Catalan conjecture

Catalan conjecture The Catalan conjecture is a theorem in number theory asserting a unique solution in integers to a simple exponential Diophantine equation. First conjectured in the 19th century, it remained unresolved through the development of algebraic number theory, Galois theory, p-adic analysis, and advances in modular forms until a proof was published in the early 21st century. The problem connects to work of Pierre de Fermat, Joseph-Louis Lagrange, Adrien-Marie Legendre, and later developments by Ernst Kummer, Leopold Kronecker, and Heinrich Weber.

Statement

The conjecture states that the Diophantine equation x^a − y^b = 1 has only the trivial solution in positive integers aside from the pair x = 3, a = 2, y = 2, b = 3. The assertion relates to properties of integers, prime numbers, and the structure of units in rings of integers of cyclotomic fields studied by Carl Friedrich Gauss, Évariste Galois, Richard Dedekind, and David Hilbert. The formulation invokes exponential equations previously considered in the context of Fermat's Last Theorem and results by Srinivasa Ramanujan, G. H. Hardy, and Louis J. Mordell.

Historical background

The conjecture was proposed by Eugène Charles Catalan in 1844 and was later discussed by Viktor Lebesgue, Alfredo Catalan (same family), and commentators in the milieu of 19th-century mathematics. Early partial results invoked methods associated with Kummer's theorem on cyclotomic fields and regular primes, influenced by the work of Ernst Kummer, Leopold Kronecker, Heinrich Weber, and later investigators such as Hans Rohrbach, Alexander Thue, and Carl Ludwig Siegel. Paths toward a proof involved advances in algebraic number theory by Emil Artin and André Weil, developments in class field theory by Emil Artin and Teiji Takagi, and techniques in Diophantine approximation pioneered by Axel Thue, Kurt Mahler, and Alan Baker. In the 20th century, the problem attracted contributions from Paul Erdős, Heinz Schraeder, Ivan Niven, and analysts applying Galois representations inspired by Jean-Pierre Serre and Barry Mazur.

Proof by Preda Mihăilescu

The proof published by Preda Mihăilescu in 2002 synthesizes earlier reductions and leverages properties of cyclotomic fields, Galois modules, and class groups studied by Kummer, Heinrich Weber, Ernst Kummer, and Richard Dedekind. Mihăilescu's approach builds on results of Robert Tijdeman, who used Baker's method in transcendental number theory, and on analytic and algebraic methods connected to Iwasawa theory developed by Kenkichi Iwasawa and Serre's conjectures addressed by Richard Taylor and Andrew Wiles. The proof employs the theory of cyclotomic units, the structure of the group of units in rings of integers as examined by Leopold Kronecker and Heinrich Weber, and the behavior of class groups and Galois modules studied by Emil Artin and John Coates. The argument resolves the remaining cases left by prior work of Robert Tijdeman, Pólya, and M. Ram Murty by exploiting fine properties of cyclotomic class groups and the so-called "Wieferich criteria" connected to investigations by Arthur Wieferich and later researchers such as Karl Wieferich and H. Hasse.

Generalizations and related results

Generalizations consider broader exponential Diophantine equations and connections to Fermat's Last Theorem proved by Andrew Wiles via modularity theorem for elliptic curves initially conjectured by Taniyama–Shimura–Weil and refined by Goro Shimura and Yutaka Taniyama. Related problems include Pillai's conjecture studied by Subbayya Sivasankaranarayana Pillai and Catalan-type equations with additive constants analyzed by Alan Baker, Kurt Mahler, and T. N. Shorey. Results on the finiteness of solutions to exponential equations have used methods from Baker's theory and transcendence theory by A. O. Gelfond and Theodor Schneider. Connections to Iwasawa theory have been explored by Kenkichi Iwasawa, Ralph Greenberg, and Barry Mazur, while work on cyclotomic fields and class numbers engages Leopold Kronecker, Ernst Kummer, and Heinrich Weber. Computational verifications and searches involved collaborations and software projects tied to institutions such as Mathematical Sciences Research Institute, Institute for Advanced Study, École Normale Supérieure, and research groups featuring Noam Elkies and John Conway.

Consequences and applications

The resolution clarified the landscape of exponential Diophantine equations and informed techniques in algebraic number theory and Diophantine approximation employed in subsequent research by Alan Baker, Gerd Faltings, and Enrico Bombieri. The methods influenced progress on related finiteness theorems such as Siegel's theorem on integral points discussed by Carl Ludwig Siegel and finiteness results by Faltings (formerly Gerd Faltings). Applications appear in the theory of cyclotomic fields, class numbers, and the study of Galois modules associated with works of Emil Artin, John Tate, and Iwasawa; these areas underpin modern research at institutions including Princeton University, University of Cambridge, and Universität Bonn. The theorem stands alongside milestones like Fermat's Last Theorem and contributes to the cumulative development of modern number theory.

Category:Number theory