Fermat's Last Theorem (1996)

Fermat's Last Theorem

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637, but was not proven until 1995 despite the efforts of many illustrious mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th. It is among the most famous theorems in the history of mathematics.
Source: Wikipedia

Simon Singh and John Lynch's film tells the enthralling and emotional story of Andrew Wiles. A quiet English mathematician, he was drawn into maths by Fermat's puzzle, but at Cambridge in the '70s, FLT was considered a joke, so he set it aside. Then, in 1986, an extraordinary idea linked this irritating problem with one of the most profound ideas of modern mathematics: the Taniyama-Shimura Conjecture, named after a y... (read more)

Fermat's Last Theorem

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637, but was not proven until 1995 despite the efforts of many illustrious mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th. It is among the most famous theorems in the history of mathematics.
Source: Wikipedia

Simon Singh and John Lynch's film tells the enthralling and emotional story of Andrew Wiles. A quiet English mathematician, he was drawn into maths by Fermat's puzzle, but at Cambridge in the '70s, FLT was considered a joke, so he set it aside. Then, in 1986, an extraordinary idea linked this irritating problem with one of the most profound ideas of modern mathematics: the Taniyama-Shimura Conjecture, named after a young Japanese mathematician who tragically committed suicide. The link meant that if Taniyama was true then so must be FLT. When he heard, Wiles went after his childhood dream again. "I knew that the course of my life was changing." For seven years, he worked in his attic study at Princeton, telling no one but his family. "My wife has only known me while I was working on Fermat", says Andrew. In June 1993 he reached his goal. At a three-day lecture at Cambridge, he outlined a proof of Taniyama - and with it Fermat's Last Theorem. Wiles' retiring life-style was shattered. Mathematics hit the front pages of the world's press. Then disaster struck. His colleague, Dr Nick Katz, made a tiny request for clarification. It turned into a gaping hole in the proof. As Andrew struggled to repair the damage, pressure mounted for him to release the manuscript - to give up his dream. So Andrew Wiles retired back to his attic. He shut out everything, but Fermat. A year later, at the point of defeat, he had a revelation. "It was the most important moment in my working life. Nothing I ever do again will be the same." The very flaw was the key to a strategy he had abandoned years before. In an instant Fermat was proved; a life's ambition achieved; the greatest puzzle of maths was no more.

Source: google.video uploader,
http://video.google.com/videoplay?docid=8269328330690408516

Fermat's last theorem

Pierre de Fermat died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book.

Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus's Arithmetica.

Fermat's Last Theorem states that

xn + yn = zn

has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote

I have discovered a truly remarkable proof which this margin is too small to contain.

Fermat almost certainly wrote the marginal note around 1630, when he first studied Diophantus's Arithmetica. It may well be that Fermat realised that his remarkable proof was wrong, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat.

In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x, y, z with
x2 + y2 = z2 such that xy/2 is a square. From this it is easy to deduce the n = 4 case of Fermat's theorem.

It is worth noting that at this stage it remained to prove Fermat's Last Theorem for odd primes n only. For if there were integers x, y, z with xn + yn = zn then if n = pq,

(xq)p + (yq)p = (zq)p.

Euler wrote to Goldbach on 4 August 1753 claiming he had a proof of Fermat's Theorem when n = 3. However his proof in Algebra (1770) contains a fallacy and it is far from easy to give an alternative proof of the statement which has the fallacious proof. There is an indirect way of mending the whole proof using arguments which appear in other proofs of Euler so perhaps it is not too unreasonable to attribute the n = 3 case to Euler.

Euler's mistake is an interesting one, one which was to have a bearing on later developments. He needed to find cubes of the form

p2 + 3q2

and Euler shows that, for any a, b if we put

p = a3 - 9ab2, q = 3(a2b - b3) then
p2 + 3q2 = (a2 + 3b2)3.

This is true but he then tries to show that, if p2 + 3q2 is a cube then an a and b exist such that p and q are as above. His method is imaginative, calculating with numbers of the form a + b√-3. However numbers of this form do not behave in the same way as the integers, which Euler did not seem to appreciate.

The next major step forward was due to Sophie Germain. A special case says that if n and 2n + 1 are primes then xn + yn = zn implies that one of x, y, z is divisible by n. Hence Fermat's Last Theorem splits into two cases.

Case 1: None of x, y, z is divisible by n.
Case 2: One and only one of x, y, z is divisible by n.

Sophie Germain proved Case 1 of Fermat's Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197. At this stage Case 2 had not been proved for even n = 5 so it became clear that Case 2 was the one on which to concentrate. Now Case 2 for n = 5 itself splits into two. One of x, y, z is even and one is divisible by 5. Case 2(i) is when the number divisible by 5 is even; Case 2(ii) is when the even number and the one divisible by 5 are distinct.

Case 2(i) was proved by Dirichlet and presented to the Paris Académie des Sciences in July 1825. Legendre was able to prove Case 2(ii) and the complete proof for n = 5 was published in September 1825. In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for Case 2(ii) which was an extension of his own argument for Case 2(i).

In 1832 Dirichlet published a proof of Fermat's Last Theorem for n = 14. Of course he had been attempting to prove the n = 7 case but had proved a weaker result. The n = 7 case was finally solved by Lamé in 1839. It showed why Dirichlet had so much difficulty, for although Dirichlet's n = 14 proof used similar (but computationally much harder) arguments to the earlier cases, Lamé had to introduce some completely new methods. Lamé's proof is exceedingly hard and makes it look as though progress with Fermat's Last Theorem to larger n would be almost impossible without some radically new thinking.

The year 1847 is of major significance in the study of Fermat's Last Theorem. On 1 March of that year Lamé announced to the Paris Académie that he had proved Fermat's Last Theorem. He sketched a proof which involved factorizing xn + yn = zn into linear factors over the complex numbers. Lamé acknowledged that the idea was suggested to him by Liouville. However Liouville addressed the meeting after Lamé and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true. Cauchy supported Lamé but, in rather typical fashion, pointed out that he had reported to the October 1847 meeting of the Académie an idea which he believed might prove Fermat's Last Theorem.

Much work was done in the following weeks in attempting to prove the uniqueness of factorization. Wantzel claimed to have proved it on 15 March but his argument

It is true for n = 2, n = 3 and n = 4 and one easily sees that the same argument applies for n > 4

was somewhat hopeful.
[Wantzel is correct about n = 2 (ordinary integers), n = 3 (the argument Euler got wrong) and n = 4 (which was proved by Gauss).]

On 24 May Liouville read a letter to the Académie which settled the arguments. The letter was from Kummer, enclosing an off-print of a 1844 paper which proved that uniqueness of factorization failed but could be 'recovered' by the introduction of ideal complex numbers which he had done in 1846. Kummer had used his new theory to find conditions under which a prime is regular and had proved Fermat's Last Theorem for regular primes. Kummer also said in his letter that he believed 37 failed his conditions.

By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular (and so Fermat's Last Theorem is true for that prime) if p does not divide the numerators of any of the Bernoulli numbers B2 , B4 , ..., Bp-3 . Kummer shows that all primes up to 37 are regular but 37 is not regular as 37 divides the numerator of B32 .

The only primes less than 100 which are not regular are 37, 59 and 67. More powerful techniques were used to prove Fermat's Last Theorem for these numbers. This work was done and continued to larger numbers by Kummer, Mirimanoff, Wieferich, Furtwängler, Vandiver and others. Although it was expected that the number of regular primes would be infinite even this defied proof. In 1915 Jensen proved that the number of irregular primes is infinite.

Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example over 1000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993.

In 1983 a major contribution was made by Gerd Faltings who proved that for every n > 2 there are at most a finite number of coprime integers x, y, z with xn + yn = zn. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Faltings' arguments.

The final chapter in the story began in 1955, although at this stage the work was not thought of as connected with Fermat's Last Theorem. Yutaka Taniyama asked some questions about elliptic curves, i.e. curves of the form y2 = x3 + ax + b for constants a and b. Further work by Weil and Shimura produced a conjecture, now known as the Shimura-Taniyama-Weil Conjecture. In 1986 the connection was made between the Shimura-Taniyama- Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrücken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space.

Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter -example to the Shimura-Taniyama-Weil Conjecture. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat's Last Theorem as a corollary to his main results. Having written the theorem on the blackboard he said I will stop here and sat down. In fact Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.

This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof. He states

The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.

In March 1994 Faltings, writing in Scientific American, said

If it were easy, he would have solved it by now. Strictly speaking, it was not a proof when it was announced.

Weil, also in Scientific American, wrote

I believe he has had some good ideas in trying to construct the proof but the proof is not there. To some extent, proving Fermat's Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.

In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties.

Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck.

In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.

On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof.

No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem.

Article by: J J O'Connor and E F Robertson
February 1996

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Fermat's_last_theorem.html]

Source: http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Fermat%27s_last_theorem.html

TRANSCRIPT, from PBS Nova Version of the documentary

"The Proof"

PBS Airdate: October 28, 1997
Go to the companion Web site

ANNOUNCER: Tonight, on NOVA. He conquered the impossible.

ANDREW WILES: Suddenly, totally unexpectedly, I had this incredible revelation.

PETER SARNAK: I was flabbergasted, excited, disturbed.

ANNOUNCER: How did this man solve an enigma that mystified the greatest minds for centuries?

ANDREW WILES: I believed I solved Fermat's Last Theorem.

ANNOUNCER: The Proof.

Major funding for NOVA is provided by the Park Foundation, dedicated to education and quality television...by the Corporation for Public Broadcasting, and viewers like you.

ANDREW WILES: Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it's dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it's all illuminated. You can see exactly where you were. At the beginning of September, I was sitting here at this desk, when suddenly, totally unexpectedly, I had this incredible revelation. It was the most—the most important moment of my working life. Nothing I ever do again will. . . I'm sorry.

STACY KEACH (NARRATOR): For seven years, Princeton professor Andrew Wiles worked in complete secrecy, struggling to solve the world's greatest mathematical problem. This obsession, which began when he was a child, would later bring him both fame and regret.

ANDREW WILES: So, I came to this. I was a ten-year-old, and one day I happened to be looking in my local public library, and I found a book on math and it told a bit about the history of this problem, that someone had resolved this problem 300 years ago, but no one had ever seen the proof. No one knew if there was a proof. And people ever since had looked for the proof. And here was a problem that I, a ten-year-old, could understand, that none of the great mathematicians in the past had been able to resolve. And from that moment, of course, I just tried to solve it myself. It was such a challenge, such a beautiful problem. This problem was Fermat's last theorem.

JOHN CONWAY: Pierre de Fermat was, by profession, a lawyer. He was Councilor to the Parliament of Toulouse in France. But, of course, that's not what he's really remembered for. What he's really remembered for is his mathematics.

STACY KEACH (NARRATOR): Pierre de Fermat was a 17th-century French mathematician who made some of the greatest breakthroughs in the history of numbers. His inspiration came from studying the Arithmetica, an Ancient Greek text.

JOHN CONWAY: Fermat owned a copy of this book, which is a book about numbers with lots of problems, which presumably, Fermat had to solve. He studied it; he wrote notes in the margins.

STACY KEACH (NARRATOR): Fermat's original notes were lost, but they can still be read in a book published by his son. It was one of these notes that was Fermat's greatest legacy.

JOHN CONWAY: And this is the fantastic observation of master Pierre de Fermat which caused all the trouble. "Cubum autem in duos cubos."

STACY KEACH (NARRATOR): This tiny note is the world's hardest mathematical problem. It's been unsolved for centuries, yet it begins with an equation so simple that children know it by heart.

CHILDREN: The square of the hypotenuse is equal to the sum of the squares of the other two sides.

JOHN CONWAY: Yeah. Well, that's Pythagoras's theorem, isn't it? That's what we all did at school. So, Pythagoras's theorem, the clever thing about it is that it tells us when three numbers are the sides of a right-angle triangle. That happens just when X squared plus Y squared equals Z squared.

ANDREW WILES: X squared plus Y squared equals Z squared. And you can ask, "Well, what are the whole number solutions of this equation?" You quickly find there's a solution 3 squared plus 4 squared equals 5 squared. Another one is 5 squared plus 12 squared is 13 squared. And you go on looking, and you find more and more. So then, a natural question is, the question Fermat raised: Supposing you change from squares. Supposing you replace the 2 by 3, by 4, by 5, by 6, by any whole number "n," and Fermat said simply that you'll never find any solutions. However far you look, you'll never find a solution.

STACY KEACH (NARRATOR): If "n" is greater than 2, you will never find numbers that fit this equation. That's what Fermat said. What's more, he said he could prove it. But instead, he scribbled a most enigmatic note.

JOHN CONWAY: Written in Latin, he says he has a truly wonderful proof, "Demonstrationem mirabilem," of this fact. And then, the last words are, "Hanc marginis exigiutas non caperet." "This margin is too small to contain it."

STACY KEACH (NARRATOR): So Fermat said he had a proof, but he never said what it was.

JOHN CONWAY: Fermat made lots of marginal notes. People took them as challenges, and over the centuries, every single one of them has been disposed of, and the last one to be disposed of is this one. That's why it's called the last theorem.

STACY KEACH (NARRATOR): Rediscovering Fermat's proof became the ultimate challenge, a challenge which would baffle mathematicians for the next 300 years.

JOHN CONWAY: Gauss, the greatest mathematician in the world. . .

BARRY MAZUR: Oh, yes. Galois. . .

JOHN COATES: Kummer, of course.

KEN RIBET: Well, in the 18th century, Euler didn't prove it.

JOHN CONWAY: Well, you know there's only been the one woman, really.

KEN RIBET: Sophie Germain.

BARRY MAZUR: Oh, there are millions. There are lots of people.

PETER SARNAK: But, nobody had any idea where to start.

ANDREW WILES: Well, mathematicians just love a challenge, and this problem, this particular problem, just looked so simple. It just looked as if it had to have a solution. And of course, it's very special because Fermat said he had a solution.

JOHN CONWAY: This thing has been there like a beacon in front of us. I mean, if you give up, you just get the feeling you've given up. It's like Everest; it won't go away. It still stays there. And so, one person can give up, but another person is still just trying to get a little bit further.

STACY KEACH (NARRATOR): The task was to prove that no numbers, other than 2, fit the equation. But when computers came along, couldn't they check each number one by one and show that none of them worked?

JOHN CONWAY: Well, how many numbers are there to be dealt with? You've got to do it for infinitely many numbers. So, after you've done it for one, how much closer have you got? Well, there's still infinitely many left. After you've done it for a thousand numbers, how many, how much closer have you got? Well, there's still infinitely many left. After you've done it for a million, well, there's still infinitely many left. In fact, you haven't done very many, have you?

STACY KEACH (NARRATOR): A computer can never check every number. Instead, what's needed is a mathematical proof.

PETER SARNAK: A mathematician is not happy until the proof is complete and considered complete by the standards of mathematics.

NICK KATZ: In mathematics, there's the concept of proving something, of knowing it with absolute certainty.

PETER SARNAK: Which—Well, it's called "rigorous proof."

KEN RIBET: Well, a rigorous proof is a series of arguments. . .

PETER SARNAK: . . .based on logical deductions. . .

KEN RIBET: Which just build one upon another. . .

PETER SARNAK: . . .step by step. . .

KEN RIBET: . . .until you get to. . .

PETER SARNAK: . . .a complete proof.

NICK KATZ: That's what mathematics is about.

STACY KEACH (NARRATOR): A proof provides a logical demonstration of why no numbers fit the equation without having to check every number. After centuries of failing to come up with such a proof, mathematicians began to abandon Fermat. In the '70s, Fermat was no longer in fashion. At the same time, Andrew Wiles was just beginning his career as a mathematician. He went to Cambridge University as a research student under the supervision of Professor John Coates.

JOHN COATES: I've been very fortunate to have Andrew as a student, and even as a research student, he was a wonderful person to work with. He had very deep ideas then, and it was always clear he was a mathematician who would do great things.

STACY KEACH (NARRATOR): But not with Fermat. Everyone thought Fermat's last theorem was impossible, so Professor Coates encouraged Andrew to forget his childhood dream and work on more mainstream math.

ANDREW WILES: The problem with working on Fermat is that you could spend years getting nothing. It's fine to work on any problem so long as it generates mathematics. Almost the definition of a good mathematical problem is the mathematics it generates, rather than the problem itself.

JOHN CONWAY: You know, not all mathematical problems are useless. Fermat's one really is useless, I think, in a certain sense. It's got no practical value whatsoever.

PETER SARNAK: If it's true, it doesn't imply anything profound, that any of us know. It doesn't lead to anything that's useful, that any of us know. It, by itself, is sort of on the outskirts. It's not what you would consider a mainstream, important, central question in modern mathematics.

ANDREW WILES: And that point, I really put aside Fermat. It's not that I forgot about it; it was always there. I always remembered it, but I realized the only techniques we had to tackle it had been around for 130 years, and it didn't seem they were really getting to the root of the problem. So, when I went to Cambridge, my advisor, John Coates, was working on Iwasawa theory and elliptic curves, and I started working with him.

STACY KEACH (NARRATOR): For Andrew's advisor, and a host of other mathematicians, elliptic curves were the "in" thing to study.

BARRY MAZUR: You may never have heard of elliptic curves, but they're extremely important.

JOHN CONWAY: OK. So, what's an elliptic curve?

BARRY MAZUR: Elliptic curves. They're not ellipses. They're cubic curves whose solution have a shape that looks like a doughnut.

PETER SARNAK: They look so simple, yet the complexity, especially arithmetic complexity, is immense.

STACY KEACH (NARRATOR): Every point on the doughnut is the solution to an equation. Andrew Wiles now studied these elliptic equations and set aside his dream. What he didn't realize was that on the other side of the world, elliptic curves and Fermat's last theorem were becoming inextricably linked.

GORO SHIMURA: I entered the University of Tokyo in 1949, and that was four years after the War, but almost all professors were tired and the lectures were not inspiring.

STACY KEACH (NARRATOR): Goro Shimura and his fellow students had to rely on each other for inspiration. In particular, he formed a remarkable partnership with a young man by the name of Utaka Taniyama.

GORO SHIMURA: That was when I became very close to Taniyama. Taniyama was not a very careful person as a mathematician. He made a lot of mistakes, but he made mistakes in a good direction, and so eventually, he got right answers, and I tried to imitate him, but I found out that it is very difficult to make good mistakes.

STACY KEACH (NARRATOR): Together, Taniyama and Shimura worked on the complex mathematics of modular functions.

NICK KATZ: I really can't explain what a modular function is in one sentence. I can try and give you a few sentences to explain. I really can't do it in one sentence.

PETER SARNAK: Oh, it's impossible.

ANDREW WILES: There's a saying attributed to Eichler that there are five fundamental operations of arithmetic: addition, subtraction, multiplication, division, and modular forms.

BARRY MAZUR: Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence seem like accidents. But they do exist.

STACY KEACH (NARRATOR): This image is merely a shadow of a modular form. To see one properly, your TV screen would have to be stretched into something called hyperbolic space. Bizarre modular forms seem to have nothing whatsoever to do with the humdrum world of elliptic curves. But what Taniyama and Shimura suggested shocked everyone.

GORO SHIMURA: In 1955, there was an international symposium, and Taniyama posed two or three problems.

STACY KEACH (NARRATOR): The problems posed by Taniyama led to the extraordinary claim that every elliptic curve was really a modular form in disguise. It became knows as the Taniyama-Shimura conjecture.

JOHN CONWAY: What the Taniyama-Shimura conjecture says, it says that every rational elliptic curve is modular, and that's so hard to explain.

BARRY MAZUR: So, let me explain. Over here, you have the elliptic world, the elliptic curves, these doughnuts. And over here, you have the modular world, modular forms with their many, many symmetries. The Shimura-Taniyama conjecture makes a bridge between these two worlds. These worlds live on different planets. It's a bridge. It's more than a bridge; it's really a dictionary, a dictionary where questions, intuitions, insights, theorems in the one world get translated to questions, intuitions in the other world.

KEN RIBET: I think that when Shimura and Taniyama first started talking about the relationship between elliptic curves and modular forms, people were very incredulous. I wasn't studying mathematics yet. By the time I was a graduate student in 1969 or 1970, people were coming to believe the conjecture.

STACY KEACH (NARRATOR): In fact, Taniyama-Shimura became a foundation for other theories which all came to depend on it. But Taniyama-Shimura was only a conjecture, an unproven idea, and until it could be proven, all the mathematics which relied on it were under threat.

ANDREW WILES: We built more and more conjectures stretched further and further into the future, but they would all be completely ridiculous if Taniyama-Shimura was not true.

STACY KEACH (NARRATOR): Proving the conjecture became crucial, but tragically, the man whose idea inspired it didn't live to see the enormous impact of his work. In 1958, Taniyama committed suicide.

GORO SHIMURA: I was very much puzzled. Puzzlement may be the best word. Of course, I was sad that—See, it was so sudden, and I was unable to make sense out of this. Some people suggested he lost confidence in himself. That may be so, but I think it was more complex. I don't really know. Confidence in himself, but not mathematically.

STACY KEACH (NARRATOR): Taniyama-Shimura went on to become one of the great unproven conjectures, a foundation for many important mathematical ideas. But what did it have to do with Fermat's last theorem?

ANDREW WILES: At that time, no one had any idea that Taniyama-Shimura could have anything to do with Fermat. Of course, in the '80s, that all changed completely.

STACY KEACH (NARRATOR): But what was the bridge between the two ideas? Taniyama-Shimura says, "Every elliptic curve is modular," and Fermat says, "No numbers fit this equation." What was the connection?

KEN RIBET: Well, on the face of it, the Shimura-Taniyama conjecture, which is about elliptic curves, and Fermat's last theorem have nothing to do with each other, because there's no connection between Fermat and elliptic curves. But in 1985, Gerhard Frey had this amazing idea.

STACY KEACH (NARRATOR): Frey, a German mathematician, considered the unthinkable. What would happen if Fermat was wrong and there was a solution to this equation after all?

PETER SARNAK: Frey showed how starting with a fictitious solution to Fermat's last equation—if, indeed, such a horrible beast existed—he could make an elliptic curve with some very weird properties.

KEN RIBET: That elliptic curve seems to be not modular. But Shimura-Taniyama says that every elliptic curve is modular.

STACY KEACH (NARRATOR): So, if there is a solution to this equation, it creates such a weird elliptic curve it defies Taniyama-Shimura.

KEN RIBET: So, in other words, if Fermat is false, so is Shimura-Taniyama. Or, said differently, if Shimura-Taniyama is correct, so is Fermat's last theorem.

STACY KEACH (NARRATOR): Fermat and Taniyama-Shimura were now linked, apart from just one thing.

KEN RIBET: The problem is that Frey didn't really prove that his elliptic curve was not modular. He gave a plausibility argument, which he hoped could be filled in by experts, and then the experts started working on it.

STACY KEACH (NARRATOR): In theory, you could prove Fermat by proving Taniyama, but only if Frey was right. Frey's idea became known as the epsilon conjecture, and everyone tried to check it. One year later, in San Francisco, there was a breakthrough.

KEN RIBET: I saw Barry Mazur on the campus, and I said, "Let's go for a cup of coffee." And we sat down for cappuccinos at this cafe, and I looked at Barry and I said, "You know, I'm trying to generalize what I've done so that we can prove the full strength of Serre's epsilon conjecture." And Barry looked at me and said, "But you've done it already. All you have to do is add on some extra gamma zero of m structure and run through your argument, and it still works, and that gives everything you need." And this had never occurred to me, as simple as it sounds. I looked at Barry, I looked at my cappucino, I looked back at Barry, and I said, "My God. You're absolutely right.

BARRY MAZUR: Ken's idea was brilliant.

KEN RIBET: And I was completely enthralled. I just sort of wandered back to my apartment in a cloud, and I sat down and I ran through my argument, and it worked. It really worked. And at the conference, I started telling a few people that I'd done this, and soon, large groups of people knew, and they were running up to me, and they said, "Is it true that you've proved the epsilon conjecture?" And I had to think for a minute, and all of a sudden, I said, "Yes. I have."

ANDREW WILES: I was at a friend's house sipping iced tea early in the evening, and he just mentioned casually in the middle of a conversation, "By the way, did you hear that Ken has proved the epsilon conjecture?" And I was just electrified. I knew that moment the course of my life was changing, because this meant that to prove Fermat's last theorem, I just had to prove Taniyama-Shimura conjecture. From that moment, that was what I was working on. I just knew I would go home and work on the Taniyama-Shimura conjecture.

STACY KEACH (NARRATOR): Andrew abandoned all his other research. He cut himself off from the rest of the world, and for the next seven years, he concentrated solely on his childhood passion.

ANDREW WILES: I never use a computer. I sometimes might scribble. I do doodles. I start trying to find patterns, really, so I'm doing calculations which try to explain some little piece of mathematics, and I'm trying to fit it in with some previous broad conceptual understanding of some branch of mathematics. Sometimes, that'll involve going and looking up in a book to see how it's done there. Sometimes, it's a question of modifying things a bit, sometimes, doing a little extra calculation. And sometimes, you realize that nothing that's ever been done before is any use at all, and you just have to find something completely new. And it's a mystery where it comes from.

JOHN COATES: I must confess, I did not think that the Shimura-Taniyama conjecture was accessible to proof at present. I thought I probably wouldn't see a proof in my lifetime.

KEN RIBET: I was one of the vast majority of people who believed that the Shimura-Taniyama conjecture was just completely inaccessible, and I didn't bother to prove it—even think about trying to prove it. Andrew Wiles is probably one of the few people on earth who had the audacity to dream that you could actually go and prove this conjecture.

ANDREW WILES: In this case, certainly the first several years, I had no fear of competition. I simply didn't think I or anyone else had any real idea how to do it. But I realized after a while that talking to people casually about Fermat was impossible, because it just generates too much interest, and you can't really focus yourself for years unless you have this kind of undivided concentration, which too many spectators would have destroyed.

STACY KEACH (NARRATOR): Andrew decided that he would work in secrecy and isolation.

PETER SARNAK: I often wondered, myself, what he was working on.

NICK KATZ: Didn't have an inkling.

JOHN CONWAY: No, I suspected nothing.

KEN RIBET: This is probably the only case I know where someone worked for such a long time without divulging what he was doing, without talking about the progress he had made. It's just unprecedented.

STACY KEACH (NARRATOR): Andrew was embarking on one of the most complex calculations in history. For the first two years, he did nothing but immerse himself in the problem, trying to find a strategy which might work.

ANDREW WILES: So, it was now known that Taniyama-Shimura implied Fermat's last theorem. What does Taniyama-Shimura say? It says that all elliptic curves should be modular. Well, this was an old problem, been around for twenty years, and lots of people had tried to solve it.

KEN RIBET: Now, one way of looking at it is that you have all elliptic curves, and then you have the modular elliptic curves, and you want to prove that there are the same number of each. Now, of course, you're talking about infinite sets, so you can't just count them, per se, but you can divide them into packets, and you can try to count each packet and see how things go. And this proves to be a very attractive idea for about thirty seconds, but you can't really get much further than that. And the big question on the subject was how you could possibly count, and in effect, Wiles introduced the correct technique.

STACY KEACH (NARRATOR): Andrew Wiles hoped to solve the problem of counting elliptic curves by converting them into something called Galois representations. Although no less complex than elliptic curves, they were easier to count. So, it was Galois representations, not elliptic curves, that Andrew would now compare with modular forms.

ANDREW WILES: Now, you might ask, and it's an obvious question, why can't you do this with elliptic curves and modular forms? Why couldn't you count elliptic curves, count modular forms, show they're the same number? Well, the answer is, people tried and they never found a way of counting them, and this was why this is they key breakthrough, that I had found the way to count not the original problem, but the modified problem. I'd found a way to count modular forms and Galois representations.

STACY KEACH (NARRATOR): This was only the first step, and already, it had taken three years of Andrew's life.

ANDREW WILES: My wife's only known me while I've been working on Fermat. I told her a few days after we got married. I decided that I really only had time for my problem and my family, and when I was concentrating very hard, and I found that with young children, that's the best possible way to relax. When you're talking to young children, they simply aren't interested in Fermat, at least at this age. They want to hear a children's story, and they're not going to let you do anything else. So, I'd found this wonderful counting mechanism, and I started thinking about this concrete problem in terms of Iwasawa theory. Iwasawa theory was the subject I'd studied as a graduate student, and, in fact, with my advisor, John Coates, I'd used it to analyze elliptic curves.

STACY KEACH (NARRATOR): Iwasawa theory, Andrew hoped, would be the key to completing his counting strategy.

ANDREW WILES: Now, I tried to use Iwasawa theory in this context, but I ran into trouble. I seemed to be up against a wall. I just didn't seem to be able to get past it. Well, sometimes when I can't see what to do next, I often come here by the lake. Walking has a very good effect in that you're in this state of concentration, but at the same time, you're relaxing; you're allowing the subconscious to work on you.

STACY KEACH (NARRATOR): Andrew struggled for months using Iwasawa theory in an effort to create something called a Class Number Formula. Without this critical formula, he would have nowhere left to go.

ANDREW WILES: So, at the end of the summer of '91, I was at a conference, and John Coates told me about a wonderful new paper of Matthias Flach, a student of his, in which he had tackled the class number formula, in fact, exactly the class number formula I needed. So, Flach, using ideas Kolyvagin, had made a very significant first step in actually producing the class number formula. So, at that point, I thought, 'This is just what I need. This is tailor-made for the problem.' I put aside the completely the old approach I'd been trying, and I devoted myself day and night to extending his result.

STACY KEACH (NARRATOR): Andrew was almost there, but this breakthrough was risky and complicated. After six years of secrecy, he needed to confide in someone.

NICK KATZ: January of 1993, Andrew came up to me one day at tea, asked me if I could come up to his office; there was something he wanted to talk to me about. I had no idea what this could be. I went up to his office. He closed the door. He said he thought he would be able to prove Taniyama-Shimura. I was just amazed. This was fantastic.

ANDREW WILES: It involved a kind of mathematics that Nick Katz is an expert in.

NICK KATZ: I think another reason he asked me was that he was sure I would not tell other people, I would keep my mouth shut. Which I did.

JOHN CONWAY: Andrew Wiles and Nick Katz had been spending rather a lot of time huddled over a coffee table at the far end of the common room working on some problem or other. We never knew what it was.

STACY KEACH (NARRATOR): To avoid any more suspicion, Andrew decided to check his proof by disguising it in a series of lectures at Princeton, which Nick Katz could attend.

ANDREW WILES: Well, I explained at the beginning of the course that Flach had written this beautiful paper and I wanted to try to extend it to prove the full class number formula. The only thing I didn't explain was that proving the class number formula was most of the way to Fermat's last theorem.

NICK KATZ: So, this course was announced. It said "Calculations on Elliptic Curves," which could mean anything. It didn't mention Fermat, it didn't mention Taniyama-Shimura. There was no way in the world anyone could have guessed that it was about that, if you didn't already know. None of the graduate students knew, and in a few weeks, they just drifted off, because it's impossible to follow stuff if you don't know what it's for, pretty much. It's pretty hard even if you do know what it's for. But after a few weeks, I was the only guy in the audience.

STACY KEACH (NARRATOR): The lectures revealed no errors, and still, none of his colleagues suspected why Andrew was being so secretive.

PETER SARNAK: Maybe he's run out of ideas. That's why he's quiet. You never know why they're quiet.

STACY KEACH (NARRATOR): The proof was still missing a vital ingredient, but Andrew now felt confident. It was time to tell one more person.

ANDREW WILES: So, I called up Peter and asked him if I could come 'round and talk to him about something.

PETER SARNAK: I got a phone call from Andrew saying that he had something very important he wanted to chat to me about. And sure enough, he had some very exciting news.

ANDREW WILES: I said, "I think you better sit down for this." He sat down. I said, "I think I'm about to prove Fermat's last theorem."

PETER SARNAK: I was flabbergasted, excited, disturbed. I mean, I remember that night finding it quite difficult to sleep.

ANDREW WILES: But, there was still a problem. Late in the spring of '93, I was in this very awkward position that I thought I'd got most of the curves being modular, so that was nearly enough to be content to have Fermat's last theorem, but there were these few families of elliptic curves that had escaped the net. I was sitting here at my desk in May of '93, still wondering about this problem, and I was casually glancing at a paper of Barry Mazur's, and there was just one sentence which made a reference to actually what's a 19th century construction, and I just instantly realized that there was a trick that I could use, that I could switch from the families of elliptic curves I'd been using. I'd been studying them using the prime three. I could switch and study them using the prime five. It looked more complicated, but I could switch from these awkward curves that I couldn't prove were modular to a different set of curves, which I'd already proved were modular, and use that information to just go that one last step. And, I just kept working out the details, and time went by, and I forgot to go down to lunch, and it got to about tea-time, and I went down and Nada was very surprised that I'd arrived so late, and then she—I told her that I believed I'd solved Fermat's last theorem. I was convinced that I had Fermat in my hands, and there was a conference in Cambridge organized by my advisor, John Coates. I thought that would be a wonderful place. It's my old hometown, and I'd been a graduate student there. It would be a wonderful place to talk about it if I could get it in good shape.

JOHN COATES: The name of the lectures that he announced was simply "Elliptic Curves and Modular Forms." There was no mention of Fermat's last theorem.

KEN RIBET: Well, I was at this conference on L functions and elliptic curves, and it was kind of a standard conference and all of the people were there. Didn't seem to be anything out of the ordinary, until people started telling me that they'd been hearing weird rumors about Andrew Wiles's proposed series of lectures. I started talking to people and I got more and more precise information. I have no idea how it was spread.

PETER SARNAK: Not from me. Not from me.

JOHN CONWAY: Whenever any piece of mathematical news had been in the air, Peter would say, "Oh, that's nothing. Wait until you hear the big news. There's something big going to break."

PETER SARNAK: Maybe some hints, yeah.

ANDREW WILES: People would ask me, leading up to my lectures, what exactly I was going to say. And I said, "Well, come to my lecture and see."

KEN RIBET: It's a very charged atmosphere. A lot of the major figures of arithmetical, algebraic geometry were there. Richard Taylor and John Coates. Barry Mazur.

BARRY MAZUR: Well, I'd never seen a lecture series in mathematics like that before. What was unique about those lectures were the glorious ideas, how many new ideas were presented, and the constancy of its dramatic build-up. It was suspenseful until the end.

KEN RIBET: There was this marvelous moment when we were coming close to a proof of Fermat's last theorem. The tension had built up, and there was only one possible punch line.

ANDREW WILES: So, after I'd explained the 3/5 switch on the blackboard, I then just wrote up a statement of Fermat's last theorem, said I'd proved it, said, "I think I'll stop there."

JOHN COATES: The next day, what was totally unexpected was that we were deluged by inquiries from newspapers, journalists from all around the world.

ANDREW WILES: It was a wonderful feeling after seven years to have really solved my problem. I'd finally done it. Only later did it come out that there was a problem at the end.

NICK KATZ: Now, it was time for it to be refereed, which is to say, for people appointed by the journal to go through and make sure that the thing was really correct. So, for two months, July and August, I literally did nothing but go through this manuscript line by line, and what this meant concretely was that essentially every day, sometimes twice a day, I would e-mail Andrew with a question: "I don't understand what you say on this page, on this line. It seems to be wrong," or "I just don't understand."

ANDREW WILES: So, Nick was sending me e-mails, and at the end of the summer, he sent one that seemed innocent at first, and I tried to resolve it.

NICK KATZ: It's a little bit complicated, so he sends me a fax, but the fax doesn't seem to answer the question, so I e-mail him back, and I get another fax, which I'm still not satisfied with. And this, in fact, turned into the error that turned out to be a fundamental error, and that we had completely missed when he was lecturing in the spring.

ANDREW WILES: That's where the problem was, in the method of Flach and Kolyvagin that I'd extended. So, once I realized that at the end of September, that there was really a problem with the way I'd made the construction, I spent the fall trying to think what kind of modifications could be made to the construction. There are lots of simple and rather natural modifications that any one of which might work.

PETER SARNAK: And every time he would try and fix it in one corner, it would sort of—Some other difficulty would add up in another corner. It was like he was trying to put a carpet in a room where the carpet had more size than the room, but he could put it in in any corner, and then when he ran to the other corners, it would pop up in this corner. And whether you could not put the carpet in the room was not something that he was able to decide.

ANDREW WILES: So, in September '93, when the proof was running into problems, Nada said to me, "The only thing I want for my birthday is the correct proof." Her birthday is on October 6. I had two or three weeks, and I failed to deliver.

NICK KATZ: I think he externally appeared normal, but at this point, he was keeping a secret from the world, and I think he must have been, in fact, pretty uncomfortable about it.

ANDREW WILES: Towards the end of November, it didn't seem to be working. I sent out an e-mail message announcing that there was a problem with this part of the argument.

JOHN CONWAY: Well, you know, we were behaving a little bit like Kremlinologists. Nobody actually liked to come out and ask him how he's getting on with the proof. So, somebody would say, "I saw Andrew this morning." "Did he smile?" "Well, yes. But he didn't look too happy."

ANDREW WILES: The first seven years I'd worked on this problem, I loved every minute of it. However hard it had been, there'd been setbacks often, there'd been things that had seemed insurmountable, but it was a kind of private and very personal battle I was engaged in. And then, after there was a problem with it, doing mathematics in that kind of rather over-exposed way is certainly not my style, and I have no wish to repeat it.

STACY KEACH (NARRATOR): After months of failure, Andrew was about to admit defeat. In desperation, he decided to ask for help, and a former student, Richard Taylor, came to Princeton.

ANDREW WILES: Richard and I spent three months at the beginning of '94 trying to analyze all the possible modifications, and at the end of that period, I was convinced that none of them was really going to give the answer. In September, I decided to go back and look one more time at the original structure of Flach and Kolyvagin to try and pinpoint exactly why it wasn't working, try and formulate it precisely. One can never really do that in mathematics, but I just wanted to set my mind to rest that it really couldn't be made to work. And I was sitting here at this desk. It was a Monday morning, September 19, and I was trying, convincing myself that it didn't work, just seeing exactly what the problem was, when suddenly, totally unexpectedly, I had this incredible revelation. I realized what was holding me up was exactly what would resolve the problem I had had in my Iwasawa theory attempt three years earlier, was—It was the most—the most important moment of my working life. It was so indescribably beautiful; it was so simple and so elegant, and I just stared in disbelief for twenty minutes. Then, during the day, I walked around the department. I'd keep coming back to my desk and looking to see if it was still there. It was still there. Almost what seemed to be stopping the method of Flach and Kolyvagin was exactly what would make horizontal Iwasawa theory. My original approach to the problem from three years before would make exactly that work. So, out of the ashes seemed to rise the true answer to the problem. So, the first night, I went back and slept on it. I checked through it again the next morning, and by eleven o'clock, I was satisfied and I went down and told my wife, "I've got it. I think I've got it. I've found it." And it was so unexpected, I think she thought I was talking about a children's toy or something and said, "Got what?" And I said, "I've fixed my proof. I've got it."

JOHN COATES: I think it will always stand as one of the high achievements of number theory.

BARRY MAZUR: It was magnificent.

JOHN CONWAY: It's not every day that you hear the proof of the century.

GORO SHIMURA: Well, my first reaction was, "I told you so."

STACY KEACH (NARRATOR): The Taniyama-Shimura conjecture is no longer a conjecture, and as a result, Fermat's last theorem has been proved. But is Andrew's proof the same as Fermat's?

JOHN CONWAY: Fermat's proof was just too big to fit into this margin. Andrew's was 200 pages long. It's not the same proof.

ANDREW WILES: Fermat couldn't possibly have had this proof. It's a 20th century proof. There's no way this could have been done before the 20th century.

JOHN CONWAY: I'm relieved that this result is now settled. But I'm sad in some ways, because Fermat's last theorem has been responsible for so much. What will we find to take its place?

ANDREW WILES: There's no other problem that will mean the same to me. I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream. I know it's a rare privilege, but if one can do this, if one can really tackle something in adult life that means that much to you, it's more rewarding than anything I could imagine.

BARRY MAZUR: One of the great things about this work is it embraces the ideas of so many mathematicians. I've made a partial list. Klein, Fricke, Hurwitz, Hecke, Dirichlet, Dedekind. . .

KEN RIBET: The proof by Langlands and Tunnell. . .

JOHN COATES: Deligne, Rapoport, Katz. . .

NICK KATZ: Mazur's idea of using the deformation theory of Galois representations. . .

BARRY MAZUR: Igusa, Eichler, Shimura, Taniyama. . .

PETER SARNAK: Frey's reduction. . .

NICK KATZ: The list goes on and on.

BARRY MAZUR: Bloch, Kato, Selmer, Frey, Fermat.

ANNOUNCER: There was another player in the Fermat game. She lived during the French Revolution and pretended to be a man in order to pursue her passion for mathematics. At NOVA's website, meet Sophie Germain at www.pbs.org.

Educators can order this show for $19.95, plus shipping and handling, by calling 1-800-949-8670. And, to learn more about how science can solve the mysteries of our world, ask about our many other NOVA videos.

JOHN CONWAY: It's like effortless; it won't go away. It still stays there.

ANDREW WILES: Well, mathematicians just love a challenge, and this problem, this particular problem, just looked so simple, it just looked as if it had to have a solution.

KEN RIBET: Andrew Wiles is probably one of the few people on earth who had the audacity to dream that you could actually go and prove this conjecture.

ANNOUNCER: NOVA is a production of WGBH, Boston.

Major funding for NOVA is provided by the Park Foundation, dedicated to education and quality television.

..by the Corporation for Public Broadcasting, and viewers like you.

This is PBS.

To learn more about this subject, you can order Fermat's Enigma, the companion book to this program, by calling 1-800-949-8670. This hardcover edition is $23 plus shipping and handling.

Source: http://www.pbs.org/wgbh/nova/transcripts/2414proof.html

Solving Fermat: Andrew Wiles
Andrew Wiles devoted much of his entire career to proving Fermat's Last Theorem, the world's most famous mathematical problem. In 1993, he made front-page headlines when he announced a proof of the problem, but this was not the end of the story; an error in his calculation jeopardized his life's work. Andrew Wiles spoke to NOVA and described how he came to terms with the mistake, and eventually went on to achieve his life's ambition.

NOVA: Many great scientific discoveries are the result of obsession, but in your case that obsession has held you since you were a child.

ANDREW WILES: I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days. I loved doing problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem—Fermat's Last Theorem. This problem had been unsolved by mathematicians for 300 years. It looked so simple, and yet all the great mathematicians in history couldn't solve it. Here was a problem, that I, a ten year old, could understand and I knew from that moment that I would never let it go. I had to solve it.

NOVA: Who was Fermat and what was his Last Theorem?

Wiles AW: Fermat was a 17th-century mathematician who wrote a note in the margin of his book stating a particular proposition and claiming to have proved it. His proposition was about an equation which is closely related to Pythagoras' equation. Pythagoras' equation gives you:

x2 + y2 = z2


You can ask, what are the whole number solutions to this equation, and you can see that:

32 + 42 = 52


and

52 + 122 = 132


And if you go on looking then you find more and more such solutions. Fermat then considered the cubed version of this equation:

x3 + y3 = z3


He raised the question: can you find solutions to the cubed equation? He claimed that there were none. In fact, he claimed that for the general family of equations:

xn + yn = zn where n is bigger than 2


it is impossible to find a solution. That's Fermat's Last Theorem.

NOVA: So Fermat said because he could not find any solutions to this equation, then there were no solutions?

Wiles AW: He did more than that. Just because we can't find a solution it doesn't mean that there isn't one. Mathematicians aren't satisfied because they know there are no solutions up to four million or four billion, they really want to know that there are no solutions up to infinity. And to do that we need a proof. Fermat said he had a proof. Unfortunately, all he ever wrote down was: "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain."

NOVA: What do you mean by a proof?

AW: In a mathematical proof you have a line of reasoning consisting of many, many steps, that are almost self-evident. If the proof we write down is really rigorous, then nobody can ever prove it wrong. There are proofs that date back to the Greeks that are still valid today.

NOVA: So the challenge was to rediscover Fermat's proof of the Last Theorem. Why did it become so famous?

AW: Well, some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they're extremely hard to solve. There's no reason why these problems shouldn't be easy, and yet they turn out to be extremely intricate. The Last Theorem is the most beautiful example of this.

NOVA: But finding a proof has no applications in the real world; it is a purely abstract question. So why have people put so much effort into finding a proof?

AW: Pure mathematicians just love to try unsolved problems—they love a challenge. And as time passed and no proof was found, it became a real challenge. I've read letters in the early 19th century which said that it was an embarrassment to mathematics that the Last Theorem had not been solved. And of course, it's very special because Fermat said that he had a proof.

NOVA: How did you begin looking for the proof?

Wiles AW: In my early teens I tried to tackle the problem as I thought Fermat might have tried it. I reckoned that he wouldn't have known much more math than I knew as a teenager. Then when I reached college, I realized that many people had thought about the problem during the 18th and 19th centuries and so I studied those methods. But I still wasn't getting anywhere. Then when I became a researcher, I decided that I should put the problem aside. It's not that I forgot about it—it was always there—but I realized that the only techniques we had to tackle it had been around for 130 years. It didn't seem that these techniques were really getting to the root of the problem. The problem with working on Fermat was that you could spend years getting nowhere. It's fine to work on any problem, so long as it generates interesting mathematics along the way—even if you don't solve it at the end of the day. The definition of a good mathematical problem is the mathematics it generates rather than the problem itself.

NOVA: It seems that the Last Theorem was considered impossible, and that mathematicians could not risk wasting getting nowhere. But then in 1986 everything changed. A breakthrough by Ken Ribet at the University of California at Berkeley linked Fermat's Last Theorem to another unsolved problem, the Taniyama-Shimura conjecture. Can you remember how you reacted to this news?

Wiles AW: It was one evening at the end of the summer of 1986 when I was sipping iced tea at the house of a friend. Casually in the middle of a conversation this friend told me that Ken Ribet had proved a link between Taniyama-Shimura and Fermat's Last Theorem. I was electrified. I knew that moment that the course of my life was changing because this meant that to prove Fermat's Last Theorem all I had to do was to prove the Taniyama-Shimura conjecture. It meant that my childhood dream was now a respectable thing to work on. I just knew that I could never let that go.

NOVA: So, because Taniyama-Shimura was a modern problem, this meant that working on it, and by implication trying to prove Fermat's Last Theorem, was respectable.

AW: Yes. Nobody had any idea how to approach Taniyama-Shimura but at least it was mainstream mathematics. I could try and prove results, which, even if they didn't get the whole thing, would be worthwhile mathematics. So the romance of Fermat, which had held me all my life, was now combined with a problem that was professionally acceptable.

NOVA: At this point you decided to work in complete isolation. You told nobody that you were embarking on a proof of Fermat's Last Theorem. Why was that?

AW: I realized that anything to do with Fermat's Last Theorem generates too much interest. You can't really focus yourself for years unless you have undivided concentration, which too many spectators would have destroyed.

NOVA: But presumably you told your wife what you were doing?

AW: My wife's only known me while I've been working on Fermat. I told her on our honeymoon, just a few days after we got married. My wife had heard of Fermat's Last Theorem, but at that time she had no idea of the romantic significance it had for mathematicians, that it had been such a thorn in our flesh for so many years.

NOVA: On a day-to-day basis, how did you go about constructing your proof?

Wiles AW: I used to come up to my study, and start trying to find patterns. I tried doing calculations which explain some little piece of mathematics. I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about. Sometimes that would involve going and looking it up in a book to see how it's done there. Sometimes it was a question of modifying things a bit, doing a little extra calculation. And sometimes I realized that nothing that had ever been done before was any use at all. Then I just had to find something completely new; it's a mystery where that comes from. I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction, I would have the same thing going round and round in my mind. The only way I could relax was when I was with my children. Young children simply aren't interested in Fermat. They just want to hear a story and they're not going to let you do anything else.

NOVA: Usually people work in groups and use each other for support. What did you do when you hit a brick wall?

Wiles AW: When I got stuck and I didn't know what to do next, I would go out for a walk. I'd often walk down by the lake. Walking has a very good effect in that you're in this state of relaxation, but at the same time you're allowing the sub-conscious to work on you. And often if you have one particular thing buzzing in your mind then you don't need anything to write with or any desk. I'd always have a pencil and paper ready and, if I really had an idea, I'd sit down at a bench and I'd start scribbling away.

NOVA: So for seven years you're pursuing this proof. Presumably there are periods of self-doubt mixed with the periods of success.

AW: Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them.

NOVA: And during those seven years, you could never be sure of achieving a complete proof.

AW: I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal. It could be that the methods needed to take the next step may simply be beyond present day mathematics. Perhaps the methods I needed to complete the proof would not be invented for a hundred years. So even if I was on the right track, I could be living in the wrong century.

NOVA: Then eventually in 1993, you made the crucial breakthrough.

Wiles AW: Yes, it was one morning in late May. My wife, Nada, was out with the children and I was sitting at my desk thinking about the last stage of the proof. I was casually looking at a research paper and there was one sentence that just caught my attention. It mentioned a 19th-century construction, and I suddenly realized that I should be able to use that to complete the proof. I went on into the afternoon and I forgot to go down for lunch, and by about three or four o'clock, I was really convinced that this would solve the last remaining problem. It got to about tea time and I went downstairs and Nada was very surprised that I'd arrived so late. Then I told her I'd solved Fermat's Last Theorem.

NOVA: The New York Times exclaimed "At Last Shout of 'Eureka!' in Age-Old Math Mystery," but unknown to them, and to you, there was an error in your proof. What was the error?

AW: It was an error in a crucial part of the argument, but it was something so subtle that I'd missed it completely until that point. The error is so abstract that it can't really be described in simple terms. Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail.

NOVA: Eventually, after a year of work, and after inviting the Cambridge mathematician Richard Taylor to work with you on the error, you managed to repair the proof. The question that everybody asks is this; is your proof the same as Fermat's?

AW: There's no chance of that. Fermat couldn't possibly have had this proof. It's 150 pages long. It's a 20th-century proof. It couldn't have been done in the 19th century, let alone the 17th century. The techniques used in this proof just weren't around in Fermat's time.

NOVA: So Fermat's original proof is still out there somewhere.

AW: I don't believe Fermat had a proof. I think he fooled himself into thinking he had a proof. But what has made this problem special for amateurs is that there's a tiny possibility that there does exist an elegant 17th-century proof.

NOVA: So some mathematicians might continue to look for the original proof. What will you do next?

Wiles AW: There's no problem that will mean the same to me. Fermat was my childhood passion. There's nothing to replace it. I'll try other problems. I'm sure that some of them will be very hard and I'll have a sense of achievement again, but nothing will mean the same to me. There's no other problem in mathematics that could hold me the way that this one did. There is a sense of melancholy. We've lost something that's been with us for so long, and something that drew a lot of us into mathematics. But perhaps that's always the way with math problems, and we just have to find new ones to capture our attention. People have told me I've taken away their problem—can't I give them something else? I feel some sense of responsibility. I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future.

NOVA: What is the main challenge now?

AW: The greatest problem for mathematicians now is probably the Riemann Hypothesis. But it's not a problem that can be simply stated.

NOVA: And is there any one particular thought that remains with you now that Fermat's Last Theorem has been laid to rest?

AW: Certainly one thing that I've learned is that it is important to pick a problem based on how much you care about it. However impenetrable it seems, if you don't try it, then you can never do it. Always try the problem that matters most to you. I had this rare privilege of being able to pursue in my adult life, what had been my childhood dream. I know it's a rare privilege, but if one can really tackle something in adult life that means that much to you, then it's more rewarding than anything I can imagine.

NOVA: And now that journey is over, there must be a certain sadness?

Wiles AW: There is a certain sense of sadness, but at the same time there is this tremendous sense of achievement. There's also a sense of freedom. I was so obsessed by this problem that I was thinking about it all the time—when I woke up in the morning, when I went to sleep at night—and that went on for eight years. That's a long time to think about one thing. That particular odyssey is now over. My mind is now at rest.

Source: http://www.pbs.org/wgbh/nova/proof/wiles.html (read less)

Fermat's Last Theorem

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637, but was not proven until 1995 despite the efforts of many illustrious mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th. It is among the most famous theorems in the history of mathematics.
Source: Wikipedia

Simon Singh and John Lynch's film tells the enthralling and emotional story of Andrew Wiles. A quiet English mathematician, he was drawn into maths by Fermat's puzzle, but at Cambridge in the '70s, FLT was considered a joke, so he set it aside. Then, in 1986, an extraordinary idea linked this irritating problem with one of the most profound ideas of modern mathematics: the Taniyama-Shimura Conjecture, named after a y... (read more)