Fermat's Last Theorem

theorem in number theory that there are no nontrivial integer solutions of xⁿ+yⁿ=zⁿ for integer n>2

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.


  • Admittedly, Fermat's Last Theorem was always called a theorem and never a conjecture. But that is unusual, and probably came about because Fermat claimed in notes that he scribbled in his copy of Diophantus's Arithmetica that he had a marvellous proof that was unfortunately too large to write in the margin of the page. Fermat never recorded his supposed proof anywhere, and his marginal comments became the biggest mathematical tease in the history of the subject. Until Andrew Wiles provided an argument, a proof of why Fermat's equations really had no interesting solutions, it actually remained a hypothesis - merely wishful thinking.
  • The Wiles proof is a master symphony of the major mathematical themes that have evolved in this century: Hecke's theory of modular forms, Artin L-functions, Grothendieck's theory of schemes, the theory of ℓ-adic representations and the Tate module, the Langlands program, Serre's p-adic modular forms, the Eichler-Shimura theory, Iwasawa theory and its generalizations by Coates to elliptic curves, Kolyvagin's Euler systems. The list is by no means exhaustive, to say the least. As Barry Mazur has so aptly put it, "Not to mention the proof, the names alone of the major contributors will not fit into any margin." It is indeed very satisfying to see the evolution of such mathematical ideas.
  • FLT deserves a special place in the history of civilization. Because of its simplicity, it has tantalized amateurs and professionals alike, and its remarkable fecundity has led to the development of large areas of mathematics such as, in the last century, algebraic number theory, ring theory, algebraic geometry, and in this century, the theory of elliptic curves, representation theory, Iwasawa theory, formal groups, finite flat group schemes and deformation theory of Galois representations, to mention a few. It is as if some supermind planned it all and over the centuries had been developing diverse streams of thought only to have them fuse in a spectacular synthesis to resolve FLT. No single brain can claim expertise in all of the ideas that have gone into this "marvelous proof". In this age of specialization, where "each one of us knows more and more about less and less", it is vital for us to have an overview of the masterpiece such as the one provided by this book.

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