In Our Time - Fermat's Last Theorem

Episode Date: October 25, 2012

Melvyn Bragg and his guests discuss Fermat's Last Theorem. In 1637 the French mathematician Pierre de Fermat scribbled a note in the margin of one of his books. He claimed to have proved a remarkable property of numbers, but gave no clue as to how he'd gone about it. "I have found a wonderful demonstration of this proposition," he wrote, "which this margin is too narrow to contain". Fermat's theorem became one of the most iconic problems in mathematics and for centuries mathematicians struggled in vain to work out what his proof had been. In the 19th century the French Academy of Sciences twice offered prize money and a gold medal to the person who could discover Fermat's proof; but it was not until 1995 that the puzzle was finally solved by the British mathematician Andrew Wiles. With:Marcus du Sautoy Professor of Mathematics & Simonyi Professor for the Public Understanding of Science at the University of OxfordVicky Neale Fellow and Director of Studies in Mathematics at Murray Edwards College at the University of CambridgeSamir Siksek Professor at the Mathematics Institute at the University of Warwick.Producer: Natalia Fernandez.

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For more details about in our time, and for our terms of use, please go to BBC.co.com. uk slash radio four. I hope you enjoy the program. Hello, on the 23rd of June 1993 at the Isaac Newton Institute in Cambridge, Andrew Wiles announced that after seven years of work it solved the most celebrated problem in mathematics, Fermat's last theorem. It turned out that Wiles had been slightly premature in making his announcement, but within a year he was able to plug the hole in his proof

and finally crack the puzzle that had foxed the most brilliant mathematicians for over 350 years. Pierre de Fermat was a 17th century French lawyer, an amateur mathematician, whose all-consuming passion was numbers. In 1637, he realized he'd made a profound discovery and scribbled in the margin of his book he was reading, I have found a truly remarkable proof which this margin is too small to contend.

It's a tantalizing footnote that began one of the most famous detective stories in the history of mathematics. But what was Fermat's theorem and why did it take the world's most accomplished mathematicians so many years to prove it? With me to discuss Fermat's theorem are Marcus Jusotoy, Professor of Mathematics and Simeonio Professor for the Public Understanding of Science at the University of Oxford, Samir-Six-Ex, Professor at the Mathematics Institute at the University of Warwick, and Vicki Neal, fellow and director of studies in mathematics at Murray Edwards College at the University of Cambridge. Marcus Sotoy, the roots of Fermas last theorem go back to Pythagoras. Can you tell us how he was obliged to those roots? Well, Pythagoras's theorem, as people will probably remember from school,

is about this equation, A squared plus B squared equals C squared. It's a do with right-angled triangles. So if you take a right angled triangle whose two smaller sides have length A and B, and the long cycle, the hypotenuse has length C, then there's this relationship between the lengths, the square of the two shorter sides, you add those together, you get the square of the longer side. Now, sometimes you can build triangles, say,

if you have the short side one unit and two units, well, the long side is rather a nasty length thing because it's the square root of five, which isn't a whole number. I think you can't square a whole number to get five, can't even square a fraction to get five. So that's not a very nice triangle. But the ancients discovered there are some nice triangles where all the sides are whole numbers.

So people might remember the triangle 3, 4, 5. So if you take 3 squared plus 4 squared, which is 9 plus 16, that's 25, which is 5 squared. So you get this beautiful triangle where all the lengths are whole numbers. So the question was, are there more triangles like that?

And people discovered that, if you take a triangle with 5, 12 and 30, 13 as the lengths, that also satisfies this beautiful equation, a square plus B squared equals C squared. So this kind of started off this search for numbers that satisfy this equation. As we'll discover, I hope, at the end of the program, if we've got time, what seems an abstract and esoteric chase ends up in massively important practical consequences. But at the very start, what Pythagoras was talking about was something according to your

is that the Egyptians and the Babylonians were practicing in engineering, nearing anyway, the right angle triangle done by knots in ropes. Exactly. If you want to build a pyramid, you want to make sure the corners are a right angle. So you need to be able to create a right angle, 90 degree angle. And they realised that this 3, 4, 5 triangle

was incredibly useful for them, because it was a very simple way. You have a piece of rope with knots in it, and you arrange it such that there are three units along one side, four along the other, five along the third length, and then you're guaranteed a right angled triangle. So amazing, this way.

was a sort of practical interest, but from thereon it became really just an intellectual interest about, oh, interesting were there other numbers which satisfy this equation? And none as we far as we know, but there's much more to be discovered about the past as we hope for this programme, more and more and more. But there's the Greek mathematician, another Greek mathematician, 8 hundred years after Pythagoras, Diophantus, he developed Pythagoras work in his book of equations called Arithmetica. And how did that take this? Yeah. How did that take it forward?

Well, Diophantus really, I mean, he's third century AD. We don't really know very much about him, but he wrote these 13 books, which is really continuing this quest to try and find interesting solutions to these equations like Pythagoras and other equations as well, which have now been called Diophantine equations in order of that work. So you might say, yeah, perhaps y squared equals X cubed minus two. Can you find any solutions to that? Now, the interesting thing, he was not only interested in whole number solutions, but also solutions which were fractions. And it's extraordinary, this guy was so far ahead of his time in asking these questions, I think. First of all, to sort of focus on these equations, just ask whether there are a whole number or fraction solutions to these. But also, he was one of the first to start to develop a sort of language, an algebraic language, of talking about X's and Y's. If you're looking at the sort of Greek text before that time, there's no talk of,

sort of an abstract algebra to represent an arbitrary number. In fact, Diophantus' work got lost for centuries and really only re-emerged during the times of the Arabic world when they were sort of re-translating these texts. And it was then that it sort of came into Europe and suddenly that Europeans discovered this beautiful sort of subject of diaphantine equations. A remarkable factor about him, as I read from what you three have written,

is that this arithmetica he did for fun. and this wasn't, didn't have any value with ropes and pyrumints. I mean, he did it for fun for his friends and these are problems. What do you think of this problem? And that became part of mathematics for many centuries and part of the Fermat theorem quest. Vicki Neal, straight to Fermat, 17th century, that's just a bit of a swoop. Can you tell us how he became interested in diophantine equations?

Yeah, let's jump several hundred years to Fermat, who was a 17th century French mathematician. and he was a mathematician in the sense of being a person who did maths, but it wasn't his profession. He hadn't studied mathematics at university or whatever. Very much the culture of the time meant that that wasn't possible. So he was a lawyer by profession and government official. And in fact, in the course of that work became sufficiently senior that he was allowed to be Pierre de Fermat, as opposed to plain old Pierre Fermat.

But in his spare time, as you do, he very naturally thought about mathematics. and he thought about a whole range of aspects of mathematics. He thought about probability, he thought about optics, the physics and mathematics of light. But he was also really excited by number theory, by the study of properties of the whole numbers. And he actually corresponded with other mathematics and some of the greatest mathematical philosophers of the time.

He wasn't burrowing away on his own in his lawyer's office at night, was he? That's right. So he was writing and he met with Mersenne, who was a monk who went around, meeting all sorts of mathematicians of the age and kind of sharing their eyes is between them in some cases whether they wanted to share them or not. And Fermat was also sitting in Toulouse

writing letters to mathematicians in England all over France, all sorts of people, saying, oh, I can do the following thing, can you? So he never said how he was doing it. He never explained how it fitted into his bigger theory or the things he discovered. He just said, here's this fact, can you prove it? Which went down very badly with some mathematicians

he felt this was rather unhelpful behaviour, rather sort of teasing them. But they didn't distrust him. They kept in correspondence and he was a valid figure at the time on the highest level. Absolutely, yes. Right, the big question, it's over to you, Vicky. What is Firmat's last theorem? Yeah, we've probably watched to talk about that. So Firmar got hold of a copy of this book of Diophantus that Marcus mentioned,

the Arithmetic. It was a translation into Latin by someone called Bachel, who, had some interest in mathematics as well as many other things. And Phamma was reading Diophantus's work. He was learning about this work of the ancient Greeks, these problems of finding whole number solutions to equations. He read about Pythagoras' theorem and the problem of finding whole number solutions to that. And we presume did what mathematicians do and said, well, can I generalise this? This is what mathematicians do all the time. So I know that I can find whole numbers square, so that one square plus another square gives a third square.

So I can write this square number as a sum of two other squares. Is it possible to do something like that with cube numbers, or fourth powers, or a hundredth powers, or a millionth powers? So is there a number that is one number times another number, sorry, one number times itself, so a cube that can be written as the sum of two other cubes. And presumably he went and did some calculations, and he tried some numerical examples. presumably he tried to adapt the work of Diophantus studying whole number of solutions to Pythagoras's equation

and he couldn't find any solutions. He couldn't find a cube that could be written as a sum of two other cubes or a 17th power that could be written as a sum of two 17th powers. So having established he couldn't find any, he presumably then went to way and tried to prove it. And all we know of this is his notes that he wrote in the very generous margin in Bachelet's translation of this work of Diophantus, where he's a asserted that it's not possible, that it's not possible to write one cube as a sum of two cubes or one fourth power as a sum of two fourth powers and so on. And then extremely provocatively, there's this remark at the end. I have a truly wonderful proof of this, but this margin is too small to contain it. Did they believe him? Well, as far as I know, it wasn't particularly published at

the time. So he wasn't making this note for publication. This was a note in his book for his own benefit. And the way it got shared with the wider world was after Fermat's death. His son published a copy of his father's book with these notes and that's when the wider world got to look at it. Before we go on, can you, I'm sorry about this. This is me being thick and that's okay by me. Can you just be clear once again what the core of Fermat's last theorem was and then we'll crack on? Absolutely. So Fermat expressed it in words and said that it's not possible to write one cube as the sum of of two cubes or one fourth powers of sum of two fourth powers and so on.

So you can square things but nothing else? Exactly. With whole numbers. With whole numbers, right. So we would usually write that with symbols. So we'd say, we'd think about the, Pythagoras's equation is x squared plus y squared equals z squared.

So we'd be thinking about equations like x cubed plus y cubed equals z cubed or x to the power four plus y to the power four equals z to the power four. And the assertion is this if you have an equation of this form, X to the N plus Y to the N equals Z to the N, there are no integer solutions for N greater than two, except things involving zero, but that sort of rubbish solutions,

boring solutions, we ignore them. Right, Samir, for a time, as Marcus said at the beginning of the programme, his theorem faded, taken up and developed as so much was, in the great Arabic Renaissance of those four or five hundred years, and then brought back into Europe. But if we can move, I'm sorry about this, to the 18th century with Leonardo Euler, the Swiss mathematician who brought a lot,

who Marcus has called his notes, perhaps the greatest mathematician that there's been. What did he bring to the table? Okay, so, excuse me, that's true. Oiler, I mean, if you ask a mathematician, give me a list of, let's say, the five greatest mathematicians of all time. Oiler will definitely be on that list for any mathematician.

Oiler was an extremely prolific mathematician. He worked across mathematics and physics, and his writings are perhaps gathered in 65 volumes. One of Oiler's interests was the mathematics of Fermat. And one of the things he did is to try to prove or disprove the many assertions that Fermat made in his letters, the challenges he, made to other mathematicians. So one of the things he looked at was Fermat's last theorem for the, I mean, maybe we should talk about X. Why was it called the last theorem? Because by the time Euler had finished with all the other assertions, that was the last thing, the last assertion that Ferma had made that was left neither proved nor disproved.

We don't know. Done the lot except for this. Exactly. Exactly. But it's perhaps the time to sort of talk about exponents. So if you multiply a number by itself seven times, then the exponent is seven. And if you multiply it by itself 11 times, the exponent is 11. So we are talking about cubes now.

A number multiplied by itself three times is not equal. or is equal to a number multiplied by itself three times, there's another number multiplied by itself three times. So the exponent is three. That's the first case, you know, the first case in which Fermat's last theorem applies, because squares, that was Pythagoras' theorem. Okay.

So this... So we moved from squares to cubes. Exactly, squares to cubes. Now, what Euler did is he looked at this, and he realized that to solve this, it's not enough to work with the usual numbers, 0, 1, 2. You need an extra symbol. And that symbol is a symbol which when you multiply it by itself three times.

When you cube it, you get 1, but it's not one of the usual numbers. It's not 0 or 1 or 2. And you take that symbol and you can calculate with it. okay and that gives you what you might think of as an enlarged domain of arithmetic

so he invented a number invented a domain if you like and new and who knew world of numbers okay and

what he looked at was factorization in that world now what is facturization If you take a number like six, you can write it as two times three, but then you can't split it further because two, you can't write it as a product of two smaller things. So two and three are primes and six isn't. Now, in that enlarged domain of arithmetic that Euler invented,

you can do factorizations as well. And Euler realized that this is what you need to prove Fermat's last theorem for cubes. Now, so Euler made a proof, but standards of proof in those days were a little bit wobbly. Okay, so there is a guess. gap in his proof that was filled later, but the proof was

essentially correct. Marcus is Soto so we've got to the end of the 18th century thanks to, thank you very much and I think it's clear so far to me anyway, so we hope. Do you

think that oilers getting to three from two to three from squared to cubed was considered this is as far as you're going to get? No, I don't think so. I think it gave some hints that there were methods that might go a lot further. So the point is that you've done one equation, x-cube plus y-cubed equals z-cubed. You've shown that that doesn't have any solutions. But unfortunately, there are infinitely many more equations out there. So you might say, well, we haven't got very far if we've only done one. And it's interesting

to think whether, you know, did firma actually have a proof? And I think that he had a method which would solve x to the four plus y to the four equals z to the four and it depends on something we now called firmat descent. I mean, one way you could do it is say, well, suppose this thing does have a solution, okay? So you're now going to try and get a contradiction. This is one of the classic tricks of the way mathematicians prove things. Suppose it has a solution. He showed how you could get, so there are three numbers, let's call them A, B and C, which satisfy this equation when you take their fourth powers, the sum of the two smaller ones add to the fourth power of the larger one. He showed a way that you could get a smaller solution, A dash, B dash and C dash.

And then he showed how you get a smaller one, a smaller one and smaller one. But these are whole numbers. So there are some point you must hit something which you can't get any smaller because, you know, one is the smallest whole number. So he said, well, if there is a solution, I can show you why I can get infinitely many solutions, each one's smaller than the last. But that's a contradiction because, you know, at some point I must hit the number one. And this idea of Fermat dissent was the key, so the first idea about how to prove this theorem.

And I think that Fermat realized he could do X to the 4 plus Y to the 4 equals Z to the 4 using this. And he thought it would generalise. And that's why he said, oh, yeah, the margin is too small for this proof. Interesting, as Vicky said, the margins in Bachelet's translation, a massive. So actually... Well, it was probably...

But even with... No, I don't mind the same... If he said he didn't have enough space, he didn't have enough space. Exactly. But so it's interesting. I think that oilers then progress showed another way of attacking this. And it kind of opened up the field for, okay, how far can we take this new domain of numbers, which are actually based on things called imaginary numbers, things like the square root of minus one. So you think, what on earth that got to do with solving these whole number equations. But it sort of opened up the possibility that maybe this thing could be proved. Samia set us off on what could be called the modern quest from Euler onwards and now we're going to meet figures in different countries of the world in Europe, in Japan, in America and in this country

who are pegging away, driving away at this problem. So it's going to be names from now. But one of the most intriguing is a woman who, in order to get her contributions thinking it all seriously, pretended she was a man for a long time in her correspondence. Can you tell us something about Sophie Germain? and what she contributed. Yes, there are regrettably few women in the story of Fermat's last zero, but I'm delighted to say that there is this one

and that the contribution she made was really very serious indeed. Sophie Germain, another French mathematician in the sense of being somebody who does mathematics, but again, not a professional mathematician in the sense that we'd understand now. She wasn't allowed to go to university anyway. Because of the culture in France at the time, women were not allowed to study maths at university. there was no way that she could have pursued it professionally. She started by finding books in her father's library when she was a teenager

and became completely engrossed in mathematics. So she used to try to study at night and her parents were very against this idea and would confiscate her clothes and her candles. So she'd be reading under the quilt with a smuggled candle and so on. At some point when the Ecole Polytechnique opened, she was about 18, she was able to get hold of the lecture notes. She wasn't allowed to be a student, but she was able to get hold of the lecture notes

and was studying and submitted her work to Legrange, who was the professor for one of these courses, but as you said, Melvin, she wasn't allowed to be a student, so she used the student in Monsieur LeBlanc, who was a student who'd left the university, but nobody had noticed,

so she could pretend to be him. And Le Grange was very impressed by these solutions and had some correspondence with her until eventually she was forced to admit who she was because they were going to meet. But that really sort of spurred her to continue with her mathematical researchers and start doing new work as well as studying. Sorry to interrupt.

What did she do on Fermat's last theorem that's relevant to what we're talking about? So she studied the work of Gauss a little bit later and he wrote one of the classic works on number theory with all sorts of new ideas. And she learned lots of mathematics related to the study of numbers from that which she used to investigate Ferma's last theorem for some particular cases. Can I turn to Samir here? Can we just pause and detail where she took the theorem,

with the help of correspondence from Gauss and so on, but nevertheless keep it in her domain, but from you at the moment. Or from you, actually. Okay. So Farma did exponent for, fourth powers. Oiler did cubes with a small gap that was

filled in later. Sophie Germain did all exponents up to 100. That sounds a very dramatic increase, doesn't it? Exactly. And what she in, her theorem is quite interesting because there's a

criterion there. There's a something that says if you want to prove first Fermat's Last theorem for a particular exponent, let's say a number, a particular number. You take that number and you do a series of operations to it and you check some things and if one of these things turns out to be true, then you know that Fermat's Last theorem is true for that exponent. So by hand calculations, she was able to check her criterion for all exponents up to 100.

But had she had computers, she could have maybe done it for a few thousand exponents. Is it possible to explain to us how she managed to do this? You talked about Euler a few minutes ago saying on any list of five of the greatest mathematicians ever lived, He was one of them. This is a woman that's vividly described by Vicki, reading mathematics under the bedclothes, and she seems to have got 96 times more stuff than he did. So what did she crack that he didn't?

I think it's partly because she came after Gauss and studied the writings of Gauss. Can you have done something about Gauss now? Okay, so Gauss is also one of the top five mathematicians of all time on anyone's list. One of his interests was number theory. So this is the mathematics of Fermat. But number theory was better understood by Gauss than, previous mathematicians.

And early on in his youth, he wrote a standard textbook on it that was used for many, many years to come. And it's still interesting to read today. And she took a lot from him, but then developed it. Exactly, yes. Gauce wasn't really interested in Fermat's last theorem. He was, because

Fermat's last theorem, it's an interesting enigma, but it's not a fundamental problem in mathematics. It has historical interest. You make that point too, Mark, that it wasn't on somebody or the other, I've gotten on a list of the 23 most important problems. To be sorry, he'll thank you very much, Hilbert. But nevertheless, it became a supreme mathematical problem

and there was a real pursuit of it, wasn't there? Yes, that's right. And so in the 19th century, the extent of the French Academy of Sciences, in 1816 and 1850 offered serious money to people who could solve it. Absolutely. The Academy de Scents was loved offering prizes

for big open problems with money and a golden medal. But as Samir said, Gauss was incredibly dismissive of this problem because what's the important thing about this equation? I could write down a whole load of equations and ask you whether they have solutions or not. There's nothing particularly important

about this. But actually, I think those prizes did sort of focus attention on the importance of this. And it kind of became this sort of trophy that people wanted to prove. But I think the other importance about Fermat's Last theorem, although I don't think there are any theorems which say, suppose Fermat's Last theorem is true, I can do this if I know that, unlike the Riemann hypothesis, for example. But the importance became, it was a fantastic catalysts

which generated most extraordinary number theory over the coming centuries. And in particular, we come to a period where mathematics, I mean, there was a mathematician who stood up at the Academy de Seons and said, I have a proof of this theorem. Lame in 1847, thought he cracked it, using sort of similar ideas to Euler, which come to the fact that if you use these kind of extended numbers, things like the square root of minus one and sort of cubes, cube roots of one and things like that, that he could actually somehow use this factorization of the equation and actually prove that this thing didn't have.

any solution. So, you know, for a few hours, people thought this had been cracked until somebody else stood up and said, actually, well, you realize you made a huge mistake. And this is a kind of classic, this is the story of Fermat's last theorem, of people saying that they proved it, and then realizing just how subtle this equation is. So it was the game element that kept it going, rather than that this will make, this will help all sorts of things in the future. I think there was no utility being thought about here. This is this is about, you know, we don't compose the magic flute because we think it's useful. We compose it because it's absolutely extraordinary, full of surprise.

And it's the same with Fermat's last year. It kept on, it could have been a very boring equation. But it wasn't. It just kept on generating more and more sort of stories and intricacies. So we come to this amazing discovery that although whole numbers, when you factorise them, so Samir gave this example, six can be written as two times three. Well, you can't write it in any other way as two whole numbers. numbers multiplied together. You know, you can't choose, I mean, if I choose a bigger number,

105 is three times five times seven. Well, you can't choose any smaller number, other numbers, like, I don't know, 11 times something else. There's only one way to break these numbers into their prime building blocks. Now, Lamey thought that that was true for this extended domain of new numbers, and that's what he used in his proof. And if it was true, he would have proved Fermat's last theorem. But it wasn't true. And it turned out, and mathematicians knew this already, and they told Lameh, I'm afraid, you know, your proof kind of works if that's true. But unfortunately, in this new domain of numbers, there are sometimes two different ways to break things apart. It's a bit like water can be made out of oxygen and two hydrogen atoms.

You can't make it out of anything else. But in these new domain of numbers, there are some other atoms which would make water. Can we briskly go over a couple more somewhere before we come with you and Vicki? Ernst Kuhner, he was chipping away at it too. So what did he bring? Okay, so Kummer was a Prussian mathematician. Actually, he started out as a school teacher, but he corresponded with other mathematicians,

and his writings were so great that eventually he was offered a professorship and took up a professorship at Berlin. So Kumar understood the, was really the first. first person to understand this failure of unique factorization. This idea that in these extended domains of arithmetic, when you try to split numbers into their indivisible atoms, as Marcus says, into primes, there is... Into prime numbers, which numbers can only be divided by themselves and one. Exactly.

Okay, if you try to split them, it doesn't have to be in one way. It could be. So six, you can only write it as two times three and, you know, no other way as a product of primes. But in these new domains, there is some confusion there. There is some failure. And Kumar found a way of measuring that failure. and that enabled him to rescue part of Lamy's proof. Can I...

He actually won the prize. They gave him the prize, although he hadn't proved about his last three, him, he won the prize of 1850 from the French Academy for his work. And they discovered he hadn't, did he give it back? No, no, they said you haven't proved this, but what your work is absolutely amazing. So here's 3,000 francs and a gold medal. So if I can say, Kumar understood exactly the limitations of what people like Oiler had done

and what Lami was trying to do, exactly how far you can stretch that argument. Thank you, Vicky. We've got Louis Maudel in 1920 and Gerd Fultings in Germany who built at this work. Can you tell us how this is moving the thing forward? Yes, so... In the 20th century. Jumping on a few more. more decades. This is somehow a very different mathematical flavour. So one of the fantastic things

about mathematics, and this is a very good example, is that you think you're solving a problem using one kind of concept, and all of a sudden you find you need ideas from all sorts of other areas of mathematics. So this is a very innocuous looking equation. It's all about whole numbers. We've already heard how imaginary numbers and the square root to minus one and all sorts of things come into play. And this is a case of using even more ideas that at first sight appear to have nothing at all to do with Fermat's equation, but they turn out to have some relevance. So one thing that one could do is to take Fermat's equation, something like x to the five plus y to the five equals z to the five, and say, well, what about, instead of thinking about just whole number

solutions, I'll imagine that these variables, X, Y and Z are complex numbers. And so you're feeding complex numbers, using these imaginary numbers, using this larger domain involving the square root of minus one, and what you get out involves complex numbers as well. And using this idea, you can turn this equation to some kind of surface. So it represents a geometrical object in some kind of slightly scary, perhaps four-dimensional space. And by thinking about this surface, one can ask questions about it. So Mordell made a conjecture that surfaces like this particular types of surface linked to this firma equation have finitely many whole number or fraction solutions. And he made this conjecture in the 20s. This was known as Modell's conjecture. And then in the

1980s, this German mathematician Gov faultings came along and proved this. So what this showed was not that there are no solutions, which is what Fermat had asserted, but at least that there are finitely many. So for any power, not just fifth powers, for any power, X to the N plus Y to the N equals said to the N, where N is greater than two, there are only finitely many solutions, which feels a bit of a cheat, because secretly we think the answer is that there is no solutions. This equation just says there are finitely many. Maybe there are 10 to the 10 to the 10 to the 10 solutions. Who knows?

But the fact that there are only finitely many is somehow a massive advance and really, really quite a profound development. So what seems to be happening to the layman over here at this side of the microphone, Marcus, is that Fermat's the interest in it is bearing up because of the increase in knowledge about the way numbers work. Because of my... It's needing the new technology of numbers for people to get to the root of it. And we enter two Japanese mathematicians,

Tanayama and Shimura, with their conjecture. Yes, well, in some ways, the really interesting equations aren't these ones of Fermat, but equations called elliptic curves, which actually Diophantus, already was looking at in the arithmetica, which are things not with Pythagoras is about squares, but what about if you take a square and a cube? So, for example, you want to know y squared equals x cubed minus two. Well, that's interesting. Are there any squares which can be written as

cubes minus two? Well, yes, there are. You can take three, five squared is 25, which is the same as three cubed, 27, minus two. So you've got a solution there. So actually, these, turn out to be much more subtle sorts of equations. And we wanted to know which white ones of these have infinitely many solutions, which have finitely many. Now, this doesn't sound like it has anything to do with Fermat's last theorem, but it will in a moment. So hold on to your seats.

No, we have got to damage. Exactly. We're going to force your position this morning. Yeah. So these two Japanese mathematicians realize, if you want to solve these rather innocuous-looking equations, a square equals a cube, actually there's a kind of amazing dictionary which translates that problem

into a completely different area called modular forms. that the number of solutions of these equations, the sort of types of solutions of these equations, are controlled by a completely different object in another area of mathematics. And so they said there's a connection, a dictionary. Every elliptic curve will have one of these modular forms, which will help you to sort out the solutions. And that was their conjecture that elliptic curves are modular forms

are somehow two sides of the same equation. But they couldn't prove it. There may be some elliptic curves, which didn't have a modular form which actually kind of controlled its behavior. So this is the kind of setup, but what does this have to do with Fermat's last theorem? Is your next question? Well, before we ask, I'm going to ask somebody to come in and, because we're getting there, and we're on schedule.

I'm going to ask him about Gerard Frye, sorry, how did he address this Japanese conjecture? And how did he, how did he push it forward? Okay, so Frye, he related, well, actually, he took Fermat's last theorem, and he was trying to argue by contradiction, which is the way mathematicians often argue things, which is to say, let's suppose that we have, that there is really a solution to, and we're trying to prove that there isn't a solution. and from that he constructed an elliptic curve, one of these objects that Marcus was talking about. And then he looked at this dictionary, which is the Tanayama- Shimura conjecture. And so this dictionary in a way relates the continuous world, the part of mathematics that's, let's say related to what's called calculus at school with the algebraic world.

where elliptic curves live. So it said take this solution to Fermat's last theorem, make an elliptic curve out of it according to a certain recipe, and then go over, this dictionary, to the continuous world. And then the object that you have there, its properties look so fantastic that somehow it's too bad to be true. It can't be there

But he somehow couldn't show that it can't be there But he was telling other mathematicians Hey look Look at this Can I just briefly Before we come to the final Before all will be revealed Because we're talking about here

Which is fascinating About a global interest We're now talking about the solution The work towards the solution Whizzing around the world we've talked about Japan, we're talking about Americans coming in, all over Europe. People are piling into this now. Even though we're told that on the list of 23 of the most fundamental problems,

it's not on the list, except he puts it on the list in other ways. I now remember, Marcus, doesn't it? Yeah, he does. He disguises it and puts it to number 10. He says, you know, we should be interested in all Diphanty equations, not just this one. The business of the globe, did that help it to move forward, the fact that so many people from different cultures were, well, the one culture of mathematics, I suppose, when you come to think of it.

Yeah, mathematics is a, very international activity. People cheerfully collaborate with mathematicians all around the world and the fact that they speak different languages doesn't really matter when they're talking about mathematics. But maybe it's important to point out that Tanayama and Shemura wasn't thinking about Femma's last theorem at all. Nothing to do with it whatsoever. So there were people working on this kind of conjecture of Taniyama and Shemura because it had really important, really profound mathematical consequences in terms of linking these worlds of elliptic curves and modular forms. Nothing whatsoever to do with Fermat's last theorem. Phelma's last theorem was not behind this

in any way at all. So there was this international kind of effort working on this problem, but not because of Fermat. The link with Fermat came when Fry noticed that perhaps one could use it, the Tanyama Shumer of conjecture to prove Fermat's last theorem. But that was somewhat later. So it was cracked by Andrew Wiles, English person. Right, how did you do it? Well, so he realised that if this observation of fries, that if there is a solution to... I mean, in 1994-95, yes.

Exactly. Well, in the mid-80s, it was realised, proved by Ribbitt and Sayre that, sure enough, if you could have a solution to Fermat's last theorem, it would create an elliptic curve which wasn't on this dictionary. It didn't have a modular form.

So suddenly Andrew Wiles realized, now I'm in the game. All I have to do is to prove this conjecture, this modularity conjecture of Tanayama Shimora that this dictionary is indeed true, that every elliptic curve must have a modular form. If thermoslars theorem is false, I've got an elliptic curve which doesn't have a modular form. So he then spent seven years, isolated up in his attic because it's such a trophy this side. He sort of wanted to prove it himself. He worked away and then realized that he, in a way, a dictionary means if you can count the number of words on one side and show that the same number of words on the other side,

you can show that there's a match. And he tried to do this sort of counting argument to sort of show why the two sides must in some sense have the same size. And he managed to do this, and he announced it at the Newton Institute in Cambridge in 1993. It was sent off to journals to be refereed, and unfortunately there was one, you know, a lot of questions always arise from a paper.

There was one mistake. And unfortunately with a proof, you know, if I want to prove that I'm related to Napoleon, one little gap in that family tree does for it. And this, you know, there was a potential this would just be yet another story. But he was able to actually patch the proof, absolutely amazingly. He came back a year later, working with one of his former PhD students, Richard Taylor, and they managed to prove this thing. And it is, I mean, I feel honoured to have been alive.

And I remember the announcement. Unfortunately, wasn't in Cambridge. I was working as a postdoc in Hebrew University. but I remember that day when it was announced, you think, wow, I was alive when that was proved. That's amazing. Finally, for our listeners, what came of this, which we all like to know, we non-mathematicians,

what came of this in, as it were, the practical world? Can you be very brief and tell us, because in your note, your last page, you said, look, after all this hundreds of years of play, these things came out for the world we live in now. How much have we got? You've got about, let me see, you've got 20 seconds at my.

most. Okay. So the mathematics that was developed in the 19th century to prove Firmar's last serum is now used in cryptography, in protecting your bank details, in protecting your passport. If Ferma hadn't wrote that comment in his margin, the world would be very, very different today. Thank you very much, Samay, Sikzek, Marker Gisotoy, Vicky Neal. Next week, we'll written by the Empress Matilda, the anarchy in England in the 12th century and all that. Thanks for listening. There are many more Radio 4 arts and discussion programmes to download for free. Find these on the website at BBC.com.uk slash Radio 4.