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John Duran
John Duran

Fermat's Last Theorem: How Simon Singh Revealed the Epic Saga of a Mathematical Mystery



Outline: # Fermat's Last Theorem: The Story of a Riddle that Confounded the World's Greatest Minds for 358 Years - ## Introduction - What is Fermat's last theorem and why is it important? - How did Pierre de Fermat state the theorem and claim to have a proof? - What are some of the failed attempts and partial results over the centuries? - How did Andrew Wiles finally prove the theorem in 1995? - ## Fermat's Last Theorem: A Simple Statement with a Complex History - How did Fermat discover the theorem while studying Diophantus's Arithmetica? - What was his famous marginal note and why did he not reveal his proof? - How did Euler, Sophie Germain, Kummer, and others contribute to the study of the theorem? - What are some of the applications and implications of the theorem in number theory and cryptography? - ## Andrew Wiles: A Childhood Dream and a Mathematical Odyssey - How did Wiles encounter the theorem as a 10-year-old boy and become fascinated by it? - How did he pursue his mathematical career at Cambridge, Princeton, and Oxford? - How did he secretly work on the theorem for seven years using advanced tools from algebraic geometry and modular forms? - How did he overcome the obstacles and challenges along the way, including a fatal gap in his proof? - ## The Proof: A Triumph of Human Ingenuity and Collaboration - How did Wiles announce his proof at a conference in Cambridge in 1993? - How did he collaborate with his former student Richard Taylor to fix the error in his proof? - How did he publish his proof in two papers in the Annals of Mathematics in 1995? - How did he receive worldwide recognition and acclaim for his achievement, including the Abel Prize in 2016? - ## Conclusion - What are some of the lessons and insights that can be drawn from the story of Fermat's last theorem and its proof? - What are some of the open questions and challenges that remain in mathematics today? - How can mathematics inspire curiosity, creativity, and passion in people of all ages and backgrounds? Article: # Fermat's Last Theorem: The Story of a Riddle that Confounded the World's Greatest Minds for 358 Years Fermat's last theorem is one of the most famous and intriguing problems in mathematics. It states that there are no positive integer solutions to the equation $x^n + y^n = z^n$ for any integer $n$ greater than 2. In other words, it is impossible to find three whole numbers that satisfy this equation when the exponent is higher than 2. For example, $3^2 + 4^2 = 5^2$ is a valid solution when $n = 2$, but there is no such solution when $n = 3$ or higher. The theorem is named after Pierre de Fermat, a 17th-century French lawyer and amateur mathematician who claimed to have a proof of it but never wrote it down. He scribbled a note in the margin of his copy of Arithmetica, a book by the ancient Greek mathematician Diophantus, saying that he had "a truly marvelous proof" of the theorem but that "the margin is too narrow to contain it". This mysterious note sparked centuries of curiosity and frustration among mathematicians who tried to find Fermat's proof or come up with their own. Many brilliant minds contributed to the quest for proving Fermat's last theorem, but none succeeded until 1995, when Andrew Wiles, a British mathematician at Princeton University, published a monumental proof that spanned over 100 pages and used sophisticated techniques from modern mathematics. Wiles had been fascinated by Fermat's last theorem since he was a 10-year-old boy and devoted seven years of his life to working on it in secrecy. His proof was hailed as one of the greatest achievements of mathematics and earned him numerous awards and honors. In this article, we will explore the story of Fermat's last theorem and its proof, from its origin in ancient Greece to its resolution in modern times. We will see how this simple statement with a complex history captivated generations of mathematicians and challenged their creativity and ingenuity. We will also see how this story illustrates the beauty, the mystery, and the joy of mathematics. ## Fermat's Last Theorem: A Simple Statement with a Complex History Fermat's last theorem is not really a theorem, but a conjecture, a statement that has not been proven. Fermat himself never published his proof, and it is widely believed that he did not have one, or that his proof was flawed or incomplete. However, he did prove some special cases of the theorem, such as when $n = 4$ or when $n$ is a prime number. He also proved some related results, such as the fact that every prime number of the form $4k + 1$ can be written as the sum of two squares. Fermat was not the first to study the equation $x^n + y^n = z^n$. In fact, he was inspired by Diophantus, who lived in Alexandria in the 3rd century AD and wrote a series of books called Arithmetica, which dealt with solving equations involving rational numbers (fractions). Diophantus was interested in finding rational solutions to the equation, not integer solutions. For example, he showed that $x^2 + y^2 = z^2$ has infinitely many rational solutions of the form $x = \fracm^2 - n^2m^2 + n^2$, $y = \frac2mnm^2 + n^2$, and $z = \fracm^2 + n^2m^2 + n^2$, where $m$ and $n$ are any rational numbers. He also considered some cases of the equation when $n = 3$ or $n = 4$, but did not find any rational solutions. Diophantus's work was largely forgotten until the 16th century, when it was rediscovered by European mathematicians. Among them was Pierre de Fermat, who lived in Toulouse and worked as a lawyer and a magistrate. Fermat was also a passionate mathematician who corresponded with other scholars and made many discoveries in number theory, geometry, calculus, and probability. He often posed problems and challenges to his friends and colleagues, but rarely revealed his methods or proofs. He wrote most of his mathematical notes in the margins of his books, which were only published after his death in 1665. One of these notes was his famous claim to have a proof of what is now known as Fermat's last theorem. He wrote it in his copy of Diophantus's Arithmetica, next to a problem that asked to find a right-angled triangle whose area is a square number. Fermat observed that this problem is equivalent to finding integer solutions to the equation $x^4 + y^4 = z^4$, which he said was impossible. He then generalized this statement to any exponent greater than 2 and wrote: "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." This tantalizing note sparked centuries of curiosity and frustration among mathematicians who tried to find Fermat's proof or come up with their own. Many attempts were made to prove or disprove the theorem, but none succeeded in covering all possible cases. Some mathematicians proved the theorem for specific values of $n$, such as when $n = 5$, $n = 7$, or $n = 11$. Others proved the theorem for infinitely many values of $n$, such as when $n$ is a regular prime (a prime number that satisfies a certain condition) or when $n$ is divisible by 4. However, these results left many gaps and exceptions that could not be resolved by existing methods. Along the way, many interesting and important results were discovered in number theory and related fields. For example, Euler proved that every even perfect number (a number that is equal to the sum of its proper divisors) is of the form $2^p-1(2^p - 1)$, where $p$ is a prime number and $2^p - 1$ is also a prime number (called a Mersenne prime). Sophie Germain proved that if $p$ is an odd prime number such that $2p + 1$ is also a prime number (called a Sophie Germain prime), then Fermat's last theorem holds for $n = p$. Kummer developed the theory of ideal numbers and showed that Fermat's last theorem holds for all regular primes (a prime number that satisfies another condition). These results also had applications and implications in cryptography, coding theory, and computer science. ## Andrew Wiles: A Childhood Dream and a Mathematical Odyssey Andrew Wiles was born in 1953 in Cambridge, England. He grew up in a family of academics and developed an early interest in mathematics. When he was 10 years old, he found a copy of Eric Temple Bell's book The Last Problem at his local library and was captivated by the story of Fermat's last theorem. He decided that he would be the one to prove it and began to study number theory on his own. Wiles pursued his mathematical education at Cambridge University, where he earned his bachelor's degree in 1974 and his doctorate in 1980 under the supervision of John Coates. He then moved to the United States and became a professor at Princeton University. He worked on various topics in number theory, such as Iwasawa theory, elliptic curves, and modular forms. He also kept his interest in Fermat's last theorem alive, but did not share it with anyone. In 1986, Wiles learned about Ribet's proof of the epsilon conjecture, which showed that Fermat's last theorem would follow from the modularity conjecture for semistable elliptic curves. This gave him a new hope and a new direction for his research. He realized that he had some of the tools and ideas needed to attack the modularity conjecture, but he also knew that it was a very difficult and risky problem. He decided to work on it secretly, without telling anyone except his wife. For the next seven years, Wiles devoted himself to proving the modularity conjecture for semistable elliptic curves. He worked in his attic office at Princeton, isolated from his colleagues and students. He faced many technical difficulties and dead ends, but he also made some breakthroughs and discoveries. He used methods from algebraic geometry, such as deformation theory and Galois representations, to link elliptic curves with modular forms. He also used results from modular forms, such as the Langlands-Tunnell theorem and the Shimura-Taniyama-Weil conjecture, to construct elliptic curves with certain properties. In June 1993, Wiles felt that he had completed his proof and decided to announce it at a conference in Cambridge, England. He gave three lectures titled "Modular Forms, Elliptic Curves and Galois Representations", which attracted a large audience of mathematicians from around the world. In the third lecture, he revealed that his main result implied the modularity conjecture for semistable elliptic curves, and hence Fermat's last theorem. The audience was stunned and amazed by his achievement. Wiles received a standing ovation and became an instant celebrity. ## The Proof: A Triumph of Human Ingenuity and Collaboration Wiles's proof of Fermat's last theorem was not yet complete. He had to write it down in detail and submit it for publication in a peer-reviewed journal. He chose the Annals of Mathematics, one of the most prestigious journals in mathematics. He sent his manuscript to the editors in August 1993 and expected it to be reviewed by several experts in the field. However, in September 1993, Wiles received some bad news. One of the referees had found a serious error in his proof. There was a gap in one of his arguments involving a certain type of object called an Euler system. Wiles had assumed that such an object existed for his elliptic curves, but he had not proved it or given a reference for it. The referee pointed out that this assumption was not justified and that there was no known way to construct such an object. Wiles was devastated by this discovery. He felt that he had failed and that his life's work was ruined. He tried to fix the error by himself, but he could not find a solution. He realized that he needed help from someone who knew more about Euler systems than he did. He contacted his former student Richard Taylor, who was then a professor at Harvard University. Taylor agreed to join him in his quest to save the proof. Together, Wiles and Taylor worked on repairing the proof for almost a year. They searched for alternative ways to prove the existence of an Euler system or to avoid using it altogether. They also consulted with other experts in number theory, such as Nick Katz, Brian Conrad, Fred Diamond, Christophe Breuil, and Ken Ribet. They faced many obstacles and setbacks, but they also made some progress and improvements. Finally, in September 1994, Wiles had a breakthrough. He realized that he could use a different type of object, called a Hecke ring, to replace the Euler system. He also found a way to relate the Hecke ring to the Galois representation of his elliptic curve, using a technique called patching. He checked his calculations and convinced himself that he had solved the problem. He called Taylor and told him the good news. Wiles and Taylor revised their proof and submitted it to the Annals of Mathematics in October 1994. The new proof was divided into two papers: the first one, by Wiles alone, contained the main result and the strategy of the proof; the second one, by Wiles and Taylor, contained the technical details and the patching method. The papers were reviewed by several referees, who confirmed that the proof was correct and complete. The papers were published in May 1995, almost two years after Wiles's announcement. The proof spanned over 100 pages and used many advanced concepts and techniques from modern mathematics. It was widely regarded as one of the greatest achievements of mathematics and a milestone in human history. Wiles received worldwide recognition and acclaim for his accomplishment. He was awarded many prizes and honors, such as the Wolf Prize, the Abel Prize, the Royal Medal, and the knighthood. ## Conclusion Fermat's last theorem is more than just a mathematical problem. It is a story of human endeavor and intellectual brilliance, spanning over three centuries and involving many cultures and disciplines. It is a story of curiosity, creativity, and passion, driven by a simple but profound question. It is a story of challenge, struggle, and perseverance, overcoming many difficulties and obstacles. It is a story of collaboration, communication, and inspiration, benefiting from the collective wisdom and knowledge of generations. The story of Fermat's last theorem also teaches us some lessons and insights about mathematics and its role in society. It shows us that mathematics is not a static or finished subject, but a dynamic and evolving one, constantly discovering new truths and exploring new frontiers. It shows us that mathematics is not an isolated or abstract activity, but a connected and relevant one, interacting with other fields of science and technology. It shows us that mathematics is not an easy or trivial pursuit, but a challenging and rewarding one, requiring hard work and dedication. The story of Fermat's last theorem also raises some questions and challenges for the future of mathematics and its practitioners. What are some of the other unsolved problems that await us in mathematics? How can we find new methods and tools to tackle them? How can we communicate our results and ideas to other mathematicians and to the public? How can we inspire more people to appreciate and enjoy mathematics? These are some of the questions that we hope to answer as we continue our mathematical journey. We hope that you have enjoyed reading this article and that you have learned something new about Fermat's last theorem and its proof. We also hope that you have been inspired by this story and that you will share it with others who might be interested in mathematics. Thank you for reading! FAQs: - Q: What is Fermat's last theorem? - A: Fermat's last theorem is a statement that there are no positive integer solutions to the equation $x^n + y^n = z^n$ for any integer $n$ greater than 2. - Q: Who was Pierre de Fermat? - A: Pierre de Fermat was a 17th-century French lawyer and amateur mathematician who claimed to have a proof of Fermat's last theorem but never wrote it down. - Q: Who was Andrew Wiles? - A: Andrew Wiles was a British mathematician who proved Fermat's last theorem in 1995 after working on it secretly for seven years. - Q: What is the modularity conjecture? - A: The modularity conjecture is a statement that every elliptic curve over Q (a type of geometric object) is modular (a type of algebraic object). - Q: Why is Fermat's last theorem important? - A: Fermat's last theorem is important because it is one of the oldest and most famous problems in mathematics, because it has many connections and applications to other areas of mathematics, and because it illustrates the beauty, the mystery, and the joy of mathematics.




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