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The Whole Story

This is it – the entire story of Fermat’s Last Theorem in a couple of thousand words. If this is a bit daunting, click here to see Fermat’s Last Theorem in 100 Words.

The Quest to Solve the World’s Most Notorious Mathematical Problem

In 1963 a 10-year old boy borrowed a book from his local library in Cambridge, England. The boy was Andrew Wiles, a schoolchild with a passion for mathematics, and the book that had caught his eye was ‘The Last Problem’ by the mathematician Eric Temple Bell. The book recounted the history of Fermat’s Last Theorem, the most famous problem in mathematics, which had baffled the greatest minds on the planet for over three centuries.

There can be no problem in the field of physics, chemistry or biology that has so vehemently resisted attack for so many years. Indeed E.T. Bell predicted that civilisation would come to an end as a result of nuclear war before Fermat’s Last Theorem would ever be resolved. Nonetheless young Wiles was undaunted. He promised himself that he would devote the rest of his life to addressing the ancient challenge.

Eric Temple Bell's classic book, which caught the eye of young Andrew Wiles

Pierre De Fermat

The 17th century mathematician Pierre de Fermat created the Last Theorem while studying Arithmetica, an ancient Greek text written in about AD 250 by Diophantus of Alexandria. This was a manual on number theory, the purest form of mathematics, concerned with the study of whole numbers, the relationships between them, and the patterns they form.

The page of Arithmetica which inspired Fermat to create the Last Theorem discussed various aspects of Pythagoras’ Theorem, which states that:

In a right-angled triangle the square of the hypotenuse is equal to the sum of the squares on the other two sides.

In other words (or rather symbols):

x2 + y2 = z2

where z is the length of the hypotenuse, the longest side, and x and y are the lengths of the other two sides.

In particular, Arithmetica asked its readers to find solutions to Pythagoras’ equation, such that x, y, and z could be any whole number, except zero. For example, 32 + 42 = 52 (i.e. 9 + 16 = 25) or 52 + 122 = 132 (i.e. 25 + 144 = 169). Fermat must have been bored with such a tried and tested equation, and as a result he considered a slightly mutated version of the equation:

x3 + y3 = z3

The equation is now said to be to the power 3, rather than the power 2. Surprisingly, the Frenchman came to the conclusion that among the infinity of numbers there were none that fitted this new equation. Whereas Pythagoras’ equation had many possible solutions, Fermat claimed that his equation was insoluble.

Fermat went even further, believing that if the power of the equation were increased further, then these equations would also have no solutions:

x3 + y3 = z3,

x4 + y4 = z4,

x5 + y5 = z5,

x6 + y6 = z6,

:

:

The mathematical short-hand for this family of insoluble equations is:

xn + yn = zn, where n is any number greater than 2.

According to Fermat, none of these equations could be solved and he noted this in the margin of his Arithmetica. To back up his theorem he had developed an argument or mathematical proof, and following the first marginal note he scribbled the most tantalising comment in the history of mathematics:

I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.

Fermat believed he could prove his theorem, but he never committed his proof to paper. After his death, mathematicians across Europe tried to rediscover the proof of what became known as Fermat’s Last Theorem. It was as though Fermat had buried an incredible treasure, but he had not written down the map. Mathematicians could not resist the lure of such an intellectual treasure and competed to find it first.

Three Centuries of Failure

Throughout the eighteenth and nineteenth centuries no mathematician could find a counter-example, a set of numbers that fitted Fermat’s equation. Hence, it seemed that the Last Theorem was true, but without a proof nobody could be as sure as Fermat seemed to be. Some of the greatest mathematicians were able to devise specific proofs for individual equations (e.g. n = 3 and n = 5), but nobody was able to match Fermat’s general proof for all equations.

The longer that the Last Theorem remained unproven, the more important it became, and the more effort was put into finding a proof. It is worth noting that finding proof was unlikely to yield any useful application, but the simple joy of solving an innocent riddle was enough to spur on generations of number theorists. Although all their attempts ended in failure, a great deal of new mathematics was inspired along the way, and it can be argued that the progress of number theory has been largely inspired by the desire to prove Fermat’s Last Theorem.

The history of Fermat’s Last Theorem is a tale of intrigue, rivalry, rich prizes, suicide and death, involving characters who became obsessed by Fermat’s accidental challenge. One of the most intriguing stories concerns the most famous prize offered for a proof of the Last Theorem. It is said that toward the end of the nineteenth century Paul Wolfskehl, a German industrialist and amateur mathematician, was on the point of suicide. Some historians claim his depression was the result of a failed romance, others believe it was due to the onset of multiple sclerosis. He appointed a date for his suicide and intended 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 research on the Last Theorem.

Suddenly, he believed he could see a way of proving the theorem, and he became engrossed in exploring his newfound strategy. After hours of algebra Wolfskehl realised that his method had reached a dead-end, but the good news was that the appointed time of his suicide had passed. Despite his failure, Wolfskehl had been reminded of the beauty and elegance of number theory, and consequently he abandoned his plan to kill himself. Mathematics had renewed his desire for life. As a way of repaying a debt to the problem which saved his life, he rewrote his will and bequeathed 100,000 Marks (worth $2 million in today’s money) to whoever succeeded in proving Fermat’s Last Theorem.

Soon after his death in 1906, the Wolfskehl Prize was announced, generating an enormous amount of publicity and introducing the problem to the general public. Within the first year 621 proofs were sent in, most of them from amateur problem-solvers, all of them flawed.

The Infinite Nightmare

One of the reasons why Fermat’s Last Theorem is so difficult to prove is that it applies to an infinite number of equations: xn + yn = zn, where n is any number greater than 2. Even the advent of computers was of no help, because, although they could be employed to help perform sophisticated calculations, they could at best deal with only a finite number of equations.

Soon after the Second World War computers helped to prove the theorem for all values of n up to five hundred, then one thousand, and then ten thousand. In the 1980′s Samuel S. Wagstaff of the University of Illinois raised the limit to 25,000 and more recently mathematicians could claim that Fermat’s Last Theorem was true for all values of n up to four million. In other words, for the first four million equations mathematicians had proved that there were no numbers that fitted any of them.

This may seem to be a significant contribution toward finding a complete proof, but the standards of mathematical proofs demand absolute confidence that no numbers fit the equations for all values of n. Even though the theorem had been proven for all values n up to four million, there is no reason why it should be true for n = 4,000,001. And if in the future supercomputers proved the theorem for all values n up to one zillion, there is no reason why it should be true for n = one zillion and one. And so on ad infinitum. Infinity is unobtainable by the mere brute force of computerised number crunching.

The mathematician’s desire for an absolute proof up to infinity may seem unreasonable, but the case of Euler’s conjecture demonstrates the necessity of unequivocal truth. The 17th century Swiss mathematician Leonhard Euler claimed that there are no whole number solutions to an equation not dissimilar to Fermat’s equation:

Euler’s equation: x4 + y4 + z4 = w4

For two hundred years nobody could prove Euler’s conjecture, but on the other hand nobody could disprove it by finding a counter-example. First manual searches and then years of computer sifting failed to find a solution. Lack of a counter-example appeared to be strong evidence in favour of the conjecture. Then in 1988 Noam Elkies of Harvard University discovered the following solution:

2,682,4404 + 15,365,6394 + 18,796,7604 = 20,615,6734

Despite all the previous evidence, Euler’s conjecture turned out to be false. In fact Elkies proved that there are infinitely many solutions to the equation. The moral of the story is that you cannot use evidence from the first million numbers to prove absolutely a conjecture about all numbers.

After three centuries of failure mathematicians were beginning to lose hope that a proof for Fermat’s Last Theorem would ever be found. When the logician David Hilbert, one of the greatest mathematicians of the 20th century, 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.”

The problem still held a special place in the hearts of number theorists, but now they viewed it in the same way that chemists thought about alchemy. Both were foolish, impossible dreams from a bygone age.

The Shimura-Taniyama Conjecture

Between 1954 and 1986 a chain of events of occurred which brought Fermat’s Last Theorem back into the mainstream. The incident which began everything happened in post-war Japan, when Yutaka Taniyama and Goro Shimura, two young academics, decided to collaborate on the study of elliptic curves and modular forms. These entities are from opposite ends of the mathematical spectrum, and had previously been studied in isolations.

Elliptic curves, which have been studied since the time of Diophantus, concern cubic equations of the form:

y2 = (x + a).(x + b).(x + c), where a, b & c can be any whole number, except zero.

The challenge is to identify and quantify the whole solutions to the equations, the solutions differing according to the values of a, b, and c.

Modular forms are a much more modern mathematical entity, born in the nineteenth century. They are functions, not so different to functions such as sine and cosine, but modular forms are exceptional because they exhibit a high degree of symmetry. For example, the sine function is slightly symmetrical because 2p can be added to any number, x, and yet the result of the function remains unchanged, i.e., sine (x) = sine (x + 2p). However, for modular forms the number x can be transformed in an infinite number of ways and yet the outcome of the function remains unchanged, hence they are said to be extraordinarily symmetric. I will not describe the transformations in any further detail because they involve relatively complicated mathematics and the numbers in question (x) are so-called complex numbers, composed of real and imaginary parts.

Despite belonging to a completely different area of the mathematics, Shimura and Taniyama began to suspect that the elliptic curves might be related to modular forms in a fundamental way. It seemed that the solutions for any one of the infinite number of elliptic curves could be derived from one of the infinite number of modular forms. Each elliptic curve seemed to be a modular form in disguise.

This apparent unification became known as the Shimura-Taniyama conjecture, reflecting the fact that mathematicians were confident that it was true, but as yet were unable to prove it. The conjecture was considered important because if it were true problems about elliptic curves, which hitherto had been insoluble, could potentially be solved by using techniques developed for modular forms, and vice versa.

Relationships between apparently different subjects are as creatively important in mathematics as they are in any discipline. The relationship hints at some underlying truth that enriches both subjects. For example, in the nineteenth century theorists and experimentalists realised that electricity and magnetism, which had previously been studied in isolation, were intimately related. This resulted in a deeper understanding of both phenomena. Electric currents generate magnetic fields, and magnets can induce electricity in wires passing close to them. This led to the invention of dynamos and electric motors, and ultimately the discovery that light itself is the result of magnetic and electric fields oscillating in harmony.

Even though the Shimura-Taniyama conjecture could not be proved, as the decades passed it gradually became increasingly influential, and by the 1970s mathematicians would begin papers by assuming the Shimura-Taniyama conjecture and then derive some new result. In due course many major results came to rely on the conjecture being proved, but these results could themselves only be classified as conjectures, because they were conditional on the proof of the Shimura-Taniyama conjecture. Despite its pivotal role, few believed it would be proved this century.

Then, in 1986, Kenneth A Ribet of the University of California at Berkeley, building on the work of Gerhard Frey of the University of Saarlands, made an astonishing breakthrough. He was unable to prove the Shimura-Taniyama conjecture, but he was able to link it with Fermat’s Last Theorem.

The link occurred by contemplating the unthinkable – what would happen if Fermat’s Last Theorem was not true? This would mean that there existed a set of solutions to Fermat’s equation, and therefore this hypothetical combination of numbers could be used as the basis for constructing a hypothetical elliptic curve. Ribet demonstrated that this elliptic curve could not possibly be related to a modular form, and as such it would defy the Shimura-Taniyama conjecture.

Running the argument backwards, if somebody could prove the Shimura-Taniyama conjecture then every elliptic curve must be related to a modular form, hence any solution to Fermat’s equation is forbidden to exist, and hence Fermat’s Theorem must be true. If somebody could prove the Shimura-Taniyama conjecture, then this would immediately imply the proof of Fermat’s Last Theorem. By proving one of the most important conjectures of the twentieth century, mathematicians could solve a riddle from the seventeenth century.

Childhood Dream, Adult Obsession

The Shimura-Taniyama conjecture had remained unproven since the 1950s and so there was little optimism that it was a realistic route to a proof of Fermat’s Last Theorem. Some mathematicians joked that, if anything, the Shimura-Taniyama conjecture was even further out of reach, because, by definition, anything that led to a proof of the Last Theorem must be impossible.

But for Wiles, anything that would lead to the Last Theorem was worth pursuing. He knew that this might be his only chance to realise his childhood dream and he had the audacity to attack the Shimura-Taniyama conjecture. As a graduate student at Cambridge University, he had concentrated on studying elliptic curves, and then as a professor at Princeton University he had continued his research, putting him in an ideal position for attempting a proof.

As he embarked on his proof, Wiles made the extraordinary decision to conduct his research in complete secrecy. He did not want the pressure of public attention, nor did he want to risk others copying his ideas and stealing the prize. In order not to arouse suspicion Wiles devised a cunning ploy that 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 curve, 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 Wiles 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. For the next seven years he worked in isolation, and his colleagues were oblivious to what he was doing. The only person who knew of his secret project was his wife – he told her during their honeymoon.

The number of elliptic curves and modular forms is infinite, and the Shimura-Taniyama conjecture claimed each elliptic curve could be matched with a modular. However, to succeed Wiles did not have to prove the full Shimura-Taniyama conjecture. Instead he only had to show that a particular subset of elliptic curves (one which would include the hypothetical Fermat elliptic curve) is modular. However, this subset is still infinite in size and it includes the majority of interesting curves.

To prove that something is true for an infinite number of cases required Wiles to pull together some of the most recent breakthroughs in number theory, and in addition invent new techniques of his own. He adopted a strategy loosely based on a method known as induction. Proof by induction can prove something for an infinite number of cases by invoking a domino toppling approach, i.e., to knock down an infinite number of dominoes, one merely has to ensure that knocking down any domino will always topple the next domino. In other words, Wiles had to develop an argument in which he could prove the first case, and then be sure that proving any one case would implicitly prove the next one.

At each stage Wiles could never be sure that he could complete his proof. He realised that even if he did have the correct strategy, the mathematical techniques required might not yet exist – he might be on the right track, but living in the wrong century. Eventually, in 1993, Wiles felt confident that his proof was reaching completion. The opportunity arose to announce his proof of a major section of the Shimura-Taniyama conjecture, and hence Fermat’s Last Theorem, at a special conference to be held at the Isaac Newton Institute in Cambridge, England. Because this was his home town, where he had encountered the Last Theorem as a child, he decided to make a concerted effort to complete the proof in time for the conference. On June 23rd he announced his seven-year calculation to a stunned audience.

His secret research programme had apparently been a success, and the mathematical community and the world’s press rejoiced. The front page of the New York Times exclaimed “At Last, Shout of ‘Eureka!’ in Age-Old Math Mystery”, and Wiles appeared on television stations around the world. People magazine even listed him among “The 25 Most Intriguing People of the Year’, alongside such luminaries as Oprah Winfrey, but the ultimate accolade came from an international clothing chain who asked the mild-mannered genius to endorse their new range of menswear.

The most unlikely consequence of Wiles’ success was consternation among fans of the TV series, Star Trek. In one particular episode (The Royale, 1989) Jean-Luc Picard, the captain of the Starship Enterprise, was seen trying to find a proof of Fermat’s Last Theorem. He was apparently unaware that it had been already been proved in the twentieth century by Andrew Wiles.

While the media circus continued, the official peer review process began. Over the summer the 200-page proof was examined line by line by a team of referees. The manuscript was split into seven chapters, and each chapter was sent to a pair of expert examiners. Wiles checked and double-checked the proof before releasing it to the referees, so he was expecting little more than the mathematical equivalent of grammatical and typographic errors, trivial mistakes that he could fix immediately. However, gradually it emerged that there was a fundamental and devastating flaw in one stage of the argument.

Essentially, the inductive argument used by Wiles could not guarantee that if one domino toppled, then so would the next. Over the course of the next year his childhood dream turned into a nightmare. Each attempt to fix the error ended in failure, each attempt to by-pass the error ended in a dead-end. And throughout this period the manuscript had only been seen by the small team of referees and Wiles himself. There were calls from the mathematics community to publish the flawed proof, which would allow others to try and fix it, but Wiles steadfastly refused. He believed that he deserved the first chance to correct a piece of work that had already taken him seven years.

After months of failure Wiles did take into his confidence Richard Taylor, a former student of his, hoping that this would give him someone to bounce ideas off, someone who could inspire him to consider alternative strategies. By September 1994 they were at the point of admitting defeat, ready to release the flawed proof so that others could try and fix it. Then on September 19th they made the vital breakthrough. Many years earlier, when he was working in secrecy, Wiles had considered using an alternative approach, but it floundered and so he had abandoned it. Now they realised that what was causing the more recent method to fail was exactly what would make the abandoned approach succeed.

Wiles recalls his reaction to the discovery: “It was so indescribably beautiful, it was so simple and so elegant. The first night I went back home and slept on it. I checked through it again the next morning and, and I went down and told me 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.’”

Fermat’s Lost Proof?

The rules of the Wolfskehl Prize demanded two years of scrutiny following publication of the proof, so it was not until June 27th 1997 that Andrew could collect his reward. When it was originally established, the Wolfskehl prize was worth $2 million dollars, but hyperinflation followed by the devaluation of the Reichsmark had reduced its value to $50,000. For Wiles, the sum of money was unimportant. His proof is the realization of a childhood dream and the culmination of a decade of concentrated effort.

Wiles’ proof of Fermat’s Last Theorem relies on verifying a conjecture born in the 1950s, which in turn shows that there is a fundamental relationship between elliptic curves and modular forms. 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’ proof of the Last Theorem can not possibly be the same as Fermat’s.

If Fermat did not have Wiles’ proof, then what did he have? The hard-headed sceptics 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 only found 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 else. 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. This was my childhood passion. There’s nothing to replace that. 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.”