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The Problem of Catalan

Yuri F. Bilu, Yann Bugeaud, and Maurice Mignotte
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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The present work, abbreviated here as BBM, is a largely self-contained proof of the Catalan Conjecture, namely that the equation \(x^p - y^q = 1\) with non-zero integer \(x\) and \(y\) and integer \(p \ge 2, q \ge 2\) has only the one solution \(3^2 - 2^3 = 1\); in other words, 8 and 9 are the only consecutive powers of integers. The book assumes a modest knowledge of algebraic number theory and of Galois theory, but proves everything else. The result, conjectured in 1842 by Eugène Catalan, was proved in 2002 by Preda Mihăilescu. He gave two proofs, one published in 2003 and requiring Baker’s theory of logarithmic forms and some computer calculation, and one published in 2006 that avoids these methods.

A rival work is Schoof’s 2008 Catalan’s Conjecture, which has the same goals and prerequisites as BBM but is half the length. The organization and approach are quite different for the two books, even though they are both presenting Mihăilescu’s 2006 proof, but I’m not enough of a connoisseur to say which is more appealing mathematically. The extra pages of BBM are occupied not by lengthier expositions, but by including a number of interesting sidelines and related results that are not needed for the proof. It also develops the Baker theory of logarithmic forms that was used in Mihăilescu’s first proof but not needed for the second proof. It also includes six appendices that quote or prove the background material about algebraic number theory and other matters that is assumed in the main exposition. Another difference is that Schoof is pitched specifically as a textbook rather than a monograph, and includes extensive exercises at the end of each chapter.

Bottom line: both are good books. I found the exposition easier to follow in Schoof, because it does a better job of outlining where it is going, and because it sticks to the main path to the proof with few sidelines. It’s also less than half the price of BMM.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

​ 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

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

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.