The extraordinary story of Fermat's Last Theorem
Andrew Wiles's solution to Fermat's Last Theorem has been called the proof of the century. It is the mathematical equivalent of splitting the atom or finding the structure of DNA. It is the culmination of a struggle which involved generations of mathematicians, and which has influenced every area of mathematics. Undoubtedly the solution is a triumph of modern mathematics, but for Wiles it also has tremendous personal meaning. It is the realisation of a childhood dream, the end of an obsession which dominated his life.
Wiles was born in Cambridge in 1953. Although strongly influenced by his father, a theologian, it was his mother, a maths teacher, who inspired his enthusiasm for numbers. The young Wiles quickly polished off the problems he was given at school, and would then amuse himself by inventing his own questions.
When he was 10, he visited his local library and borrowed The Last Problem, by Eric Temple Bell. The book was a history of Fermat's Last Theorem, a mathematical problem that had plagued mathematicians since the 17th century.
Wiles still remembers how he felt the moment he was introduced to the Last Theorem: "It looked so simple, and yet all the great mathematicians in history couldn't solve it. Here was a problem that I as a 10-year-old could understand, and I knew from that moment that I would never let it go. I had to solve it."
Pierre de Fermat was born in 1601 in south-west France. Encouraged by his father to pursue a career in local government, Fermat became a councillor at the Toulouse Chamber of Petitions. In his spare time he studied mathematics, a hobby which would earn him a place in history.
Known as the Prince of Amateurs, this 17th-century genius discovered the laws of probability and laid the foundations of calculus, subjects that have since revolutionised science. But Fermat's greatest contribution was to the purest and most elegant of mathematical researches - number theory, a discipline that concentrates on the properties of simple whole numbers.
To the outsider, problems in number theory seem like pointless riddles with no application, but for practitioners, these conundrums are the ultimate intellectual test.
For centuries, number theorists have delighted in trying to outwit each other, and Fermat was no exception. Having solved a particular problem, he would write to other mathematicians, asking if they had the ingenuity to match his solution.
These challenges, and the fact that he would never reveal his own calculations, caused others a great deal of frustration. René Descartes called Fermat a "braggart", and the Englishman John Wallis referred to him as "that damned Frenchman". Fermat took particular pleasure in taunting his cousins across the Channel.
Fermat invented his most famous challenge while studying an ancient Greek mathematical text, Arithmetica. One chapter described the properties of a particular equation: x2 + y2 = z2. The equation is closely related to Pythagoras's theorem, and simply asks for three numbers (x, y, z), such that the first number squared (x * x) added to the second number squared (y * y) equals the third number squared (z * z). There are many possible solutions, such as: 32 +42 = 52, ie, 9 + 16 = 25.
One night Fermat considered a slightly perverted form of the equation, and wondered if it was possible to find a cubed number added to another cubed which equalled a third cubed number. Eventually he came to the conclusion that there are, in fact, no solutions to this equation: x3 + y3 = z3.
Furthermore, Fermat believed that unless the equation was squared, finding a solution would be impossible. In the margin of his Arithmetica he wrote down that there are no solutions to the equation: xn + yn = zn, where "n" represents any number greater than two.
The claim that among the infinity of numbers there is none that fits the equation was shocking, but Fermat believed he could back it up with a rigorous proof. Next to the marginal note outlining the theory, the mischievous genius jotted down an additional comment that would haunt generations of mathematicians: "I have a truly marvellous proof of this proposition which this margin is too narrow to contain." It was only after his death that these tantalising marginal notes were discovered, and so began a marathon effort to rediscover Fermat's proof.
Fermat made quite a habit of casually jotting down theorems without the accompanying proof, and in the years after his death there was a concerted effort to rediscover his proofs. After a century, all the theorems were successfully proved, except for the one concerning the equation: xn + yn = zn. Dubbed Fermat's Last Theorem, it became an increasingly valuable trophy for ambitious mathematicians - prizes were offered, careers were sacrificed, and rivalries flourished.
In 1742 Leonhard Euler, the greatest number theorist of the 18th century, became so frustrated that he asked his friend Clêrot to search Fermat's house in case some vital scrap of paper still remained. No clues were found.
In the 19th century the most significant progress was made by Sophie Germain, a woman who, living in an era of male chauvinism, took on the identity of a man in order to conduct her research. Although Germain's ideas eventually reached a dead-end, they generated new techniques that became valuable in tackling other problems. This was often the way with the Last Theorem - failed attempts to prove it spawned new areas of mathematical research.
At the turn of the century, Paul Wolfskehl, a German industrialist and amateur mathematician, bequeathed the equivalent of DM 100,000 for whoever could prove the Last Theorem. The story of the prize begins with Wolfskehl's obsession with a beautiful woman whose identity has never been established. The woman rejected Wolfskehl and he was left in such a state of despair that he decided to commit suicide. He appointed a date on which to shoot himself through the head at the stroke of midnight.
In the hours before his planned suicide, Wolfskehl visited his library and began reading about the latest ideas concerning the Last Theorem. Suddenly, he believed he could see a way of proving the theorem, and became engrossed in exploring his strategy. By the time Wolfskehl realised his method was flawed, the appointed time of his suicide had passed.
Because he had been reminded of the beauty and elegance of number theory, Wolfskehl abandoned his plan to kill himself. The prize he left in his will was his way of repaying a debt to the problem that saved his life.
When it was announced in 1908, the great value of the Wolfskehl Prize - about £1million in today's terms - generated an enormous amount of publicity. Within the first year, amateur problem-solvers sent in 621 proofs, all of them flawed. Interest among serious academics, though, was in decline. After more than 200 years of failure, many felt that there were more important questions to be addressed.
When the logician David Hilbert was asked why he never attempted a proof of the 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."
During this century the problem still held a special place in the hearts of number theorists, but they treated Fermat's Last Theorem in the same way that chemists treated alchemy. Both were foolish romantic dreams from a past age.
'Since I first met Fermat's Last Theorem as a child, it's been my greatest passion," recalls Andrew Wiles. "I don't think many of my school friends caught the mathematics bug so I didn't discuss it with my contemporaries. But I did have a teacher who had done research in mathematics and he gave me a book about number theory - that gave me some clues about how to start tackling it. To begin with, I worked on the assumption that Fermat didn't know too much more about mathematics than I did."
Throughout his teenage years the obsession continued. Wiles studied how Euler, Germain and others had tackled the problem and then, at university, began to see if the ideas his lecturers taught him could offer him a new insight. In 1975 Andrew Wiles became a researcher at Cambridge University, and, having joined the ranks of professional mathematicians, was forced to work on more respectable and contemporary problems. For the next decade he temporarily surrendered his dream.
In Japan in the Fifties, a breakthrough occurred that would ultimately resurrect interest in the Last Theorem. Goro Shimura and Yutaka Taniyama began to examine so-called modular forms, a variety of objects which are special because of their immensely high level of symmetry and complexity.
The duo came to the realisation that the ingredients for constructing different modular forms were encoded in the solutions to a particular set of equations, known as elliptic equations. This was the equivalent of realising that the instructions for building life were encoded in strands of DNA.
Although this was a beautiful hypothesis, nobody could prove it was true for every elliptic equation and every modular form. Crucially, if the Taniyama-Shimura conjecture could be proven, it could be used to prove dozens of other conjectures. It became one of the holy grails of 20th-century number theory.
For 30 years there was no progress, and then, in 1986, Ken Ribet, a professor at the University of California, Berkeley, was attending the International Congress of Mathematicians when he had a revelation: "I found the crucial ingredient that had been missing. 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. 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 said to myself, this absolutely has to work."
Ribet had not proved the Taniyama-Shimura conjecture, but he had shown that if somebody could prove it, then Fermat's Last Theorem would automatically follow. His argument began by wondering what would happen if the Last Theorem was false, and then working out the consequences.
One consequence was that Taniyama-Shimura would also be false. He ran the argument backwards and concluded that if Taniyama-Shimura were true, Fermat must be true.
"There were thousands of mathematicians at the International Congress," recalled Ribet, "and I sort of casually mentioned to a few people that the Taniyama-Shimura conjecture implies Fermat's Last Theorem. It spread like wildfire and soon large groups of people knew, and they were running up to me asking, 'Is it really true you've found a link to Fermat's Last Theorem?' And I had to think for a minute, and all of a sudden I said, 'Yes, I have'."
For three-and-a-half centuries the Last Theorem had been a curious and impossible riddle on the edge of mathematics. Now Ribet had brought it centre stage. The most important mathematical problem from the 17th century was coupled to the most significant problem of the 20th.
Initially, there was renewed hope, but then the reality of the situation dawned. Mathematicians had been trying to prove Taniyama-Shimura for 30 years and they had failed. Why should they make any progress now?
Even Ribet was pessimistic: "I was one of the vast majority of people who believed that the Taniyama-Shimura conjecture was completely inaccessible. 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."
'It was one evening at the end of the summer of 1986 when I was sipping iced tea at the house of a friend," recalls Wiles. "Casually, in the middle of a conversation, he 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. I just knew that I would go home and work on the Taniyama-Shimura conjecture.
"No one had any idea how to approach it but at least it was mainstream mathematics. I could try to 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."
By this time Wiles was a professor at Princeton University. He abandoned any work that was not directly relevant to the Last Theorem and stopped attending the round of conferences and colloquia, and reduced his lecturing and tutoring to a bare minimum.
In his attic study he attempted to develop a strategy for proving the Taniyama-Shimura conjecture: "I used to come up and start trying to find patterns. I tried doing calculations that 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 of any use at all. Then I just had to find something new - it's a mystery where that comes from."
From the moment he embarked on the proof, Wiles made the remarkable decision to work in complete isolation and secrecy. "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 that he might find himself, having completed the bulk of the proof, still missing the final element. If news of his breakthroughs were to leak out, there would be nothing to stop a rival mathematician building on his work, completing the proof and stealing the prize.
Even Ribet, who had linked Fermat and Taniyama-Shimura, was 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. 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 ask for insights, they ask for feedback.
"When you talk to other people you get a pat on the back, people tell you that what you've done is important. It's sort of nourishing, and if you cut yourself off from this then you are doing something that's probably psychologically very odd."
The only person who was aware of Wiles's secret was his wife, Nada. "My wife's only known me while I've been working on Fermat," says Wiles. "I told her on our honeymoon, just a few days after we got married. At that time she had no idea of the romantic significance Fermat had for mathematicians, that it had been such a thorn in our flesh for so many years."
First, Wiles had to prove the Taniyama-Shimura conjecture - every single elliptic equation can be correlated with a modular form. "What one would naively 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 a 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."
For this problem, computers would be useless. Although a computer could check an individual case in a few seconds, checking an infinity of cases would take eternity. Instead, what was required was a logical step-by-step argument: "I carried this problem around in my head 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."
Wiles is a mild-mannered man, softly spoken and somewhat hesitant: no one could have guessed at the passion that was driving him. In many ways his approach harked back to that of Fermat himself - he was isolated, secretive and obsessed. His only concern apart from the Last Theorem was his family: "The only way I could relax was when I was with my children. Young children simply aren't interested in Fermat."
After a year, Wiles decided to adopt a general strategy based on proof by induction. Induction is an immensely powerful form of proof, because it can allow a mathematician to prove that a statement is true for an infinite number of cases in just two steps. The first step is to prove that the statement is true for the number one. The next step is to show that if the statement is true for any particular number, then it must also be true for the next. Hence, if the statement is true for the number one then it must be true for the number two: and if it is true for the number two then it must be true for the number three, and so on. In two steps the mathematician has tackled an infinite number of cases.
Another way to think of proof by induction is to imagine the infinite number of cases as an infinite line of dominoes. Knocking the dominoes down one by one would take an infinite amount of time and effort, but proof by induction allows mathematicians to knock them all down by knocking down the first one.
Two years after embarking on the proof, Wiles discovered the key to achieving the first step in his inductive argument, hidden in the work of Evariste Galois, a tragic hero from 19th-century France.
Galois's life was dominated by politics as much as mathematics. The prodigy published his first paper at the age of 17, but soon found that the academic establishment was rejecting him because of his ardent republican views.
At 20, Galois abandoned mathematics and joined a republican branch of the militia known as the "Friends of the People". On Bastille Day 1831, Galois marched through Paris dressed in the militia's uniform, a gesture of defiance that earned him a six-month prison sentence.
After his release, Galois became involved in a romance with a mysterious woman by the name of Stéphanie-Félicie Poterine du Motel. There are no clues as to how the affair started, but its end is well documented.
Stéphanie was already engaged to a gentleman by the name of Pescheux d'Herbinville, who uncovered his fiancée's infidelity. D'Herbinville was furious and, being one the finest shots in France, immediately challenged Galois to a duel at dawn. Galois was well aware of his challenger's reputation. During the evening prior to the confrontation, he wrote down all his theorems, hoping that the colleagues who had rejected him would reconsider his work. Within the complex algebra were occasional references to "Stéphanie" or "une femme" and exclamations of despair - "I have not time, I have not time!"
The next morning, in an isolated field, Galois and d'Herbinville faced each other at 25 paces armed with pistols. D'Herbinville was accompanied by seconds, Galois stood alone. The pistols were raised and fired. D'Herbinville still stood, Galois was hit in the stomach. He lay helpless on the ground. The victor calmly walked away, leaving his opponent to die.
Conspiracy theorists suggested that d'Herbinville was not a cuckolded fiancé but a government agent, and that Stéphanie was not just a lover but a scheming seductress. Either way, one of the world's greatest mathematicians was killed at the age of 20, having studied mathematics for only five years.
Galois's notes, written prior to his death, describe a concept known as group theory, a powerful tool capable of cracking previously insoluble problems. Wiles realised that he could exploit Galois's groups to topple the first domino in his inductive proof.
Getting this far had required enormous determination. Wiles describes his experience of doing mathematics in terms of a journey through a 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 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 momentary, sometimes over a period of a day or two, is the culmination of, and couldn't exist without, the many months of stumbling around in the dark that precede it."
By 1990 Wiles found himself in what seemed the darkest room of all. He had been exploring it for almost two years without making any progress. "I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal.
"It could well be that the methods needed to take the next step were simply beyond present-day mathematics. Perhaps the methods I needed would not be invented for 100 years. So even if I was on the right track, I could be living in the wrong century."
To topple all the remaining dominoes, Wiles tried to adapt a method known as Iwasawa theory. Although the approach seemed to offer some hope, it could not guarantee that every single domino would topple and, after a year of failing to make the technique work, Wiles decided to look for an alternative method. Having been a virtual recluse in Princeton for five years, he decided it was time to get back into circulation to find out the latest mathematical gossip.
In 1991 he headed north to Boston to attend a conference on elliptic equations. Wiles was welcomed by colleagues from around the world, who were delighted to see him after such a long absence. They were still unaware of what he had been working on and Wiles was careful not to give away any clues.
Initially, nothing was relevant to Wiles's plight, but an encounter with the Cambridge mathematician John Coates proved fruitful: "Coates mentioned to me that a student of his named Matheus Flach was writing a beautiful paper in which he was analysing elliptic equations. He was building on a recent method due to Kolyvagin and, although I knew I would have to further develop this so-called Kolyvagin-Flach method, it looked as though it was tailor-made for me. I devoted myself night and day to extending Kolyvagin-Flach."
"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 caught my attention. It mentioned a 19th-century construction, and I suddenly realised that I should be able to use that to make the Kolyvagin-Flach method work completely. I went on into the afternoon and I forgot to go down for lunch. 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 before I went downstairs. Nada was very surprised that I'd arrived so late. Then I told her - I'd solved Fermat's Last Theorem."
After seven years of intense effort Wiles had proved the Taniyama-Shimura conjecture, and in so doing had proved Fermat's Last Theorem. On June 23, 1993 he attended a seminar at the Sir Isaac Newton Institute in Cambridge, to give a lecture describing the proof. Although the contents of the lecture were supposed to be secret, rumours were circulating: "The press had already got wind of the lecture, but fortunately they were not there. None the less, there were plenty of people in the audience who were taking photographs, and the director of the Institute certainly had come well prepared with a bottle of Champagne. There was a typical dignified silence while I read out the proof and I then just wrote up the statement of Fermat's Last Theorem. I said, 'I think I'll stop there'. Then there was sustained applause."
While mathematicians were spreading the good news via e-mail, TV crews and science reporters descended upon the Newton Institute, all demanding interviews with the "greatest mathematician of the century". The New York Times exclaimed "At Last Shout of 'Eureka!' in Age-Old Math Mystery" and the headline on the front page of Le Monde read Le théor¸m de Fermat enfin résolu.
Overnight, Wiles became the most famous, in fact the only famous, mathematician in the world, and People magazine listed him among "The 25 most intriguing people of the year", along with Princess Diana and Oprah Winfrey. An international clothing chain asked the mild-mannered Wiles to endorse its new range of menswear.
While the media circus continued and mathematicians made the most of being in the spotlight, the serious work of checking the proof was under way. Although Wiles's lecture had outlined his calculation, this did not qualify as official peer review. Academic protocol demands that any mathematician submits a complete manuscript to a respected journal, which then sends it to a team of referees who examine the proof line by line. Wiles had to spend the summer anxiously waiting for the referees' report.
Wiles returned to Princeton. The referees sent him a stream of questions to clarify certain points, but he was expecting little more than the mathematical equivalent of grammatical or typographic errors.
Then, in August, he received a query which forced him to think seriously about one of the logical links in his argument: "For a while it seemed to be as trivial as the other problems, but by September I began to realise that this wasn't just a minor difficulty but a fundamental flaw. It was an error in a crucial part of the argument involving the Kolyvagin-Flach method, but it was something so subtle that I'd missed it completely.
"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."
Only a few weeks earlier, newspapers around the globe had dubbed Wiles the most brilliant mathematician in the world. Now he was faced with the humiliation of having to admit he had made a mistake.
Before confessing to the error he decided to try to fill in the gap. "I decided to go straight back into my old mode and shut myself off from the outside world. For a long time I would think that the fix was just around the corner, but as time went by it seemed that the problem just became more intransigent."
As the months passed, news of the error began to leak out, and there was growing pressure to reveal the details so that others could try to fix it. Wiles adamantly refused. He was not prepared to give away his dream. But after six months he realised that he needed somebody who could inspire him to explore more lateral approaches. He needed somebody who was an expert in manipulating the Kolyvagin-Flach method and who could also keep the details of the problem secret.
Wiles decided to invite Richard Taylor, a Cambridge lecturer, to Princeton to work alongside him. Taylor was one of the referees responsible for verifying the proof and a former student of Wiles, and so could be doubly trusted.
During the summer of 1994 Wiles and Taylor struggled with the error, but 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. 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.
While Taylor re-examined alternative methods Wiles decided to spend September looking one last time at the structure of the Kolyvagin-Flach method to try to pinpoint exactly why it was not working. "I was sitting at my desk one Monday morning, September 19, examining the Kolyvagin-Flach method. It wasn't that I believed I could make it work, but I wanted at least to explain why it didn't.
"Suddenly, totally unexpectedly, I had this incredible revelation. It was so indescribably beautiful, it was so simple and so elegant. I realised that although the Kolyvagin-Flach wasn't working completely, it was all I needed to complete my original Iwasawa theory approach, from three years earlier. So out of the ashes of Kolyvagin-Flach seemed to rise the true answer.
"I couldn't understand how I'd missed it and I just stared at it in disbelief for 20 minutes. During the day I walked around the department, and I'd keep coming back to my desk to see if 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."
Iwasawa Theory on its own had been inadequate. The Kolyvagin-Flach method on its own was inadequate. Together they complemented each other perfectly. When Wiles recounts these moments, the memory is so powerful that he is moved to tears.
This was not only the fulfilment of his dream, but - having been pushed to the brink of submission - he had fought back to prove his genius to the world. The previous 14 months had been the most painful, humiliating and depressing 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.' "
Next month Wiles receives the 90-year-old Wolfskehl Prize, officially marking the end of the greatest mathematical challenge in history (inflation and deflation mean it is now worth about £30,000). Yet there remains a question. The proof is a masterpiece of modern mathematics, which leads to the 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 Arithmetica, and Wiles's 100 pages of dense mathematics certainly fulfils 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 hard-headed sceptics believe that Fermat's Last Theorem was the result of a rare moment of weakness by the 17th-century genius. They claim that although Fermat wrote, "I have discovered a truly marvellous proof", he had in fact found only a flawed proof.
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 17th-century techniques, and would have involved an argument so cunning that it has eluded everybody. Indeed, there are plenty of mathematicians who believe that they can still achieve fame and glory by discovering Fermat's original proof.
As far as Wiles is concerned, the battle to prove Fermat is over: "There's no other problem that will mean the same to me. I had therare privilege of being able to pursue my childhood dream. Having solved this problem there's certainly a sense of loss, but at the same time there is a tremendous sense of freedom. That particular odyssey is now over. My mind is at rest."
© Copyright Telegraph Group Limited 1997.