Catalan's conjecture

Catalan's conjecture is the statement that the only solution in integers greater than 1 to the exponential Diophantine equation x^a − y^b = 1 with a,b > 1 is 3^2 − 2^3 = 1. The problem connects to classical topics in Fermat-type equations, links to algebraic number theory through cyclotomic fields and Galois modules, and culminated in a proof by Preda Mihăilescu in 2002 now known as Mihăilescu's theorem.

Statement

The conjecture asserts that the unique pair of consecutive perfect powers (powers larger than 1) is given by 9 and 8, i.e. 3^2 and 2^3. This formulation ties to earlier work on the Catalan sequence and to problems studied by Lagrange, Legendre, and Germain in the context of integer solutions of exponential equations. It is an explicit instance of an exponential Diophantine equation in the tradition of Fermat and Galois-inspired algebraic approaches that later engaged researchers at institutions such as the École Polytechnique and the University of Göttingen.

History and context

The conjecture was proposed by Eugène Catalan in 1844, following investigations into perfect powers by contemporaries including Euler and Gauss. Early partial results were obtained by Lebesgue and Hensel who used p-adic methods and cyclotomic techniques connected to work by Kummer on unique factorization in cyclotomic fields. During the late 19th and 20th centuries, mathematicians such as Tijdeman, Chevalley, and Zassenhaus contributed bounds and finiteness results inspired by the Thue–Siegel–Roth theorem lineage, while connections to the abc conjecture and to results of Faltings (formerly Faltings's theorem) influenced strategy. Progress employed methods from algebraic number theory developed by figures like Weber and Hilbert, and analytic techniques traced to Selberg and Weil. Important milestones include Tijdeman's 1976 finiteness theorem using linear forms in logarithms, building on work by Baker and Yu, and computational verifications by teams at institutions such as CNRS and universities including University of Cambridge and University of Bonn.

Proof (Mihăilescu's theorem)

Preda Mihăilescu announced a proof in 2002 that established the conjecture as a theorem, using a synthesis of algebraic methods rooted in cyclotomic units and Galois module structure. The strategy built on earlier frameworks by Frey and Ribet in relating Diophantine problems to algebraic structures, and on techniques from Iwasawa theory developed by Iwasawa and Coates. Mihăilescu's argument exploited properties of cyclotomic fields studied by Kummer and properties of units in these fields as investigated by Weber and Washington, making crucial use of the theory of primitive roots and Galois module criteria previously considered by Pearson and Zassenhaus. Key steps reduced the problem to the nonexistence of certain combinatorial configurations of cyclotomic units, invoking results shaped by the work of Serre and Shimura on modularity paradigms and by contributions from Tate on cohomological methods. The final publication refined arguments that had origins in techniques employed by Grothendieck-influenced algebraic number theory and in explicit logarithmic form estimates from Baker.

Consequences and related results

Mihăilescu's theorem settled a long-standing question posed in the era of Cauchy and Abel, influencing further research on exponential Diophantine equations including Catalan-type problems and Pillai's conjecture examined by Pillai. It sharpened understanding of perfect powers, feeding into work on the abc conjecture by Oesterlé and Masser and affecting studies of Diophantine finiteness like Siegel's theorem and Faltings's theorem. The theorem intersects with computational number theory projects at ERC-funded centers and with explicit methods advanced by researchers including Bennett, Andreescu, and Elkies. It also provided context for later explorations in modular approaches related to the proof of Fermat's Last Theorem by Wiles and Taylor, and motivated refinements in effective bounds developed by Murty and Bombieri.

Generalizations and open problems

Generalizations consider equations x^a − y^b = c for fixed integers c (Pillai-type equations) and broader families such as Brocard, Nagell–Ljunggren, and Lebesgue–Nagell problems studied by Nagell and Abel-inspired lines. Open problems include effective versions of finiteness statements treated by Baker and quantitative aspects tied to the abc conjecture by Oesterlé and Masser, and the complete classification of near-misses between perfect powers pursued by researchers like Riesel and Mihăilescu's collaborators. Further directions relate to the arithmetic of cyclotomic units in Iwasawa towers treated by Greenberg and to explicit lower bounds for linear forms in logarithms studied by Baker and Nesterenko. These problems remain central to contemporary research programs at institutions such as the Max Planck Institute for Mathematics, Institute for Advanced Study, and major universities worldwide.

Category:Number theory