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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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.- 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.
Bibliographische Angaben
- Autoren: Yuri F. Bilu , Yann Bugeaud , Maurice Mignotte
- 2016, Softcover reprint of the original 1st ed. 2014, XIV, 245 Seiten, Masse: 15,5 x 23,6 cm, Kartoniert (TB), Englisch
- Verlag: Springer, Berlin
- ISBN-10: 3319362550
- ISBN-13: 9783319362557
Sprache:
Englisch
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