Bilu Y.F., Bugeaud Y., Mignotte M. The Problem of Catalan

New York: Springer, 2014. — 251 p.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 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 very modest background: a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory.A 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