Catalan's Conjecture
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
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Ren´e Schoof
Catalan’s Conjecture
123
Ren´e Schoof Universit`a di Roma Tor Vergata Italia [email protected]
Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Universit`a degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley ´ Claude Sabbah, CNRS, Ecole Polytechnique Endre S¨uli, University of Oxford Wojbor Woyczynski, Case Western Reserve University
ISBN: 978-1-84800-184-8 DOI: 10.1007/978-1-84800-185-5
e-ISBN: 978-1-84800-185-5
British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2008933674 Mathematics Subject Classification (2000): 11D41; 11D61; 11R18 c Springer-Verlag London Limited 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Whilst we have made considerable efforts to contact all holders of copyright material contained in this book, we have failed to locate some of them. Should holders wish to contact the Publisher, we will be happy to come to some arrangement with them. Printed on acid-free paper Springer Science+Business Media springer.com
Preface
A first draft of this book was written in August 2003. It was based on two sets of lecture notes by Yuri Bilu. In the following years, the draft was used in student seminars in Rome, on the island of Minorca, and in Regensburg. With the benefit of these experiences, a new version was written. It is the basis of this book. The notes contain a complete proof of Catalan’s conjecture. To read the first few chapters requires little more than a basic mathematical background and some knowledge of elementary number theory. The other chapters involve Galois theory, some more algebraic number theory, and a little bit of commutative algebra [1]. The basic facts from the arithmetic of cyclotomic fields are discussed in the text. This material can also be found in the textbooks Introduction to Cyclotomic Fields by L.C. Washington [5
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