Neverending Fractions An Introduction to Continued Fractions ( c March 13, 2014: do not circulate) Jonathan Borwein, Alf van der Poorten, Jeffrey Shallit, and Wadim Zudilin
Contents Preface page ix 1 Some preliminaries from number theory 1 1.1 Divisibility in Z. Euclidean algorithm 1 1.2 Primes 5 1.3 Fibonacci numbers and the complexity of the Euclidean algorithm 8 1.4 Approximation of real numbers by rationals 11 1.5 Farey sequences 16 Notes 21 2 Continued fractions 23 2.1 A π-overview of the theory 23 2.2 Finite continued fractions and the Euclidean algorithm 25 2.3 Algebraic theory of continued fractions 27 2.4 Relations for continuants of a continued fraction 30 2.5 Continued fraction of a real number 32 2.6 A taste of Diophantine approximation 35 2.7 Equivalent numbers 37 2.8 Continued fraction of a quadratic irrational 40 2.9 The Euler Lagrange theorem 43 2.10 Examples of non-periodic continued fractions 45 2.11 The continued fraction for e 48 2.12 The order of approximation of e by rationals 49 2.13 Bounded partial quotients 52 2.14 In the footsteps of Maillet 54 Notes 61 v
vi Contents 3 Metric theory of continued fractions 64 3.1 Partial quotients of a number as functions of that number 65 3.2 Growth of partial quotients of a typical real number 67 3.3 Approximation of almost all real numbers by rationals 70 3.4 Gauss Kuzmin statistics 75 Notes 77 4 Quadratic irrationals through a magnifier 80 4.1 Continued fractions of algebraic numbers 80 4.2 Quadratic irrationals revisited 81 4.3 Units and Pell s equation 86 4.4 Negative expansions and the many surprises of the number 163 91 Notes 94 5 Hyperelliptic curves and Somos sequences 97 5.1 Two surprising allegations 97 5.2 Continued fractions in function fields 99 5.3 Units and torsion 103 5.4 The elliptic case 108 5.5 Higher genus 112 Notes 113 6 From folding to Fibonacci 114 6.1 Folding 114 6.2 Zaremba s conjecture 117 6.3 On to Fibonacci 118 6.4 Folding and rippling 120 Notes 126 7 The integer part of qα + β 127 7.1 Inhomogeneous Diophantine approximation 127 7.2 Lambert series expansions of generating functions 134 7.3 High-precision fraud 143 Notes 144 8 The Erdős Moser equation 147 8.1 Arithmetic and analysis of the equation 147 8.2 Asymptotics and continued fraction for log 2 152 8.3 Efficient ways of computing continued fractions 154 8.4 Bounds for solutions 158 Notes 159
Contents vii 9 Irregular continued fractions 161 9.1 General theory 161 9.2 Euler continued fraction 164 9.3 Gauss continued fraction for the hypergeometric function 166 9.4 Ramanujan s AGM continued fraction 169 9.5 An irregular continued fraction for ζ(2) = π 2 /6 173 9.6 The irrationality of π 2 179 Notes 184 Appendix A Selected continued fractions 188 References 200 Index 209
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
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 (1942 2010) 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