Catalan's Conjecture

• The Case “q = 2”.- The Case “p = 2”.- The Nontrivial 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.- Semisimple Group Rings.- The Plus Argument II.- The Density Theorem.- Thaine’s Theorem.

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. Gouvea, 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)