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 Mihilescu. 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 Mihilescu'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.

Extensively covers the Catalan Conjecture and Mihailescu's subsequent proof Includes thorough exposition of cyclotomic fields Provides a bridge between number theory and classical analysis Includes supplementary material: sn.pub/extras

Inhalt
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.- Mihilescu's Ideal.- The Real Part of Mihilescu'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.