Yuri F. Bilu, Yann Bugeaud, Maurice Mignotte

The Problem of Catalan (eBook, PDF)

• Format: PDF

Produktbeschreibung

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 other words, 3² – 2³ = 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 Mihăilescu’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.

Produktdetails

• Verlag: Springer International Publishing
• Erscheinungstermin: 9. Oktober 2014
• Englisch
• ISBN-13: 9783319100944
• Artikelnr.: 43788714

Inhaltsangabe

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.

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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.