a different shaped grass so both instruments start with the approximation of the sine wave but then let's add on a frequent a sine wave which is vibrating three times as fast so it goes down in the middle now it's going to pull the middle of the graph down and the tops get pushed up and so instead of getting a sawtooth shape the clarinet with these different harmonics gets a much more square shape onto its graph for the sound okay what on earth is all this music got to do with the primes well what Riemann discovered is the primes also have strange harmonics hiding behind them which helped correct Gauss's guess so Gauss's guess you can think of a little bit like the tuning fork it's the first approximation to the primes but what Riemann discovered using some very powerful mathematics involving something called the the zeta function complex numbers by this a map act of mathematical alchemy he produced essentially these sine waves which when you add them on to the Gauss's guess they kind of push and pull Gauss's guess in a similar way to the clarinet emerging the sound the clarinet emerging from the tuning fork adding on these harmonics actually correct Scouters guessed such you get an exact form so there are slight variations on sine waves and essentially if you know a little bit about where this is coming from these essentially each note corresponds to something called a zero of the Riemann zeta-function and if you add on all of these harmonics on to Gauss's guess you get an exact formula for the Prime's so I'm going to show you a little animated version of Riemann's formula for the primes now this if I had to choose one if eyes cast out on a desert island I had to choose one formula take take with me on that island it probably would be this formula of riemann so it's my favorite formula so i wanted to show you an animated version of it um so what Riemann's formula says is that okay want to count the number of primes that's the blue staircase here going up here that is equal to galsses guess which uses the logarithm function in these dice that's this yellow graph here especially a slight improvement that riemann made but that's not exactly it then we have to add on the sine waves that riemann found using the zeta function so the sine ways we're going to add on these sine waves and then you get an exact formula so I'm going to show you an animation where here's Gauss's guess and they're going to add on a sine wave the the harmonics that Riemann discovered one by one so remember they're basically waves which go up and down like this so I'm going to add them onto the graph so when the graph is going down it's going to pull the yellow graph down because it's saying okay you've slightly overestimated the number of primes come down a bit when the sine wave goes up you have to push the graph up which says no you've underestimated now push it up a little bit so I'm going to add on one at a time these harmonics that Riemann discovered and gradually you'll see the graph being pushed and pulled being approximated better okay a little bit more here pull it down here and after you add on about a hundred of these harmonics this is the prime numbers from 1 to 100 after 100 of these harmonics you've managed to push and pull the graph until you've got a pretty good match for the number of primes and if you add on the infinitely many of these harmonics you'll get a precise formula for the number of primes so now with these somehow hidden inside these harmonics are exactly where this prime number of dice is landing I mean with Gauss's guess you couldn't tell there were no Prime's between 23 and 29 but somehow the harmonics have hidden inside them the way the prime number dice are distributing the Prime's now there are two important pieces of information your iPod nee-san about a frequency when it's building up the sound of a band one is the frequency of the harmonic but also how loud you want that frequency to play is it a very loud contribution or very small one so riemann essentially looks at the same thing for the primes he made a graph and said okay well let's plot on the the the the vertical the the frequency of the notes so that's essentially how fast it's vibrating and on the the horizontal here we're going to chart how loud the harmonic is so that's really the amplitude of the wave how big is the sine wave is there a loud one it's a very small one when Riemann plotted ten of these notes he did calculations he managed to calculate which where ten of these notes would lie he found that they weren't scared all over the place with some notes playing higher than a louder than others some making big log contributions he found that all the notes were lining up in some straight line all the notes seemed to be in some perfect balance they were all lining up with the same kind of volume now Riemann believed this couldn't just be a coincidence there must be that all the notes would be slowing on this line and Riemann made a prediction it's called the Riemann hypothesis that all of these notes would be playing at the same volume now for 150 years we've been trying to prove that Riemann was right about these notes the volume of these notes are somehow in this perfect balance and we haven't been able to prove it it is our greatest unsolved problem it's one of the reasons that the the clays chose it as one of the millennium problems it probably is I would say the greatest of those seven millennium problems and what the amazing thing is the Riemann hypothesis if it's true there's this pattern in the primes in some ways it will explain to us why we're not sorry there's a pattern in the notes it was sometime how I explained to us why there aren't any patterns in the primes because if there was a note off that line it would say that one of the notes is playing much louder than any of the other notes and Enrico Barbieri describes it really nicely he says this would be like listening to an orchestra where everything's in a beautiful balance and then suddenly one note comes in one instrument a tuber comes in and blasts out the rest of the orchestra and all you hear is the tuba now if that were true it would really force a very strong pattern on the primes and say there are low surprise over here and hardly any here so the fact of the notes are somehow all playing at the same volume all playing the same role it's kind of distributing the primes in a very fair way with no sort of patterns emerging in them so a pattern in the music will in sense explain why there are no patterns in the primes another way to interpret the Riemann hypothesis is that the primes are a little by litte bit like the molecules of gas in this room now I don't exactly know where each molecule of gas is but I do know that if I go into the corner here I'm not going to suddenly find a vacuum and I'll die and collapse because I've got no oxygen and there's a concentration somewhere else so the Riemann hypothesis will say something very similar to the way that for the primes about the way the molecules of gas are distributed here the Riemann hypothesis doesn't really help you to tell where the primes are what it does say is the primes are fairly distributed around the universe of numbers so you won't find somewhere where there are hardly any Prime's and actually says very equivalently that these prime number dice are a fair set of dice so the errors that they're making either way of the theoretical number and what you'd expect from a fair set of dice and so it's very intriguing that the randomness of the primes I said at the beginning of that a mathematician is a pattern searcher Nature gave us these numbers which didn't seem to have any pattern to them at all just noisy numbers but by changing our perspective looking at things in a new way going through the Gauss's dice with these logarithms through the music of riemann we suddenly found where the real pattern is the pattern is in this music are not in the primes themselves so it's really the triumph of the mathematician somehow over nature and now if you want to discover a little bit more about how far we've got in the in the search to prove this theorem the strange connections now with quantum physics the frequencies of these any of these notes of Riemann's seem to share a lot in common with frequencies and energy levels in quantum physics on so here's your guide to winning with the clay million if you want now I study the primes really because I think they're beautiful because there's something very Universal about them if there was another lecture going on and the other side of the galaxy we might all look very different our chemistry might be different our biology but the primes would still be the pro same primes they'll be the same numbers we're talking about but now there's a very practical reason for study the primes and that's why the problem that I set right at the beginning to crack these numbers into the prime constituents this problem is now at the heart of the codes which are being used on the Internet to keep credit cards secure actually this code was developed here at MIT by three mathematicians who realized that the primes you can use the fact that we don't really understand the primes to build a code that is really hard to crack so every time you go onto the internet and visit a web site and you want to send them your credit card what you're getting is a public kind of a telephone number for that website something like this number here and your computer does a calculation with your credit card and this public number here scrambles the credit card number up sends it to the website to unscramble that calculation you don't need to know this big number here the two primes which built that number those Prime's are kept very secret by the website they're what undo the calculation so ed what our gentleman here did he won a bottle of champagne but watch out because he might be after your credit cards because if he's got a clever way now how did you actually you presumably got your little machine just to check one prime after another until it hits 307 and then suddenly the thing breaks into twos but Google exactly well what's Google doing Google is basically doing exactly that okay well yeah he's definitely onto your credit-card side watch it huh essentially the only clever ways we know so far as mathematicians we have slightly sophisticated more sophisticated ways is it just tribe one prime after another this number is small enough that Google or even somebody on their pocket calculator would crack it very quickly but of course websites are using numbers with now a couple of hundred digits and to be able to use that method to crank that into primes will take you the lifetime of the universe to be able to actually find the primes which build it but maybe we are actually found a proof of the Riemann hypothesis it might tell us many more things about the primes whenever you prove something it doesn't just prove your theorem it generally shows you many more things beyond that as well so it's a possibility that the Riemann hypothesis may give us an enough insight to be able to to build a prime number spectrometer which helped us to crack these numbers in seconds and I'm going to end with a sequence from another of my favorite math movie Sam called sneakers in sneakers there's a mathematician called dr. Yannick who has indeed found a way to find the primes which build all of the numbers he's programmed it into a little black box then the evil ben kingsley gets the black box shoots the mathematician and it's never good for mathematicians we are they get mad or we get shot by the end of the movie this one we get shot but then Robert Redford of River Phoenix rescue back this little black box they plug again to their computer and this is the kind of thing you'll be able to do if you understand the primes well enough air traffic control system okay mother oh my gosh target systems are based on mathematical problems so complex it cannot be solved without a key Janek must have figured out a way to solve those problems without the key and he hardwired to intervention turn it off so it's a Code Breaker No it's become no more in a surprise announcement the Republican National Committee has revealed it is bankrupt a spokesman for the party said they had plenty of money in their accounts last week but today they just don't know where the money has gone but not everybody is going begging Amnesty International Greenpeace and the United Negro College Fund announced record earnings this week mostly two large anonymous donation well it could only happen in the movies couldn't it all could it thank you very much Marcus we probably have time for two questions yes your heart right she has afro accent yes yes a prime number sieve yes exactly that's uh thank you I'll remember that yeah question the back do I think you'll be solved in our lifetime that's a very good question um it's always hard to make predictions about mathematical problems I think when I cited my research I would have thought Fermat's Last Theorem was never going to be proved and then suddenly a new idea comes along and and you've gotta weigh in I would say that we're probably just one big idea away from proving the Riemann hypothesis now I don't know when that idea is going to come but but in some ways we've got a good a good idea of how we should be proving it there's a similar theorem called the Riemann hypothesis for curves which is not about counting primes but counting solutions to equations and we've been able to prove that we built a machinery which helps us to prove that particular theorem now we've believed that we've got to build the same sort of machinery for the primes and we'll find a way into it and in a way the connections with quantum physics are telling us the sort of mathematics that we should be using it's the sort of mathematics which gets used in quantum physics which is basically if you know what these technical terms is is an operator and looking at the eigenvalues of that operator like a big matrix and the matrix the way it sort of pushed simple space will have something to do with these prime so so in some ways I would expect to see a proof before we celebrate two hundred years of Riemann's prediction France next year it will be one hundred and fifty years since Riemann made that prediction next year is big year for Darwin the Origin of Species was published in 1859 but also the origin of the primes Riemann's paper was published in the same year as well let's give Marcus a big hand thanks

Channel: PoincareDuality

Views: 152,746

Rating: 4.8577075 out of 5

Keywords: harvard, math, Harvard University, mathematics, clay, institute, public, lecture, MIT, P=NP, Problem, Computer, Millennium, Prize, Problems

Id: PgqEaUT8Qo0

Length: 60min 7sec (3607 seconds)

Published: Wed Nov 23 2011