Catalan's Conjecture

  • René Schoof

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-ix
  2. René Schoof
    Pages 1-8
  3. René Schoof
    Pages 9-11
  4. René Schoof
    Pages 13-16
  5. René Schoof
    Pages 17-20
  6. René Schoof
    Pages 21-31
  7. René Schoof
    Pages 33-40
  8. René Schoof
    Pages 41-46
  9. René Schoof
    Pages 47-53
  10. René Schoof
    Pages 55-64
  11. René Schoof
    Pages 65-68
  12. René Schoof
    Pages 69-75
  13. René Schoof
    Pages 77-84
  14. René Schoof
    Pages 85-90
  15. René Schoof
    Pages 91-94
  16. René Schoof
    Pages 95-106
  17. René Schoof
    Pages 107-115
  18. Back Matter
    Pages 117-124

About this book

Introduction

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.

Keywords

Algebra Arithmetic Catalan's conjecture algebraic number theory diophantine equations number theory proof theorem

Authors and affiliations

  • René Schoof
    • 1
  1. 1.Università di Roma Tor VergataRomeUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-84800-185-5
  • Copyright Information Springer-Verlag London 2008
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-84800-184-8
  • Online ISBN 978-1-84800-185-5
  • Series Print ISSN 0172-5939
  • About this book