Catalan's Conjecture

Author:   René Schoof
Publisher:   Springer London Ltd
Edition:   2009 ed.
ISBN:  

9781848001848


Pages:   124
Publication Date:   02 December 2008
Format:   Paperback
Availability:   Awaiting stock   Availability explained
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Catalan's Conjecture


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Overview

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 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.

Full Product Details

Author:   René Schoof
Publisher:   Springer London Ltd
Imprint:   Springer London Ltd
Edition:   2009 ed.
Dimensions:   Width: 15.50cm , Height: 0.70cm , Length: 23.50cm
Weight:   0.454kg
ISBN:  

9781848001848


ISBN 10:   1848001843
Pages:   124
Publication Date:   02 December 2008
Audience:   College/higher education ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Paperback
Publisher's Status:   Active
Availability:   Awaiting stock   Availability explained
The supplier is currently out of stock of this item. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out for you.

Table of Contents

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.

Reviews

"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. Gouvêa, 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)"


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)


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