"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." "ANDREW WILES" "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." "This the story of one man's obsession with the world's greatest mathematical problem." "For seven years, professor Andrew Wiles worked in complete secrecy, creating the calculation of the century." "It was a calculation which brought him fame and regret." "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,but 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." "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 tried to solve." "He studied this ;" "he wrote notes in the margins." "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." "And this is the fantastic observation of master Pierre de Fermat which caused all the trouble." ""Cubum autem in duos cubos."" "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 of by heart." "The square of the hypotenuse is equal to the sum of the squares of the other two sides." "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." "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." "You will never find numbers that fit this equation." "If n is greater than 2, that's what Fermat said." "What's more, he said he could prove it." "In a moment of brillance, he scribbled the following mysterious note." "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." "So Fermat said he had a proof, but he never said what it was." "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." "Rediscovering Fermat's proof became the ultimate challenge, a challenge which would baffle mathematicians for the next 300 years." "Gauss, the greatest mathematician in the world..." "Oh, yeah." "Galois..." "Kummer, of course." "Well, in the 18th century, Euler didn't prove it." "Well, you know there's only been the one woman, really." "Sophie Germain." "Oh, there are millions." "There are lots of people." "But, nobody had any idea where to start." "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." "Mathemeticians had to proof that no numbers fit the equation." "but with the invent of computers couldn't they check each number one by one and show that none of them fit it ?" "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?" "A computer can never check every number." "Instead, what's needed is a mathematical proof." "A mathematician is not happy until the proof is complete and considered complete by the standards of mathematics." "In mathematics, there's the concept of proving something, of knowing it with absolute certainty." "Which ..." "Well, it's called "rigorous proof."" "That's mathematics is about." "A proof is a sort of reason, it explains why no numbers fit the equation without having to check every number." "After centuries of failing to find a proof, mathematicians began to abandon Fermat." "In favour of more serious maths." "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." "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." "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." "When I went to Cambridge, my advisor John Coates was working on Iwasa theory on elliptic curves and I started working working with him." "elliptic curves were the "in" thing to study." "but perversely, elliptic curves are neither the ellipses nor curves." "You may never have heard of elliptic curves but they're extremely important." "OK." "So, what's an elliptic curve?" "Elliptic curves." "They're not ellipses." "They're cubic curves whose solution have a shape that looks like a doughnut." "They look so simple, yet the complexity, especially arithmetic complexity, is immense." "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." "I entered the University of Tokyo in 1949, and that was four years after the War, almost all professors were tired and the lectures were not inspiring." "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." "(Utaka Taniyama)" "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." "Together, Taniyama and Shimura worked on the complex mathematics of modular functions." "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." "Oh it's impossible !" "There's a saying attributed to Eichler that there are five fundamental operations of arithmetic :" "addition, subtraction, multiplication, division, and modular forms." "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." "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." "In 1955, there was an international symposium, and Taniyama posed two or three problems." "The problems posed by Taniyama led to the extraordinary claim that every elliptic curve was really a modular form in disguise." "It became known as the Taniyama-Shimura conjecture." "What the Taniyama-Shimura conjecture says, it says that every rational elliptic curve is modular, and that's so hard to explain." "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." "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." "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, an idea, and until it could be proved, all the maths which relied on it were under threat." "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." "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." "I was very much puzzled." "Puzzlement may be the best word." "Of course, I was sad but see, it was so sudden, and I was unable to make sense out of this." "Taniyama-Shimura went on to become one of the great unproven conjectures." "But what did it have to do with Fermat's last theorem ?" "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." "Taniyama-Shimura says, "Every elliptic curve is modular,"" "and Fermat says, "No numbers fit this equation."" "What was the connection ?" "[music] One way or another" "I'm gonna find you" "I'm gonna get get get ..." "One way or another" "I'm gonna w*** you." "I'm get get get you" "One way or another" "I'm gonna see you" "I wanna wanna meet meet your one day" "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." "Frey, a German mathematician, considered the unthinkable." "What would happen if Fermat was wrong and there was a solution to this equation after all ?" "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." "That elliptic curve seems to be not modular." "But Shimura-Taniyama says that every elliptic curve is modular." "So, if there is a solution to this equation, it creates such a weird elliptic curve it defies Taniyama-Shimura." "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." "Fermat and Taniyama-Shimura were now linked, apart from just one thing." "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." "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." "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." "Ken's idea was brilliant." "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." "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." "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." "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." "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." "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." "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." "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." "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." "Andrew's trick was to transform the elliptic curves into something called Galois representations." "which would make counting easier." "Now it was a question of compairing modular forms with Galois representations not elliptic curves." "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." "This was only the first step, and already, it had taken three years of Andrew's life." "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," "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." "Iwasawa theory was suppose to help create something called a Class Number Formula." "But several months past and the Class Number Formula remained out of reach." "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." "Andrew was almost there, but this breakthrough was risky and complicated." "After six years of secrecy, he needed to confide in someone." "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." "It involved a kind of mathematics that Nick Katz is an expert in." "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." "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." "In order not to arose any more suspicion," "Andrew decided to check his proof by disguising it in a course of lectures, which Nick Katz could them attend." "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." "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." "The lectures revealed no errors, and still, none of his colleagues suspected why Andrew was being so secretive." "Maybe he's run out of ideas." "That's why he's quiet." "You never know why they're quiet." "The proof was still missing a vital ingredient, but Andrew now felt confident." "It was time to tell one more person." "So, I called up Peter and asked him if I could come 'round and talk to him about something." "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." "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."" "I was flabbergasted, excited, disturbed." "I mean, I remember that night finding it quite difficult to sleep." "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 that" "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." "The name of the lectures that he announced was simply "Elliptic Curves and Modular Forms."" "There was no mention of Fermat's last theorem." "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." "Not from me." "Not from me." "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."" "Maybe some hints, yeah." "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."" "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." "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." "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." "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."" "The next day, what was totally unexpected was that we were deluged by inquiries from newspapers, journalists from all around the world." "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." "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."" "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." "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." "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." "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." "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." "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."" "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." "In desperation, Andrew called for help he invited his former student, Richard Taylor, to work with him in Princeton." "But no solution came, after year of failure, he was ready to abandon his flawed proof." "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 19th, 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."" "I think it will always stand as one of the high achievements of number theory." "It was magnificent." "It's not every day that you hear the proof of the century." "Well, my first reaction was, "I told you so."" "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?" "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." "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?" "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, it's more rewarding than anything I could imagine." "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..." "The proof by Langlands and Tunnell..." "Deligne, Rapoport, Katz..." "Mazur's idea of using the deformation theory of Galois representations..." "Igusa, Eichler, Shimura, Taniyama..." "Frey's reduction..." "The list goes on and on." "Bloch, Kato, Selmer, Frey, Fermat."