The Problem of Catalan by Yuri F. BiluThe Problem of Catalan by Yuri F. Bilu

The Problem of Catalan

byYuri F. Bilu, Yann Bugeaud, Maurice Mignotte

Hardcover | October 27, 2014

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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 MihÄilescu. In other words, 32- 23= 1 is the only solution of the equationxp-yq= 1 in integersx, y, p, qwithxy' 0 andp, q2.

In this book we give a complete and (almost) self-contained exposition of MihÄilescu's work, which must be understandable by a curious university student, not necessarily specializing in Number Theory. We assume a very modest background:a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory.

Title:The Problem of CatalanFormat:HardcoverDimensions:245 pages, 23.5 × 15.5 × 0.01 inPublished:October 27, 2014Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3319100939

ISBN - 13:9783319100937

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Table of Contents

An Historical Account.- Even Exponents.- Cassels' Relations.- Cyclotomic Fields.- DirichletL-Series and Class Number Formulas.- Higher Divisibility Theorems.- Gauss Sums and Stickelberger's Theorem.- MihÄilescu's Ideal.- The Real Part of MihÄilescu'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 FiniteG-Modules.- Appendix F: Radical Extensions.