Beyond Fermat's last theorem | Science | The Guardian Beyond Fermat's last theorem Keith Devlin on the 21st century uses of a 17th century puzzle Wednesday 21 July 1999 07.48 EDT First published on Wednesday 21 July 1999 07.48 EDT Share on Messenger Close When British mathematician Andrew Wiles proved Fermat's last theorem in 1994, he ended a saga that had begun in the middle of the 17th century. But like a good storyteller, he left unanswered a tantalising question. Now, five years later, four mathematicians have managed to answer that question. Brian Conrad and Richard Taylor of Harvard University, Christophe Breuil of the Universit de Paris-Sud, and Fred Diamond of Rutgers University announced recently that they had finally managed to build on Wiles's work to prove the Shimura-Taniyama conjecture. To understand what this means, it's best to start at the beginning of the Fermat story. Pierre de Fermat - a French lawyer whose hobby was mathematics - was one of the most famous mathematicians of all time.

In 1670, looking through his father's papers five years after his death, his son came across a note scribbled in the margin of a textbook. The note claimed that the equation xn + yn = zn has no (whole-number) solutions for any power n greater than 2. (Mathematicians ignore the trivial solutions that arise when one of the unknowns is allowed to be zero. For n = 2, the equation is related to Pythagoras's theorem about right-angle triangles, and has many whole number solutions; x = 3, y = 4, z = 5 being one such.) So began a saga that continued until late in 1994, as mathematician after mathematician, professional and amateur, attempted to produce a proof - and failed. It was its very resistance to proof that led to the theorem's fame, and attracted many amateur maths buffs to have a go. Experts all agree that there is little likelihood of a proof using high school techniques, however. Wiles's proof involved some remarkable new results that have opened up a completely new area of mathematics. Wiles started with an idea put forward by the Japanese mathematician Yutaka Taniyama in 1955.

Taniyama was studying elliptic curves; these are curves having an equations of the form y2 = x3 + Ax2 + Bx + C. Taniyama suggested that every elliptic curve has a (fairly technical) property known as modularity. It was a bold claim, with very little to back it up. But in 1971, another Japanese mathematician, Goro Shimura, showed that Taniyama's suggestion was true at least for a very special class of elliptic curves. With this partial evidence in hand, Taniyama's original suggestion gained credibility, and became known as the Shimura-Taniyama conjecture. At the time, no one suspected there would be any connection between this very abstract conjecture and Fermat's last theorem. But, in 1986, German mathematician Gerhard Frey noticed that Fermat's last theorem might be a direct consequence of the Shimura-Taniyama Conjecture.The American mathematician Kenneth Ribet proved Frey right: the Shimura-Taniyama conjecture implied Fermat's last theorem. Ribet's remarkable result was electrifying.

The Shimura-Taniyama conjecture concerned geometric objects about which a great deal was known. Indeed, by then there was good reason to believe the conjecture. There were also some obvious - but daunting - ways to set about finding a proof. At last, mathematicians had a powerful framework with which to approach the last theorem. Among those who took up the challenge was Andrew Wiles, who by then was a professor of mathematics at Princeton University in New Jersey. Wiles had been fascinated with Fermat's last theorem since childhood, when he had attempted to solve the problem using high school mathematics. For the next seven years, Wiles concentrated his efforts on finding a way to prove the Shimura-Taniyama conjecture. By 1991, he felt sure he could prove not the entire conjecture, which applies to all elliptic curves, but a special case of the conjecture that applies to elliptic curves of a particular kind. In 1993, after a further two years effort, he eventually succeeded in doing just that.

Believing that the class of elliptic curves for which his proof worked included those necessary to deduce the last theorem, in June 1993 Wiles announced he had solved Fermat's last theorem. He was wrong. By December of that year, he had to admit that his argument did not seem to work for the "right" elliptic curves. Though everyone agreed that his achievement was one of the most significant advances in number theory in the 20th century, it appeared that he was destined to follow in the footsteps of the many illustrious mathematicians , including perhaps Fermat himself, who had dared rise to the challenge in Fermat's marginal note. Fermat's last theorem remained unproved. Several months of silence followed, while Wiles retreated to his Princeton home to try to make his argument work. In October 1994 came the announcement that he had succeeded. All the experts agreed he was right. That was not the end of the story. Wiles's new proof showed that "many" elliptic curves are modular, as did his original, faulty proof. And this time, among the elliptic curves for which his proof worked were the ones required to provide Fermat's last theorem.

The question remained, however: Was every elliptic curve modular, as the full Shimura-Taniyama conjecture proposed? Before Wiles's work, most experts had come to the conclusion that the full conjecture was probably true, but no one had the slightest idea how to prove it. Indeed, Ribet believed it would not be proved within his lifetime. So great was Wiles's work, however, that, after his 1994 proof had been acknowledged to be correct, those same experts felt it would not be long before the new methods were strengthened to prove the full conjecture. The hunt was on. It took just five years to resolve the matter. Building on Wiles's work, Conrad, Taylor, Breuil, and Diamond have now shown that Taniyama was entirely correct: every elliptic curve is modular. This marks the start of a whole new era of research in number theory. For whereas Fermat's last theorem itself was little more than a curiosity from the 17th century, having no significant consequences, the Shimura-Taniyama conjecture is a rich and powerful result offering the potential for a great many wide-ranging applications.

For number theorists, the proof of the Shimura-Taniyama conjecture is a fitting end to the 20th century that appropriately sets the stage for the new millennium.  Keith Devlin is dean at the school of science, Saint Mary's College of California, Moraga. Wiles biography is a counterexample to Fermat's Last Theorem, then the elliptic curve y2 = x(x - an)(x + bn) cannot be modular, thus violating the Shimura - Taniyama conjecture. This result set the stage for Wiles's work. In fact Wiles abandoned all his other research when he heard what had been proved and, for seven years, he concentrated solely on attempting to prove the Shimura - Taniyama conjecture, knowing that a proof of Fermat's Last Theorem then followed. Wiles said:- ... after a few years I realised that talking to people casually about Fermat was impossible because it generated too much interest and you cannot focus yourself for years unless you have this kind of undivided concentration which too many spectators would destroy... In fact married life was a rather restricted affair for Wiles who said:- ... my wife has only known me while I have been working on Fermat .

I told her a few days after I got married. I decided that I really only had time for my problem and my family and while I was concentrating very hard then I found with young children that it was the best possible way to relax. When you're talking to young children they're simply not interested in Fermat ... In 1988 Wiles went to Oxford University where he spent two years as a Royal Society Research Professor. While in Oxford he was elected, in 1989, a Fellow of the Royal Society . In [ 1 ] the course of his research is described:- Using Mazur's deformation theory of Galois representations, recent results on Serre's conjecture on the modularity of Galois representations, and deep arithmetical properties of Hecke algebras, Wiles (with one key step due jointly to Wiles and R Taylor) succeeded in proving that all semistable elliptic curves defined over the rational numbers are modular. Although less than the full Shimura - Taniyama conjecture, this result does imply that the elliptic curve given above is modular, thereby proving Fermat's Last Theorem. In fact the path to the proof was not as smooth as suggested by this description.

In 1993 Wiles told two other mathematicians that he was close to a proof of Fermat's Last Theorem. He filled what he thought were the remaining few gaps and gave a series of lectures at the Isaac Newton Institute in Cambridge ending on 23 June 1993. At the end of the final lecture he announced he had a proof of Fermat's Last Theorem. When the results were written up for publication, however, a subtle error was discovered. Wiles said:- ... the first seven years I had worked on this problem I loved every minute of it however hard it had been. There had been setbacks, things which 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 overexposed way is certainly not my style, I certainly have no wish to repeat it... Wiles worked hard for about a year, helped in particular by R Taylor referred to above, and by 19 September 1994, having almost given up, he decided to have one last try:- ... suddenly, totally unexpectedly, I had this incredible revelation.

It was the most important moment of my working life. Nothing I ever do again ... 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 round the department. I'd keep coming back to my desk to see it was still there - it was still there. In 1994 Wiles was appointed Eugene Higgins Professor of Mathematics at Princeton. His paper which proves Fermat's Last Theorem is Modular elliptic curves and Fermat's Last Theorem which appeared in the Annals of Mathematics in 1995. From 1995 Wiles began to receive many honours for this outstanding piece of work. He was awarded the Schock Prize in Mathematics from the Royal Swedish Academy of Sciences and the Prix Fermat from the Université Paul Sabatier. In 1996 he received further awards included the Wolf Prize and was elected as a foreign member to the National Academy of Sciences of the United States, receiving its mathematics prize. Wiles said:- ... 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 its a rare privilege but I know if one can do this it's more rewarding than anything one can imagine. In [ 1 ] his worked is summed up:- Wiles's work is highly original, a technical tour de force, and a monument to individual perseverance. In addition to the prizes and awards mentioned above, Wiles has continued to receive many honours for his outstanding work. In 1998, not being eligible for the award of a Fields medal on the grounds that he was over forty years of age, the International Mathematical Union presented him with a silver plaque at the International Congress of Mathematicians held that year in Berlin. He gave an evening lecture at the Congress which attracted an audience of over 2300. In the same year of 1998 he was awarded the King Faisal Prize and in the following year received the Clay Research Award. He received the award from the Clay Mathematics Institute on 10 May 1999 in Cambridge, Massachusetts:- ... for his role in the development of number theory.

In the same year he was honoured by having "asteroid 9999 Wiles" named after him. In 2000, Andrew Wiles became "Sir Andrew Wiles" when he was made a Knight Commander of the Order of the British Empire by the Queen. He received the Pythagoras Award in Crotone, Italy, in 2004 and, in the following year, Wiles received the Shaw Prize. Sir Run Run Shaw, a leader of the Hong Kong media industry, established this Prize in 2002. The first award was made in 2004 to:- ... individuals, regardless of race, nationality and religious belief, who have achieved significant breakthrough in academic and scientific research or application, and whose work has resulted in a positive and profound impact on mankind. The 2005 Shaw Prize for Mathematical Sciences was made to Wiles:- ... for his proof of Fermat's last theorem. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page Andrew Wiles Biography – Life of English Mathematician Nationality British Sir Andrew Wiles is a British mathematician.

He is best known for providing a proof of Fermat’s Last Theorem, something which had eluded science for more than three centuries. Wiles is a specialist in number theory, and he is a research professor at the University of Oxford. He was knighted in 2000 in recognition of his achievements, and Oxford’s Mathematical Institute building is named in his honor. Wiles has also received many other awards and honors, including a plaque presented in 1998 because he was slightly too old to receive the Fields Medal, the most prestigious prize in mathematics, but one restricted to the under-40s. Early Life Wiles was born on April 11, 1953, in Cambridge, England. He had an academic background from the start: his father held the post of Oxford’s Regius Professor of Divinity, and later worked at Ridley Hall at Cambridge as the college’s chaplain. Andrew was educated in Cambridge, first at King’s College School and later at The Leys. His fascination with Fermat’s Last Theorem dates back to his school days. When he was just ten, he found a book about it in his local public library.

Even at that age, he could understand the theorem itself, which states that for an equation in which a, b, and c are raised to the same power, n, the sum of the first two, cannot equal the third for any integer above two. Wiles’ attention was caught by the fact that such a simple-sounding equation remained unsolved after more than 300 years, despite Pierre Fermat’s famous claim that he had found a proof in the 17th century – but had no space in the margin of his notebook to set it down. Nevertheless, Wiles quickly came to realize that his mathematical knowledge was not sufficient to solve the problem and for more than two decades, it lay dormant in his mind. In 1986, however, he noted the proof of the related epsilon conjecture by Ken Ribet. At the age of 33, Wiles resolved to restart his work on Fermat’s Last Theorem. Professional Career By the time he returned to Fermat’s Last Theorem, Wiles had become a respected mathematician. He had obtained his bachelor’s degree from Oxford in 1974, and six years later earned a Ph.D. at Cambridge.

He then moved to the United States, spending a short period at New Jersey’s Institute for Advanced Study before accepting a job at Princeton as a professor. In the mid-1980s, he spent several years in France, at one point acting as a Guggenheim Fellow. In 1988, he returned to Oxford for a short spell before going back to Princeton. He remained based in the U.S. for most of his subsequent career, but eventually returned, once again, to Oxford in 2011. Graduate Work and Research Wiles produced some significant research during his graduate work. As far back as 1975, under the guidance of John Coates, he worked on the Iwasawa theory and the complex multiplication of elliptic curves. With Barry Mazur, he came up with a generalization of a principal conjecture of the Iwasawa theory that extended its application from rational to totally real fields. By 1986, having followed the progress of the likes of Ken Ribet, Jean-Pierre Serre, and Gerhard Frey, Wiles became convinced that it was possible to prove Fermat’s Last Theorem.

He was particularly well placed to do so, as the modularity theorem he needed to employ was itself connected with the study of elliptic curves. Proving Fermat’s Last Theorem In the 1980s, most mathematicians felt that Fermat’s Last Theorem, despite its importance, was extremely hard to prove. Coates himself used the word “impossible” at one point when discussing the idea of proving it. Wiles himself was among a small minority who believed that a proof was realistic. For more than six years, he spent all the time he had available for research on this one problem. He was very careful not to let anyone else know what he was doing, and told only his wife the full story of his research. To cover his tracks, he intermittently released carefully chosen snippets of his earlier work. Wiles Goes Public At a conference in Cambridge in 1993, Wiles went public for the first time, causing a sensation which reached even the general public. However, a flaw in his reasoning was demonstrated a few months later, and for more than a year he was unable to make the necessary adjustments that would rescue his achievement.

He admitted that he was close to abandoning the search by September 1994, when he finally realized how to get around the flaw in his original paper. Again, the world’s media crowded in. Wiles was interviewed by the BBC and the resulting program was shown on PBS in the United States. Leave a Reply Andrew Wiles Andrew Wiles Andrew John Wiles (born April 11 , 1953 ) is a British mathematician . In one of the great success stories in the history of mathematics , Andrew Wiles (with help from Richard Taylor) proved Fermat's Last Theorem in 1994 . Before this result, Andrew Wiles had done outstanding work in number theory . In work with John Coates he obtained some of the first results on a famous conjecture of Birch and Swinnerton-Dyer, and he also did important work on the main conjecture of Iwasawa theory . He was and is a professor at Princeton University . Fermat's Last Theorem (FLT) asserts that there are no positive natural numbers x, y, and z such that in which n is a natural number greater than 2.

Wiles' odyssey began in 1985 when Ken Ribet, inspired by an idea of Gerhard Frey, proved that FLT would follow from another conjecture of Taniyama and Shimura , to the effect that every elliptic curve can be parametrized by modular forms . Though less familiar than Fermat's Last Theorem, the Taniyama-Shimura conjecture is the more significant of the two, because it touches on truly deep currents in number theory. No one had any idea how to prove it. Working in absolute secrecy, and sharing his ideas and progress only with Nicholas Katz, another professor of mathematics at Princeton, Wiles eventually developed a proof of the Taniyama-Shimura conjecture, and of hence FLT. The proof is a tour de force introducing many new ideas. Wiles was uncharacteristically dramatic in revealing the proof. He arranged to give three lectures at the Newton Institute in June of 1993 . He did not announce the topic of the lectures in advance, and as the audience and the world became aware of where the lectures were headed, the audience swelled so that the third lecture was to an overpacked room.

In the coming months, the manuscript of the proof was circulated only to a small number of mathematicians while the world awaited. The first version of the proof depended on the construction of an object called an Euler system, and this aspect proved problematical, so the final version of the proof differed from his original one. This difficulty was overcome with the help of Richard Taylor. The final version of Wiles' proof was published in the Annals of Mathematics 141 ( 1995 ), p. 443-551, together with another, supporting article by Wiles and Taylor titled " Ring -theoretic properties of certain Hecke algebras" (Annals of Mathematics 141 (1995), p. 553-572). Wiles was awarded a Wolf Prize in mathematics, in 1996. External link NOVA - Official Website | Andrew Wiles on Solving Fermat NOVA Andrew Wiles on Solving Fermat Posted 11.01.00 NOVA Andrew Wiles devoted much of his career to proving Fermat's Last Theorem, a challenge that perplexed the best minds in mathematics for 300 years.

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. In this interview, Wiles recounts how he came to terms with the mistake, and eventually went on to achieve his life's ambition. Anyone who thinks that mathematics doesn't involve passion and emotion should hear directly from Andrew Wiles. Enlarge Photo credit: © WGBH Educational Foundation A childhood dream 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 10 year old, could understand, and I knew from that moment that I would never let it go. I had to solve it. Who was Fermat and what was his Last Theorem? 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. So Fermat said because he could not find any solutions to this equation, then there were no solutions? 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." What do you mean by a proof? 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.

"I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal." So the challenge was to rediscover Fermat's proof of the Last Theorem. Why did it become so famous? 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. 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? 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. On Fermat's Trail How did you begin looking for the proof?

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. 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? 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. 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. 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. Keeping a secret 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? 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. But presumably you told your wife what you were doing? 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. "There's no problem that will mean the same to me." On a day-to-day basis, how did you go about constructing your proof?

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. Alone and in the dark Usually people work in groups and use each other for support.

What did you do when you hit a brick wall? 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. So for seven years you're pursuing this proof. Presumably there are periods of self-doubt mixed with the periods of success. 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. And during those seven years, you could never be sure of achieving a complete proof. 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. For eight years, Wiles was utterly obsessed with the challenge of Fermat's Last Theorem. Enlarge Photo credit: © WGBH Educational Foundation A Fitful breakthrough Then eventually in 1993, you made the crucial breakthrough. 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. 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? 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.

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? 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. So Fermat's original proof is still out there somewhere. 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. A melancholy triumph So some mathematicians might continue to look for the original proof. What will you do next? 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. "My mind is now at rest." What is the main challenge now? The greatest problem for mathematicians now is probably the Riemann Hypothesis. But it's not a problem that can be simply stated. And is there any one particular thought that remains with you now that Fermat's Last Theorem has been laid to rest?

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. And now that journey is over, there must be a certain sadness? 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. This feature originally appeared on the site for the NOVA program The Proof .

Related Links Essay on Andrew Wiles and His Contributions to Modern Math | Custom-essay.net Essay on Andrew Wiles and His Contributions to Modern Math Free essays Essay on Andrew Wiles and His Contributions to Modern Math In the current paper I will write about Andrew Wiles and share some brief facts about his life and the contributions he has made to science of math. Sir Andrew John Wiles (was born April 11, 1953, Cambridge, UK, Knight-Commander of the Order of the British Empire in 2000) – is English and American mathematician and professor of mathematics at Princeton University and head of its Department of Mathematics, member of the Scientific Council of the Institute Clay Mathematics. He received his bachelor’s degree in 1974 at Merton College, Oxford. Academic career began in the summer of 1975 under the guidance of Professor John Coates Claire College at Cambridge University, where he received his doctorate. In the period from 1977 to 1980, Wiles served as a research assistant at the college and associate professor Clare at Harvard University.

Together with John Coates, he worked on the arithmetic of elliptic curves with complex multiplication methods Iwasawa theory. In 1982, Wiles moved from Britain to the United States, according to Who is Andrew Wiles? (2010). One of the highlights of his career was proof of Fermat’s Last Theorem in 1993 and the detection technique, which made it possible to complete the proof by his former graduate student, Robert Taylor, in 1994. Work on Fermat’s theorem, he began in the summer of 1986 after Ken Ribet proved the connection of semistable elliptic curves (a special case of Theorem Taniyama – Shimura) and Fermat’s theorem. The basic idea of such a connection belongs to Gerhard Fry, German mathematician, according to Andrew Wiles on solving Fermat (2000). The work of Wiles Fermat’s Last Theorem has been reflected in the musical “Grand Tango Farm” Lessner and Rosenblum. Wiles and his work are mentioned in the episode “Facets” series “Star Trek: Deep Space Nine”.

Sir Andrew John Wiles is a British mathematician, who gained fame in 1993 by the demonstration of Fermat’s last theorem. Wiles was able to prove Fermat’s Last Theorem from the connection, outlined by Frey, and demonstrated by Ken Ribet in 1985, that a demonstration of the so-called Taniyama-Shimura conjecture would lead directly to a proof of Fermat’s last theorem. In summary, the Taniyama-Shimura conjecture states that every elliptic curve can be associated unambiguously with a mathematical object called a modular fashion. If Fermat’s last theorem were false, then there would be an elliptic curve such that it can not associate with any modular form, and therefore the Taniyama-Shimura conjecture would be false. Therefore, the Taniyama-Shimura proves Fermat’s last theorem. The proof of the Taniyama-Shimura already assumed an important challenge as it was one of the so-called Langlands program, which aims to unify areas of mathematics apparently unrelated to each other.

Wiles spent 8 years following the demonstration of Ribet in isolation working on the problem, which is an unusual way of working in mathematics, where it is common for mathematicians from around the world share their thoughts often. To avoid suspicion, Wiles was publishing articles regularly, as any mathematician of any university in the world. Wiles’s odyssey began in 1986 when Ken Ribet, an idea inspired by Gerhard Frey showed that UTF result as a consequence of the Taniyama-Shimura conjecture, because each one could be of elliptic curves parametrized by modular forms. Being less singular UTF, the Taniyama-Shimura conjecture is broader because it involves very fundamental ideas of number theory. Nobody had any idea how to prove it. Working in secrecy and sharing your progress only with Nicholas Katz, a professor of mathematics at Princeton, has developed Wiles proof of the Taniyama-Shimura, and from it the UTF. The proof is difficult and introduces many new ideas. Wiles was quite dramatic in the presentation of evidence. In June 1993, not advertising the topics in advance, he has scheduled three lectures at the Newton Institute.

The audience and the world were eager to know its contents. In the following months, the manuscript of the statement was circulated only among a small number of mathematicians. The first version of the proof depended on building a system of Euler and this aspect proved difficult, resulting in a final version of the statement than the original. This difficulty was overcome with the help of Richard Lawrence Taylor. The fictionalized story of the statement of UTF is presented in detail in the book by Simon Singh: “Fermat’s Last Theorem”, Editora Record (BR), 1998. The book version was the television documentary series on the science of the BBC “Horizon”. Fermat’s Last Theorem (or Fermat’s last theorem) – is one of the most popular theory of mathematics, and its condition is formulated on the conceptual level of secondary education, and proof of many mathematicians have searched for more than three hundred years. It was finally proved in 1995 by Andrew Wiles, according to Andrew Wiles- Mathematics (2011).

Wiles has been awarded several major prizes in mathematics and science: Junior Whitehead Prize of the LMS (1988); Fellow of the Royal Society (1989); Schock Prize (1995); Fermat Prize (1995); Wolf Prize (1995/6); NAS Award in Mathematics from the National Academy of Sciences (1996); Royal Medal (1996); Ostrowski Prize (1996); Cole Prize (1997); Wolfskehl Prize (1997) – A silver plaque from the International Mathematical Union (1998) recognizing his achievements, in place of the Fields Medal, which is restricted to those under 40 (Wiles was born in 1953 and proved the theorem in 1994); King Faisal Prize (1998); Clay Research Award (1999); Shaw Prize (2005); Pythagoras Award (Croton, 2004), according to Andrew John Wiles (2009).

Public honors are: The asteroid 9999 Wiles was named for Wiles in 1999; Wiles was appointed to the rank of Knight Commander of the Order of the British Empire in the United Kingdom in 2000. All in all, it should be noted that Sir Andrew John Wiles, who is an English and American mathematician, professor of mathematics at Princeton University, head of its Department of Mathematics, member of the Scientific Council Clay Mathematics Institute, received bachelor degree in 1974 at Merton College of Oxford University. His career began in the summer of 1975 under the leadership of Professor John Coates of Clare College in Cambridge University where he received the degree and PhD. In the period from the year 1977 – 1980 Wiles hold office junior fellow at Clare College and professor at Harvard University. In 1982 he moved to the United States.

One of the most important events in his career was his application, he has shown great theorem Fermat in 1993 and the publication of elegant method that allows proof to be completed in 1994, professional work of the Great Fermat theorem by early summer of 1986. It can be said that the contribution of Andrew Wiles was considerable and very important for the society. 13/04/2013 Fermat's last theorem Fermat's last theorem Version for printing 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. There is a statue of Fermat and his muse in his home town of Toulouse: (Click it to see a larger version) 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 . The Bernoulli number Bi is defined by x/(ex - 1) = Bi xi /i! 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.

Question: What did British mathematician Andrew Wiles prove in 1993?
Answer: Fermat's Last Theorem