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Sir Andrew Wiles is a living legend and a significant contributor to mathematics. Among his most notable accomplishments was his successful attempt to solve Fermat's Last Theorem. He is currently a Royal Society Research Professor at Oxford University. The paper will investigate Sir Wiles' bibliography, his prominent contributions, and the consequences of his contributions to mathematical education.

Sir Wiles was born on April 11, 1953, in Cambridge, England. Maurice Frank Wiles and Patricia Wiles are his parents. At the time of his birth, his dad was a Religious Professor of Divinity at Oxford University, and later - from 1952 to 1955) as a Chaplain at Ridgley Hall (Singh, Simon, 7). Since his father was a present member at Cambridge, Wiley was born and raised in the same locality, where at the age of ten, he stumbled upon and became obsessed with the then Uncovered Fermat’s Last Theorem (Singh, Simon, 18). From a very young age, Wiles was a brilliant mathematician. At the tender age of 10, he was able to comprehend most mathematical theories, and in particular intrigued by the Fermat’s Last Theorem: He had stumbled upon a book called “The Last Problem” written by E. T. Bell. He decided that since “Fermat didn’t know any more sophisticated mathematics than he did. … he was going to try it (finding the proof of the theory) (Singh, Simon, 28).”

He attended The Kings College School as well as the Leys School both located in the region. In 1974, he joined Oxford’s Merton College for the Bachelor’s degree, and under the guidance of Professor John Coates Claire at Cambridge’s Clare College he received Ph.D (Basti, Mehran, 94). in 1980. Between the period 1977 and 1980, Sir Wiles taught as a research assistant at the college Clare College and an associate professor at the University of Harvard. From 1981, he joined he joined and stayed at New Jersey’s Institute for Advance Studies and became a professor at the Princeton University.

As he advanced his education, he continues exploring the theory and formulating ideas to prove the Fermat’s Last Theorem (Singh, Simon, 28).” By 1986, he has found a connection between the theory and the Serre’s ε-conjecture. With the idea intact, the presented it a conference in Cambridge in 1993. However, at the time, there were quite some shortcomings in exploring the whole theory. Richard Taylor, his ex-student, intrigued by the unit joined Sir Wiles and together supported the proof, which was published in Annals of Mathematic (Singh, Simon, 40)”. With the first step towards solving the theory, Wiles became a familiar figure, becoming a member of the United States National Academy of Science.

Influential Contributions to Mathematics

Though attributed and famously known for his contribution in Fermat’s Last Theorem, Sir Wiles worked on various outstanding number theory problems (the Birch and Swinnerton-Dye conjectures, the Shimura-Taniyama-Weil conjecture, and the principle of Iwasawa Theory). In the Fermat's Last Theorem, he stated that “there do not exist any positive integer solutions of xn + yn = zn for n>2.” This was a problem posted in the 14th Century by Diophantus, and Fermat claimed to have solved it in the 17th century (though he did not provide the proof, with the claim of insufficient room)

Many mathematicians before Wiles had tried to provide proof of the theorem, but none could come up with a comprehensive solution (Derbyshire, J, 129). Different from them, Wiles starts off by stating the famous Latin Fermat’s quote, about the narrowness of the margin, and using the recent history and technology, proceeded to the solution of the problem. In seven years, Sir Wiles concentrated primarily on the subject, giving little attention to other things. He studies the works of Barry Mazur, Karl Rubin, Gerhard Frey, Jean-Pierre Serre, among other great mathematicians in the field. When he finally had the solution to the problem, he documented the first lecture in 1993 titles “Modular Forms, Elliptic Curves, and Galois Representations.” The genius ideas and presentation wowed the present members and created a sensation. However, he was still a long way in filling proving reason to the gaps in his theory, and with help from Richard Taylor in 1995, the theory was soaring towards completion. Aside from the article “Annals of Mathematics,” he provided an additional article, “Ring-Theoretic Properties of Certain Hecke Algebras.” The contributions which led to him being knighted in 2000 (Singh, Simon, 125).

Proving the Fermat’s Last Theorem.

As previously shown, the theorem states “no natural number x, y, and z exist such that x3 + y3= z3. That is, the sum of the two cubes cannot be a cube. Pierre de Fermat, in 1637, had stated in the article Arithmetica written by Diophantus of Alexandria (250 AD) that “I have discovered a remarkable proof [of this theorem], but this margin is too small to contain it (Singh, Simon, 124)” Centuries later, no one had been able to debunk this statement, therefore, not able to prove or disprove it.

According to Wiles, when Pierre De Fermat wrote the ideas, he doubts that Fermat himself could have solved it. He notes that the culture established by the mathematicians of Fermat’s time did not necessarily require him to provide his proof. In fact, he thinks that there is not mathematical tool existing at the time, which could have aided him in finding the answer to the problem. Sir Wiles stated that if he could have had the tools during his 15 years of fascination, he could have solved the theorem earlier (Henshaw, John M and Steven Lewis, 78) He came at the right time when the tools were developed, and thus he was able to provide the book of proof.

While at college, Wiles decided to stop proving the theorem for a while. It had become an addiction at the time, and he thought that staying away from solving it for a while would help him concentrate on his studies and other passions. However, in 1985, an important event that reversed his decision happened. Gerhard Frey’s Ken Ribet demonstrated that UTF resulted from the consequence of Taniyama-Shimura conjecture since each could be parametrized molecular forms of the elliptic curves (Singh, Simon, 67). The Taniyama-Shimura conjecture, being a less singular UTF, involved fundamental ideas of number theory, but due to its broad nature, it was not yet solved; until then.

Sir Wiles, noting the progress of Gerhard Frey, was able to connect the Fermat’s Last Theorem to the Taniyama-Shimura conjecture, and found a path that would lead him to proof the theorem. The Taniyama-Shimura conjecture concluded that each “elliptic curve could be associated unambiguously with modular fashion.” With that idea, Wiles noted that if the Fermat’s Last Theorem were not true, then an elliptic curve would appear, such that, it would dissociate with any modular form, rendering the Taniyama-Shimura conjecture false (Singh, Simon, 70). Therefore, the Taniyama-Shimura conjecture would prove the Fermat’s Last Theorem.

The Taniyama-Shimura proof had already challenged the Langland’s program; one that sought to unify all the separate branches of mathematics. For the next eight years, Wiles spent time dedicating himself to proofing the theorem. Following the demonstrations of Ribet, he isolated himself and worked on the problem sharing his thoughts through constant publications. With progress, he started working in solitude guiding his secrets and sharing his ideas to his professor of Mathematics at Princeton, Nicholas Katz. During this time, Sir Wiles developed the Wiles proof of the Taniyama-Shimura and through it, further demonstration of the UTF (Singh, Simon, 72). The proofs were complex in nature and often introduced new problems and ideas.

The solution of the Taniyama-Shimura was a new theory of elliptic curves, and as Wiles experienced, and by proving the theory, another idea was formed. As ideas after ideas came up, they were used to develop different concepts pertaining the Fermat’s Last Theorem. Wiles was back on his quest, and seven years later, he proved the theorem, taking another year to find the complete solution to the Fermat’s Last Theorem. Within the period, in 1993, with the help of a computer, Wiles and Richard Taylor ran models in the computer that proved the n < 40000000, and later, the two used algebraic geometry to prove the theorem before its publication in 1995 (Singh, Simon, 88).

The Effect of Sir Andrew Willy’s Proof to Mathematics

Since the ancient times, mathematicians have strived to formulate proofs and come up with theorems to provide solutions to numerical enigmas. During the next half of the nineteenth century, it was common to have mathematical problems printed in the popular press, next to anagrams and puzzles. The aim was to have the public aware of the problems and provide alternative genius methods for solving these problems that had rattled the brains of other mathematicians.

In the dawn of the 20th Century, (8th August 1900), David Hilbert, a famous German Mathematician, presented a list of not-yet-solved mathematical problem – later named the Hilbert problems - to the current members of the International Congress of Mathematics (Derbyshire, 32). 100 years later on 24th May 2000, Clay Mathematics Institute introduced the Millennium problems, at the Millennium Conference: They were seven yet-solved mathematics problems. Each of the unsolved question having a prize of one million dollars placed on them (Derbyshire, 37). Both presentations have had a significant impact on the twentieth-century mathematics.

Over the centuries, mathematicians had dedicated themselves to finding the proof to the Fermat’s Last Theorem. Among them is Euler, who was able to find the proof of the theory up until n=3; Andrien-Marie Legendre, whose proof went to n=5; and later on Sophie German, with her discovery of the probable truth for n being equal to a particular prime number p, such that 2p + 1 is a prime number (Singh, Simon, 109).

A ten-year-old Ander Wales, in the library the school library, stumbled on the question and by a deeper interest dedicated the latter part of his life solving the answer. While striving to address the problem, he shed light on the complex field of mathematics, by utilizing algebraic geometry tools, that Galois Theory, the elliptic curves, and modular forms theory; all of which were vital in solving the Fermat’s Last Theory (Derbyshire, 32).

Although for many other mathematicians, pure motivation is the primary drive towards solving unsolved theories, the game and the incentives that come along with it play a part. The million dollar price that comes along with the solution motivates the public to find answers to the problem. In proving the theorem, Sir Wiles not only had all the fame and the money coming his way, but he also solved other mathematics problems as well. For example, Goro Shimura and Yutaka Taniyama, the Japanese mathematicians had come up with a conjecture in the 1950s, that there was a relationship between an elliptical equation and modular forms. After the works of Gerhard Frey, Ken Riber and Barry Mazue, the conjecture was proved, and so was the Fermat's Last Theorem by Wilis. The combination of great minds and the passing of ideas lay a platform in the mathematics field today.

The value of finding interest, and seeking other people with the same interest is a belief promoted math. The education system looks into achievements like those of Wiles in driving a culture of collaboration in solving problems. With every idea that comes from one person, another person gets an idea to expound on the next idea that they have. It is a culture of interdependence and interrelationship between different areas of knowledge to finding answers to questions and problems.

In the history of knowledge, the unification of different areas of study has been unknown. In mathematics, pure and applied mathematics had previously been treated as completely separate. With the advancement in theories and proofs, these branches have been used intertwiningly to provide insight to ideas and knowledge in one branch just like the other. In the same manner, the proof of the Fermat’s Last Theorem has had an impact on the modern mathematics. It opened a new era and method of solving problems in number theory.

Similar techniques used by Wiles – like combining different approaches to solve problems – have given the mathematics world a robust framework in attacking problems. Theories that may seem useless at one time can be transformed into useful ideas later. Before Wiles solved the problem, Fermat’s Last Theorem was not entirely helpful (Henshaw, John M and Steven Lewis, 78) in the computer and software engineering; the theorem has turned out to be an essential part of the modern cryptography and computer security system.

Honors

Due to his contribution in Mathematics, Sir Wiles has received a host of accolades throughout his life (Sample, Ian). In 1900 he was awarded the Junior Whitehead Prize of the LMS. In 1989, the Fellow of Royal Society. Later in 1995, he was awarded the Schock Prize. Then the Fermat Prize in 1995. He went on to win the Wolf and the Ostrowski Prize in 1996, and the Cole prize in 1997. He went through other accolades before being accorded the Pythagoras Award in 2004. Apart from those, he received other public honors: His appointment in 2000 to the rank of “Knight Commander of the Order of the British Empire” was the highlight. In 2016, Sir Wiles was awarded the Abel Prize (an equivalent of the Nobel Prize), following his “stunning proof of the Fermat’s Last Theorem (Sample, Ian). With these achievements, Andrew Wiley stands amongst the great mathematicians throughout history, and in the process creating a legacy for himself, while changing the mathematics environment forever.

Works Cited

Basti, Mehran. DNA Of Mathematics. 1st ed., Victoria, Friesenpress, 2014,.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.

Henshaw, John M and Steven Lewis. An Equation for Every Occasion. 1st ed., Baltimore, John Hopkins Press University Press, 2014,.

Sample, Ian. "Abel Prize Won By Oxford Professor For Fermat's Last Theorem Proof". The Guardian, 2016, https://www.theguardian.com/science/2016/mar/15/british-mathematician-andrew-wiles-abel-prize-fermats-last-theorem-proof.

Singh, Simon. Fermat's Last Theorem. 1st ed., London, Harper Perennial, 2011,.

May 17, 2023

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