Fermat's Last Theorem

by Simon Singh

  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.’

  Duelling with Infinity

  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.’

  In order to find a solution, Wiles adopted his usual approach to solving difficult problems. ‘I sometimes write scribbles or doodles. They’re not important doodles, just subconscious doodles. I never use a computer.’ In this case, as with many problems in number theory, computers would be of no use whatsoever. The Taniyama–Shimura conjecture applied to an infinite number of equations and, although a computer could check an individual case in a few seconds, it could never check all cases. Instead what was required was a logical step-by-step argument which would effectively give a reason and explain why every elliptic equation had to be modular. To find the proof Wiles relied solely on a piece of paper, a pencil and his mind. ‘I carried this thought around in my head basically 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. Without distraction I would have the same thing going round and round in my mind.’

  After a year of contemplation Wiles decided to adopt a general strategy known as induction as the basis for his proof. 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 by only proving it for just one case. For example, imagine that a mathematician wants to prove that a statement is true for every counting number up to infinity. The first step is to prove that the statement is true for the number 1, which presumably is a fairly straightforward task. The next step is to show that if the statment is true for the number 1 then it must be true for the number 2, and if it is true for the number 2 then it must be true for the number 3, and if it is true for the number 3 then it must be true for the number 4, and so on. More generally, the mathematician has to show that if the statement is true for any number n, then it must be true for the next number n + 1.

  Proof by induction is essentially a two step process:

  (1) Prove that the statement is true for the first case.

  (2) Prove that if the statement is true for any one case, then it must be true for the next case.

  Another way to think of proof by induction is to imagine the infinite number of cases as an infinite line of dominoes. In order to prove every case it is necessary to find a way of knocking down every one of the dominoes. Knocking them 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 just knocking down the first one. If the dominoes are carefully arranged, then knocking down the first domino will knock down the second domino, which will in turn knock down the third domino, and so on to infinity. Proof by induction invokes the domino effect. This form of mathematical domino-toppling allows an infinite number of cases to be proved by just proving the first one. Appendix 10 shows how proof by induction can be used to prove a relatively simple mathematical statement about all numbers.

  The challenge for Wiles was to construct an inductive argument which showed that each of the infinity of elliptic equations could be matched to each of the infinity of modular forms. Somehow he had to break the proof down into an infinite number of individual cases and then prove the first case. Next, he had to demonstrate that, having proved the first case, all the others would topple. Eventually he discovered the first step to his inductive proof hidden in the work of a tragic genius from nineteenth-century France.

  Evariste Galois was born in Bourg-la-Reine, a small village just south of Paris, on 25 October 1811, just twenty-two years after the French Revolution. Napoleon Bonaparte was at the height of his powers, but the following year saw the disastrous Russian campaign, and in 1814 he was driven into exile and replaced by King Louis XVIII. In 1815 Napoleon escaped from Elba, entered Paris and reclaimed power but within a hundred days he was defeated at Waterloo and forced to abdicate once again in favour of Louis XVIII. Galois, like Sophie Germain, grew up during a period of immense upheaval, but whereas Germain shut herself away from the turmoils of the French Revolution and concentrated on mathematics, Galois repeatedly found himself at the centre of political controversy, which not only distracted him from a brilliant academic career, but also led to his untimely death.

  In addition to the general unrest which impinged on everybody’s life, Galois’s interest in politics was inspired by his father, Nicolas-Gabriel Galois. When Evariste was just four years old his father was elected mayor of Bourg-la-Reine. This was d
uring Napoleon’s triumphant return to power, a period when his father’s strong liberal values were in keeping with the mood of the nation. Nicolas-Gabriel Galois was a cultured and gracious man and during his early years as mayor he gained respect throughout the community, so even when Louis XVIII returned to the throne he retained his elected position. Outside of politics, his main interest seems to have been the composition of witty rhymes, which he would read at town meetings to the delight of his constituents. Many years later this charming talent for epigrams would lead to his downfall.

  At the age of twelve Evariste Galois attended his first school, the Lycée of Louis-le-Grand, a prestigious but authoritarian institution. To begin with he did not encounter any courses in mathematics and his academic record was respectable but not outstanding. However, one event occurred during his first term which would influence the course of his life. The Lycée had previously been a Jesuit school and rumours began to circulate suggesting that it was about to be returned to the authority of the priests. During this period there was a continual struggle between republicans and monarchists to sway the balance of power between Louis XVIII and the people’s representatives, and the increasing influence of the priests was seen as an indication of a shift away from the people and towards the King. The students of the Lycée, who in the main had republican sympathies, planned a rebellion but the director of the school, Monsieur Berthod, uncovered the plot and immediately expelled the dozen or so ringleaders. The following day when Berthod demanded a demonstration of allegiance from the remaining senior scholars, they refused to drink a toast to Louis XVIII, whereupon another hundred students were expelled. Galois was too young to be involved in the failed rebellion and so remained at the Lycée. Nevertheless, watching his fellow students being humiliated in this way only served to inflame his republican tendencies.

  It was not until the age of sixteen that Galois enrolled in his first mathematics class, a course which would, in the eyes of his teachers, transform him from a conscientious pupil into an unruly student. His school reports show that he neglected all his other subjects and concentrated solely on his new found passion:

  This student works only in the highest realms of mathematics. The mathematical madness dominates this boy. I think it would be best for him if his parents would allow him to study nothing but this. Otherwise he is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments.

  Galois’s desire for mathematics soon outstripped the capacity of his teacher, and so he learnt directly from the very latest books written by the masters of the age. He readily absorbed the most complex of concepts, and by the time he was seventeen he published his first paper in the Annales de Gergonne. The path ahead seemed clear for the prodigy, except that his own sheer brilliance was to provide the greatest obstacle to his progress. Although he obviously knew more than enough mathematics to pass the Lycée’s examinations, Galois’s solutions were often so innovative and sophisticated that his examiners failed to appreciate them. To make matters worse Galois would perform so many calculations in his head that he would not bother to outline clearly his argument on paper, leaving the inadequate examiners even more perplexed and frustrated.

  The young genius did not help the situation by having a quick temper and a rashness which did not endear him to his tutors or anybody else who crossed his path. When Galois applied to the Ecole Polytechnique, the most prestigious college in the land, his abruptness and lack of explanation in the oral examination meant that he was refused admission. Galois was desperate to attend the Polytechnique, not just because of its academic excellence but also because of its reputation for being a centre for republican activism. One year later he reapplied and once again his logical leaps in the oral examination only served to confuse his examiner, Monsieur Dinet. Sensing that he was about to be failed for a second time and frustrated that his brilliance was not being recognised, Galois lost his temper and threw a blackboard rubber at Dinet, scoring a direct hit. Galois was never to return to the hallowed halls of the Polytechnique.

  Undaunted by the rejections, Galois remained confident of his mathematical talent and continued his own private researches. His main interest concerned finding solutions to equations, such as quadratic equations. Quadratic equations have the form

  The challenge is to find the values of x for which the quadratic equation holds true. Rather than relying on trial and error mathematicians would prefer to have a recipe for finding solutions, and fortunately such a recipe exists:

  Simply by substituting the values for a, b and c into the above recipe one can calculate the correct values for x. For instance, we can apply the recipe to solve the following equation:

  By putting the values of a, b and c into the recipe, the solution turns out to be x = 1 or x = 2.

  The quadratic is a type of equation within a much larger class of equations known as polynomials. A more complicated type of polynomial is the cubic equation:

  The extra complication comes from the additional term x3. By adding one more term x4, we get the next level of polynomial equation, known as the quartic:

  By the nineteenth century, mathematicians also had recipes which could be used to find solutions to the cubic and the quartic equations, but there was no known method for finding solutions to the quintic equation:

  Galois became obsessed with finding a recipe for solving quintic equations, one of the great challenges of the era, and by the age of seventeen he had made sufficient progress to submit two research papers to the Academy of Sciences. The referee appointed to judge the papers was Augustin-Louis Cauchy, who many years later would argue with Lamé over an ultimately flawed proof of Fermat’s Last Theorem. Cauchy was highly impressed by the young man’s work and judged it worthy of being entered for the Academy’s Grand Prize in Mathematics. In order to qualify for the competition the two papers would have to be re-submitted in the form of a single memoir, so Cauchy returned them to Galois and awaited his entry.

  Having survived the criticisms of his teachers and rejection by the Ecole Polytechnique Galois’s genius was on the verge of being recognised, but over the course of the next three years a series of personal and professional tragedies would destroy his ambitions. In July of 1829 a new Jesuit priest arrived in the village of Bourgla-Reine, where Galois’s father was still mayor. The priest took exception to the mayor’s republican sympathies and began a campaign to oust him from office by spreading rumours aimed at discrediting him. In particular the scheming priest exploited Nicolas-Gabriel Galois’s reputation for composing clever rhymes. He wrote a series of vulgar verses ridiculing members of the community and signed them with the mayor’s name. The elder Galois could not survive the shame and the embarrassment which resulted and decided that the only honourable option was to commit suicide.

  Evariste Galois returned to attend his father’s funeral and saw for himself the divisions that the priest had created in the village. As the coffin was being lowered into the grave, a scuffle broke out between the Jesuit priest, who was conducting the service, and supporters of the mayor, who realised that there had been a plot to undermine him. The priest suffered a gash to the head, the scuffle turned into a riot, and the coffin was left to drop unceremoniously into its grave. Watching the French establishment humiliate and destroy his father served only to consolidate Galois’s fervent support for the republican cause.

  Upon returning to Paris, Galois combined his research papers well ahead of the competition deadline and submitted the memoir to the secrerary of the Academy, Joseph Fourier, who was supposed to pass it on to the judging committee. Galois’s paper did not offer a solution to the quintic problem but it did offer a brilliant insight and many mathematicians, including Cauchy, considered that it was a likely winner. To the shock of Galois and his friends, not only did he fail to win the prize, but he had not even been officially entered. Fourier had died a few weeks prior to the judging and, although a stack of competition entries was passed on to the committee,
Galois’s memoir was not among them. The memoir was never found and the injustice was recorded by a French journalist.

  Last year before March 1st, Monsieur Galois gave to the secretary of the Institute a memoir on the solution of numerical equations. This memoir should have been entered in the competition for the Grand Prize in Mathematics. It deserved the prize, for it could resolve some difficulties that Lagrange had failed to do. Monsieur Cauchy had conferred the highest praise on the author about this subject. And what happened? The memoir is lost and the prize is given without the participation of the young savant.

  Le Globe, 1831

  Galois felt that his memoir had been deliberately lost by a politically biased Academy, a belief that was reinforced a year later when the Academy rejected his next manuscript, claiming that ‘his argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigour’. He decided that there was a conspiracy to exclude him from the mathematical community, and as a result he neglected his research in favour of fighting for the republican cause. By this time he was a student at the Ecole Normale Supérieure, a slightly less prestigious college than the Ecole Polytechnique. At the Ecole Normale Galois’s notoriety as a trouble-maker was overtaking his reputation as a mathematician. This culminated during the July revolution of 1830 when Charles X fled France and the political factions fought for control in the streets of Paris. The Ecole’s director Monsieur Guigniault, a monarchist, was aware that the majority of his students were radical republicans and so confined them to their dormitories and locked the gates of the college. Galois was being prevented from fighting alongside his brothers, and his frustration and anger were compounded when the republicans were eventually defeated. When the opportunity arose he published a scathing attack on the college director, accusing him of cowardice. Not surprisingly, Guigniault expelled the insubordinate student and Galois’s formal mathematical career was at an end.