The Problem of Catalan

Nonfiction, Science & Nature, Mathematics, Number Theory, Algebra
Cover of the book The Problem of Catalan by Yann Bugeaud, Maurice Mignotte, Yuri F. Bilu, Springer International Publishing
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Author: Yann Bugeaud, Maurice Mignotte, Yuri F. Bilu ISBN: 9783319100944
Publisher: Springer International Publishing Publication: October 9, 2014
Imprint: Springer Language: English
Author: Yann Bugeaud, Maurice Mignotte, Yuri F. Bilu
ISBN: 9783319100944
Publisher: Springer International Publishing
Publication: October 9, 2014
Imprint: Springer
Language: English

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 Mihăilescu. In other words, 32 – 23 = 1 is the only solution of the equation xpyq = 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 Mihăilescu’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.

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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 Mihăilescu. In other words, 32 – 23 = 1 is the only solution of the equation xpyq = 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 Mihăilescu’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.

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