Neverending Fractions

Transcription

1 Neverending Fractions Despite their classical nature, continued fractions are a neverending research area, with a body of results accessible enough to suit a wide audience, from researchers to students and even amateur enthusiasts. Neverending Fractions brings these results together, offering fresh perspectives on a mature subject. Beginning with a standard introduction to continued fractions, the book covers a diverse range of topics, including elementary and metric properties, quadratic irrationals and more exotic topics such as folded continued fractions and Somos sequences. Along the way, the authors reveal some amazing applications of the theory to seemingly unrelated problems in number theory. Previously scattered throughout the literature, these applications are brought together in this volume for the first time. A wide variety of exercises guide readers through the material, and this will be especially helpful to readers using the book for self-study; the authors also provide many pointers to the literature.

2 AUSTRALIAN MATHEMATICAL SOCIETY LECTURE SERIES Editor-in-chief: Professor C. Praeger, School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia Editors: Professor P. Broadbridge, School of Engineering and Mathematical Sciences, La Trobe University, Victoria 3086, Australia Professor M. Murray, School of Mathematical Sciences, University of Adelaide, SA 5005, Australia Professor J. Ramagge, School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia Professor M. Wand, School of Mathematical Sciences, University of Technology, Sydney, NSW 2007, Australia 1. Introduction to Linear and Convex Programming, N. CAMERON 2. Manifolds and Mechanics, A. JONES, A. GRAY & R. HUTTON 3. Introduction to the Analysis of Metric Spaces, J. R. GILES 4. An Introduction to Mathematical Physiology and Biology, J. MAZUMDAR 5. 2-Knots and their Groups, J. HILLMAN 6. The Mathematics of Projectiles in Sport, N. DE MESTRE 7. The Petersen Graph, D. A. HOLTON & J. SHEEHAN 8. Low Rank Representations and Graphs for Sporadic Groups, C. E. PRAEGER & L. H. SOICHER 9. Algebraic Groups and Lie Groups, G. I. LEHRER (ed.) 10. Modelling with Differential and Difference Equations, G. FULFORD, P. FORRESTER & A. JONES 11. Geometric Analysis and Lie Theory in Mathematics and Physics, A. L. CAREY & M. K. MURRAY (eds.) 12. Foundations of Convex Geometry, W. A. COPPEL 13. Introduction to the Analysis of Normed Linear Spaces, J. R. GILES 14. Integral: An Easy Approach after Kurzweil and Henstock, L. P. YEE & R. VYBORNY 15. Geometric Approaches to Differential Equations, P. J. VASSILIOU & I. G. LISLE (eds.) 16. Industrial Mathematics, G. R. FULFORD & P. BROADBRIDGE 17. A Course in Modern Analysis and its Applications, G. COHEN 18. Chaos: A Mathematical Introduction, J. BANKS, V. DRAGAN & A. JONES 19. Quantum Groups, R. STREET 20. Unitary Reflection Groups, G. I. LEHRER & D. E. TAYLOR 21. Lectures on Real Analysis, F. LÁRUSSON 22. Representations of Lie Algebras, A. HENDERSON

3 Australian Mathematical Society Lecture Series: 23 Neverending Fractions An Introduction to Continued Fractions JONATHAN BORWEIN University of Newcastle, New South Wales ALF VAN DER POORTEN Macquarie University, Sydney JEFFREY SHALLIT University of Waterloo, Ontario WADIM ZUDILIN University of Newcastle, New South Wales

4 University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. Information on this title: / c Jonathan Borwein, The Estate of Alf van der Poorten, Jeffrey Shallit and Wadim Zudilin 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by CPI Group Ltd, Croydon CR0 4YY A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Borwein, Jonathan M. Neverending fractions: an introduction to continued fractions / Jonathan Borwein, University of Newcastle, New South Wales, Alf van der Poorten, Macquarie University, Sydney, Jeffrey Shallit, University of Waterloo, Ontario, Wadim Zudilin, University of Newcastle, New South Wales. pages cm (Australian Mathematical Society lecture series; 23) Includes bibliographical references and index. ISBN (alk. paper) 1. Continued fractions. 2. Processes, Infinite. 3. Fractions. I. Van Der Poorten, A. J. II. Shallit, Jeffrey Outlaw. III. Zudilin, Wadim. IV. Title. V. Title: Continued fractions. QA295.B dc ISBN Paperback Additional resources for this publication at Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

5 Contents Preface page ix 1 Some preliminaries from number theory Divisibility in Z. Euclidean algorithm Primes Fibonacci numbers and the complexity of the Euclidean algorithm Approximation of real numbers by rationals Farey sequences 16 Notes 21 2 Continued fractions A π-overview of the theory Finite continued fractions and the Euclidean algorithm Algebraic theory of continued fractions Relations for continuants of a continued fraction Continued fraction of a real number A taste of Diophantine approximation Equivalent numbers Continued fraction of a quadratic irrational The Euler Lagrange theorem Examples of non-periodic continued fractions The continued fraction for e The order of approximation of e by rationals Bounded partial quotients In the footsteps of Maillet 54 Notes 61 v

6 vi Contents 3 Metric theory of continued fractions Partial quotients of a number as functions of that number Growth of partial quotients of a typical real number Approximation of almost all real numbers by rationals Gauss Kuzmin statistics 75 Notes 77 4 Quadratic irrationals through a magnifier Continued fractions of algebraic numbers Quadratic irrationals revisited Units and Pell s equation Negative expansions and the many surprises of the number Notes 94 5 Hyperelliptic curves and Somos sequences Two surprising allegations Continued fractions in function fields Units and torsion The elliptic case Higher genus 112 Notes From folding to Fibonacci Folding Zaremba s conjecture On to Fibonacci Folding and rippling 120 Notes The integer part of qα + β Inhomogeneous Diophantine approximation Lambert series expansions of generating functions High-precision fraud 143 Notes The Erdős Moser equation Arithmetic and analysis of the equation Asymptotics and continued fraction for log Efficient ways of computing continued fractions Bounds for solutions 158 Notes 159

7 Contents vii 9 Irregular continued fractions General theory Euler continued fraction Gauss continued fraction for the hypergeometric function Ramanujan s AGM continued fraction An irregular continued fraction for ζ(2) = π 2 / The irrationality of π Notes 184 Appendix A Selected continued fractions 188 References 200 Index 209

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9 Preface This book arose from many lectures the authors delivered independently at different locations to students of different levels. Theory is a scientific name for story. So, if the reader somehow feels uncomfortable about following a theory of continued fractions, he or she might be more content to read the story of neverending fractions. The queen of mathematics number theory remains one of the most accessible parts of significant mathematical knowledge. Continued fractions form a classical area within number theory, and there are many textbooks and monographs devoted to them. Despite their classical nature, continued fractions remain a neverending research field, many of whose results are elementary enough to be explained to a wide audience of graduates, postgraduates and researchers, as well as teachers and even amateurs in mathematics. These are the people to whom this book is addressed. After a standard introduction to continued fractions in the first three chapters, including generalisations such as continued fractions in function fields and irregular continued fractions, there are six topics chapters. In these we give various amazing applications of the theory (irrationality proofs, generating series, combinatorics on words, Somos sequences, Diophantine equations and many other applications) to seemingly unrelated problems in number theory. The main feature that we would like to make apparent through this book is the naturalness of continued fractions and of their expected appearance in mathematics. The book is a combination of formal and informal styles. The aforementioned applications of continued fractions are, for the most part, not to be found in earlier books but only in scattered scientific articles and lectures. We have included various remarks and exercises but have been sparing with the latter. In the topics chapters we do not always give full details. Needless to say, all topics can be followed up in the end notes for each chapter and through the references. ix

10 x Preface We would like to thank many colleagues for useful conversations during the development of this book, especially Mumtaz Hussain, Pieter Moree and James Wan. We are also deeply indebted to the copy-editor Susan Parkinson whose incisive and tireless work on the book has enhanced its appearance immeasurably. Finally, Alf van der Poorten ( ) died before this book could be brought to fruition. He was both a good friend and a fine colleague. We offer this book both in his memory and as a way of bringing to a more general audience some of his wonderful contributions to the area. Chapters 4, 5 and 6 originate in lectures Alf gave in the last few years of his life and, for matters of both taste and necessity, they are largely left as presentations in his unique and erudite style. Alfred Alf Jacobus van der Poorten (16 May 1942 in Amsterdam 9 October 2010 in Sydney) A full mathematical biography of Alf is to be found in the 2013 volume dedicated to his memory [31]. Jon Borwein, Jeffrey Shallit, and Wadim Zudilin Newcastle and Waterloo