I'd like to start with a question. What exactly is mathematics?
What is it that mathematicians in the building here at the University of Oxford are doing all day up in their offices?
I think most people have some sort of idea of what it is that the physicists and the chemists and the biologists are doing.
Chemists mixing chemicals together, blowing things up, the physicists smashing things together in their collider.
What is it? What is it? All mathematicians do.
I think actually if you ask somebody on the street, most people's perception is that somehow we're doing some mathematics is about multiplication,
percentages, long division and as research mathematicians, well, perhaps we're in long division to a lot of decimal places.
I think perhaps, you know, we'd been put out of a job by a computer now,
but I think that's really a failure of the way that we teach mathematics at school.
I think in some sense we teach mathematics a little bit like we teach a musical instrument as if all
you're allowed to play are scales and arpeggios and you never actually hear a real piece of music.
And so that's what I'm going to have to do, say, in this lecture, is to play you some of the music that really excites me about mathematics.
And I think there's actually quite a lot in common between mathematics and music.
And when people ask me, So what is mathematics?
What is a mathematician? I tend to answer by saying, Well, I think a mathematician is a pattern searcher.
Mathematics is the science of patterns, trying to understand a kind of logic and order in the kind of chaotic world around us all.
A similar to the way that music is distinguished from noise,
from beings that patterns against the kind of chaos that you might hear, which is white noise.
So I would say that mathematics is a science of pattern searching.
So I'm going to start off with a few patterns to try and warm you up mathematically, to explore the idea of try and find some underlying logic.
So rather like those puzzles you probably had at school when you're in school,
I'm going to give you a sequence of numbers, and what I want you to do is to try and find the pattern in these numbers.
So I'm going to start over with a few easy ones.
If you've done a mathematics degree or a study for mathematics, were you not allowed to play on the first two?
Okay. So you can join in a bit later. But I'm not ongoing mathematics.
We will. What's the next number in the sequence? One, three, six, ten, 15, 21, 28.
36. Exactly right.
So these are the triangular numbers. Very simply, what you're doing is adding one plus two plus three or four plus five.
And we call these the triangular numbers because you can view them very graphically.
If you take the number of stones you need to make a triangle,
when you add an extra layer on each time and the number of sevens you'll need is given by this sequence, the triangle numbers.
And we understand these numbers incredibly well. For example, we have a formula which will tell you,
say the hundreds of triangular number on this list without having to add up all the numbers from 1 to 100.
And in fact, you can get it if you put together two of these triangles, you can make a rectangle counting things.
And a rectangle is very easy. So the formula is just half the number of things in that rectangle.
Okay. The next sequence, if you read The Da Vinci Code, you know the last page on this one.
I but what's the next sequence? Three things sequence. What's the next number in this sequence?
34. Exactly. And how do you get that set? I mean, it's 21, but I guess.
So there's a it's a a C series of numbers which grows out of the previous numbers in the sequence.
And these are a very famous set of numbers to the Fibonacci numbers.
So you get a sense, boy, 18, 13 and 21, 21, 34 gives you 55, and these are somehow nature's favourite numbers.
You find them all over the natural world. So for example, you take a flower and you count the number of petals in that flower.
Then invariably it's a number in the film, an orgy sequence, or sometimes you get double the number,
you sort of get two copies of the flower, one on top of each other.
And if it isn't the number in the Fibonacci sequence, that's because the petals falling off your flower,
which isn't how mathematicians get round exceptions.
So we we understand these numbers quite well, although there are still some mysteries about these numbers.
We have a formula to calculate. Calculate the Fibonacci numbers.
You don't have to Internet 170 number. You don't have to add up all the pairs all the way up to the hundredth number.
And it's a rather nice one which involves this magic number two, the golden ratio, which expresses a sort of perfect proportions in all its in nature.
So this is a very famous sequence actually discovered not by Stephen Archie verse,
but by Indian musicians and poets who realise these numbers also count rhythms that you can make.
Okay, what about next sequence? One, two, four, 16, 32.
Well, 32 is one off. So that actually the next number in the sequence is actually 31.
So this is a warning to you that this is why mathematics is rather exciting,
because sometimes you think you know where something's going and then it can take a rather dramatic turn in another direction.
And it's these surprises, I think, which I find so wonderful about mathematics.
So let me explain to you why things was equally good in answer as 30 to 32, you think it's just has a two.
But actually these numbers are describing something called the Circle Division numbers.
So what are these? Well, it only take a circle and I put one point on the circle.
I just have one region. But if I put another point and they join them up, then actually I can get I can divide the circle up into two regions.
So 2.4 divide seven into two regions. Now for the third points, if I join all of those, I get a triangle in the middle and four regions out round now.
So three regions around the outside. So about 1 to 4 regions put forth points on join up all the points.
You get this little envelope in the middle, four triangles and all regions around the outside eight regions.
So you think I started the pattern? Now I know what's going to happen next.
Put another dots on your circle. I'm sure now you've got five points.
Join them all up. You have this pentagram in the Pentagon in the mental count, the number of regions, 16 ranges.
And so by now you're absolutely convinced you know what's going to happen when you add the sixth point.
But here's the surprise it has to which controlling number of regions.
So where do you even try this later? If you put six dots, however you arrange in the maximum number of regions you can get surprisingly only 31.
So it is pairs of two, which is controlling this sequence.
And actually another formula that's slightly more complicated is you have to take the number of dots,
race to the power for calculate this polynomial and this gives you the number of regions you get.
So if I want to put 100 dots, this formula will tell me how how many regions all these I'm so a warning here.
They're all surprise. Now, there is the surprises, I think, which make mathematics such an exciting subject.
It's like a piece of music where suddenly you think you know where it's heading and then the composer takes you on completely different journey.
Okay, so warming you are. But here's a more challenging sequence.
So if you've got a master's degree or a professor of maths here at Oxford, you're allowed to join in on this sequence.
What's the next number in the sequence? Ten, nine, ten, 11, 13, 16.
A little quieter. You thought it was also easy?
Triangular number seven asked. You don't call Lenovo's.
The math professors. There are a few maths professors here. You're allowed to play.
5 to 5 active members. What about the teams in the nine?
A little the beginning, Megan. Well, I must admit, if you do manage to get that, 26 is the next number on the sequence.
I recommend you buy a lottery ticket this coming weekend because they've learned about the lottery ticket numbers for the 28th of September last year.
So another warning hat. So pick your battles carefully.
In mathematics, not everything does have concerns,
although even randomness we have managed to apply mathematics to be able to spot structures even in random sequences like this.
Unfortunately, I don't have a formula for these. If I did, I would be here talking to you now.
I'll be on my tropical island, enjoying myself with some. Okay.
So the last was two, three, five, seven, 11, 13, 17, 19.
23. Exactly. And these are the prime numbers.
Probably the most important numbers in the whole of mathematics.
And these are the individual numbers, a number which I can't abide by anything except itself.
And one. So 23 is the next number that we have wanted to be count up to 29.
Now, I would say of all the numbers in mathematics, these are the most important.
They're the most fundamental numbers yet, as I hope to explain to you,
they are also the most challenging sequence of numbers in the whole of all subjects.
These numbers are so important because they literally build all of our numbers.
If we take a number like 105, is that a final it now?
Clearly it's divisible by five. Exactly. So divided by five, I get down to 21 times five.
21 is that point. Now, that's three times seven, but now three times seven times five.
And these are now individual numbers. These are prime numbers.
These are the primes which build that number 140.
I you can do this with any number. Take your telephone number.
It's either going to be indivisible in prime. And if it is, your lucky is always off to one of those respects.
Or you can divide it and you can keep on dividing, dividing, dividing.
And so you get down into this.
So for me, as a mathematician, these prime numbers are honestly like the building blocks of myself in the absence of arithmetic,
because to me they're a little bit like the absence in the periodic table.
The case study is one of the building of the periodic table, solving puzzles able to date to new elements assigned here.
These are the items which build the whole of the molecular moment.
For me, competition and the primes that in my periodic table.
Yes, you look at the primes, the sequence of numbers, these primes build the subject to looking for patterns.
If I look for patterns on these numbers, there seems to be none.
There are two. I've written the primes as a piece and I really think that primes like myself.
So the whole piece, every time it runs over a prime number.
But this seems to be a subject since I have been drinking a lot of caffeine or need to go into the
cognac department because it seems very hard to predict when the next speech is going to occur.
You go 17, 19, 23. They want to be at 29 and a double be there.
And actually, I think many of us opticians feel that actually there's more in common between the prime numbers and those lottery ticket numbers.
Now, between the primes and the Fibonacci and the triangular numbers,
we don't have some magic formula which helps us to calculate where the next point is going to be.
And for me, these points represents one of the biggest challenge for the patent search.
Now, who do you think were the first to actually discover the primes?
And, you know, it is the first to discover primes. The ancient Greeks.
They certainly did some fantastic stuff. But no, they were the first to actually.
Egyptian Egyptians. They were one of the first to do mathematics.
They calculated PI, for example, but no, they were the first to say which one it was.
So now that's where they also did some great math. But at this point in time, when they were first discovered, the Brazilian Babylonians,
no doubt about it, is only the Chinese are going through pretty much the whole of the planet.
All you need is a little insects, which actually just drive away warring mathematicians.
This and this little insect actually discovered that the planes were incredibly effective for its evolutionary survival.
So this is called an extremely strange voice like landed in the forests of North America.
But this particular scholar has a 17 year old's say it on the ground, doing absolutely nothing for 17 years.
And then in the 17th year, all of these cicadas emerge en masse.
And here's the sound of one of these cicadas.
You have to multiply this by several million of these which appear in reports residency levels absolutely
unbearable because they cause you why they so these they they make like eggs and they after six weeks,
six weeks, they all die. For us as Colombian for another 17 years for the next generation of women around us.
Extraordinary lifestyle. First of all, how they managed to count the 17 years gone by.
There's nothing really in the Maxim cycle which has a 17 year period.
So it's sunspots, 11 years, but it's so the fact they even count 17 is pretty impressive.
Very rarely do any appear early or length. But what's curious to me is I 17 that's one of these prime numbers is that importance.
What is seems to be important because there's another species in another area of North America which has a 30 year life cycle.
So you only get these 30 or 70. There's never any 12 year cycle or 14, 15, 16 or 18.
So what is it about these primes, which seems to be in some way helping these cicadas?
Well, we're not really too sure, but we have a theory of this, Mary,
is that maybe there was a phrase around that used to sort of compete to the bee to try and write it
for us at the same time as the Second and the the predator used to appear periodically as well.
And is it called? Is that a plane noise? Sounds like a king out of sync with this predator.
Much better than the second. Is it a non flying license?
Let's take an example. Suppose a predator appears every six years in all forest nesting cicadas.
You've chosen nine years who appear every nine years in the forest.
Well, six and nine. They share a divide.
Very common. So, in fact, you find that it's every second time.
The second is a man pair. Yeah, 18 is a number.
There is a six and nine. And so every 18 is like the cycle of the predator and it very quickly gets wiped out.
But if you change the cicadas lifecycle just a little bit, let's change it to 70 is a prime.
So actually it's appearing more often in the forests and maybe opening itself up to even more attack by the Predator.
But no, because salmon is now prime, it means that six and seven,
those two keep out of sync until it's only a year 42 that they meet for the first time.
So it seems as if the primes gave those had a prime lifecycle has an evolutionary advantage to why you couldn't avoid the predator.
And in the forest in America over the 70 year cycle, it seems like there's quite a competition for who to find the brain.
So maybe the the predators go back in sync found that seven and then as it works harder to find a new prime.
And so I think there's a message here you know your maths you survive in this world.
We have no evidence of this predator, but it's quite a curious life cycle.
So one could say that these cicadas, which still exist in North America, there was a big brood this summer which came out some on the East Coast.
I got a chance to go and see some of these 13 year cicadas a few years ago appearing in the Baltimore region.
They're absolutely extraordinary. And my if I say one insect and the oldest on it, it would be this one.
But, yes, I think if you're going to say which culture really started to understand the problems, and I always think your first answer was that right.
I think the ancient Greeks are really credited with the first explorations.
All the problems and the first real beginning of mathematics,
I think got an analytic subject the ability to improve things with 100% certainty, which is somehow what makes my body so different.
I think to the other sciences and in particular, one of my favourites proves that All the time was about primes by Euclid.
You see the chemists, they have this thing in the periodic table and they have in devised any chemistry classroom or lab and
it's up there on the wall and you've got a list of just all the atoms which are the middle molecules.
So maybe enormous know and I suppose we should have a sort of periodic table prime primes should just write the primes in the table.
And when we need to know something about primes and God can solve it,
but you can show you that's trying to produce a poster with all the primes on voice you've been writing for hour because he proved that the primes,
they never run out. There are infinitely many of these building blocks of mathematics.
But how do you have enough spine on a human?
Beings proved that something will go on to exist.
Well, it was amazing that along the way he came up with and I want to show it to you,
because I think it captures the power of lunch because something is amazing is intimacy.
So his first assumption was, well, supposing we're only following you, perhaps you could write down a list of all the crimes.
Suppose there are there are actually only two of the 43.
Perhaps using those, you could make all those. You know, why didn't we make 5.3 on five times 768 making the different combination?
Perhaps you can make all numbers by not lying easy to you that show you always be missing a number.
How do you do this? We'll take all of those building blocks and multiply them together and then enact a genius us to add ones that numbers.
Okay, so here's a new novel. What primes is that divisible by?
It's got to be. It is why it's either prime or whatever it is. So we said the only primes are in all this, so it must be divisible by one.
That is fine. That is. Which one is it? Well, it's being constructed, so you always get related to one.
This is Euclid's clever trick. So 17, 17.
Divide that. Now you get remainder one.
So it means that most of Mr. Primes this number is either the Prime or Philosophise surprise, which are not on the list.
So you might say, okay, well then why are you asking?
And so you add some more Brian to the list and then you say, Well, that's still number.
You can't build out the prime units because he plays the same trick again.
So I have. However long you try,
Euclid is able to show that you must have missed him on a finite supply and would never be able to build all numbers at this point.
Little bit argument shows the primes as they go on.
So with mathematicians, we have it much harder than the chemists and you see a gradual down on the table.
We've got to find some other way to understand this sequence of numbers.
Now, this thing you might think, oh, maybe you could come up with a clever way to actually find primes,
because you can multiply that together and add one and got a new primes.
Maybe that's a good way of learning. But fortunately, that doesn't always work.
If you do two things three times, fine.
Add one, all that surprising, but very quickly you multiply two times three all the way out to 13, for example, and add one that's multiplying.
It's not busy widening of the prime solve to 13, but it is not prime itself.
And you're right, it is an open question.
You will make your name in mathematical history and you can prove that in the afternoon you get some primes out of this consumption.
We don't know that. So as she's ever since you leave spring, let me have the clients go over forever.
We just don't know where they are. We don't have a formula for how to guarantee you a prime.
Now, having some form is some help in finding some of the biggest crimes we know so far.
So a few years ago, there's something called the Great Internet Mersenne Prime Search,
which you can take possibly you can download a piece of software which you get some computer files on my computer.
Sitting up here on the desk is actually running this program, always in the background, looking for new primes.
And in fact, the person who found this prime, they broke the record having no records, Spergel says.
Could you find a prime with more than 10 million digits, that kind of challenge out there?
And somebody had offered a prize of $100,000, if you can find it.
Now, this is a very good formula for finding big primes.
It was first discovered by a French mathematician, which is why this is called the Great Internet Mersenne Primes.
So you raise two to the power, in this case, 43,112,609.
That's an incredibly reasonable number. But then you take one of that number and suddenly this number then becomes prime.
And it was the first prime to be discovered. You pass this 10 million digit and the person $100,000.
The finding is but is subsequently that was discovered in 2009.
So that's quite a big gap.
Just recently, the beginning of 2013, an even bigger coin was found using the same program and this is now over 17 million digits.
If I read this out, allows you to find it would be here for a couple of months to find out.
But essentially that but thanks to you, we know that there are primes because we want this is a net and that's the best we know so far.
But it's always going to be a big one. So I see there's a prize at $200,000 to the first person to find it a prime with a billion digits,
although I think an interesting will probably use out the whole money if you try to use to your computer an Amazon Prime search to find it.
But I think part of this is just how little we understand about these numbers, because if I asked you to find a billion digits Fibonacci number,
you could read easily,
use the fact that what we know happened in large numbers to find out or a triangular number or a swing of the prime base numbers.
And also we just don't know where they are beyond here, complete mystery.
So this is the challenge, how we find a way to understand how these primes,
the building blocks of the whole of mathematics are laid out through the universe and numbers.
So for centuries, really trying to find formulas or patterns and we just can't seem to get anywhere.
Now our online is quotes, which comes from one of the experts in fine numbers at Princeton and Rico on the air.
And I think that rather sums up one of the traits of being a good mathematician.
He said, When things get too complicated, sometimes my sense is simple wonder how do I ask the right question?
And is that skill to do lateral thinking, which I think makes a great research mathematician.
So we were trying to find the formula to try to find a pattern, but maybe we needed to.
Also, we knew about the primes. I think there's a way to understand what their secrets are.
And it was, in fact, one of the greatest mathematicians, I think in history of all subjects,
Carl Friedrich Gauss, who often knew question about plants and found a hidden pattern.
It's kind of curious. Gauss clearly was going to be a great mathematician from a very young age.
He was a mathematical diary where he recorded his discovery of a useful way to construct a 17 sided figure for his 15th birthday.
He given a book of logarithms, researched that kind of thing you might get from birthdays.
But what was curious was in the back of this book of logarithms was a table of prime numbers.
And Gauss. He understood the rooms, but the prime numbers began to sense.
And he was saying, well, know, what is the secret to the numbers that are in his title?
He couldn't find names, and then he thought it was something different about these numbers.
And he also knew the question is new, this new question which revealed the person rather remarkably,
it revealed a pattern between the price, the faculty school and the logarithms at the front of his book.
So what was the new question?
Well, the new question was, okay, so trying to find where the next time is, let's try to count how many primes there are.
Now, you might say, well, isn't that a bit of a silly thing to do? Because Euclid already proved that there are many, many frauds.
So how can you count them but now say no, I don't know how many primes they're all saying the first ten numbers were two three,
five, 734 primes in the first ten numbers. I see.
My guess is counting one was counted as a prime number.
And that's how you I've seen it in his book of logarithms one in the 19th beginning of the 19th century, with Stewart on it as a prime.
But because we now believe primes for the most important thing,
rob them of their building blocks, they make new numbers, and then you multiply it by one.
You don't get anything new. So it's a little bit like the fact you say so actually we don't say one seven now, so we say Ready for Prime ten.
And that whole piece that I showed you will be a twin 29 times, all the way up to 125 times as a single 6117.
Okay. As in, how far can you predict how many primes you expect to get?
Hundreds of thousand, for example. And you can draw a graph of this.
So sort of a graph which counts the number of points and what I call it counts a staircase in the prime.
So every time you take a step upwards. So, for example, 101 is a prime number.
So you've got 25 rises and 100. When you got to 101, you take a new step up.
So recorded inside this graph is all the information about where the primes up, but now says, okay,
we got obsessed with the minutiae of when you step so this say to step backwards and see if we
can see any overarching patterns the way the staircase grows as they climb higher and higher.
And when he started to look at this question, he's found this constant beginning to match.
And he made this connection with Lawrence.
Now, I just want to check that you all are up to speed with remembering what a rate I mean, how many people used to set of all tangles and errors.
So yes, it will people now putting up their hands.
I should also admits to using a small table because the lower the tables were basically the calculator of their day.
They enable add engineers or navigators to speed up a calculation because this is
the one thing you just have to remember after the logarithm is logarithm times.
Multiplication, intimidation and multiplying two numbers together is quite complicated.
But he did say he did ask for an easy and then can use a little to back so that's
all you have to remember that logarithms to understand this connection the calcite.
So Kells started to analyse the data that he has. This is a lot more data than Gauss had available to him, but they didn't have.
The time was enough to detect this pattern in the number of primes you get as you count higher and higher.
So so let's look at it. So the middle column tells you the number of primes less than the number in the column on the left hand side.
So for example, under a thousand, there are 168 primes.
But the more interesting thing when Posner is in the last column,
because the last quantum record is the proportion of primes amongst all numbers to a thousand.
It's also a thousand with a heart, he prime says.
In a thousand, that's approximately one in six number.
Up to a thousand is at a prime. So we saw from 120 flights are waiting for numbers, up to 100 to prime.
But it's also getting rarer. So up to a thousand is one in six.
And this is one of them all. Scotland is recording. It's recording the proportion of primes to all numbers.
So for example, let's take I had in London.
So I want to talk about numbers about ten, 10 million.
So what was the chance that my London telephone number would be a prime number?
Well, that last one tells you up to 10 million. That's a one in 15 chance that my telephone number is going to be prime.
And I lived a few years ago, and I had to change my phone number.
My telephone number. And I, I phoned up the telephone company and said, I need to be telephone number.
And then she the woman who had got to give a number and I very quickly chatted on my computer and it wasn't prime.
I never remember that. Could I have another one? So she tried another one and I'll pull it out and it is still on prime.
And we went through this column trying to make sure she just got herself images and then thought,
I'm going to be the next door which comes out and it's almost going to have an even number.
But this tells me we would have to go through this 50 times and one of those might be the product.
So I'm still here.
So the last column is where the pattern that Gauss discovers emerges, because all the while the Prime Minister isn't there watching the beginning.
So it's all the same as the sequence we had. Sometimes things need a bit of time to settle down.
This one is a little sound around ten 10,000. So there's a one in 8.1 chance that a number is prime around 10,000.
And when you multiply the first one by ten go 10000 to 100000.
The chance of getting a decline goes down. So it's now one in 10.49.
It's 2.3 to the chances. If I what I want you to think about is to think about this a little bit like that in nature.
Use some dice to actually choose the problems.
And this last column records the number of signs on the dice that you need when you're trying to choose the crimes around a thousand.
So why don't you say so? So you could use a cube. A six is faced shape with one side of the opinion.
And to get a sequence of numbers, which looks very much like primes around a thousand, you could use this and just circle numbers every time.
Get a P and you get a sequence of numbers to very much like with primes around a thousand,
but as in plain white, the number of signs on the dice increases.
So we see that there are around 10,008.1 and that's a bit like having an octahedron.
So this is an eight face figure. Just one side of the prime.
And this is the one that might be deciding primes around 10,000.
And as you go further up, so on here, around a million, you've got a 12.7.
So there's like 12 sided dice. So don't count Hedren.
This might be choosing the crimes around 10,000.
So so the trick is, well, how can you predict the number of sides in the dice as you climb higher and lower angles?
Notice this concept every time you multiply the first column by ten, the number of signs in the dice seems to be going up by 2.3 each time.
8.1 to 10.4 you add 2.3 multiply by ten again 10.4 to 12.7 adding 2.3.
Now you get 15.0 to 17.4. That's just a random rounding up.
This happens and essentially this pattern just keeps on going the way the primes do now,
although on a small scale they look very random when you take a step back because
they seem to have this incredible pattern being just sort of chosen by dice numbers.
Size is decided by adding 2.3 every time you multiply by ten.
But here's the connection with the lower because remember logarithms turn
multiplication into addition by multiplying the best column by ten I add 2.3.
So the number size of the dice is the lower rhythm function, which is controlling the number of signs in the dice as you climb higher and higher.
That was Gauss's belief. So you want to know the probability that the number is prime that is is lower than function, which doesn't.
So in some sense logarithms tell you the numbers on the dice.
So if I want to use a very large number, the probability that is prime is one in logarithm of that number.
I don't believe that this wasn't you know, we got to be careful because as I shown, this sequence can have a pattern.
It looks like powers of two when I was doing these numbers for you and it suddenly went to 31 and 32.
So you have to be careful. But 100 years off, the Gauss made this prediction.
It was in fact confirmed that this is the right model for the.
Trying set out in this incredibly useful way, however far you go up in the universe, anomalous.
And in fact, you can use this model now to make a prediction about the number of crimes that you'll see.
Because, for example, if my friend is three times out of six, you find that this says things being prime numbers.
So I can use the fact that I know how many sides there are on the dice to predict how many crimes I should get because I call it paranoia.
And so you can use this gauss. This is amazing. I just.
I think it's rather amazing that a 15 year old boy, Gauss,
discovered this extraordinary pattern that seems to be hiding behind waypoints in our lifeline slightly refines this approximation.
And so what he got was the overarching trend of the statements of the problem as you climb higher and higher.
Now, this is pretty good. Good enough for an engineer, but not good enough for a mathematician.
We don't have things to be precise. So how can we get from gauss's guess over arching sort of positing that line?
So an exact formula to the number of primes because actually from 1000 to 1030, there aren't five primes.
There are actually I think only four or six is slightly different.
So you always think it's an error. And of course, if you throw it, you're not going to expect to get exactly a six times landing on the wrong side.
So how do we work out the errors in these times? We've got a good sense of the pattern, but how do we correct this?
Well, it wasn't.
Gauss began as a student, then Harem, who came up with an extraordinary discovery about how to correct the errors that Gauss assessment will make.
And the best way, I think, to explain Reading's ideas, which are full of extraordinary mathematics involving complex numbers,
imaginary numbers, I would say function Z, the functions.
But actually I think the way you get to the heart of what Raymond did and to understand the what we now call the Riemann hypothesis,
the biggest problem in the whole of mathematics is the music,
because actually what Freeman discovered was the hidden music underlying the finds,
which helps you to understand ebb and flow of these dances and choosing the primes.
So what I do. It's. I could explain to you a little bit about how musical theory works.
We understand it's a lot of science. The ideas that we've developed.
So actually the building of some numbers, applying the building blocks of music discovered by for a reasonably similar period.
This is how the tuning fork makes. So there's a sound.
But, you know, I've got a tuning fork here. So that is actually the purest sort of sound because if you are cool, that sound on a similar scale,
what you get is a perfect sine wave and well for a start it is a sideways eruption, the building blocks of all sounds.
So for example,
if I take a different instrument so upon in so there's the eye on the ball in I'm playing an I on hear this is I have to borrow only my twin violin.
So this is why this is the less half size. So it's the same.
But it's very difficult to use them. So what's going on here? I mean, if I were to reproduce, I see my computer doesn't have all these.
Well, how does the computer manage to make the sound of a body come out of the speakers?
Well, what we understood is the volume is made up not of just a single tuning for a that you heard earlier, but many of you.
So there are many sideways that build up to the sound of evolving.
So I think if you recall the sound of on and on a citizen, you get this kind of jagged sawtooth growth, but you can build this straw out of sideways.
So essentially the same waves that you get are all the sound waves that fit the length of the morning.
So the annual hearing is actually the very sort of big sine wave that's like racing between the beach and the the body.
But we can also fix another sine wave inside that which is vibrating twice as fast and as half the wavelength.
So it's sort of vibrating in these two places. So that's called the first call warning, and that's also the period that notes the same time.
You're also hearing any sine wave that can fit inside the length of this violin.
So, in fact, you're hearing ones that have half the wavelength of the wavelength.
And it's the culmination of all these tuning forks which combined make you hear the sound move on it.
So we can do this radically by combining. So here's the sine wave.
This is the the, the a of the tuning. But now I'm going to add on some other notes as well.
I'm going to add on the first time. So that's a sign letter which is going to vibrate.
So it's going to what it's going to do is push and see. Yes.
So if I add on another sign which vibrates twice as fast as going up there and then down here.
So what's going to happen is that when I add the graph, it's going to push this little graph up,
but underneath here is negative and it's going to pull this side of the graph down.
So what I'm going to do is add all of these lines and the first one, I'm going to add more and more sideways.
And as we add on more of this, so I might add more of this line.
Let's run this little animation. You see that the sound of the tuning pool gradually changes in the sawtooth shape of the body.
And this is really what the speakers are doing when they reproduce the sound of a violin.
They're just raising it all of these different sine wave scanner tricks here.
And hearing this shape, which is the shape of the body that I'm a trumpeter rather than a violinist.
And harmonics are these things like absolutely essential to the way a trumpet works.
Because as trumpet so an early now you didn't have any vowels at all.
You couldn't change the length of the piece of string, as it were, the length of the tune.
You had to create new notes by just putting more energy through the the point, and that will create the higher harmonics.
So, for example, I don't have to change the length of the point, but I can get many notes out of this.
What I'm doing is getting the different sideways, which fits inside the trumpet.
But. Uh, so that's the sound, although essentially all those different sine waves that can fit in that length of point and so thrashy.
I mean, there are only eight different combinations of the vowels here.
I mean, you see some harmonics every time I play the trumpet.
So I don't think you'd be surprised if you have a switch that's gonna.
One, two, three, four, five. I prefer.
I. So I had a bit of time to warm up, but I didn't change the instruments.
So you're going to need different combinations of sideways to create the sound of that instrument.
So, for example, I'm going to take the clarinets. So the sound of the clarinet is used here while I travel with Chinese.
It's impossible to close down, as you can see now.
And you see you got these kind of square like coronations of a castle.
And this is because you're getting different sorts of sideways signs here.
So again, you got basically the shooting pool, which is so that the sun is going to be open at one end and closed in the other.
So you get sort of whole the sine wave and then the next sine wave that appears inside here is actually one which is vibrates three times as follows.
So you get this, this shape. And so you're not hearing all this the same sine wave of all that is fine,
but about half of them actually only there is this sort of odd number, wavelets.
And if you combine these now, you've got a very different shape graph.
So basically start with the tuning for sine wave and all of these extra functions.
In this case, the same way this library did three times. So it goes up, then down, then up again.
So that's going to pull the middle of the graph down, the science up.
So when I run is an add on the harmonics, I get a different shape going.
So the different combination of sine wave produces a square function and this is the sound of the parent.
And in fact, every time I see if I was basically just taking advantage of the sound of the orchestra being broken down into the sideways,
the different frequencies to make up the the graph of that sound.
So this is the idea or how this going to sound because in a way,
what you should think of as the the status of the problems is that it's the music we're trying to capture that down to.
The girl's got a good approximation. He's got this graph which is kind of like that.
Well, that's a bit like the sine wave or the tuning fork is the first approximation.
Riemann's were using the z function and complex numbers.
He managed to find some rather weird waves harmonics hiding behind which you add these harmonics along the different frequencies.
They help you to correct galaxies, get gas and get a true formula for the number of points.
So these are what are called the series of the Riemann Z definition.
But essentially what they give you a graphs which are waves and rather sits like sideways and a little more complicated.
And so the first harmonic is at the third, they're actually many of these on this is the 29th and the 50th.
But inside here, the waves are kind of encoded.
The errors the palace is making when he's making his guess, the number of points.
So I have to add these on Gauss's case to get a formula for the number of points you see here.
What I want to show you now is reminds me of the number of times I set this course out, identifying and identifying choices.
It's actually a cicada. It was called out in the desert island, and I had to take one mathematical formula with me.
I think it will be this looming agreement because I think it is absolutely magical.
So I'm going to say this graphical form or other similar to the the graphs, the music.
So the idea is this we're trying to get a formula for the number of primes.
That's the blue states gauss's case.
Using the logarithm is a good estimate, but it's not exact.
If I want these waves of regions which can sometimes add a little bit all or sometimes type in the way encoded inside,
there are the errors that Gauss is making.
So what I'm going to do is run for you this little animation. So galaxies, gas is a yellow graph.
The blues that I use is the graph we want to capture by adding all these waves which push and pull the graph wrong.
In the same way as the clarinets emerge from the tuning fork on a violin version that you bought by adding the wrong frequencies.
If you add all these types of regions, it gradually pushes and pulls Gauss's case until even over hundreds of these harmonics,
these little waves pushing and pulling the graph, you see the staircase beginning to emerge.
The subtlety of where the primes really are is encoded by these waves that we discover these zeros of the Riemann ze function.
So already, by a hundred ways you're able to tell, oh, there are no primes between 23 and 29, which you couldn't tell with Gauss's guess.
So encoded inside here is really how the pines are laid out.
And it tells us something rather amazing because although the planes outwardly look like they have no pattern at all,
we saw some with his music when remote analyses. He found an amazing pattern hiding behind music.
Because if you think about the same ways, what's important for a CD player is, first of all, the frequency of those waves, but also the amplitude.
How do all that frequencies contribute? The amplitude is how big the wave is.
And Sarina, he quoted a graph which recorded for each of these waves.
What frequency? I want to ask you have the labels so you pulse the frequency will be on the vertical and horizontal will be allowed.
The amazing discovery he made was this The frequencies all changing the volume of these waves is playing.
That seems to be exactly the same. And so this is all of these that seem to be lying on the magic straight line.
They're going to be scattered all over the place. But the music seems to have this amazing pattern inside.
So they lie on this thing, which is will read his critical line.
I really believe it's going to just be a coincidence that first hand all have his just perfect balance too.
And he made this conjecture. He believed that all of the notes that he discovered they would all be playing in this perfect kind of balance.
And this is what we call the Riemann hypothesis. We are not being able to prove it is one of the biggest open problems in the whole of our subject.
And it's really this incredible pattern in the music actually explains outwardly why the primes do look so random.
If one of these notes was off this line, it would be it.
Since playing much louder than all of the other notes. Read the material slightly, very nicely, he said.
It'd be like listening to an orchestra and then suddenly the tuba comes in and blasts the rest of the orchestra out.
So all you can hear is the tuba. So it's really like all this is is false.
And there's a note off the line. It would imply an incredible presence on the client.
So it's one of those notes controlling how the primes are distributed in a way you can interpret this.
So I think this is rather this is called the Riemann hypothesis.
So it says that all of these notes are playing at the same volume, and the Riemann hypothesis sort of explains why you don't see any presence.
In fact, it will predict for these prime number dice point on all these offensive to all a distributing the primes fairly amongst all of the numbers.
The Riemann hypothesis actually is capturing exactly that.
But the error controlled the error is how big this wave is.
The error that remains predicting is exactly what you would expect from a fair set of dice.
So reminisce hypothesis is capturing in some sense the passage of the primes.
Although they are random, they are as fairly distributed through the universe of numbers as we could hope for.
I often quite like to describe the Riemann hypothesis as almost describing the atoms of molecules in the molecules of air.
Well, we don't know exactly precisely where every single molecule is,
but I do know that if I go to one corner of this room, I'm not going to suddenly find a vacuum and I'm going to die.
I know that the molecules are fairly distributed through this route and there's a chance that
they might be able to use their two parts the way that we think the gas is distributed.
It's not important where each one is. I don't know that there's they're pretty evenly distributed.
And for the primes very often as a mathematician, I'm going to need to know where exactly each point is.
What I need to know is that fairly soon it says I'm climbing up through the universe in numbers.
I'm not going to suddenly find a vacuum without any problems at all.
And that's what the Riemann hypothesis very often helps us. Well, let us kind of ask the question where all of the primes and it tells us enough
about how the primes are distributed to make progress in understanding numbers.
And I think it's an amazing because in a way, you know, nature gave in these numbers the primes and outwardly seem to have no patterns at all.
But the primes are the building blocks of the subject looking for patterns.
But mathematicians, we found some clever ways to look at these primes.
A new way to look through Google gal. They changed into the the logarithm controlling the number of signs in the dice.
Then through the music to be able to discover this kind of hidden music where suddenly this very strong pattern,
the reason believes, is there controls the way that points outwardly look.
So somehow the power of the mathematician is finally finding some order inside these numbers.
Although we still looking for an explanation about why those notes sound like they did.
Is it $1,000,000 prize offered by the Clay Institute for anybody who can prove the primes are really behaving as we even believes they do?
Now I discovered that I study the primes because I think that useful.
It's somehow trying to get to the eternal truths of the universe.
But there are also now very practical reasons for understanding the primes in France.
One of the challenges I gave you is getting these 105 primes.
You managed to pull that together in a business. Three, five, five, nine, seven.
Why isn't that a challenge for Young? And I say offer a price and offer a bottle of champagne to anyone who can crack these numbers on the Leonardo,
which is not prime number 100 with 6000. Okay. Well, I'll give you this one right at 99 one 1911.
I have a feeling that I actually. Okay. So that's a challenge.
Okay. 9,999,911.
You have to find the primes which builds that normal prime number.
So it is made out of sodium chloride and you have to pull it apart.
And so I did offer a bottle of champagne was in the fridge, but it's beginning to warm out the lights.
So the first person to discover two primes to build the number will win the ball.
Now, why is this important?
Well, actually, this problem is now the heart of all the codes that are being used on the Internet to protect your credit cards.
So essentially, if you send your credit cards to a company online and you want to keep that credit card secure, they send you an on the line.
It's actually a little bigger than this. Your computer on a calculation with this number and your credit card number says it all across the wires.
But to do that calculation, you need two primes, which below is number to be able to undo the calculation.
So criminals, this is public on every poll, on any Internet websites is selling you something that would be a public number.
If you can write that public number into the two primes, you'll be able to break every credit card that's going to that website.
So the number is slightly bigger than nine.
And just on to his is the kind of size that we cannot crack at the moment.
So it is a $200,000 prize for anyone who could crack this number.
So if you don't like champagne with money, then this one.
But essentially, you know, the whole of Internet cryptography is based on the fact that we don't really understand these numbers at all.
Chemists, they have a spectrometer which divides chemicals into the constituents, and we do not have a prime number spectrometer in mathematics.
So for me, I think since, you know, I started with question what is mathematics?
And for me, think what makes mathematics a living,
breathing substance is actually these unsolvable Riemann hypothesis and the challenge of trying to find some pattern in the primes.
So we had a chance for some questions. We have some microphones for you, and when you get to my phone, you can stand up for the cameras.
So this is way where you're playing your parts in this television production because we are probably,
since the ensemble is be giving incredibly intelligent questions.
So please help me to live up to that billing.
And any questions from you about problems in mathematics?
You put your hand up now get him on TV. The first one is always the most nerve wracking one in any society we have to dismantle to stand up.
How do you verify that very large price to ensure that they are prime?
Yeah, that's a very good question. There we've chosen these very large primes because they have a very particular property, the Mersenne primes.
So let me solve the Mersenne primes. How do you verify that they are actually prime numbers?
They're a very special sort of number because they have to surpass something at minus one.
And we discovered a way to test their finality, which is very different from just a random number.
In fact, it's something called the Lucas Lima test.
And it's what actually your program will be running on as it's running on your laptop.
And essentially, it's sort of test something much simpler.
It turns the formality.
This is about whether a certain number divides another number in a sequence, which is rather like the Fibonacci sequence and the Lucas Lane numbers.
You have to build these. So it's a very constructive thing which which proves whether that number is prime.
We already know. And it's something that the ancient Greeks knew that if you go to the power of something,
that thing must be prime in order to the power of it minus long have any chance of.
But what about that sort of primes? I mean, for example, every time an alter, if you will,
have in your warnings small columns which are actually probably generating primes when they're being used in order to find to create cryptography.
And so how do we actually find primes of, say, 100 digits, which is sort of a couple of hundred pages?
What do you need to create some these primes that use on the Internet?
And then often it's just a probabilistic we have a way of testing the probability that a number is prime,
which is generally good enough, and it involves something called modular arithmetic.
So we know that primes have a very particular property or clock arithmetic, and it's something that was discovered by Thoma and Euler.
That's and that's what's being used to test whether a number is has a good chance of being prime.
Because one of the problems is that you have to really take one step to find all the numbers up to the square root of the number.
And that's why you're trying to do that with the number I chose, which adds a prize of $200,000.
Just you wouldn't be able to find a prime position. That particular number you're trying to prove that isn't prime by finding primes to builds it.
And so there are some attacks out there which involve quantum computers and we do a lot of quantum computing here in all cases.
Instead of testing one line after another, a quantum Q computer that could potentially test all primes in one go.
Because what happens in quantum physics when you take advantage of it's an electron wrong being just here or here,
on or off can be in two places at the same time.
And you can use that essentially in a quantum computer to to put the quantum computer into a state subsidy test,
all primes in one go and collapses into the states where the primes are the ones that divide that number.
So we have a software that has a hardware. Yes, it creates a quantum computer.
Good enough. I think we can crack 15 into three times five. But I think you from the last.
It's not the best question down here. And yet we've got the mind to.
Send us your beer formula that you will be able to find all the prime numbers.
So obviously a really complicated formula, but you can either be wrong.
That's a good question because it sort of raises the question about what is similar to the formula.
Do you want what are you looking for so you can already prove that that works the same polynomial for me I have.
That's a nice polynomial formula which told you the number of simple division office it was.
You raise numbers to the power four and do it more know, but we can already prove that you can't have a polynomial which will give you all the time.
You always have to be some numbers which it produces. Which. All right.
Well then why not? Found an amazing formula which produce primes much more frequently than you might expect.
We sent out A, B and squared plus and 41.
You run that and try on this and in that you get amazing number of primes coming out of that format.
But you might say, okay, well maybe it is a different format.
And I'm saying, you know, Riemann's formula is does give you a way of understanding the primes,
but it's not a formula that can be used, for example, to find a prime with a billion digits.
But it's a formula that can help you. And so that's a little bit like trying to sing the highest note.
Well, that's what music is. So so the search for the big primes, I think is a little bit serve, and that's not really what mathematics is about.
Much more interesting to me is to try and find what's the overarching way all the pines are laid out in the universe and and read this formula.
Although now to tell you the biggest the prime has been in digits.
It tells us, I think, a lot more. And it tells us about the underlying patterns in the primes.
All that is on another high note. You know, as a Christian, every.
All right. What progress, if any, has been made towards improving the Riemann hypothesis?
Well, there was some exciting progress made about a couple of decades ago when there was an amazing connection with physics.
Because one question you can ask about the Riemann hypothesis is about amplitude of those waves.
But the other thing I quoted was the frequency. So you might ask,
is there any sort of logic to the frequencies that you find of the notes that we have that tell you how the problems are distributed?
And somebody made a statistical analysis of the sort of where these harmonics are happening.
And what they discovered is that you have a random property at the Institute for the Study in Princeton.
So now one of the important things in science is bringing people together from many different disciplines.
This is what we try to do here in Oxford, and that physicists talk to this mathematician asking, What do you be doing?
And he said, I've been doing this kind of statistical analysis of the way the frequencies occur, and this is what was the formula.
And then physicist. And that's exactly the same formula that we're using to predict energy levels in long atoms like uranium and nuclear.
So these. So it's very difficult to solve the equations for the energy levels, but they have this math.
And the pattern was just such a bizarre one that you just can't believe that to optimise it.
So it's the most exciting progress on the Riemann hypothesis has really been.
That's mainly the notes that Riemann discovered.
Those frequencies share a lot in common that the mathematics that we need to understand them
is actually the same mathematics that we need to understand energy levels in quantum physics,
and those are controlled by all races, eigenvalues and matrices, basically.
So the hunt now really is, is for well, can we find one operator on line one in quantum physics,
which tells you what the energy levels are, which sort of controls the primes energy.
And although we haven't actually come up with what an operator is that's looking at the
primes as almost a physical model has enabled us to discover things about prime citizens,
mathematicians that we've been unable to crack.
So I think it's an exciting time when we're getting this cross-disciplinary interaction, which is helping us to see new ways to look again.
It's coming back to Bulgari. Perhaps we need to ask a different question and you find a new way in.
There's a question of back. Thank you.
So if you think about a sawtooth wave, square wave or something like that,
it's quite clear that you can dramatise it either in terms of an infinite some of its
harmonics like you show or in terms of something like its period in its amplitude.
One of those is actually very simple and the other is very hard.
And I can't help but think that maybe our understanding of the running hypothesis is only
through these harmonics because we haven't seen perhaps the simple way that's missing.
Do you think that's a fair analogy? Oh, I think that is a fair analogy.
And that's what often happens is you can get very boxed in by a particular way of looking at things.
And that's why I think always you need to be open to a new way to look at a subject.
There's a I don't know whether you've seen this psychological experiments where you're showing a video and there are two basketball teams,
one in black, one in white, both possible between each other.
And you're asked to count the number of times the black team passes the ball and you're sold the red herring.
Oh, men and women counted differently and you want to be count in.
You get about 17, you see about 17 and and then experimenting.
Also, did you see the gorilla walk across the street, bang his chest and walk off?
And the first time I saw this video, I did this experiment at the Royal Society know in the Navy at every junction office,
and they rerun the program showing exactly what everybody looked at.
There were a few Tallahassee because they had been concentrating. So I'm going to count how many times the ball guys to the video.
One, two, three, four, five, six, seven. And and I think this is a good reminder that in all of the subjects that we studied, science, mathematics,
that we can get very fixated on a particular way of looking at a subject such that we miss just some obvious gorilla across that.
So I think that's, you know, we have over very senses this we do believe this is the way to unlock the secrets of primes through Raven's work.
But every now and again, somebody comes up with a slightly different perspective on it.
Absolutely. I think that's always the hope that maybe there's a different way to look at where where all of this becomes amazingly simple.
But, you know, we've been looking at this thing for a long time and looking at,
you know, it is the biggest crime and nobody's come up with a different way.
So at the moment, I think we stopped living in Freeman's house, but which I think is a beautiful house.
I think it's a beautiful place to be. But but I think we should always be open to the possibility that somebody might
come from totally that field and show us a new way of looking at things,
which reveals a simplicity. We've just. Let's have a question here and a question here.
You. So there are some other conjecture.
Is that in hindsight going to conjecture is or the dream prior conditions and that if
we assume that the Riemann hypothesis or if we can make more progress about it soon,
do you think proving the Riemann hypothesis, he can be some kind of master key to crack all the other conjectures?
And that's a very good question. And in some ways the answer is probably no,
because very often we think we know how claims are laid out because and and a lot of the mathematics only do I have the beginning of my theorem.
Assume the Riemann hypothesis is true, which is what we all believe.
Then I can make this progress and do things. But similar a lot of problems of our times.
And even if you know the Riemann hypothesis, since they they don't tell you about goldbach conjecture,
for example, some of these conjectures have progress based on the Riemann hypothesis.
So in a way it is the golden key to everything.
In particular, it probably is the golden key to understanding how to crack these codes based on primes.
I mean, I think it's always fun to set up the the the importance of the Riemann hypothesis because it could bring down the Internet.
But actually, you know, if I ask for money and I'll say no, I bet you that when reading a hypothesis,
it proves that it probably won't give rise to a complete collapse of in business.
But again, you know, the Riemann hypothesis we know doesn't tell you anything about well,
it does help some algorithms for cracking numbers, but it won't apply already tell you how to crack numbers into into primes.
But on the other hand, you know, we never know what's the proof will tell us.
And I think that's often why we study these things. This building here in Oxford is named after Andrew Wiles, who proves Fermat's Last Theorem.
Now, Fermat's Last Theorem was a separate equation, actually, and was like the idea was at the end has a solution.
But what we really care about is the mathematics that school generated.
We understood so much deeper about our subject because of the calculus fall back home.
So I don't think anybody is used to suppose that that's not true.
Therefore I'm going to do this. But the mathematics generated was extraordinary, and I suspect that when we eventually prove the Riemann hypothesis,
it won't necessarily be the Riemann hypothesis, which will be the golden key.
But the way we proved it,
which will actually give us far greater insights into things like over 20 primes and maybe even cracking numbers is that prime mover?
You know the question here. Okay.
Of course, that was the same time. Any other questions?
I think that's great. Thank you very much for taking part and everyone is coming later.
You can ask the question in a second to make sure it's not from the vice.
Then thank you very much for being a great audience.