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Solved and Unsolved Problems in Number Theory

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Solved and Unsolved Problems in Number Theory CONTENTS PAGE PREFACE....................................... Chapter I FROM PERFECT KGXIBERS TO THE QUADRATIC RECIPROCITY LAW SECTION 1. Perfect Xumbcrs .......................................... 1 2 . Euclid ............ ............................. 4 3. Euler’s Converse Pr ................ ............... 8 SECOND EDITION 4 . Euclid’s Algorithm ....... ............................... 8 5. Cataldi and Others...... ............................... 12 6 . The Prime Kumber Theorem .............................. 15 Copyright 0.1962. by Daniel Shanks 7 . Two Useful Theorems ...................................... 17 Copyright 0. 1978. by Daniel Shanks 8. Fermat. and 0t.hcrs........................................ 19 9 . Euler’s Generalization Promd ............................... 23 10. Perfect Kunibers, I1 ....................................... 25 11. Euler and dial.. ........................... ............. 25 Library of Congress Cataloging in Publication Data 12. Many Conjectures and their Interrelations.................... 29 Shanks. Daniel . 13. Splitting tshe Primes into Equinumerous Classes ............... 31 Solved and unsolved problems in number theory. 14. Euler’s Criterion Formulated ...... ....................... 33 15. Euler’s Criterion Proved .................................... 35 Bibliography: p. Includes index . 16. Wilson’s Theorem ......................................... 37 1. Numbers. Theory of . I . Title. 17. Gauss’s Criterion ................................... 38 [QA241.S44 19781 5E.7 77-13010 18. The Original Lcgendre Symbol .. ..................... 40 ISBN 0-8284-0297-3 19. The Reciprocity Law .......... ..................... 42 20 . The Prime Divisors of n2 + a ... ..................... 47 Chapter I1 Printed on ‘long-life’ acid-free paper THE CSDERLTISG STRUCTURE 21 . The Itesidue Classes as an Invention ......................... 51 Printed in the United States of America 22 . The Residue Classes 3s a Tool .............................. 55 23 . The Residue Classcs as n Group............................. 59 24 . Quadratic Residues ....................................... 63 V vi Solved and Unsolved Problems in Number Theory Contents vii SECTION PAGE SECTION PAGE 25 . Is the Quadratic Recipro&y Law a Ilerp Thcoreni? ........... 6.2 61 . Continued Fractions for fi............................... 180 26 . Congruent.id Equations with a Prime Modulus ................ 66 62 . From Archimedes to Lucas ................................. 188 27 . Euler’s 4 E’unct.ion......................................... 68 63 . The Lucas Criterion ....................................... 193 28 . Primitive Roots with a Prime i\Iodulus . ........... 71 64 . A Probability Argument .................................... 197 29 . mpas a Cyclic Group ................ ........... 73 65. Fibonacci Xumbers and the Original Lucas Test .............. 198 30 . The Circular Parity Switch ........... ........... 76 31 . Primitive Roots and Fermat Xumhcrs., ........... 78 32 . Artin’s Conjectures .................. ........... 80 Appendix to Chapters 1-111 33 . Questions Concerning Cycle Graphs .... ........... 83 SUPPLEMENTARYCOMMENTS. THEOREMS. AND EXERCISES 201 34 . Answers Concerning Cycle Graphs..... ........... 92 ......... 35 . Factor Generators of 3% ................................... 98 Chapter IV 36 . Primes in Some Arithmetic Progressions and a General Divisi- bility Theorem .......... ............................ 104 PROGRESS 37 . Scalar and Vect.or Indices ................................... 109 SECTION 38 . The Ot.her Residue Classes ....... ...................... 113 66. Chapter I Fifteen Years Later .................... 217 39 . The Converse of Fermat’s Theore ................ 67 . Artin’s Conjectures, I1 ......................... 222 40 . Sufficient Coiiditiorls for Primality........................... 118 68. Cycle Graphs and Related Topics ...................225 69 . Pseudoprimes and Primality ...................... 226 Chapter 111 70 . Fermat’s Last “Theorem, ” I1 ..................... 231 71. Binary Quadratic Forms with Negative Discriminants ......233 PYTHAGOREAKISM AKD ITS MAXY COKSEQUESCES 72. Binary Quadratic Forms with Positive Discriminants .......235 41 . The Pythagoreans ................... ............ 121 73 . Lucas and Pythagoras ......................... 237 42 . The Pythagorean Theorein ........... ................ 123 74. The Progress Report Concluded ...................238 43 . The 42 and the Crisis .......................... 44 . The Effect upon Geometry ................................. 127 45 . The Case for Pythagoreanism ......................... 46 . Three Greek Problems ..... .................. Appendix 47 . Three Theorems of Fermat ................................. 142 STATEMENTON FUNDAMENTALS ....................... 239 48 Fermat’s Last’ “Theorem” ....... ....................... 144 . TABLEOF DEFINITIONS............................ 241 49 . The Easy Case and Infinite Desce ....................... 147 REFERENCES................................... 243 50 . Gaussian Integers and Two Applications . ........... 149 51 . Algebraic Integers and Kummer’s Theore ........... 1.51 INDEX...................................... 255 52 . The Restricted Case, S Gcrmain, and Wieferich ... 53. Euler’s “Conjecture” . ................................ 157 54 . Sum of Two Squares ....................... ...... 159 55 . A Generalization and Geometric Xumber Theory .............. 161 56 . A Generalization and Binary Quadratic Forms..... ........ 165 57 . Some Applicat.ions.............................. 58 . The Significance of Fermat’s Equation ............ 59 . The Main Theorem ........... .......................... 174 60 . An Algorithm ................................. ..... 178 PREFACE TO THE SECOND EDITION The Preface to the First Edition (1962) states that this is “a rather tightly organized presentation of elementary number theory” and that I “number theory is very much a live subject.” These two facts are in I conflict fifteen years later. Considerable updating is desirable at many places in the 1962 Gxt, but the needed insertions would call for drastic surgery. This could easily damage the flow of ideas and the author was reluctant to do that. Instead, the original text has been left as is, except for typographical corrections, and a brief new chapter entitled “Pro- gress” has been added. A new reader will read the book at two levels-as it was in 1962, and as things are today. Of course, not all advances in number theory are discussed, only those pertinent to the earlier text. Even then, the reader will be impressed I with the changes that have occurred and will come to believe-if he did not already know it-that number theory is very much a live subject. The new chapter is rather different in style, since few topics are developed at much length. Frequently, it is extremely hrief and merely gives references. The intent is not only to discuss the most important changes in sufficient detail but also to be a useful guide to many other topics. A propos this intended utility, one special feature: Developments in the algorithmic and computational aspects of the subject have been especially active. It happens that the author was an editor of Muthe- matics of Cmptation throughout this period, and so he was particu- ~ larly close to most of these developments. Many good students and professionals hardly know this material at all. The author feels an obligation to make it better known, and therefore there is frequent emphasis on these aspects of the subject. To compensate for the extreme brevity in some topics, numerous references have been included to the author’s own reviews on these topics. They are intended especially for any reader who feels that he must have a second helping. Many new references are listed, but the following economy has been adopted: if a paper has a good bibliogra- phy, the author has usually refrained from citing the references con- I tained in that bibliography. The author is grateful to friends who read some or all of the new chapter. Especially useful comments have come from Paul Bateman, Samuel Wagstaff, John Brillhart, and Lawrence Washington. DANIELSHANKS December 1977 ix PREFACE TO THE FIRST EDITION It may be thought- that the title of this book is not well chosen since the book is, in fact, a rather tightly organized presentation of elementary number theory, while the title may suggest a loosely organized collection of problems. h-onetheless the nature of the exposition and the choice of topics to be included or. omitted are such as to make the title appropriate. Since a preface is the proper place for such discussion we wish to clarify this matter here. i Much of elementary number theory arose out of the investigation of ~ three problems ; that of perfect numbers, that of periodic decimals, and that of Pythagorean numbers. We have accordingly organized the book into three long chapters. The result of such an organization is that motiva- tion is stressed to a rather unusual degree. Theorems arise in response to previously posed problems, and their proof is sometimes delayed until an appropriate analysis can be developed. These theorems, then, or most of them, are “solved problems.” Some other topics, which are often taken up in elementary texts-and often dropped soon after-do not fit directly I into these main lines of development, and are postponed until Volume 11. I Since number theory is so extensive, some choice of topics is essential, and while a common criterion used is the personal
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