Fermat's Last Theorem
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| Fermat's Last Theorem | |
|---|---|
 Public domain · source | |
| Name | Fermat's Last Theorem |
| Caption | Pierre de Fermat |
| Discovered | 17th century |
| Proved | 1994 |
| Prover | Andrew Wiles |
| Field | Number theory |
Fermat's Last Theorem is a theorem about the nonexistence of nontrivial integer solutions to the equation x^n + y^n = z^n for integer n > 2. Stated in the 17th century, the assertion became a central problem in number theory and attracted work from mathematicians across Europe and beyond, eventually culminating in a proof announced in 1994 by Andrew Wiles after links to elliptic curve research and the Taniyama–Shimura–Weil conjecture were established.
Statement and historical background
The statement was written by Pierre de Fermat in the margin of a copy of Arithmetica, claiming a "marvelous proof" that would not fit the margin; this note prompted commentary and study by figures such as Blaise Pascal, Leonhard Euler, Joseph-Louis Lagrange, and Adrien-Marie Legendre. During the 18th and 19th centuries, work by Euler on n=3, Sophie Germain's methods involving auxiliary primes, and Dirichlet and Legendre's treatment of n=5 advanced special cases; later contributions by Kummer using ideals in ring theory and cyclotomic fields framed the problem in the language used in Algebraic number theory and influenced David Hilbert and contemporaries. The problem shaped research agendas in institutions including the Royal Society, the Académie des Sciences, and universities such as University of Cambridge and University of Göttingen.
Attempts and partial results
Efforts included elementary proofs for specific exponents by Fermat for n=4 via infinite descent, and for n=3 and n=5 by Euler, Sophie Germain, Dirichlet, and Legendre. Kummer's introduction of ideal theory addressed regular primes and produced results for many exponents; this work intersected with studies by Richard Dedekind, Ernst Eduard Kummer, and later Heinrich Weber. 20th-century advances involved Gerd Faltings' theorem (formerly Mordell conjecture) on rational points, and contributions from Gerhard Frey, who proposed linking certain semistable elliptic curves to hypothetical counterexamples; Jean-Pierre Serre formulated the epsilon conjecture (later refined), and Ken Ribet proved the crucial link from Frey curve constructions to the Taniyama–Shimura–Weil conjecture, narrowing the path to a full proof that connected researchers at centers like Princeton University, Cambridge University, and Institut des Hautes Études Scientifiques.
Proof by Andrew Wiles
Andrew Wiles announced a proof in 1993 that built on work by Gerhard Frey, Jean-Pierre Serre, and Ken Ribet by proving cases of the Taniyama–Shimura–Weil conjecture for semistable elliptic curves. Wiles' strategy used techniques from modular forms, Galois representations, and Iwasawa theory and relied on contributions by collaborators including Richard Taylor. After a gap was found in the initial argument by peer reviewers at Princeton University and the wider mathematical community, Wiles and Richard Taylor repaired the proof; the corrected proof was published in 1995 and accepted by journals such as Annals of Mathematics. Wiles was subsequently recognized by awards including the Abel Prize and the Royal Society's honors, and his work influenced research at institutions like Harvard University and University of Oxford.
Mathematical implications and consequences
The proof unified areas of algebraic geometry, arithmetic geometry, and analytic number theory, accelerating progress on problems involving modularity and the arithmetic of elliptic curves studied by mathematicians such as Jean-Pierre Serre, Serre's conjectures, Barry Mazur, and Ken Ribet. It propelled work on the Langlands program by highlighting deep links between Galois representations and automorphic forms, influencing researchers affiliated with the Institute for Advanced Study and universities across Europe and North America. The resolution also affected computational approaches in number theory practiced at places like Los Alamos National Laboratory and spurred advances in algorithmic verification used in projects at Mathematical Sciences Research Institute.
Generalizations and related conjectures
Related problems include the Catalan conjecture (proved by Preda Mihăilescu), the Beal conjecture, and generalizations considered in the context of Diophantine equations and Mordell–Weil theorem, linking to the Birch and Swinnerton-Dyer conjecture and broader aspects of the Langlands program. Work by Gerd Faltings, Andrew Wiles, Richard Taylor, and others continues to inform conjectures connecting modular forms, automorphic representations, and the arithmetic of abelian varieties studied at institutions like École Normale Supérieure and Max Planck Institute for Mathematics. The legacy also influences modern curricula at universities including Princeton University, University of Cambridge, and Massachusetts Institute of Technology where advanced courses explore these interconnected topics.