The Problem of Catalan
(Sprache: Englisch)
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...
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Klappentext zu „The Problem of Catalan “
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, 32 - 23 = 1 is the only solution of the equation xp - yq = 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.
Inhaltsverzeichnis zu „The Problem of Catalan “
- 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
- Mih?ilescu's Ideal
- The Real Part of Mih?ilescu'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
Bibliographische Angaben
- Autoren: Yuri F. Bilu , Yann Bugeaud , Maurice Mignotte
- 2014, XIV, 245 Seiten, Masse: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer, Berlin
- ISBN-10: 3319100939
- ISBN-13: 9783319100937
- Erscheinungsdatum: 27.10.2014
Sprache:
Englisch
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