Synopses & Reviews
Eugène Charles Catalan made his famous conjecture - that 8 and 9 are the only two consecutive perfect powers of natural numbers - in 1844 in a letter to the editor of Crelle's mathematical journal. One hundred and fifty-eight years later, Preda Mihailescu proved it. Catalan's Conjecture presents this spectacular result in a way that is accessible to the advanced undergraduate. The first few sections of the book require little more than a basic mathematical background and some knowledge of elementary number theory, while later sections involve Galois theory, algebraic number theory and a small amount of commutative algebra. The prerequisites, such as the basic facts from the arithmetic of cyclotomic fields, are all discussed within the text. The author dissects both Mihailescu's proof and the earlier work it made use of, taking great care to select streamlined and transparent versions of the arguments and to keep the text self-contained. Only in the proof of Thaine's theorem is a little class field theory used; it is hoped that this application will motivate the interested reader to study the theory further. Beautifully clear and concise, this book will appeal not only to specialists in number theory but to anyone interested in seeing the application of the ideas of algebraic number theory to a famous mathematical problem.
Review
From the reviews: "Catalan's Conjecture is very readable, direct, and full of insight. It manages to completely explain a significant recent result 'from scratch' in a mere 120 pages, quite an achievement these days. It makes for delightful reading for any number theorist, but it would also be an excellent way to learn some algebraic number theory. One could easily build an undergraduate seminar around it; I'm sure students would enjoy it and learn a lot. It's an excellent book." (Fernando Q. Gouvêa, The Mathematical Association of America, April, 2009) "In the monograph under review, the author gives a complete proof of Mihailescu's theorem in about one hundred pages. The exposition is self-contained ... . This monograph is very carefully written, with a lot of useful remarks, comments and exercises." (Yann Bugeaud, Mathematical Reviews, Issue 2009 k) "The present volume by Schoof ... with a debt to notes of Y. Bilu, details an essentially self-contained, comparatively elementary approach. The theory of cyclotomic fields ... happily plays a decisive role here. This work offers a talented mathematics major the perfect basis for a capstone experience: the whole story of a major recent breakthrough settling a very old problem by ingeniously applying a theory of independent importance. ... Summing Up: Highly recommended. Upper-division undergraduates and above." (D. V. Feldman, Choice, Vol. 47 (2), October, 2009)
Synopsis
In 1844, Eugène Charles Catalan conjectured that 8 and 9 are the only two consecutive perfect powers of natural numbers, and it was proved more than 150 years later. This book presents this spectacular result in a highly accessible way.
Table of Contents
Introduction.- The case "q = 2".- The case "p = 2".- The non-trivial solution.- Runge's method.- Cassels' theorem.- An obstruction group.- Small p or q .- The Stickelberger ideal.- The double Wieferich criterion.- The minus argument.- The plus argument I.- Semi-simple group rings.- The plus argument II.- The density theorem.- Thaine's theorem.- Appendix: Euler's theorem.- Bibliography.- Index.