Fermat’s Last Theorem

In 1963, Andrew Wiles, a 10-year old boy with a passion for mathematics, borrowed a book from his local library in Cambridge, England. The book that caught his eye was ‘The Last Problem’, by Eric Temple Bell. The book recounted the history of Fermat’s Last Theorem, a problem which had remained unsolved for over three centuries. In fact, E.T. Bell predicted that civilization would come to an end as a result of a thermonuclear war before the Last Theorem would be resolved.

Despite this, the young Wiles remained undeterred. He promised himself that he would devote the rest of his life to solving this ancient challenge…

 

Pierre de Fermat is perhaps the most famous number theorist to have ever lived. What most people don’t know about Fermat is that he was, in fact, a lawyer and only an amateur mathematician (hence his nickname ‘Prince of Amateurs’). Even more surprising is the fact that he published only one mathematical paper in his life, which was an anonymous article written as an appendix to a colleague’s book.

Fermat created the Last Theorem while studying Arithmetica, an ancient Greek text dating back to 250 AD. Arithmetica was a manual on number theory, the form of mathematics concerned with the study of whole numbers (0, 1, 2, 3…), the relationships between them and the patterns they form.

Fermat was inspired to create the Last Theorem while reading about the Pythagoras Theorem, which states that:

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

x2 + y2 = z,

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

Arithmetica encouraged its readers to find solutions to Pythagoras’ equation, such that x, y and z could be any whole number, except 0. Fermat, instead, considered a slightly modified version of the equation:

x3 + y3 = z3

Surprisingly, Fermat came to the conclusion that there were no numbers that fit this equation. While Pythagoras’ equation had an infinite number of solutions, the Frenchman claimed that his equation unsolvable. Fermat even went further, believing that if the power of the equation were increased further, then these equations would still have no solutions. This led to Fermat stating that the equation xn + yn = zhas no real solutions when n is any number greater than 2.

Fermat claimed that he had a proof to this equation, but the space in the margins of Arithmetica was insufficient to fully demonstrate it. He believed he could prove his problem, but never committed his proof to paper. After his death in 1665, mathematicians across Europe tried to rediscover the proof of ‘The Last Theorem’.

 

Throughout the eighteenth and nineteenth century, no mathematician could disprove Fermat’s theorem by finding a set of numbers that fit his equation. This led people to believe that the theorem is true, but without proof, nobody could prove that it is absolutely true. Mathematicians were able to devise proofs for specific equations (such as n = 4 or n = 5), but nobody was able to find a proof for all the equations.

The longer the theorem remained unsolved, the more important it became, and the more effort was put into finding a proof. In 1742, Leonard Euler, the greatest number theorist of the eighteenth century, became so frustrated with the problem that he asked his friend to search Fermat’s house in case some vital scrap of paper still remained. Unfortunately, no clues were found.

In the nineteenth century, the most significant progress was made by Sophie Germain, a woman who took on the identity of a man in order to conduct her research. Although her ideas eventually reached a dead end, they generated new techniques that were invaluable in solving other problems.

Towards the end of the century, Paul Wolfskehl, a German Industrialist and amateur mathematician, offered a price of DM 100,000 for whoever could prove the Last Theorem. When it was announced in 1908, the great value of the Wolfskehl Price (about $ 1 million in today’s currency) generated an enormous amount of publicity. Within the first year, amateur problem-solvers sent in 621 proofs, all of them flawed. The interest in serious mathematicians, however, was on the decline. After more than 200 years of failure, many felt that there were more important questions to be answered.

 

During the 1950s, a major breakthrough occurred in Japan that rekindled interest in the Last Theorem. Goro Shimura and Yutaka Taniyama began to examine ‘modular forms’, a group of objects that are special because of their high level of symmetry and complexity. They realized that modular forms were encoded in solutions to a particular set of equations, known as elliptical equations.

Although this was a brilliant hypothesis, nobody could prove that it was true for every modular form and elliptical equation. If the Taniyama-Shimura conjecture, as it was called, could be proven, it could be used to prove many other conjectures.

For 30 years, there was no progress, until Ken Ribet, a professor at the University of California, Berkeley, found a connection between the conjecture and Fermat’s theorem. He couldn’t prove the conjecture himself, but showed that if anyone did, then the Last Theorem would follow. Ribet had effectively linked the most important mathematical problem of the 17th century to the most significant problem of the 20th.

Initially, there was renewed hope. Fermat’s problem could be solved. But then, the reality of the situation dawned – mathematicians had been trying to prove the conjecture for the last 30 years. Why would they make progress now?

 

 

Throughout his teenage years, Wiles studied how Euler, Germain, and others had tackled the problem. When he was in college, he began to see if the ideas his lecturers taught him would offer a new insight. In 1975, he became a researcher at Cambridge University and was forced to work on more respectable and contemporary problems. As a result, he had to stop trying to find a proof for the theorem for the next decade.

By the time Ribet proved the link between the Taniyama-Shimura conjecture and the Last Theorem, Wiles was a professor at Princeton University. He then abandoned any work that didn’t have to do with this problem and stopped attending the circuit of conferences and reduced his lecturing and tutoring to a bare minimum.

In his attic study, Wiles began to attempt to prove the theorem. He had made the unconventional decision to work in secrecy. In the modern day, most mathematicians worked together to solve problems, and Wiles’ approach was a throwback to the days of Pythagoras. Most of the mathematical community was unaware of Wiles’ research, and the only person aware of his secret was his wife, Nada.

 

First, Wiles had to prove the Taniyama-Shimura conjecture – every single modular form had to be linked to an elliptical equation. For this type of problem, computers were useless. Although computers could check an individual case in a few seconds, they wouldn’t be able to check an infinite number of cases. Instead, Wiles need to devise a step-by-step argument.

After a year, Wiles decided to adopt a general strategy based on proof by induction. Induction allows mathematicians to prove that a statement is true using just two steps:

  1. Prove the statement is true for the first number in the sequence
  2. Prove that if the statement is true for one number, it must be true for the next one as well

This is similar to knocking down dominoes. Once you knock down the first domino, the others follow. Two years after embarking on the proof, Wiles discovered the key to knocking down the first domino.

 

Evariste Galois was a mathematical prodigy and published his first paper at the age of 17. Prior to his tragic death at the age of 20, he described a concept known as group theory, a tool that could crack previously unsolvable problems. Wiles realized that he could use group theory to topple the first domino.

To topple all the remaining dominoes, Wiles to tried to adopt a method known as Iwasawa theory. However, this approach could not guarantee that every single domino would topple, and after a year of trying to make the technique work, Wiles decide to look for another method. He decided that it was time to go back into circulation and find out more about the latest mathematical inventions.

In 1991, Wiles a conference in Boston about elliptical equations. Initially, nothing was useful, until Wiles met with his old mentor from Cambridge, John Coates. Coates mentioned that one of his students, Matheus Flach, was writing a paper on elliptical equations that built on a recent method developed by Kolyvagin. Wiles then devoted himself day and night to extending the Kolyvagin-Flach model.

 

After seven years of intense effort, Wiles proved the Taniyama-Shimura conjecture, effectively proving Fermat’s Last Theorem. On June 23rd, 1993, he attended a seminar at the Sir Isaac Newton Institute at Cambridge to give a lecture describing the proof. Overnight, Wiles became the most famous mathematician in the world and was listed among “The 25 most intriguing people of the year” by People magazine.

While the media circus continued, the serious work of checking the proof was already underway. Although Wiles’ lecture had outlined his calculation, this did not qualify as an official peer review. Wiles had to spend his summer anxiously waiting for the report on his proof.

Occasionally, the referees asked Wiles a few questions about certain parts of the proof that they didn’t understand. In August, Wiles received a query which seemed to be as trivial as the other queries he received. However, upon closer inspection, he realized that it was an error in a crucial part of the argument involving the Kolyvagin-Flach model. There was no guarantee that the model would work in the way Wiles wanted it to work.

 

Before confessing to the error, Wiles decided to try and fill in the gap. He shut himself off from the outside world again. However, as time passed, the problem seemed to be getting more and more difficult to solve.

As the months rolled by, news of the error began to leak out and there was a growing pressure to reveal the details so others could try to fix it. Wiles steadfastly refused. He was unwilling to give up on his dream and let someone else take the credit for his life’s work.

However, by the summer of 1994, Wiles was ready to give up. After eight years of struggle, he was prepared to admit defeat. But he was convinced to continue working on the problem for just another month.

One day, Wiles had an incredible revelation. He realized that he could fix the error by combining the Kolyvagin-Flach model with the Iwasawa theory, the model he used originally. On its own, the Iwasawa theory had been inadequate, as had the Kolyvagin-Flach model. Together, however, they complemented each other perfectly.

 

Wiles proof was finally released to the public, and the next month, he received the 90-year-old Wolfskehl prize, officially marking the end of the greatest mathematical challenge ever. However, there still remains a question. Wiles proof relies heavily on modern mathematics. This leads to the conclusion that Wiles’ proof is different from that of Fermat’s. Fermat wrote that his proof would not fit in the margins of Arithmetica, similar to Wiles’ proof, but it is highly improbable that he invented modular forms, the Taniyama-Shimura conjecture, Galois Groups and the Kolyvagin-Flach model 300 years before anyone else did.

 

Some mathematicians believe that Fermat did have a genuine proof, while others believed he had developed a flawed proof. But, as far as Wiles is concerned, the battle to prove the theorem is over. He had fulfilled his childhood dream, and his mind was finally at rest.

 

For more information on Fermat’s Last Theorem, read Simon Singh’s book ‘Fermat’s Enigma’.

 

 

 

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