Fermat's Last Theorem (30 page)

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.'

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.'

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.'

Another motivation for Wiles's secrecy must have been his craving for glory. He feared the situation arising whereby he had completed the bulk of the proof but was still missing the final element of the calculation. At this point, if news of his breakthroughs were to leak out, there would be nothing stopping a rival
mathematician building on Wiles's work, completing the proof and stealing the prize.

In the years to come Wiles was to make a series of extraordinary discoveries, none of which would be discussed or published until his proof was complete. Even close colleagues were oblivious to his research. John Coates can recall exchanges with Wiles during which he was given no clues as to what was going on: ‘I remember saying to him on a number of occasions, “It's all very well this link to Fermat's Last Theorem but it's still hopeless to try and prove Taniyama–Shimura.” I think he just smiled.'

Ken Ribet, who completed the link between Fermat and Taniyama–Shimura, was also completely unaware of Wiles's clandestine activities. ‘This is probably the only case I know where someone worked for such a long time without divulging what he was doing, without talking about the progress he was making. It's just unprecedented in my experience. In our community people have always shared their ideas. Mathematicians come together at conferences, they visit each other to give seminars, they send e-mail to each other, they talk on the telephone, they ask for insights, they ask for feedback – mathematicians are always in communication. When you talk to other people you get a pat on the back; people tell you that what you've done is important, they give you ideas. It's sort of nourishing and if you cut yourself off from this, then you are doing something that's probably psychologically very odd.'

In order not to arouse suspicion Wiles devised a cunning ploy which would throw his colleagues off the scent. During the early 1980s he had been working on a major piece of research on a particular type of elliptic equation, which he was about to publish in its entirety, until the discoveries of Ribet and Frey made him change his mind. Wiles decided to publish his research bit by bit,
releasing another minor paper every six months or so. This apparent productivity would convince his colleagues that he was still continuing with his usual research. For as long as he could maintain this charade, Wiles could continue working on his true obsession without revealing any of his breakthroughs.

The only person who was aware of Wiles's secret was his wife, Nada. They married soon after Wiles began working on the proof, and as the calculation progressed he confided in her and her alone. In the years that followed, his family would be his only distraction. ‘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.'

In order to prove Fermat's Last Theorem Wiles had to prove the Taniyama–Shimura conjecture: every single elliptic equation can be correlated with a modular form. Even before the link to Fermat's Last Theorem mathematicians had tried desperately to prove the conjecture, but every attempt had ended in failure. Wiles was acquainted with the failures of the past: ‘Ultimately what one would naïvely have tried to do, and what people certainly did try to do, was to count elliptic equations and count modular forms, and show that there are the same number of each. But nobody has ever found any simple way of doing that. The first problem is that there are an infinite number of each and you can't count an infinite number. One simply doesn't have a way of doing it.'