Fermat's Last Theorem

by Simon Singh

  The Missing Link

  During the autumn of 1984 a select group of number theorists gathered for a symposium in Oberwolfach, a small town in the heart of Germany’s Black Forest. They had been brought together to discuss various breakthroughs in the study of elliptic equations, and naturally some of the speakers would occasionally report any minor progress that they had made towards proving the Taniyama–Shimura conjecture. One of the speakers, Gerhard Frey, a mathematician from Saarbrücken, could not offer any new ideas as to how to attack the conjecture, but he did make the remarkable claim that if anyone could prove the Taniyama–Shimura conjecture then they would also immediately prove Fermat’s Last Theorem.

  When Frey got up to speak he began by writing down Fermat’s equation:

  Fermat’s Last Theorem claims that there are no whole number solutions to this equation, but Frey explored what would happen if the Last Theorem were false, i.e. that there is at least one solution. Frey had no idea what his hypothetical, and heretical, solution might be and so he labelled the unknown numbers with the letters A, B and C:

  Frey then proceeded to ‘rearrange’ the equation. This is a rigorous mathematical procedure which changes the appearance of the equation without altering its integrity. By a deft series of complicated manoeuvres Frey fashioned Fermat’s original equation, with the hypothetical solution, into

  Although this rearrangement seems very different from the original equation, it is a direct consequence of the hypothetical solution. That is to say if, and it is a big ‘if, there is a solution to Fermat’s equation and Fermat’s Last Theorem is false, then this rearranged equation must also exist. Initially Frey’s audience was not particularly impressed by his rearrangement, but then he pointed out that this new equation was in fact an elliptic equation, albeit a rather convoluted and exotic one. Elliptic equations have the form

  but if we let

  then it is easier to appreciate the elliptical nature of Frey’s equation.

  By turning Fermat’s equation into an elliptic equation, Frey had linked Fermat’s Last Theorem to the Taniyama–Shimura conjecture. Frey then pointed out to his audience that his elliptic equation, created from the solution to the Fermat equation, is truly bizarre. In fact, Frey claimed that his elliptic equation is so weird that the repercussions of its existence would be devastating for the Taniyama–Shimura conjecture.

  Remember that Frey’s elliptic equation is only a phantom equation. Its existence is conditional on that fact that Fermat’s Last Theorem is false. However, if Frey’s elliptic equation does exist, then it is so strange that it would be seemingly impossible for it ever to be related to a modular form. But the Taniyama–Shimura conjecture claims that every elliptic equation must be related to a modular form. Therefore the existence of Frey’s elliptic equation defies the Taniyama–Shimura conjecture.

  In other words, Frey’s argument was as follows:

  (1) If (and only if) Fermat’s Last Theorem is wrong, then Frey’s elliptic equation exists.

  (2) Frey’s elliptic equation is so weird that it can never be modular.

  (3) The Taniyama–Shimura conjecture claims that every elliptic equation must be modular.

  (4) Therefore the Taniyama–Shimura conjecture must be false!

  Alternatively, and more importantly, Frey could run his argument backwards:

  (1) If the Taniyama–Shimura conjecture can be proved to be true, then every elliptic equation must be modular.

  (2) If every elliptic equation must be modular, then the Frey elliptic equation is forbidden to exist.

  (3) If the Frey elliptic equation does not exist, then there can be no solutions to Fermat’s equation.

  (4) Therefore Fermat’s Last Theorem is true!

  Gerhard Frey had come to the dramatic conclusion that the truth of Fermat’s Last Theorem would be an immediate consequence of the Taniyama–Shimura conjecture being proved. Frey claimed that if mathematicians could prove the Taniyama–Shimura conjecture then they would automatically prove Fermat’s Last Theorem. For the first time in a hundred years the world’s hardest mathematical problem looked vulnerable. According to Frey, proving the Taniyama–Shimura conjecture was the only hurdle to proving Fermat’s Last Theorem.

  Although the audience was impressed by Frey’s brilliant insight, they were also struck by an elementary blunder in his logic. Almost everyone in the auditorium, except Frey himself, had spotted it. The mistake did not appear to be serious: nonetheless as it stood Frey’s work was incomplete. Whoever could correct the error first would take the credit for linking Fermat and Taniyama–Shimura.

  Frey’s audience dashed out of the lecture theatre and headed for the photocopying room. Often the importance of a talk can be gauged by the length of the queue waiting to run off copies of the lecture. Once they had a complete outline of Frey’s ideas, they returned to their respective institutes and began to try and fill in the gap.

  Frey’s argument depended on the fact that his elliptic equation derived from Fermat’s equation was so weird that it was not modular. His work was incomplete because he had not quite demonstrated that his elliptic equation was sufficiently weird. Only when somebody could prove the absolute weirdness of Frey’s elliptic equation would a proof of the Taniyama–Shimura conjecture then imply a proof of Fermat’s Last Theorem.

  Initially mathematicians believed that proving the weirdness of Frey’s elliptic equation would be a fairly routine process. At first sight Frey’s mistake seemed to have been elementary and everyone who had been at Oberwolfach assumed that it was going to be a race to see who could shuffle the algebra most quickly. The expectation was that somebody would send out an e-mail within a matter of days describing how they had established the true weirdness of Frey’s elliptic equation.

  A week passed and there was no such e-mail. Months passed and what was supposed to be a mathematical mad dash was turning into a marathon. It seemed that Fermat was still teasing and tormenting his descendants. Frey had outlined a tantalising strategy for proving Fermat’s Last Theorem, but even the first elementary step, proving that Frey’s hypothetical elliptic equation was not modular, was baffling mathematicians around the globe.

  To prove that an elliptic equation is not modular, mathematicians were looking for invariants similar to those described in Chapter 4. The knot invariant showed that one knot could not be transformed into another, and Loyd’s puzzle invariant showed that his 14–15 puzzle could not be transformed into the correct arrangement. If number theorists could discover an appropriate invariant to describe Frey’s elliptic equation, then they could prove that, no matter what was done to it, it could never be transformed into a modular form.

  One of those toiling to prove and complete the connection between the Taniyama–Shimura conjecture and Fermat’s Last Theorem was Ken Ribet, a professor at the University of California at Berkeley. Since the lecture at Oberwolfach, Ribet had become obsessed with trying to prove that Frey’s elliptic equation was too weird to be modular. After eighteen months of effort he, along with everybody else, was getting nowhere. Then, in the summer of 1986, Ribet’s colleague Professor Barry Mazur was visiting Berkeley to attend the International Congress of Mathematicians. The two friends met up for a cappuccino at the Café Strada and began sharing bad luck stories and grumbling about the state of mathematics.

  Eventually they started discussing the latest news on the various attempts to prove the weirdness of Frey’s elliptic equation, and Ribet began explaining a tentative strategy which he had been exploring. The approach seemed vaguely promising but he could only prove a very minor part of it. ‘I sat down with Barry and told him what I was working on. I mentioned that I’d proved a very special case, but I didn’t know what to do next to generalise it to get the full strength of the proof.’

  Professor Mazur sipped his cappuccino and listened to Ribet’s idea. Then he stopped and stared at Ken in disbelief. ‘But don’t you see? You’ve already done it! All you have to do is add some gamma-zero of (M
) structure and just run through your argument and it works. It gives you everything you need.’

  Ribet looked at Mazur, looked at his cappuccino, and looked back at Mazur. It was the most important moment of Ribet’s career and he recalls it in loving detail. ‘I said you’re absolutely right – of course – how did I not see this? I was completely astonished because it had never occurred to me to add the extra gamma-zero of (M) structure, simple as it sounds.’

  It should be noted that, although adding gamma-zero of (M) structure sounds simple to Ken Ribet, it is an esoteric step of logic which only a handful of the world’s mathematicians could have concocted over a casual cappuccino.

  ‘It was the crucial ingredient that I had been missing and it had been staring me in the face. I wandered back to my apartment on a cloud, thinking: My God is this really correct? I was completely enthralled and I sat down and started scribbling on a pad of paper. After an hour or two I’d written everything out and verified that I knew the key steps and that it all fitted together. I ran through my argument and I said, yes, this absolutely has to work. And there were of course thousands of mathematicians at the International Congress and I sort of casually mentioned to a few people that I’d proved that the Taniyama–Shimura conjecture implies Fermat’s Last Theorem. It spread like wildfire and soon large groups of people knew; they were running up to me asking, Is it really true you’ve proved that Frey’s elliptic equation is not modular? And I had to think for a minute and all of a sudden I said, Yes, I have.’

  Fermat’s Last Theorem was now inextricably linked to the Taniyama–Shimura conjecture. If somebody could prove that every elliptic equation is modular, then this would imply that Fermat’s equation had no solutions, and immediately prove Fermat’s Last Theorem.

  For three and half centuries Fermat’s Last Theorem had been an isolated problem, a curious and impossible riddle on the edge of mathematics. Now Ken Ribet, inspired by Gerhard Frey, had brought it centre stage. The most important problem from the seventeenth century was coupled to the most significant problem of the twentieth century. A puzzle of enormous historical and emotional importance was linked to a conjecture that could revolutionise modern mathematics. In effect, mathematicians could now attack Fermat’s Last Theorem by adopting a strategy of proof by contradiction. To prove that the Last Theorem is true, mathematicians would begin by assuming it to be false. The implication of being false would be to make the Taniyama–Shimura conjecture false. However, if Taniyama–Shimura could be proven true, then this would be incompatible with Fermat’s Last Theorem being false, therefore it, too, would have to be true.

  Frey had clearly defined the task ahead. Mathematicians would automatically prove Fermat’s Last Theorem if they could first prove the Taniyama–Shimura conjecture.

  Initially there was renewed hope but then the reality of the situation dawned. Mathematicians had been trying to prove Taniyama–Shimura for thirty years and they had failed. Why should they make any progress now? The sceptics believed that what little hope there was of proving the Taniyama–Shimura conjecture had now vanished. Their logic was that anything that might lead to a solution of Fermat’s Last Theorem must, by definition, be impossible.

  Even Ken Ribet, who had made the crucial breakthrough, was pessimistic: ‘I was one of the vast majority of people who believed that the Taniyama–Shimura conjecture was completely inaccessible. I didn’t bother to try and prove it. I didn’t even think about trying to prove it. Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove this conjecture.’

  6

  The Secret Calculation

  An expert problem solver must be endowed with two incompatible qualities – a restless imagination and a patient pertinacity.

  Howard W. Eves

  ‘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 he told me that Ken Ribet had proved the 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. I just knew that I would go home and work on the Taniyama–Shimura conjecture.’

  Over two decades had passed since Andrew Wiles had discovered the library book that inspired him to take up Fermat’s challenge, but now, for the first time, he could see a path towards achieving his childhood dream. Wiles recalls how his attitude to Taniyama–Shimura changed overnight: ‘I remembered one mathematician who’d written about the Taniyama–Shimura conjecture and cheekily suggested it as an exercise for the interested reader. Well, I guess now I was interested!’

  Since completing his Ph.D. with Professor John Coates at Cambridge, Wiles had moved across the Atlantic to Princeton University where he himself was now a professor. Thanks to Coates’s guidance Wiles probably knew more about elliptic equations than anybody else in the world, but he was well aware that even with his enormous background knowledge and mathematical skills the task ahead was immense.

  Most other mathematicians, including John Coates, believed that embarking on the proof was a futile exercise: ‘I myself was very sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn’t see it proved in my lifetime.’

  Wiles was aware that the odds were against him, but even if he ultimately failed in proving Fermat’s Last Theorem he felt his efforts would not be wasted: ‘Of course the Taniyama–Shimura conjecture had been open for many years. No one had had any idea how to approach it 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. I didn’t feel I’d be wasting my time. So the romance of Fermat which had held me all my life was now combined with a problem that was professionally acceptable.’

  The Attic Recluse

  At the turn of the century the great logician David Hilbert was asked why he never attempted a proof of Fermat’s Last Theorem. He replied, ‘Before beginning I should have to put in three years of intensive study, and I haven’t that much time to squander on a probable failure.’ Wiles realised that to have any hope of finding a proof he would first have to completely immerse himself in the problem, but unlike Hilbert he was prepared to take the risk. He read all the most recent journals and then played with the latest techniques over and over again until they became second nature to him. Gathering the necessary weapons for the battle ahead would require Wiles to spend the next eighteen months familiarising himself with every bit of mathematics which had ever been applied to, or had been derived from, elliptic equations or modular forms. This was a comparatively minor investment, bearing in mind that he fully expected that any serious attempt on the proof could easily require ten years of single-minded effort.

  Wiles abandoned any work which was not directly relevant to proving Fermat’s Last Theorem and stopped attending the never-ending round of conferences and colloquia. Because he still had responsibilities in the Princeton Mathematics Department, Wiles continued to attend seminars, lecture to undergraduates and give tutorials. Whenever possible he would avoid the distractions of being a faculty member by working at home where he could retreat into his attic study. Here he would attempt to expand and extend the power of the established techniques, hoping to develop a strategy for his attack on the Taniyama–Shimura conjecture.

  ‘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 realised 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.

  ‘Basically it’s just a matter of thinking. Often you write something down to clarify your thoughts, but not necessarily. In particular when you’ve reached a real impasse, when there’s a real problem that you want to overcome, then the routine kind of mathematical thinking is of no use to you. Leading up to that kind of new idea there has to be a long period of tremendous focus on the problem without any distraction. You have to really think about nothing but that problem – just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconscious appears to take over and it’s during that time that some new insight comes.’

  From the moment he embarked on the proof, Wiles made the remarkable decision to work in complete isolation and secrecy. Modern mathematics has developed a culture of cooperation and collaboration, and so Wiles’s decision appeared to hark back to a previous era. It was as if he was imitating the approach of Fermat himself, the most famous of mathematical hermits. Wiles explained that part of the reason for his decision to work in secrecy was his desire to work without being distracted: ‘I realised 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.’