Title Problem of Catalan / Yuri F. Bilu, Yann Bugeaud, Maurice Mignotte

Published Switzerland : Springer, 2014

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Description 1 online resource Contents 880-01 An Historical Account -- Even Exponents -- Cassels' Relations -- Cyclotomic Fields -- Dirichlet L-Series and Class Number Formulas -- Higher Divisibility Theorems -- Gauss Sums and Stickelberger's Theorem -- Mihăilescu?s Ideal -- The Real Part of Mihăilescu?s Ideal -- Cyclotomic units -- Selmer Group and Proof of Catalan's Conjecture -- The Theorem of Thaine -- Baker's Method and Tijdeman's Argument -- Appendix A: Number Fields -- Appendix B: Heights -- Appendix C: Commutative Rings, Modules, Semi-Simplicity -- Appendix D: Group Rings and Characters -- Appendix E: Reduction and Torsion of Finite G-Modules -- Appendix F: Radical Extensions 880-01/(S Machine generated contents note: 1. Historical Account -- 1.1. Catalan's Note Extraite -- 1.2. Particular Cases -- 1.3. Cassels' Relations -- 1.4. Analysis: Logarithmic Forms -- 1.5. Algebra: Cyclotomic Fields -- 1.6. Numerical Results -- 1.7. Final Attack -- 2. Even Exponents -- 2.1. Equation xp = y2 + 1 -- 2.2. Units of Real Quadratic Rings -- 2.3. Equation x2 -- y" = 1 with Q [≥] 5 -- 2.4. Cubic Field Q(3[√]2) -- 2.5. Equation x2 -- y3 = 1 -- 3. Cassels' Relations -- 3.1. Cassels' Divisibility Theorem and Cassels' Relations -- 3.2. Binomial Power Series -- 3.3. Proof of the Divisibility Theorem -- 3.4. Hyyro's Lower Bounds -- 4. Cyclotomic Fields -- 4.1. Degree and Galois Group -- 4.2. Integral Basis and Discriminant -- 4.3. Decomposition of Primes -- 4.4. Units -- 4.5. Real Cyclotomic Field and the Class Group -- 4.6. Cyclotomic Extensions of Number Fields -- 4.7. General Cyclotomic Fields -- 5. Dirichlet L-Series and Class Number Formulas -- 5.1. Dirichlet Characters and L-Series -- 5.2. Dedekind ζ-Function of the Cyclotomic Field -- 5.3. Calculating L(1, χ) for χ [≠] 1 -- 5.4. Class Number Formulas -- 5.5. Composite Moduli -- 6. Higher Divisibility Theorems -- 6.1. Most Important Lemma -- 6.2. Inkeri's Divisibility Theorem -- 6.3. Deviation: Catalan's Problem with Exponent 3 -- 6.4. Group Ring -- 6.5. Stickelberger, Mihailescu, and Wieferich -- 7. Gauss Sums and Stickelberger's Theorem -- 7.1. Stickelberger's Ideal and Stickelberger's Theorem -- 7.2. Gauss Sums -- 7.3. Multiplicative Combinations of Gauss Sums -- 7.4. Prime Decomposition of a Gauss Sum -- 7.5. Proof of Stickelberger's Theorem -- 7.6. Kummer's Basis -- 7.7. Real and the Relative Part of Stickelberger's Ideal -- 7.8. Proof of Iwasawa's Class Number Formula -- 8. Mihailescu's Ideal -- 8.1. Definitions and Main Theorems -- 8.2. Algebraic Number (x -- ζ) -- 8.3. qth Root of (x -- ζ) -- 8.4. Proof of Theorem 8.2 -- 8.5. Proof of Theorem 8.4 -- 8.6. Application to Catalan's Problem I: Divisibility of the Class Number -- 8.7. Application to Catalan's Problem II: Mihailescu's Ideal vs Stickelberger's Ideal -- 8.8. On the Real Part of Mihailescu's Ideal -- 9. Real Part of Mihailescu's Ideal -- 9.1. Main Theorem -- 9.2. Products of Binomial Power Series -- 9.3. Mihailescu's Series (1 + ζ T)/q -- 9.4. Proof of Theorem 9.2 -- 10. Cyclotomic Units -- 10.1. Circulant Determinant -- 10.2. Cyclotomic Units -- 11. Selmer Group and Proof of Catalan's Conjecture -- 11.1. Selmer Group -- 11.2. Selmer Group as Galois Module -- 11.3. Units as Galois Module -- 11.4. q-Primary Cyclotomic Units -- 11.5. Proof of Theorem 11.5 -- 12. Theorem of Thaine -- 12.1. Introduction -- 12.2. Preparations -- 12.3. Proof of Theorem 12.2 -- 12.4. Reduction of a Multiplicative Group Modulo a Prime Ideal -- 12.5. Reduction of a Multiplicative Group Modulo a Prime Number and Proof of Theorem 12.3 -- 13. Baker's Method and Tijdeman's Argument -- 13.1. Introduction: Thue, Gelfond, and Baker -- 13.2. Heights in Finitely Generated Groups -- 13.3. Almost nth Powers -- 13.4. Effective Analysis of Classical Diophantine Equations -- 13.5. Theorem of Schinzel and Tijdeman and the Equation of Pillai -- 13.6. Tijdeman's Argument -- Appendix A Number Fields -- A.1. Embeddings, Integral Bases, and Discriminant -- A.2. Units, Regulator -- A.3. Ideals, Factorization -- A.4. Norm of an Ideal -- A.5. Ideal Classes, the Class Group -- A.6. Prime Ideals, Ramification -- A.7. Galois Extensions -- A.8. Valuations -- A.9. Dedekind ζ-Function -- A.10. Chebotarev Density Theorem -- A.11. Hilbert Class Field -- Appendix B Heights -- Appendix C Commutative Rings, Modules, and Semi-simplicity -- C.1. Cyclic Modules -- C.2. Finitely Generated Modules -- C.3. Semi-simple Modules -- C.4. Semi-simple Rings -- C.5. "Dual" Module -- Appendix D Group Rings and Characters -- D.1. Weight Function and the Norm Element -- D.2. Characters of a Finite Abelian Group -- D.3. Conjugate Characters -- D.4. Semi-simplicity of the Group Ring -- D.5. Idempotents -- Appendix E Reduction and Torsion of Finite G -Modules -- E.1. Telescopic Rings -- E.2. Products of Telescopic Rings -- E.3. Elementary Divisors and Finitely Generated Modules -- E.4. Reduction and Torsion -- Appendix F Radical Extensions -- F.1. Field Generated by a Single Root -- F.2. Kummer's Theory -- F.3. General Radical Extensions -- F.4. Equivariant Kummer's Theory Summary In 1842 the Belgian mathematician Eugène Charles Catalan asked whether 8 and 9 are the only consecutive pure powers of non-zero integers. 160 years after, the question was answered affirmatively by the Swiss mathematician of Romanian origin Preda Mihăilescu. In this book we give a complete and (almost) self-contained exposition of Mihăilescu?s work, which must be understandable by a curious university student, not necessarily specializing in Number Theory. We assume very modest background: a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory Analysis wiskunde mathematics getallenleer number theory algebra Mathematics (General) Wiskunde (algemeen) Notes Online resource; title from PDF title page (EBSCO, viewed October 16, 2014) Subject Consecutive powers (Algebra) Problem solving. Problem Solving MATHEMATICS -- Algebra -- Intermediate. Problemas-Resolución Consecutive powers (Algebra) Problem solving Form Electronic book Author Bugeaud, Yann, 1971- author. Mignotte, Maurice, author. ISBN 9783319100944 3319100947 3319100939 9783319100937

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