Diophantine equations

Diophantine equations
Gustavb · CC BY-SA 3.0 · source
Name	Diophantine equations
Caption	Example of integer solutions
Field	Number theory
Introduced	Antiquity
Notable	Pierre de Fermat; Srinivasa Ramanujan; Yuri Matiyasevich

Contents

Diophantine equations are polynomial equations for which integer or rational solutions are sought, studied within Number theory and connected to problems in Algebraic geometry, Combinatorics, Cryptography, and Computational complexity theory. They arose in antiquity alongside problems treated in works by Diophantus of Alexandria, and were developed through contributions by Pierre de Fermat, Leonhard Euler, Carl Friedrich Gauss, Srinivasa Ramanujan, and Yuri Matiyasevich, with modern impact on Andrew Wiles's proof of the Taniyama–Shimura conjecture and links to the Hilbert's tenth problem. Research on these equations connects to institutions such as the Institut des Hautes Études Scientifiques, the Princeton University, the University of Cambridge, and publications like the Annals of Mathematics and Acta Arithmetica.

Overview and Definitions

A Diophantine equation is classically defined as a polynomial equation whose solutions are required to lie in the integers or rationals, a notion treated in texts by Diophantus of Alexandria, Pierre de Fermat, Joseph-Louis Lagrange, Carl Friedrich Gauss, and David Hilbert. Typical examples include linear Diophantine equations, exponential Diophantine equations, and homogeneous forms as studied by Srinivasa Ramanujan, Ernest Nagel, Harvey Friedman, and Alexander Grothendieck in related settings. The scope includes equations over rings of integers in number fields, which appear in work by Ernst Kummer, Heinrich Weber, Emmy Noether, and Richard Dedekind, and it intersects conjectures from Tate conjecture contexts and the Birch and Swinnerton-Dyer conjecture.

Problems seeking integer solutions date to Diophantus of Alexandria and later were transformed by Pierre de Fermat into challenges like Fermat's Last Theorem, whose proof involved research by Ernst Kummer, Gerd Faltings, Andrew Wiles, and Richard Taylor. Systematic study progressed in treatises by Joseph-Louis Lagrange and Carl Friedrich Gauss, continued through the 19th and 20th centuries by Srinivasa Ramanujan, Louis Mordell, Harold Davenport, Kurt Gödel, and culminated in Matiyasevich's solution to Hilbert's tenth problem building on work by Julia Robinson, Martin Davis, and Hilbert. Institutional support across the University of Göttingen, École Normale Supérieure, University of Oxford, and Massachusetts Institute of Technology fostered developments linking these problems to conjectures such as the Taniyama–Shimura conjecture and tools like the Theory of elliptic curves.

Classes include linear equations as studied by Carl Gustav Jacob Jacobi and Adrien-Marie Legendre; quadratic forms treated by Adrien-Marie Legendre and Joseph-Louis Lagrange; exponential equations explored by Mihăilescu's theorem's antecedents in works of Carl Ludwig Siegel and Alan Baker; Thue equations developed by Axel Thue and extended by Klaus Roth; elliptic curve equations central to work by André Weil, John Tate, Barry Mazur, and Vojta conjecture-related researchers; and higher-degree Diophantine varieties analyzed with techniques from Alexander Grothendieck and Jean-Pierre Serre.

Solution methods draw on algebraic number theory from Ernst Kummer and Richard Dedekind, descent methods used by Pierre de Fermat and modernized by John Tate and Gerd Faltings, the method of Thue–Siegel–Roth influenced by Axel Thue and Klaus Roth, transcendence techniques from Alan Baker, and modular methods pioneered in the context of Fermat's Last Theorem by Andrew Wiles and Ken Ribet. Other tools include the theory of heights associated with Enrico Bombieri and Paul Vojta, diophantine approximation from Alexander Ostrowski and Aleksandr Khinchin, p-adic methods tied to Kenkichi Iwasawa and Jean-Pierre Serre, and computational algebra systems developed at institutions like Massachusetts Institute of Technology and Centre national de la recherche scientifique.

Decidability and complexity issues were crystallized by David Hilbert's list and resolved in part by Yuri Matiyasevich's theorem building on work by Julia Robinson, Martin Davis, and Hilary Putnam, linking Diophantine decidability to Turing machine computations and Gödel's incompleteness theorems. Algorithms for solving special classes use lattice reduction methods from Hendrik Lenstra, Arjen Lenstra, H. W. Lenstra, Jr., and Michał Karpinski, modular techniques exploited by Noam Elkies and John Cremona, and computer-assisted proofs exemplified by work at Princeton University and University of Washington.

Key results include Fermat's Last Theorem proved by Andrew Wiles with contributions from Richard Taylor; Matiyasevich's theorem resolving Hilbert's tenth problem using work by Julia Robinson, Martin Davis, and Hilary Putnam; Mordell's conjecture proved by Gerd Faltings; the proof of the Taniyama–Shimura conjecture for semistable elliptic curves by Andrew Wiles and Richard Taylor; Baker's effective results in transcendence theory from Alan Baker; and Mazur's results on rational isogenies by Barry Mazur with influence on computational aspects developed by John Cremona and Noam Elkies.