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Mathematicians Encyclopedia Entry 1779004265

** This encyclopedia entry is dedicated to the life and work of a renowned mathematician, whose groundbreaking contributions to the field of **number theory** have left an indelible mark on the world of mathematics. **CONTENT:** ### Overview The mathematician in question is none other than **Andrew Wiles**, a British mathematician who rose to international fame in 1994 for solving **Fermat's Last Theorem** (FLT), a problem that had gone unsolved for over 350 years. Wiles' work on FLT not only showcased his exceptional mathematical prowess but also demonstrated the power of **pure mathematics** to tackle seemingly intractable problems. Wiles' journey to solving FLT was a long and arduous one, spanning over seven years of intense focus and dedication. His work built upon the foundations laid by other mathematicians, including **Pierre de Fermat** and **Leonhard Euler**, who had made significant contributions to the field of number theory. Wiles' solution to FLT was a masterclass in mathematical elegance, using a combination of **modular forms** and **elliptic curves** to prove the theorem. ### History/Background Andrew Wiles was born on April 11, 1953, in Cambridge, England. He developed a passion for mathematics at an early age and went on to study at Clare College, Cambridge, where he earned his undergraduate degree in mathematics. Wiles then pursued his graduate studies at the University of Oxford, where he earned his Ph.D. in mathematics under the supervision of **John Coates**. Wiles' work on FLT began in the late 1980s, when he was a professor at Princeton University. He spent the next seven years working in secret, sharing his progress with only a handful of colleagues. In 1993, Wiles presented his proof of FLT to a gathering of mathematicians at the Isaac Newton Institute in Cambridge, but the proof was met with skepticism due to a flaw in the argument. Wiles spent the next year revising his proof, and in 1994, he presented a corrected version to the mathematical community. ### Key Information * **Fermat's Last Theorem**: Wiles' solution to FLT states that there are no integer solutions to the equation \(a^n + b^n = c^n\) for \(n > 2\). * **Modular forms**: Wiles used modular forms to construct a **Galois representation** that linked the solutions to FLT with the properties of elliptic curves. * **Elliptic curves**: Wiles used elliptic curves to prove the **modularity theorem**, which states that every elliptic curve over the rational numbers can be associated with a modular form. * **Ribet's work**: Wiles built upon the work of **Karl Rubin** and **Gerhard Frey**, who had shown that FLT was related to the **Taniyama-Shimura conjecture**. * **The proof**: Wiles' proof of FLT is a 100-page manuscript that uses a combination of modular forms, elliptic curves, and Galois representations to prove the theorem. ### Significance Wiles' solution to FLT has had a profound impact on the field of mathematics, demonstrating the power of pure mathematics to tackle seemingly intractable problems. The proof has also led to significant advances in our understanding of **number theory**, **algebraic geometry**, and **representation theory**. Wiles' work has also inspired a new generation of mathematicians to pursue careers in pure mathematics. His proof of FLT has been hailed as one of the greatest achievements in mathematics in the 20th century, and it continues to inspire mathematicians and scientists around the world. **INFOBOX:** - Name: Andrew Wiles - Type: Mathematician - Date: April 11, 1953 - Location: Cambridge, England - Known For: Solving Fermat's Last Theorem **TAGS:** Andrew Wiles, Fermat's Last Theorem, Number Theory, Pure Mathematics, Modular Forms, Elliptic Curves, Galois Representations, Taniyama-Shimura Conjecture, Mathematical History.