Wiles' proof of Fermat's Last Theorem

Andrew Wiles's proof of Fermat's Last Theorem is a proof of the modularity theorem for semistable elliptic curves by Andrew Wiles, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the Modularity Theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, seen as virtually impossible to prove using current knowledge. Wiles first announced his proof on Wednesday 23 June 1993 at a lecture in Cambridge entitled "Elliptic Curves and Galois Representations." However, in September 1993 the proof was found to contain an error. One year later, on Monday 19 September 1994, in what he would call "the most important moment of [his] working life," Wiles stumbled upon a revelation, "so indescribably beautiful... so simple and so elegant," that allowed him to correct the proof to the satisfaction of the mathematical community. The correct proof was published in May 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.

The proof itself is over 150 pages long and consumed seven years of Wiles's research time.John Coates described the proof as one of the highest achievements of number theory, and John Conway called it the proof of the century. For solving Fermat's Last Theorem, he was knighted, and received other honours such as the 2016 Abel Prize.

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