In Our Time - Prime Numbers

Episode Date: January 12, 2006

Melvyn Bragg and guests discuss prime numbers: 2, 3, 5, 7, 11, 13, 17 … This sequence of numbers goes on literally forever. Recently, a team of researchers in Missouri successfully calculated the highest prime number - it has 9.1 million digits. For nearly two and a half thousand years, since Euclid first described the prime numbers in his book Elements, mathematicians have struggled to write a rule to predict what comes next in the sequence. The Swiss mathematician Leonhard Euler feared that it is "a mystery into which the human mind will never penetrate." But others have been more hopeful... In the middle of the nineteenth century, the German mathematician Bernhard Riemann discovered a connection between prime numbers and a complex mathematical function called the 'zeta function'. Ever since, mathematicians have laboured to prove the existence of this connection and reveal the rules behind the elusive sequence. What exactly are prime numbers and what secrets might they unlock about our understanding of atoms? What are the rules that may govern the prime sequence? And is it possible that the person who proves Riemann's Hypothesis may bring about the collapse of the world financial system? With Marcus du Sautoy, Professor of Mathematics and Fellow of Wadham College at the University of Oxford; Robin Wilson, Professor of Pure Mathematics at the Open University and Gresham Professor of Geometry; Jackie Stedall, Junior Research Fellow in the History of Mathematics at Queen's College, Oxford.

This BBC podcast is supported by ads outside the UK. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio 4. I hope you enjoy the program. Hello. 2, 3, 5, 7, 11, 13, 17, blast off. These are prime numbers. I could continue, but we could be here for some time.

Last month, the team of researchers in Missouri successfully calculated the highest latest prime number, and it is 9.1 million digits. For nearly 2.5,000 years since Euclid first described the prime numbers in his book, Elements, mathematicians have struggled to write a rule to predict what comes next in the sequence. The Swiss mathematician Leonard Euler feared that it's a mystery into which the human mind will never penetrate. But others have been more hopeful. In the middle of the 19th century, the German mathematician Bernard Riemann discovered a connection between prime numbers and a complex mathematical function called the Zeta function.

Ever since, mathematicians have laboured to prove the existence of this connection and reveal the rules behind the elusive sequence. What exactly are prime numbers and what secrets might they unlock about our understanding of atoms? What are the rules that may govern the prime sequence? With me to discuss prime numbers, a Marcus Gisotoy, Professor of Mathematics and Fellow of Wardham College at the University of Oxford, Robin Wilson, Professor of Pure Mathematics at the Open University and Gresham Professor of Geometry, and Jackie Staddle, Junior Research Fellow in the History of Mathematics at Queen's College, Oxford.

Marcus Chesota, what are prime numbers and what is their significance? Prime numbers, probably everybody remembers from school, are these numbers which are not divisible by anything. So you can't divide them a number like 17. I can't write as two smaller numbers multiplied together. but 15, for example, I can write as 3 times 5, so that's not a prime number. Why they're so important to me as a mathematician are that they're the building blocks of all numbers. So if you take any number, it can be built by multiplying these prime numbers together.

So a number like 105 is built by multiplying 3 times 5 times 7. So the primes are a little bit like the atoms of arithmetic. They're the sort of hydrogen and oxygen of the world of mathematics. And that's why they're so important to the mathematician, because they're a little bit like the periodic table for a mathematician. The periodic table in chemistry lists 109 chemical elements from which you can build all molecules. Well, the primes, 2, 3, 5 and 7,

they're like the hydrogen, helium and lithium of the world of mathematics. So the significance is to mathematicians in order to further their progress as mathematicians, mostly. We're constantly finding we're having to go back to understanding the primes, because primes build numbers, from numbers you get mathematics, from mathematics you get the whole of science. So in a sense, they're right down there at the bedrock of our subject. So quite often a mathematician will be exploring a subject will find they'll need to know things about the primes in order to make progress.

I mean, I've got lots of theorems which depend on certain properties of the primes being true in order to be able to prove them. Can you tell us about the occurrence of prime numbers in the natural world? Yeah, that's very interesting. In fact, mathematicians weren't the first to discover the importance. of the primes. It was in fact a curious little insect in the forest in North America. It's a cicada. It has a very strange life cycle. This cicada hides underground for 17 years, a prime number, doing absolutely nothing. Then after 17 years, the cicadas emerge en masse into the forest and

sort of party away. They sing, they eat the leaves, they have sex, they lay eggs. And then after six weeks of partying, the forest is so loud that the residents in North America have to move out. But after six weeks, they all die, and the forest goes quiet again for another 17 years. So is it just a coincidence that it's 17 that they're hiding underground? But it seems not there's another species which hides underground for 13 years, and another for seven years. Two more prime numbers. Two more prime numbers, exactly. So what is it about the primes that are helping these cicadas?

Well, we're not too sure, actually, but we have a hypothesis. There might be a predator in the forest that also appears periodically. Now, if the predator chooses, say, six years for every time it appears, then the cicada that appears every seven years will keep out of sync much more often than a cicada that appears every eight years, for example, or every nine years. But the cicada that appears every seven years will meet the predator for the first time in year 42. 42 is six times seven. But the cicada who appears every nine years, for example, well, in year 18, both the predators,

predator and the cicada are there in the forest, both divisal by six and nine. So it seems like those primes were literally the key to the evolutionary survival of these cicadas, that those that chose a non-prime number year were knocked out because they kept on getting in sync with the predator. Robin Wilson, in the third century BC, we come to the great Greek mathematician Euclid. Yes, the most important thing that Euclid did in his elements, which of course is probably the most important mathematics book of all time, which was used constantly for about 2,000 years. It's largely a book about geometry,

but in the middle of it, there's a whole section about numbers. And talking about prime numbers, his most important statement was to prove that there are infinitely many primes. In other words, if you try to write out a list of all the primes, as you said earlier, you have to go on forever. There aren't just a finite number of them.

How did he prove that? It was very, very ingenious, and in fact I think mathematicians regarded it is probably one of the most beautiful proofs in the whole of mathematics. The idea is to do it by contradiction. You assume that there are only a finite number of primes and then show that from that finite number of primes, in fact, you can construct another one.

And the way he did it is, supposing you only knew that the primes two, three and five, okay? If you multiply them together and then add one, you get 31, which is another prime. if you take 2 and 3 and 5 and 7 and multiply them together and add 1 you get 211 which is another prime so it seems that they always get another prime you've got to be slightly careful because sometimes you don't get a prime if you take all the primes up to 13 and add 1 you get 30,031 which looks like a prime but in fact it's 59 times 509 but the point is if you assume they're only a finite number of them and you multiply them together and then add 1

your new number will not be divisible by any of the primes in your list. Therefore, there must be at least one other prime in the list. Therefore, they must be infinite. And therefore, each time you've got to find that number, there's another one, and therefore they must be infinite, yes. Jackie Settle. Euclid made his discoveries about prime numbers while looking for perfect numbers. Why were the reefs interested in perfect numbers?

Well, perhaps I should explain what a perfect number is. A perfect number is one where you can take all its divisors or factors and add them together and you get back to the number you started with. For instance, six is the simplest one probably. Six divides by one, by two, by three. If you add one and two and three together, you get back to six. The next perfect number is 28, with divisors one, two, four, seven, fourteen. Again, if you add those together, you get back to 28.

Now these are very special numbers and they're quite rare. This is a very special property. And for some reason, people were interested in perfect numbers and how to find them. And it comes about that you can find perfect numbers if you can find a particular kind of prime first. And to find those special primes, you have to take what I might call the two series. You start from two and you double it to get four, double again to get eight, double again to get 16 and so on. You look at those numbers in the two series. theories, if you can find a prime by subtracting one from one of them, that will lead you to a perfect number.

Now Euclid knew this, I don't know how he knew it, but he knew this, that you can use these particular primes to build perfect numbers. There's another Greek mathematician, if I can pronounce him, eratosthenes, devised a remarkably neat theory for listing prime numbers. Can you tell us about that? It's not really a theory. It's a method, and it's nice and simple. Eratosthenes lived slightly. later than Euclid. He was the librarian at Alexandria. And also known as a geographer, he was one of the people who first made an estimate for the circumference of the earth. But also he looked at prime numbers and said you can find the prime numbers by eliminating other numbers if you like. So if you make a list of your numbers, one, two, three, four, five, six, seven, then you go along your list and you cross out everything that's divisible by two.

You cross out all the even numbers. And then you go along and cross out everything that's divisible by three. and then everything that's divisible by five. And when you've done that, what you've got left are prime numbers. So it's like a sieve. It's called the sieve of eratosthenes. You sieve out all the numbers that have some factors and what you're left within your net are the primes. Marcus is Soto.

And then for about 2,000 years, Botheus sort of carried through the ideas of But Uteutli, does he understand it? But there wasn't a great deal of interest in prime numbers until the 17th century, when you have two Frenchmen turning. Why the gap, first of all, why the lack of... loss of interest? I don't think it's so much a loss of interest, but a lack of tools. I think we really had to wait, and we'll see this over the discussion. Each generation brought a new

sort of mathematics and a new way of looking at things, which gave people new tools to be able to talk about these numbers. I mean, the point about these numbers, there's this infinite list of numbers, but a mathematician at heart, what we do is we're pattern searches. If you look at a sequence of these prime numbers, there don't seem to be any patterns to them at all. seem extremely random. And so I think those two millennium are spent sort of seeing that there are no patterns and just not knowing quite what to do.

And this is what happened, I think, in the sort of 17th century, is that suddenly the age of equations came when people understood that equations were quite good ways of generating sequences of numbers. So that's why we get something like Mezen, who's a French monk, who was quite obsessed with numbers, also obsessed with music. He came up with a theory of harmonics.

and he actually took these numbers that Jackie's already mentioned where you multiply two together a lot of times, take one off that number, and perhaps you'll get a prime that way. And these numbers are now called Mersen primes after Mersen's sort of discovery that this seemed quite a good way to construct primes. In fact, this discovery of the 9.1 million digit number

that is a prime is a prime of this shape. You have to multiply two together, I think, about 30 million times or something, and then take one off that number. So this seemed to be quite a good way to generate primes, but it doesn't always give you a prime. Fermat also came up with a way of generating primes. He multiplied two together a lot of times and added one. So we come to an age when people were starting to explore the power of equations to actually find the primes for you. Robin Wilson, it was in the 18th century the Swiss mathematician Leonard Euler, who first proved a link, as I understand it,

which came to be known as the Zeta function between the Zeta function and prime numbers. Can you tell us, can you tell me, please, what the Zeta function is and what the link is? It's rather technical, and I'll try and make it as simple as I can. There was an outstanding problem which no one could solve, and it was to do with adding the reciprocals of the squares of the numbers. Now, what this means, the counting numbers are 1, 2, 3, 4, 5 and so on.

Their squares are 1, 4, 9. 16 and so on. The reciprocals of these are one, a quarter, a ninth, a 16th, a 25th and so on. Fractions are sort of like... They're all the fractions, yes. Reciprocum is one over. Now, if you add them all together, it was known that they added together to give an actual number, somewhere around 1.6. But no one could actually work out what that number was until Euler came along. and then he did two amazing things. The first was that using rather dubious mathematics,

he actually came up with the right answer. He showed that the sum of that series, the sum of the one over the squares of the natural numbers, is the number pi squared over six. Now, pi is the circle number. It's the ratio of the circumference of a circle to the diameter. Why should pi come in with the sum of the series? You'll have to tell me.

Well, as I say, it was all highly mysterious, okay? But the other thing that he did was he linked what we've been talking about, which has nothing to do with primes. He linked it very, very strongly with the primes. And the link with the primes is as follows. We're talking about the squares of the numbers. So if you take, for example, two and square, two squared, let's divide that by one less.

So that's four over three. take the next prime number three squared subtract one, eight so let's look at nine over eight next prime 25 square it you get 25 over 24

so if you take the product of all these things you take 4 over 3 9 over 8 25 over 24 if you keep on sort of taking the square of a prime and divide it by one less

and multiply them all together amazingly you get the same thing you get pi squared over 6 he got a direct link between the original sum we were talking about, where you're adding together the reciprocals of the squares and the prime numbers. And that was completely unsuspected.

Why should there be a link between the first problem which had nothing to do with primes but generally very closely involved with the primes? Anyway, the crucial thing is that first of all he took a problem which seemed to have nothing to do with primes and he linked it with primes and then he extended it.

We've been talking about adding one over the squares, he did a similar thing for the fourth powers. And there he got added them all together, and he got something like pi to the fourth over 90. Now, Oilo was not shy of doing hard work, so he did it for the sixth powers, the eighth powers. He went up to the 26th powers. And then from this, he then introduced what you mentioned, the Zeta function, which is rather crucial. So what he did was he said, okay, go back to our original problem, you're squaring numbers, you're taking one over them, adding them together.

We'll call that zeta of two. The next one is you're taking the fourth powers, taking one over them, adding them together, we'll call that zita of four. And in general, if you take all the natural numbers, all the whole numbers, and raise them to any power, say the kth power, and add them together, you call it zita of k.

So what he produced out of this was a thing called the zeta function, which arose directly out of adding together all those fractions. But as we've seen, he also linked it very, very closely with the primes. Does that, Jackie Stadol, does that take, did that take the, obviously took the study forward? And in the 19th century, Germany became the major European mathematical centre. And we arrived at Frederick Gauss. What part did he take up would have been discovered by Euler? No, he didn't actually.

he moved off on a slightly different tack. Gouse was a child prodigy really. He was a terrific calculator even as a child, very interested in mathematics and in numbers and calculation. And as a young boy at school, he had a table of prime numbers and simply spent a lot of time looking at these numbers and trying to see some sort of pattern in them. And what he did was counting primes, really.

He looked at how many primes you have in the first hundred numbers, say, how many you have in the next hundred, how many you have in the first thousand. And what you discover is that as you go through higher and higher numbers, the primes begin to thin out a bit. They become less dense as you go further on. There are always infinitely many of them. You'll always find another one.

But it takes you longer, on average, to find the next one. So that was his discovery. And he was able to be a bit more precise than that. He was able to give an estimate for what we might call the density of primes at any point. So if you look at the numbers around a million, for instance, He found that the density of primes is one over the logarithm of a million. This is something he discovered by simply looking and calculating. And I think we don't often think of mathematics as an experimental subject,

but just as with Euler, a lot of mathematics is done by pure calculation, pure slog, lots and lots of hard work, looking at the patterns that emerge, making a hypothesis, testing it out. Marks Gossi-Riemann was Gausie's pupil, and he made an enormous contribution, as I understand it. Definitely. And I think he almost made it by chance. I mean, experimental work is very important in mathematics and also being lucky, I think is as well. And I think it was, I don't think anyone expected this CETA function to really be as helpful as it has been in understanding the primes. I think I like a model of what Gauss discovered is to say that the primes, they look very random. So it looks like nature chose them by throwing a dice and deciding at each throw whether a number is prime or not.

I mean, obviously you can't make a prime not a prime by throwing a dice, but it produces quite a good model for the way the primes look. And what Gauss discovered was that there are 25 primes less than 100. So that means there's a one in four chance that a number from 1 to 1.000. to 100 is a prime. So nature is using a sort of four-sided dice to choose primes up to 100. When you get to a thousand, they've thinned out a bit more. There's only one in six chance that a number is prime. So it looks like nature's now used a six-sided dice, say with five sides blank, one side with a P on it, which says prime. And so you throw the

dice and you find that you get one in six numbers randomly distributed, and that looks rather like the primes. So Gauss discovered how the number of sides on the dice is growing, as you count higher and higher through the universe of numbers. So they thin out, so the number of sides and the dice is growing and growing. So that gave him a way to estimate how many primes you'll expect to see as you count further and further through the universe of numbers. This is actually called the prime number conjecture was proved 100 years later by Hademeyer and de la Vallé Pousson.

Mathematicians, though, do not like inexact things. We're obsessed with things being precise and exact. So Gauss has got us a good first guess at roughly how many primes there are as you count higher and higher. But mathematicians want to know exactly how that prime number diced landed and know precisely how many primes there are. And this is what Riemann discovered. His use of the Zeta function enabled him to understand, in a sense, how that prime number dice landed. And he discovered this almost by chance. He was working on this function, the Riemann Zeta function.

Actually, it wasn't called the Riemann Zeta function then, of course. it was just called the Zeta function. And you can think of this function. Robin's already alluded to the fact that you put in a number, you do a little calculation, perhaps don't worry what that calculation is, and you get out an answer. Feed in two, do some calculation,

out comes pi squared over six. So this is what a function is. You feed in numbers, and it's like a little calculating program, and it outputs answers. He took some new sort of numbers that were on the mathematical block at the time, which were called imaginary or complex numbers.

Now, these are sort of like two-dimensional numbers. They came out of trying to solve equations like, what's the square root of minus one? Well, most of us don't know any numbers whose squares are minus one. But at that sort of time in mathematics, we decided to invent a new number called an imaginary number, which could solve that equation. Riemann took these new numbers and said,

what happens if we feed them into the Zeta function? and amazingly he got a sort of, you can think of, most people probably know what a graph is. If you feed in numbers, the height of the graph tells you the answer to that function. Well, he produced a three-dimensional graph. You feed in these two-dimensional numbers, into the zeta function, out pops a number, and then that's the height of this landscape. Jackie Settle, can you tell us then how this developed into what we now know as the Riemann hypothesis?

Yes, well, in exploring this three-dimensional landscape that Marcus is to talking about, you look at it and it has mountains and valleys. It's like looking at a piece of land really. It never dips below sea level, but sometimes it goes down to sea level. And what Reeman was interested in were the points where it dips to sea level. It goes right down to zero. The height of the land goes down to zero. This only happens at isolated points. But what he discovered when he calculated some of these points was that they all lay beautifully on a straight line. And the real part, in each case, the real part of the number where the zero

happens is a half, it's exactly a half. So the imaginary part can be anything you like, but the real part always seems to be exactly a half, which makes the numbers lie on this lovely straight line, this north-south line. So Reamon's hypothesis was that all the zeros are going to lie on this line.

He couldn't prove it. It was, he did a lot of calculation. We now know that he did more calculation in his rough notes than he ever published. Again, lots of calculations. but his hypothesis was that this was always going to happen. I mean, these zeros are so important for this landscape because they're a little bit, Riemann discovered that they're the DNA of these imaginary complex landscapes.

So in a sense, if you know where this landscape falls to sea level, it tells you everything else about the landscape. I mean, it's like from that DNA you can generate the rest of the landscape. That's why these points at sea level, these zeros, where the function output zero are so important. It builds the landscape. Now, what does that have to do with primes? Well, primes also helped to build that landscape.

As Robin described, the function that you use to build the Riemann Zeta function is also built out of primes. And this is Riemann's amazing discovery, that, well, there must be a link between these two things. If they both build the same thing, this beautiful complex landscape, the zeros of the DNA of this thing, but the primes built the function that generated it, there must be a link between the two. And that's Riemann's great discovery. Robin, do you want to add to that? Yes, I wanted to still say how it actually carried on because I'd like to introduce the English mathematician, G.H. Hardy.

20th century. Yes, Hardy was probably the greatest English mathematician of the first half of the 20th century. And Hardy in particular got interested in the Riemann hypothesis. As Jackie said, the hypothesis is do all these zeros, these dips in the landscape, do they all occur on a north-south line? and Hardy showed that in fact infinitely many of them do that if you look at this line you'll find the first zero at 14.1 or something

and the next one at 20 or something and the next one at 20 something and they occur from time to time but there in fact that there are infinitely many of them on the line this doesn't mean that you've got all of them because of course the fact you've got infinitely many of them may mean that you have a few which are not on the line or you may have infinitely many not on the line

but he showed that in fact that infinitely many of them are on the line. So that gave good support to the hypothesis. In mathematics, proof is absolute. And just a single zero not on that line basically throws the whole thing out the window. I think we perhaps should say what the Rehmann hypothesis, why all these points being on the line, what it says for prime numbers, because why is it so important? Riemann discovered that if they're on this line,

it means that these dice that Gauss was using to choose the primes are actually a fair set of dice. They're distributing the primes fairly throughout the universe of numbers, rather randomly, but at least they're distributed fairly. If the Riemann hypothesis is false and there's a zero off that line, it means those dice are very biased.

It's forcing lots of primes in some areas and no primes in others. They're a bit like, primes are a little bit like molecules in a room. We're not quite sure where each molecule is, but we know that they're fairly distributed around the room. There isn't a vacuum in one corner of our studio and a concentration of molecules in another. The Riemann hypothesis, if it's true,

will say that the primes are distributed rather like the molecules in this room, rather fairly distributed, but still quite random. So can you give us an idea of the listeners who've stayed hanging on, many of them, I suspect, by their fingertips as I am doing, completely intrigued by this outer space thought that you mathematicians go in for. And there's a whole world out there that we talk about, really,

but are there any other uses that you can see might come up with? Well, this is, I think, the most exciting bit, the new bit of the story, which is these primes seem to have nothing to do with physics at all, but an amazing meeting. Most work in a mathematics department gets done over T when you talk to people. And at T at Princeton, the Institute for Advanced Study, a mathematician, Hugh Montgomery, was just chatting idly with a physicist, Freeman Dyson about what he'd been talking about in the number theory seminar.

and he spotted a pattern that he found in the way the zeros are laid out on Remen's critical line. And he seemed to think that the zeros didn't like to be close together. They sort of repelled each other. And when he showed Freeman Dyson this model, Freeman Dyson said, oh my God, that's exactly the way that energy levels are laid out in large atoms like uranium and urbium. And there's such special patterns that there couldn't just be a coincidence that the way the zeros are laid out on this line

look exactly the same as sort of spectra. that you get in energy levels in large atoms like uranium. So we suddenly realized there had to be a connection between these two. It's a bit like an archaeologist seeing designs in one area in Mexico and then in Egypt, and they're exactly the same. There's got to be some connection between these two cultures. So now we think that the way to understand Reimann's hypothesis is to use the mathematics of quantum physics,

this sort of sign that we've been given is telling us that the mathematics of quantum physics might be the maths that should be used to understand why the zeros are on this line. What if it is proved, Jackie said on, what if it isn't proved? Well, in one sense, it doesn't matter either way, because either way we'll get some very interesting mathematics out of it. I mean, all the evidence really now is pointing to the fact that it's true,

but we never know that it may be false. But if it's false, it'll be false for an interesting reason, and mathematicians will explore those reasons. And this has happened before in mathematics, that something that was thought to be true for a very long time is overturned, and you get a... a whole new development taking place. This happened in geometry, for instance, in the 19th century,

when you suddenly get completely new kinds of geometry that no one had thought of. It would be the same if the Riemann hypothesis wasn't true. If it is true, in proving that it's true, and I think it will be proved eventually if it is true, then that will also lead to new thoughts and new mathematics. Well, thank you for making this listener have an inkling of understanding of something very difficult and deep. And thank you very much to Robin Wilson, Jackie Stiddle,

Marcus Uso-Toy. Our next week's program will be about relativism, the philosophical school of thought that challenged the notion of absolute truth. Thanks for listening. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programs about history, science and philosophy at BBC.com.com.com.com.com.