Fermat's last theorem

Fermat's last theorem (sometimes abbreviated as FLT and also called Fermat's great theorem) is one of the most famous theorems in the history of mathematics. It states that:

The proof

Using sophisticated tools from algebraic geometry (in particular elliptic curves and modular forms), Galois theory and Hecke algebras, the English mathematician Andrew Wiles, from Princeton University, with help from his former student Richard Taylor, devised a proof of Fermat's last theorem that was published in 1995 in the journal Annals of Mathematics.

Related Topics:
Algebraic geometry - Elliptic curve - Modular form - Galois theory - Hecke algebra - Andrew Wiles - Princeton University - Richard Taylor - Annals of Mathematics

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In 1986, Ken Ribet had proved Gerhard Frey's epsilon conjecture that every counterexample an + bn = cn to Fermat's last theorem would yield an elliptic curve defined as:

Related Topics:
Ken Ribet - Gerhard Frey

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y^2 = x(x - a^n)(x + b^n),

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which would provide a counterexample to the Taniyama-Shimura conjecture.

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This latter conjecture proposes a deep connection between elliptic curves and modular forms.

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Andrew Wiles and Richard Taylor were able to establish a special case of the Taniyama-Shimura conjecture sufficient to exclude such counterexamples arising from Fermat's last theorem.

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The story of the proof is almost as remarkable as the mystery of the theorem itself. Wiles spent seven years working out nearly all the details by himself and with utter secrecy (except for a final review stage for which he enlisted the help of his Princeton colleague, Nick Katz). When he announced his proof over the course of three lectures delivered at Cambridge University on June 21-23 1993, he amazed his audience with the number of ideas and constructions used in his proof. Unfortunately, upon closer inspection a serious error was discovered: it seemed to lead to the breakdown of this original proof. Wiles and Taylor then spent about a year trying to revive the proof. In September 1994, they were able to resurrect the proof with some different, discarded techniques that Wiles had used in his earlier attempts.

Related Topics:
Nick Katz - Lecture - Cambridge University - 1993 - 1994

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