Bilu / Bugeaud / Mignotte The Problem of Catalan

2014
ISBN: 978-3-319-10094-4
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 245 Seiten, eBook

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 Mihailescu. In other words, 3 2 – 2 3 = 1 is the only solution of the equation x p – y q = 1 in integers x, y, p, q with xy ? 0 and p, q = 2. In this book we give a complete and (almost) self-contained exposition of Mihailescu’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.

Zielgruppe

Research

Autoren/Hrsg.

Bilu, Yuri F.

Bilu, Yuri

Bugeaud, Yann

Mignotte, Maurice

Inhaltsverzeichnis

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.- Mihailescu’s Ideal.- The Real Part of Mihailescu’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.