France, law, and everything math. These were the ingredients chosen to create the perfect ideal life. But Pierre de Fermat has decided to add an extra ingredient to his lifetime, A MARGINAL NOTE. Thus the last theorem was born! Using every single brain cell mathematicians, mathematicians and mathematicians have dedicated their lives to finding proof and solving a 300 year old problem.

Our story starts in the nineteenth century Paul Wolfskehl, 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 realized 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.

So now, who is Fermat?

Pierre de Fermat (17 August 1601 – 12 January 1665) studied civil law at the University of Orleans and progressed in his judicial career path until reaching a comfortable position in the Parliament of Toulouse, which allowed him to spend his spare time on his great love: mathematics.

In the afternoons, Fermat would put the law to one side and dedicate himself to deepening his mathematical investigations. He studied the treatises of the scholars of classical Greece and combined those old ideas with the new methods of algebra.

In this way, Fermat found mathematical problems with which he challenged by letter other intellectuals such as Descartes and Pascal.

The result of his quarrels through the post with the philosopher René Descartes inspired Newton and Leibniz to develop infinitesimal calculus.

Later, in 1654, a writer and professional gambler asked the mathematician Blaise Pascal for help in fairly distributing the money wagered on an interrupted game of dice, based on the scores obtained until then.

Pascal challenged Fermat to solve the problem and together they succeeded, thereby laying the foundation of probability theory.

Over the years that was Fermat’s style; he would come up with theorems and send them to other mathematicians without providing proof, so basically he’s challenging them to find the proof.

He would write stuff like “I can provide this, but I have to feed the cat”

Sometime in his career his interest switched to number theory: field of math concerned with the study of whole numbers, the relationships between them, and the patterns they form.

Not a lot of other mathematicians shared that interest, but he kept on going. One day he was going through the book Arithmetica of Diophantus the Greek mathematician of the 3rd century AD

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

In particular, Arithmetica asked its readers to find solutions to Pythagoras’ equation, such that c, b, and a could be any whole number, except zero.

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:

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 insolvable.

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

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 tantalizing comment in the history of mathematics:

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.

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.

Soon after Wolfskehl’s 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.

One of the reasons why Fermat’s Last Theorem is so difficult to prove is that it applies to an infinite number of equations: x^n+ y^n = z^n, 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.

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.

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.

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. 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.

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 realized 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 program 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’, but the ultimate accolade came from an international clothing chain who asked the mild-mannered genius to endorse their new range of menswear.

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 realized 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:

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 cannot possibly be the same as Fermat’s.

If Fermat did not have Wiles’ proof, then what did he have? The hard-headed skeptics 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 marvelous 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: 

References:

Pierre de Fermat

Math history

Fermat and the greatest problem in the history of mathematics

Fermat’s last theorem, the whole story

Smithsonian