# Andrew Wiles

Andrew Wiles is a mathematician from England. He is most famous for solving Fermat’s Last Theorem, which had not been solved for over 350 years after Fermat wrote it.

**Early Life**

Andrew Wiles was born on April 11, 1953 in Cambridge, England. Wiles had always been interested in mathematics and after finishing his homework, he would often write his own math problems which he would then proceed to solve. He would also often go to the library looking for books on mathematical subjects. One day, when he was ten years old, Wiles took a book on Fermat’s Last Theorem out of the library. The theorem fascinated Wiles and he was amazed that such a simple sounding theorem had not been solved for over three hundred years, even though Fermat claimed he had found the proof (which was never written down). Wiles set out to solve the problem even though he was only ten.

Wiles thought the theorem would be easy to solve and he attempted to come up with a solution through his teens,but he realized that he wasn’t ready to attempt the task because he didn’t have the necessary mathematical skills. He set the problem aside for two decades while he continued his education in mathematics.

**Education**

Wiles began his education at King’s College School and The Leys School. After he graduated from these schools, he went to Merton College in Oxford. He received a bachelor’s degree majoring in mathematics in 1974. Once he had his bachelor’s degree, Wiles began to pursue his PhD at Clare College in Cambridge. He received his PhD in 1980.

Wiles did not try to solve Fermat’s Last Theorem for his doctorate because he realized that he could end up working for years and never get anywhere. Instead, he decided to focus on the work being done on the Iwasawa theory of elliptic curves by his advisor John Coates.

Wiles was a Junior Research fellow at Clare College, Cambridge as well as a Benjamin Peirce Assistant Professor at Harvard University while he was pursuing his PhD. After he received his PhD, he spent some time in Bonn at the Sonderforschungsbereich Theoretische Mathematik. At the end of 1981, Wiles moved to the United States to work at Princeton where he was offered a professorship.

Wiles stayed at Princeton until 1985 when he was offered a Guggenheim Fellowship in Paris. Wiles worked at Oxford University in 1988 for two years before going back to Princeton in 1990.

Wiles was still interested in Fermat’s Last Theorem and he now felt that he had the necessary mathematical skill to take the problem on. From 1986 to 1993, he focused on finding the proof to the theorem.

**What is Fermat’s Last Theorem?**

Fermat’s Last Theorem was written by a French mathematician by the name of Pierre de Fermat. He was a leading mathematician in the 1600s and before his death in 1665, he made a number of contributions to a variety of different areas in mathematics, such as analytic geometry, probability, differential calculus, and optics. He is most famous for his last theorem.

Fermat rarely published his work. He only published one article during his lifetime and that was anonymous. He would write letters about his work to friends and this is where most of the knowledge about his work comes from.

After his death, Fermat’s friends were concerned that all of his work would be lost since he never published anything. Fermat’s son, Samuel, decided to collect all of Fermat’s letters, papers, and other mathematical writings so that they could be published. This was how Fermat’s Last Theorem was found.

Samuel found the theorem written in the margins of another book (an ancient Greek text titled *Arithmetica*written by Diophantus). The theorem stated that there are no positive whole numbers for x, y, and z in the equation x^{n} + y^{n} = z^{n} if n is larger than two.

Next to the theorem Fermat wrote a small note which read, “*I have discovered a truly remarkable proof which this margin is too small to contain*.” This proof was never written down and it took over 350 years for the answer to be found.

**The Answer**

Wiles began working on a proof for Fermat’s Last Theorem in secret. He didn’t tell anyone about what he was working on (other than his wife) because he felt that too many interested observers would stop him completely concentrating on the problem.

Wiles found the answer to the problem in the work of other mathematicians. The first step to Wiles’s solution was in the work by two Japanese mathematicians, Yutaka Taniyama and Goro Shimura. In the 1950s, Taniyama noticed that every elliptic curveseemed to be related to a modular form. Elliptic curves are objects used in algebraic geometry. Taniyama’s work was continued by Shimura and the Taniyama-Shimura conjecture was developed. A conjecture is different than a theory. A conjecture is an interesting idea that has not been proven although there is a lot of evidence in favour of the idea.

**The Taniyama–Shimura Conjecture**

The Taniyama-Shimura conjecture became an important idea in mathematics and a lot of work used the conjecture even though it had not been proven. In 1984, a meeting of mathematicians was held in Germany. At this meeting, the mathematicians were discussing elliptic equations and some of the speakers discussed the Taniyama-Shimura conjecture and the progress they had made towards proving it. Gerhard Frey, one of the speakers, made the claim that if the Taniyama-Shimura conjecture was proved then that would also prove Fermat’s Last Theorem.

To prove his point, Frey wrote Fermat’s Last Theorem on the board.Again, Fermat’s theorem states that there are no positive whole numbers for x, y, and z in the equation x^{n} + y^{n} = z^{n} if n is larger than two.Frey made the assumption that this was not true and that there was at least one solution (i.e. a set of whole positive numbers) that worked with the equation. Frey used this assumption to rearrange the equation. This is often done in mathematics and even though it changes the way the equation looks, it does not affect the equation’s integrity.

By rearranging Fermat’s equation, Frey turned it into an elliptic equation. By doing this, Frey linked Fermat’s Last Theorem with the Taniyama–Shimura conjecture.

Frey then pointed out that the elliptic equation he came up with by assuming that Fermat’s Last Theorem was true was so strange that it would be impossible for it to be connected to a modular form (i.e. the Taniyama-Shimura conjecture was false).

Frey showed that if Fermat’s Last Theorem were false and there was a set of numbers that could fit Fermat’s equation, then the elliptic equation that Frey came up with would exist. Since this elliptic equation can never be modular, then the Taniyama-Shimura conjecture, which states that every elliptic equation must be modular, is false.

In other words, if the Taniyama-Shimura conjecture is proven to be true (i.e. every elliptic equation is modular), then Frey’s elliptic equation cannot exist. If Frey’s elliptic equation cannot exist, then there is no solution to Fermat’s equation and Fermat’s Last Theorem must be true.

If someone could prove that the Taniyama-Shimura conjecture is true, then Fermat’s Last Theorem would also be true.

People were excited by Frey’s work but they quickly spotted an error in Frey’s calculations. This error needed to be corrected before Frey’s connection of the two equations would work. The problem was that Frey had not proven that his elliptic equation was so strange that it could not be connected to a modular form. At first, the mathematicians that heard Frey’s talk thought that it would be easy to prove this, but it turned into the work of many months.

In 1986, a professor at the University of California at Berkeley, Ken Ribet, finally succeeded in proving the absolute strangeness of Frey’s equation. Ribet had proven that Frey’s equation was not modular and as a result had linked it with Fermat’s Last Theorem.

Mathematicians now had a new strategy for attempting to prove Fermat’s Last Theorem. If the Taniyama-Shimura conjecture were true, then Fermat’s Last Theorem would have to be true as well. It was impossible for one to be true and the other to be false.

The only problem was that mathematicians had been trying to prove the Taniyama–Shimura conjecture for the last thirty years and had been unable to do it. Even though connecting the two ideas together was exciting, it offered no clues as to how to actually prove the Taniyama–Shimura conjecture. In fact, once the two ideas had been connected, most mathematicians assumed that the Taniyama-Shimura conjecture would be just as impossible to solve as Fermat’s Last Theorem.

When Wiles heard that Ribet had proven the connection between the Taniyama–Shimura conjecture and Fermat’s Last Theorem, he became very excited. Wiles had been studying elliptical equations for years as part of his PhD program and he had continued studying them while working in Princeton.

Wiles spent eighteen months reviewing and studying all of the published works and mathematical methods that were connected to elliptic equations or modular forms. Wiles became completely focused on the problem and even stopped attending conferences and meetings while he worked on the problem.

Wiles worked on the problem in complete secrecy to avoid distractions. He was also worried that if some of his work became public, then another mathematician could complete the work before he could and the other mathematician would get the recognition for solving the problem. Wiles worked for years and although he made a number of important discoveries over the course of his work, he planned to never publish them or even discuss them until he had proven Fermat’s Last Theorem.

To hide what he was working on, Wiles took the research he had been working on before Ribet’s discovery and published minor parts of it every six months. This convinced his colleagues that he was continuing his regular research.

**A Scare**

Wiles had already spent years on the problem and while he had made a number of significant breakthroughs in the field of mathematics, he still had not solved Fermat’s Last Theorem.

Wiles thought he had been beaten to the punch in 1988 when he read that a Japanese mathematician, Yoichi Miyaoka, claimed to have solved Fermat’s Last Theorem. Miyaoka claimed that he had solved the theorem through the use of differential geometry. Once Miyaoka published his work, mathematicians began to examine it closely and found that there was an error and that Miyaoka had not actually proven the theorem.

After three years of work, Wiles was able to show how the nineteenth-century French mathematician Evariste Galois’s concept of group theory could be used with elliptic equations.Wiles was able to break elliptic equations down into an infinite number of pieces. He then showed that the first part of each equation had to be modular.

Wiles was still struggling with part of his proof, so he decided to break his silence and ask for help from Nick Katz, a fellow mathematician who also worked at Princeton. Wiles needed help with some difficult mathematics (the Kolyvagin-Flach method) that Wiles was adapting to use in his proof. He did not have a lot of familiarity with the method and wanted to make sure he was using it correctly.

Since it was such a big part of the proof and so complicated, Wiles and Katz agreed that Wiles should hold a series of lectures for graduate students to explain his adaption of the Kolyvagin-Flach method without mentioning the Taniyama–Shimura conjecture or Fermat’s Last Theorem. This would allow Katz to sit in on the lectures and check the work being presented by Wiles. After listening to the lectures, Katz agreed that Wiles adaptation of the Kolyvagin-Flach method worked.

That’s all Wiles needed to hear. After some more work, Wiles was ready to announce that he had solved the problem.

**The Discovery**

Wiles decided to present his discovery at a conference in Cambridge. There were a lot of rumour about what Wiles was planning to present, and many people came up to Wiles to ask about his presentation. Wiles simply told them to come to the presentation to find out what he was going to say.

Over the course of three lectures, Wiles presented the work he had done over the past seven years. In the final lecture, he presented his proof to Fermat’s Last Theorem to loud applause.

The news spread around the world and was a major headline in a number of newspapers. *People*magazine even named Wiles as one of “The 25 most intriguing people of the year.”

**A Problem**

While Wiles was preparing his proof for publication, he came across an error that threatened to invalidate Wiles’s whole proof. Wiles was determined to solve the error himself so he held off publicizing the paper for six months.

In September 1994, Wiles finally figured out the problem with help from a former student, Richard Taylor. Wiles then published the proof in the May 1995 issue of *Annals of Mathematics*. The whole issue was focused on Wiles’s work.

It took two years before his work was checked because the work was too complex for most mathematicians. Once it was checked, the mathematical world agreed, Wiles had solved Fermat’s Last Theorem.

For solving the theorem, Wiles was awards a prize that had been set up by German industrialist Paul Wolfskehl in 1907 (100,000 marks) and in 2016, Wiles was awarded the Abel Prize for solving Fermat’s Last Theorem.