Fermat's Last Theorem

Topics: Fermat's Last Theorem, Diophantine equation, Number theory Pages: 5 (1603 words) Published: March 13, 2012
Fermat's Last Theorem

Fermat's Last Theorem states that no three positive integers, for example (x,y,z), can satisfy the equation x^n+y^n=z^n if the integer value of n is greater than 2. Fermat's Last Theorem is an example a Diophantine equation(Weisstein). A Diophantine equation is a polynomial equation in which the solution must be an integers. These equations came from the works of Diophantus who was a mathematician who worked methods on solving these equations. Fermat's Last Theorem was based on Diophantus's work. A more common Diophantine equation would be Pythagorean Theorem, where the solution would be the the Pythagorean triples(Weisstein). However, unlike Pythagorean Theorem, Fermat's Last Theorem has no practical real world applications.

Fermat had scribbled on the margin of Arithmetica, the book that inspired his theorem, that he had a proof that would not fit on the margin of a book. From the 1600's-mid 1900's this proof remained unsolved. It was eventually solved by Andrew Wiles. Andrew Wiles as a child always loved math, he would always make up problems and challenge himself. His greatest challenge was when he stumbled upon Fermat's Last Theorem at the local library. The problem for Wiles was that on the margin, Fermat did not write the actual proof to the theorem, just that he had a brilliant idea which was too big for a margin to hold. Wiles had to rediscover Fermat's proof, however he had the information of the other mathematicians who attempted to solve this theorem over the centuries. Ever since Wiles was a teenager, he remained determined to solve this theorem. When solving the equation seemed impossible, a breakthrough by Ken Ribet linked Fermat's Last Theorem with The Taniyama-Shimura Conjecture, or the Modular Theorem(Koch). Once he realized this he immediately regained hope, if he was able to solve this then the next step would be Fermat's Last Theorem. Wiles continuously worked on creating a proof, he worked in complete isolation and told almost nobody about his work in fear of attracting attention. Every day he would think of different methods of creating a proof, it was always on his mind. For many years, Wiles was completely obsessed with the equation and slowly made progress towards creating a proof. There was no way to tell if his progress would actually help him solve the problem. But eventually in 1993 Wiles solved Fermat's Last Theorem. Unknown to the public, Wiles had made an error on a major part of the proof. The error was so subtle and abstract, it took a year to revise the proof. The proof is completely different from Fermat's original proof. The proof Wiles came up with was more than one hundred pages long and required techniques that were not available at the time of Fermat. Wiles can finally relax now that he solved his childhood obsession, but he still continues to challenge himself with math problems. Finally Andrew Wiles' mind is at rest(Koch).

Pierre De Fermat was born in France, he was a lawyer and an amateur mathematician. Fermat was born into a wealthy family and attended the university of Bordeaux. Fermat was a married man who had five children. Fermat was a busy lawyer and kept math a hobby, never publishing his proofs. His greatest idea, Fermat's Last Theorem, was not announced by Fermat, but by his son who had stumbled upon his fathers notes in a book. Fermat is one of the fathers of analytic geometry and probability theory. Fermat contributed in the field of optics, light travel, and calculus("About," ).

Unfortunately Fermat's Last Theorem itself has no real world applications, it is simply a theoretical problem. Despite not having any application it helped prove many other ideas that...
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