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The Problem of Catalan

Buch | Hardcover
XIV, 245 Seiten
2014 | 2014
Springer International Publishing (Verlag)
978-3-319-10093-7 (ISBN)

Lese- und Medienproben

The Problem of Catalan - Yuri F. Bilu, Yann Bugeaud, Maurice Mignotte
CHF 74,85 inkl. MwSt

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.

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.

Erscheint lt. Verlag 27.10.2014
Zusatzinfo XIV, 245 p. 3 illus.
Verlagsort Cham
Sprache englisch
Maße 155 x 235 mm
Gewicht 555 g
Themenwelt Mathematik / Informatik Mathematik Algebra
Mathematik / Informatik Mathematik Arithmetik / Zahlentheorie
Mathematik / Informatik Mathematik Wahrscheinlichkeit / Kombinatorik
Schlagworte Algebra • algebraic number theory • Catalan Conjecture • Cyclotomic Fields • Galois Theory
ISBN-10 3-319-10093-9 / 3319100939
ISBN-13 978-3-319-10093-7 / 9783319100937
Zustand Neuware
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