The Problem of Catalan

By: Yuri F. Bilu and Yann Bugeaud and Maurice Mignotte

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Synopsis: 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 MihÄfilescu. In other words, 32 - 23 = 1 is the only solution of the equation xp - yq = 1 in integers x, y, p, q with xy ≠ 0 and p, q ≥ 2. In this book we give a complete and (almost) self-contained exposition of MihÄfilescu's work, which must be understandable by a curious university student, not necessarily specializing in Number Theory. We assume a very modest background:a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory.