Fermat's Last Theorem

by Simon Singh

  The death of the Taniyama–Shimura conjecture would have devastating repercussions throughout number theory, because for two decades mathematicians had tacitly assumed its truth. In Chapter 5 it was explained that mathematicians had written dozens of proofs which began with ‘Assuming the Taniyama–Shimura conjecture’, but now Elkies had shown that this assumption was wrong and all those proofs had simultaneously collapsed. Mathematicians immediately began to demand more information and bombarded Elkies with questions, but there was no response and no explanation as to why he was remaining tight-lipped. Nobody could even find the exact details of the counter-example.

  After one or two days of turmoil some mathematicians took a second look at the e-mail and began to realise that, although it was typically dated 2 April or 3 April, this was a result of having received it second or third hand. The original message was dated 1 April. The e-mail was a mischievous hoax perpetrated by the Canadian number theorist Henri Darmon. The rogue e-mail served as a suitable lesson for the Fermat rumour-mongers, and for a while the Last Theorem, Wiles, Taylor and the damaged proof were left in peace.

  That summer Wiles and Taylor made no progress. After eight years of unbroken effort and a lifetime’s obsession Wiles was prepared to admit defeat. He told Taylor that he could see no point in continuing with their attempts to fix the proof. Taylor had already planned to spend September in Princeton before returning to Cambridge, and so despite Wiles’s despondency, he suggested they persevere for one more month. If there was no sign of a fix by the end of September, then they would give up, publicly acknowledge their failure and publish the flawed proof to allow others an opportunity to examine it.

  The Birthday Present

  Although Wiles’s battle with the world’s hardest mathematical problem seemed doomed to end in failure, he could look back at the last seven years and be reassured by the knowledge that the bulk of his work was still valid. To begin with Wiles’s use of Galois groups had given everybody a new insight into the problem. He had shown that the first element of every elliptic equation could be paired with the first element of a modular form. Then the challenge was to show that if one element of the elliptic equation was modular, then so must the next piece be modular, and so must they all be modular.

  During the middle years Wiles wrestled with the concept of extending the proof. He was trying to complete an inductive approach and had wrestled with Iwasawa theory in the hope that this would demonstrate that if one domino fell then they all would. Initially Iwasawa theory seemed powerful enough to cause the required domino effect but in the end it could not quite live up to his expectation. He had devoted two years of effort to a mathematical dead end.

  In the summer of 1991, after a year in the doldrums, Wiles encountered the method of Kolyvagin and Flach and he abandoned Iwasawa theory in favour of this new technique. The following year the proof was announced in Cambridge and he was proclaimed a hero. Within two months the Kolyvagin–Flach method was shown to be flawed, and ever since the situation had only worsened. Every attempt to fix Kolyvagin–Flach had failed.

  All of Wiles’s work apart from the final stage involving the Kolyvagin–Flach method was still worthwhile. The Taniyama–Shimura conjecture and Fermat’s Last Theorem might not have been solved; nevertheless he had provided mathematicians with a whole series of new techniques and strategies which they could exploit to prove other theorems. There was no shame in Wiles’s failure and he was beginning to come to terms with the prospect of being beaten.

  As a consolation he at least wanted to understand why he had failed. While Taylor re-explored and re-examined alternative methods, Wiles decided to spend September looking one last time at the structure of the Kolyvagin–Flach method to try and pinpoint exactly why it was not working. He vividly remembers those final fateful days: ‘I was sitting at my desk one Monday morning, 19 September, examining the Kolyvagin–Flach method. It wasn’t that I believed I could make it work, but I thought that at least I could explain why it didn’t work. I thought I was clutching at straws, but I wanted to reassure myself. Suddenly, totally unexpectedly, I had this incredible revelation. I realised that, although the Kolyvagin–Flach method wasn’t working completely, it was all I needed to make my original Iwasawa theory work. I realised that I had enough from the Kolyvagin–Flach method to make my original approach to the problem from three years earlier work. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem.’

  Iwasawa theory on its own had been inadequate. The Kolyvagin–Flach method on its own was also inadequate. Together they complemented each other perfectly. It was a moment of inspiration that Wiles will never forget. As he recounted these moments, the memory was so powerful that he was moved to tears: ‘It was so indescribably beautiful; it was so simple and so elegant. I couldn’t understand how I’d missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I’d keep coming back to my desk looking to see if it was still there. It was still there. I couldn’t contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.’

  This was not only the fulfilment of a childhood dream and the culmination of eight years of concerted effort, but having been pushed to the brink of submission Wiles had fought back to prove his genius to the world. The last fourteen months had been the most painful, humiliating and depressing period of his mathematical career. Now one brilliant insight had brought an end to his suffering.

  ‘So the first night I went back home and slept on it. I checked through it again the next morning and by 11 o’clock I was satisfied, and I went down and told my wife, “I’ve got it! I think I’ve found it.” And it was so unexpected that she thought I was talking about a children’s toy or something, and she said, “Got what?” I said, “I’ve fixed my proof. I’ve got it.’”

  The following month Wiles was able to make up for the promise he had failed to keep the previous year. ‘It was coming up to Nada’s birthday again and I remembered that last time I could not give her the present she wanted. This time, half a minute late for our dinner on the night of her birthday, I was able to give her the complete manuscript. I think she liked that present better than any other I had ever given her.’

  Epilogue

  Grand Unified Mathematics

  A reckless young fellow from Burma,

  Found proofs of the theorem of Fermat

  He lived then in terror,

  Of finding an error,

  Wiles’s proof, he suspected, was firmer!

  Fernando Gouvea

  This time there was no doubt about the proof. The two papers, consisting of 130 pages in total, were the most thoroughly scrutinised mathematical manuscripts in history and were eventually published in Annals of Mathematics (May 1995).

  Once again Wiles found himself on the front page of the New York Times, but this time the headline ‘Mathematician Calls Classic Riddle Solved’ was overshadowed by another science story – ‘Finding on Universe’s Age Poses New Cosmic Puzzle.’ While journalists were slightly less enthusiastic about Fermat’s Last Theorem this time around, the matematicians had not lost sight of the true significance of the proof. ‘In mathematical terms the final proof is the equivalent of splitting the atom or finding the structure of DNA,’ announced John Coates. ‘A proof of Fermat is a great intellectual triumph and one shouldn’t lose sight of the fact that it has revolutionised number theory in one fell swoop. For me the charm and beauty of Andrew’s work has been that it has been a tremendous step for number theory.’

  During Wiles’s eight-year ordeal he had brought together virtually all the breakthroughs in twentieth-century number theory and incorporated them in one almighty proof. He had created completely new mathematical techniques and combined them with traditional ones in ways that had never been considered possible. In doing so he had opened up new lines of attack on a whole host of other problems. According to Ken Ribet
the proof is a perfect synthesis of modern mathematics and an inspiration for the future: ‘I think that if you were lost on a desert island and you had only this manuscript then you would have a lot of food for thought. You would see all of the current ideas of number theory. You turn to a page and there’s a brief apperance of some fundamental theorem by Deligne and then you turn to another page and in some incidental way there’s a theorem by Hellegouarch – all of these things are just called into play and used for a moment before going on to the next idea.’

  While science journalists eulogised over Wiles’s proof of Fermat’s Last Theorem, few of them commented on the proof of the Taniyama–Shimura conjecture that was inextricably linked to it. Few of them bothered to mention the contribution of Yutaka Taniyama and Goro Shimura, the two Japanese mathematicians who back in the 1950s had sown the seeds for Wiles’s work. Although Taniyama had committed suicide over thirty years earlier, his colleague Shimura was there to witness their conjecture proved. When asked for his reaction to the proof, Shimura gently smiled and in a restrained and dignified manner simply said, ‘I told you so.’

  Like many of his colleagues, Ken Ribet feels that proving the Taniyama–Shimura conjecture has transformed mathematics: ‘There’s an important psychological repercussion which is that people now are able to forge ahead on other problems that they were too timid to work on before. The landscape is different, in that you know that all elliptic equations are modular and therefore when you prove a theorem for elliptic equations you’re also attacking modular forms and vice versa. You have a different perspective of what’s going on and you feel less intimidated by the idea of working with modular forms because basically you’re now working with elliptic equations. And, of course, when you write an article about elliptic equations, instead of saying that we don’t know anything so we’re going to have to assume the Taniyama–Shimura conjecture and see what we can do with it, now we can just say that we know the Taniyama–Shimura conjecture is true, so therefore such and such must be true. It’s a much more pleasant experience.’

  Via the Taniyama–Shimura conjecture Wiles had unified the elliptic and modular worlds, and in so doing provided mathematics with a shortcut to many other proofs – problems in one domain could be solved by analogy with problems in the parallel domain. Classic unsolved elliptic problems dating all the way back to the ancient Greeks could now be reexamined using all the available modular tools and techniques.

  Even more important, Wiles had made the first step toward Robert Langlands’s grander scheme of unification – the Langlands program. There is now a renewed effort to prove other unifying conjectures between other areas of mathematics. In March 1996 Wiles shared the $100,000 Wolf Prize (not to be confused with the Wolfskehl Prize) with Langlands. The Wolf Committee was recognising that while Wiles’s proof was an astounding accomplishment in its own right, it had also breathed life into Langlands’s ambitious scheme. Here was a breakthrough that could lead mathematics into the next golden age of problem-solving.

  Following a year of embarrassment and uncertainty the mathematical community could at last rejoice. Every symposium, colloquium, and conference had a session devoted to Wiles’s proof, and in Boston mathematicians launched a limerick competition to commemorate the momentous event. It attracted this entry:

  ‘My butter, garçon, is writ large in!’

  A diner was heard to be chargin’,

  ‘I had to write there,’

  Exclaimed waiter Pierre,

  ‘I couldn’t find room in the margarine.’

  E. Howe, H. Lenstra, D. Moulton

  The Prize

  Wiles’s proof of Fermat’s Last Theorem relies on verifying a conjecture born in the 1950s. The argument exploits a series of mathematical techniques developed in the last decade, some of which were invented by Wiles himself. The proof is a masterpiece of modern mathematics, which leads to the inevitable conclusion that Wiles’s proof of the Last Theorem is not the same as Fermat’s. Fermat wrote that his proof would not fit into the margin of his copy of Diophantus’s Arithmetica, and Wiles’s 100 pages of dense mathematics certainly fulfills this criterion, but surely the Frenchman did not invent modular forms, the Taniyama–Shimura conjecture, Galois groups, and the Kolyvagin–Flach method centuries before anyone else.

  If Fermat did not have Wiles’s proof, then what did he have? Mathematicians are divided into two camps. The hardheaded skeptics believe that Fermat’s Last Theorem was the result of a rare moment of weakness by the seventeenth-century genius. They claim that, although Fermat wrote ‘I have discovered a truly marvellous proof,’ he had in fact found only a flawed proof. The exact nature of this flawed proof is open to debate, but it is quite possible that it may have been along the same lines as the work of Cauchy or Lamé.

  Other mathematicians, the romantic optimists, believe that Fermat may have had a genuine proof. Whatever this proof might have been, it would have been based on seventeenth-century techniques, and would have involved an argument so cunning that it has eluded everybody from Euler to Wiles. Despite the publication of Wiles’s solution to the problem, there are plenty of mathematicians who believe that they can still achieve fame and glory by discovering Fermat’s original proof.

  Although Wiles had to resort to twentieth-century methods to solve a seventeenth-century riddle, he has nonetheless met Fermat’s challenge according to the rules of the Wolfskehl committee. On June 27, 1997, Andrew Wiles collected the Wolfskehl Prize, worth $50,000. Fermat’s Last Theorem had been officially solved.

  Wiles realises that in order to give mathematics one of its greatest proofs, he has had to deprive it of its greatest riddle: ‘People have told me that I’ve taken away their problem, and asked if I could give them something else. 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. Perhaps that’s always the way with math problems. We just have to find new ones to capture our attention.’

  But what next will capture Wiles’s attention? Not surprisingly for a man who worked in complete secrecy for seven years, he is refusing to comment on his current research, but whatever he is working on, there is no doubt that it will never fully replace the infatuation he had with Fermat’s Last Theorem. ‘There’s no other problem that will mean the same to me. This was my childhood passion. There’s nothing to replace that. I’ve solved it. I’ll try other problems, I’m sure. Some of them will be very hard and I’ll have a sense of achievement again, but there’s no other problem in mathematics that could hold me the way Fermat did.

  ‘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 you can tackle something in adult life that means that much to you, then it’s more rewarding than anything imaginable. Having solved this problem there’s certainly a sense of loss, but at the same time there is this tremendous sense of freedom. I was so obsessed by this problem that for eight years I was thinking about it all the time – when I woke up in the morning to when I went to sleep at night. That’s a long time to think about one thing. That particular odyssey is now over. My mind is at rest.’

  Appendices

  Appendix 1. The Proof of Pythagoras’ Theorem

  The aim of the proof is to show that Pythagoras’ theorem is true for all right-angled triangles. The triangle shown above could be any right-angled triangle because its lengths are unspecified, and represented by the letters x, y and z.

  Also above, four identical right-angled triangles are combined with one tilted square to build a large square. It is the area of this large square which is the key to the proof.

  The area of the large square can be calculated in two ways.

  Method 1: Measure the area of the large square as a whole. The length of each side is x + y. Therefore, the area of the large square = (x + y)2.

  Method 2: Measure the area of each element of the large square. The area of each triangle is 1�
��2xy, i.e.1⁄2 × base × height. The area of the tilted square is z2. Therefore,

  area of large square = 4 × (area of each triangle) + area of tilted square

  Methods 1 and 2 give two different expressions. However, these two expressions must be equivalent because they represent the same area. Therefore,

  The brackets can be expanded and simplified. Therefore,

  The 2xy can be cancelled from both sides. So we have

  which is Pythagoras’ theorem!

  The argument is based on the fact that the area of the large square must be the same no matter what method is used to calculate it. We then logically derive two expressions for the same area, make them equivalent, and eventually the inevitable conclusion is that x2 + y2 = z2, i.e. the square on the hypotenuse, z2, is equal to the sum of the squares on the other two sides, x2 + y2.

  This argument holds true for all right-angled triangles. The sides of the triangle in our argument are represented by x, y and z, and can therefore represent the sides of any right-angled triangle.

  Appendix 2. Euclid’s Proof that √2 is Irrational

  Euclid’s aim was to prove that √2 could not be written as a fraction. Because he was using proof by contradiction, the first step was to assume that the opposite was true, that is to say, that √2 could be written as some unknown fraction. This hypothetical fraction is represented by p⁄q, where p and q are two whole numbers.