Preface

In 1842 the Belgian mathematician Eugène Charles Catalan asked whether 8 and 9 are the only consecutive pure powers of nonzero integers. One hundred and sixty years after, the question was answered affirmatively by the Swiss mathematician of Romanian origin Preda Mihăilescu. In other words, 3 2 D 1 is the only solution of the equation x y D 1 in integers x; y; p; q with xy ¤ 0 and p; q 2. Since 2002, the different steps of the proof have been presented by various authors; see, for instance, the expository articles [9, 10, 82]. Complete proofs appeared in monographs by Schoof [124] and Cohen [25, 26]. 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 such as ideal decomposition, units (including the Dirichlet unit theorem), ideal classes, and finiteness of the class group. From the ramification theory we use only one basic fact: a prime number is ramified in a number field if and only if it divides the discriminant. All necessary facts from algebraic number theory are gathered (without proofs) in Appendix A. We do not assume any knowledge about cyclotomic fields; everything needed is defined or proved in the book. With our minimalistic approach, some omissions were inevitable. For instance, an experienced reader can notice that many arguments in this book have an obvious non-Archimedean flavor. Nevertheless, we resisted the temptation of broader use of the language of (non-Archimedean) valuations. Our main motivation was that a matured reader will easily reveal the non-Archimedean context wherever it is hidden, but abusing the non-Archimedean language may create problems for a less knowledgeable reader. Another example is restricting to commutative groups and rings in Appendices C and D. Of course, certain results from these appendices (like the theorem of Maschke) extend to noncommutative case as well. We, however, assume commutativity, because this makes the arguments technically simpler and is sufficient for our purposes.