User:Timothy Clemans/Fermat's Last Theorem

Fermat's Last Theorem

Fermat's Last Theorem states that if n is greater than two, then the equation an + bn = cn has no solutions in non-zero integers a, b, and c. Interest in the problem began in the year 1637 when the French mathematician and jurist, Pierre de Fermat, wrote in a margin of his copy of volume two of Diophantus' Arithmetica that he had written a proof of this, which was marvelous and too long to fit into the margin. An elementary proof had never been found including the one proposed by Fermat and it became one of the greatest unsolved problems in the history of mathematics. In the past, many of the world's greatest mathematicians worked on the problem. The problem thru the years has been studied by amateurs. There are those who are still trying to create an elementary proof. A large amount of modern mathematics evolved from attempts by experts to prove Fermat's Last Theorem.

In number theory, Fermat’s Last Theorem states that if n is a positive integer greater than or equal to three, then the equation, an + bn = cn, has no solutions in non-zero integers a, b, and c. This was conjectured in 1637 by the French mathematician and jurist, Pierre de Fermat, in the form of a marginal note. He also said that he had proved it, however his proof has never been discovered.

In number theory, Fermat's Last Theorem states that if n is greater than two, then there are no solutions to the equation an + bn = cn in non-zero integers a, b, and c. The issue of proving this was introduced by the 17th century French mathematician and jurist Pierre de Fermat. In 1637 he made a marginal note among many others in his copy of Diophantus' Arithmetica where he asserts that he has a truly marvelous proof and it was too large to fit into the margin. His proof and orginal notes have never been found, however he did write down a proof of case n = 4. The problem of proving Fermat's Last Theorem became one of the greatest unsolved probblems in the history of mathematics.

Much of modern number theory has its roots in attempts to prove Fermat's Last Theorem. Fermat introduced the idea of infinite descent, which is one of the main tools in the study of Diophantine equations, in his proof of case n = 4. Modern number theory began to be greatly influenced by work on Fermat's Last Theorem after sophisticated concepts of algebraic number theory and the theory of L-functions were applied to Fermt's Last Theorem by Ernst Eduard Kummer. The work on Fermat's Last Theorem has had major affect on the research being conducted in mathematics, because the proof of Fermat's Last Theorem showed that some of the greatest mysteries in mathematics maybe resolved.

Most of the proof of Fermat's Last Theorem was put together by the algebraic numbber theorist Andrew Wiles of Princeton University who began professional study of Fermat's Last Theorem and the mathematics that he would need in the late part of summer of 1986. In the summer of 1986 Ken Ribet proved following an insight of Gerhard Frey and Jean-Pierre Serre's careful study of certain mod l Galois representations, that if every semistable elliptic curve with rational coeffients is modular, then Fermat's Last Theorem can not be false. Fermat's Last Theorem was proved in the October of 1994 by Andrew Wiles and Richard Taylor.