My Numbers, My Friends: Popular Lectures on Number Theory Paulo Ribenboim Springer My Numbers, My Friends Paulo Ribenboim My Numbers, My Friends Popular Lectures on Number Theory Paulo Ribenboim Department of Mathematics and Statistics Queen’s University Kingston, Ontario K7L 3N6 Canada Mathematics Subject Classification (2000): 11-06, 11Axx Library of Congress Cataloging-in-Publication Data Ribenboim, Paulo My numbers, my friends / Paulo Ribenboim p. cm. Includes bibliographical references and index. ISBN 0-387-98911-0 (sc. : alk. paper) 1. Number Theory. I. Title QA241.R467 2000 612’.7— dc21 99-42458 c 2000 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Av- enue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-387-98911-0 Springer-Verlag New York Berlin Heidelberg SPIN 10424971 Contents Preface xi 1 The Fibonacci Numbers and the Arctic Ocean 1 1 Basicdefinitions................... 2 A. Lucassequences............... 2 B. SpecialLucassequences............ 3 C. Generalizations................. 3 2 Basicproperties................... 5 A. Binet’sformulas............... 5 B. DegenerateLucassequences........ 5 C. Growth and numerical calculations . 6 D. Algebraicrelations............... 7 E. Divisibility properties ............ 9 3 Prime divisors of Lucas sequences ......... 10 A. The sets P(U), P(V ), and the rank of appearance.................... 10 B. Primitive factors of Lucas sequences . 17 4 PrimesinLucassequences............. 26 5 Powers and powerful numbers in Lucas sequences . 28 A. Generaltheoremsforpowers........ 29 B. Explicit determination in special sequences . 30 vi Contents C. Uniform explicit determination of multiples, squares, and square-classes for certain families of Lucas sequences . 35 D. Powerful numbers in Lucas sequences . 41 2 Representation of Real Numbers by Means of Fibonacci Numbers 51 3 Prime Number Records 62 4 Selling Primes 78 5 Euler’s Famous Prime Generating Polynomial 91 1 Quadraticextensions................ 94 2 Ringsofintegers................... 94 3 Discriminant..................... 95 4 Decompositionofprimes.............. 96 A. Propertiesofthenorm........... 96 5 Units.......................... 100 6 Theclassnumber.................. 101 A. Calculationoftheclassnumber...... 103 B. Determination of all quadratic fields with classnumber1................ 106 7 Themaintheorem.................. 108 6 Gauss and the Class Number Problem 112 1 Introduction..................... 112 2 HighlightsofGauss’life................ 112 3 Briefhistoricalbackground............. 114 4 Binaryquadraticforms............... 115 5 The fundamental problems . ............ 118 6 Equivalenceofforms................. 118 7 Conditional solution of the fundamental problems . 120 8 Proper equivalence classes of definite forms . 122 A. Anothernumericalexample........ 126 9 Proper equivalence classes of indefinite forms . 126 A. Anothernumericalexample........ 131 10 Theautomorphofaprimitiveform........ 131 11 Composition of proper equivalence classes of primitiveforms.................... 135 Contents vii 12 Thetheoryofgenera.................. 137 13 The group of proper equivalence classes of primitiveforms.................... 143 14 Calculationsandconjectures............ 145 15 The aftermath of Gauss (or the “math” after Gauss)......................... 146 16 Formsversusidealsinquadraticfields........ 146 17 Dirichlet’sclassnumberformula.......... 153 18 Solution of the class number problem for definite forms........................... 157 19 The class number problem for indefinite forms . 161 20 Morequestionsandconjectures.......... 164 21 Many topics have not been discussed ....... 168 7 Consecutive Powers 175 1 Introduction..................... 175 2 History........................ 177 3 Specialcases..................... 178 4 Divisibility properties ................ 190 5 Estimates....................... 195 A. The equation aU − bV =1 ......... 196 B. The equation Xm − Y n =1 ........ 197 C. The equation XU − Y V =1 ........ 201 6 Final comments and applications ......... 204 8 1093 213 A. Determination of the residue of qp(a) . 216 B. Identities and congruences for the Fermat quotient.................... 217 9 Powerless Facing Powers 229 1 Powerfulnumbers.................. 229 A. Distributionofpowerfulnumbers..... 230 B. Additiveproblems.............. 232 C. Differenceproblems............. 233 2 Powers......................... 235 A. Pythagorean triples and Fermat’s problem . 235 B. VariantsofFermat’sproblem....... 238 C. TheconjectureofEuler........... 239 D. The equation AXl + BY m = CZn .... 240 viii Contents E. Powersasvaluesofpolynomials...... 245 3 Exponentialcongruences.............. 246 A. TheWieferichcongruence......... 246 B. Primitivefactors............... 248 4 Dreammathematics................. 251 A. The statements ............... 251 B. Statements .................. 252 C. Binomials and Wieferich congruences . 254 D. Erd¨os conjecture and Wieferich congruence . 257 E. Thedreaminthedream.......... 257 √ √ 2 10 What Kind of Number Is 2 ? 271 0 Introduction..................... 271 1 Kindsofnumbers.................. 271 2 Hownumbersaregiven............... 276 3 Briefhistoricalsurvey................ 284 4 Continuedfractions................. 287 A. Generalities.................. 288 B. Periodiccontinuedfractions........ 289 C. Simple continued fractions of π and e . 291 5 Approximationbyrationalnumbers........ 295 A. Theorderofapproximation........ 295 B. TheMarkoffnumbers............ 296 C. Measuresofirrationality.......... 298 D. Order of approximation of irrational algebraicnumbers.............. 299 6 Irrationalityofspecialnumbers.......... 301 7 Transcendental numbers ............... 309 A. Liouville numbers . ............ 310 B. Approximation by rational numbers: sharpertheorems.............. 311 C. Hermite, Lindemann, and Weierstrass . 316 D. AresultofSiegelonexponentials..... 318 E. Hilbert’s7thproblem............ 320 F. TheworkofBaker.............. 321 G. TheconjectureofSchanuel......... 323 H. Transcendence measure and the classificationofMahler........... 328 8 Final comments ................... 331 Contents ix 11 Galimatias Arithmeticae 344 Index of Names 361 Index of Subjects 369 Preface Dear Friends of Numbers: This little book is for you. It should offer an exquisite intel- lectual enjoyment, which only relatively few fortunate people can experience. May these essays stimulate your curiosity and lead you to books and articles where these matters are discussed at a more technical level. I warn you, however, that the problems treated, in spite of be- ing easy to state, are for the most part very difficult. Many are still unsolved. You will see how mathematicians have attacked these problems. Brains at work! But do not blame me for sleepless nights (I have mine already). Several of the essays grew out of lectures given over the course of years on my customary errances. Other chapters could, but probably never will, become full-sized books. The diversity of topics shows the many guises numbers take to tantalize∗ and to demand a mobility of spirit from you, my reader, who is already anxious to leave this preface. Now go to page 1 (or 127?). Paulo Ribenboim ∗Tantalus, of Greek mythology, was punished by continual disappointment when he tried to eat or drink what was placed within his reach. 1 The Fibonacci Numbers and the Arctic Ocean Introduction There is indeed not much relation between the Fibonacci numbers and the Arctic Ocean, but I thought that this title would excite your curiosity for my lecture. You will be disappointed if you wished to hear about the Arctic Ocean, as my topic will be the sequence of Fibonacci numbers and similar sequences. Like the icebergs in the Arctic Ocean, the sequence of Fibonacci numbers is the most visible part of a theory which goes deep: the theory of linear recurring sequences. The so-called Fibonacci numbers appeared in the solution of a problem by Fibonacci (also known as Leonardo Pisano), in his book Liber Abaci (1202), concerning reproduction patterns of rab- bits. The first significant work on the subject is by Lucas, with his seminal paper of 1878. Subsequently, there appeared the classical papers of Bang (1886) and Zsigmondy (1892) concerning prime divisions of special sequences of binomials. Carmichael (1913) published another fundamental paper where he extended to Lucas se- quences the results previously obtained in special cases. Since then, I note the work of Lehmer, the applications of the theory in primality tests giving rise to many developments. 2 1. The Fibonacci Numbers and the Arctic Ocean The subject is very rich and I shall consider here only certain aspects of it. If, after all, your only interest is restricted to Fibonacci and Lucas numbers, I advise you to read the booklets by Vorob’ev (1963), Hoggatt (1969), and Jarden (1958). 1 Basic definitions A. Lucas sequences