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Introduction to Number Theory INTRODUCTION TO NUMBER THEORY DANIEL E. FLATH AMS CHELSEA PUBLISHING INTRODUCTION TO NUMBER THEORY 10.1090/chel/384.H INTRODUCTION TO NUMBER THEORY DANIEL E. FLATH AMS CHELSEA PUBLISHING 2010 Mathematics Subject Classification. Primary 11-01. For additional information and updates on this book, visit www.ams.org/bookpages/chel-384 Library of Congress Cataloging-in-Publication Data Names: Flath, Daniel E., author. Title: Introduction to number theory / Daniel E. Flath. Other titles: Number theory Description: [2018 edition]. | Providence, Rhode Island : American Mathematical Society, 2018. | Series: AMS Chelsea Publishing [series] ; 384 | Originally published: New York : Wiley, 1989. | Includes bibliographical references and indexes. Identifiers: LCCN 2018014214 | ISBN 9781470446949 (alk. paper) Subjects: LCSH: Number theory. | AMS: Number theory – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA241 .F59 2018 | DDC 512.7–dc23 LC record available at https://lccn.loc.gov/2018014214 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 1989 held by Daniel E. Flath. All rights reserved. Reprinted by the American Mathematical Society, 2018 Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 232221201918 To Laura Preface When I came to Singapore I was given a fourth year undergraduate Honours Number Theory course. I decided to teach Gauss’s immortal Disquisitiones Arithmeticae. This book is the result. On historical and mathematical grounds alike number theory has earned a place in the curriculum of every mathematics student. This is a textbook for an advanced undergraduate or beginning graduate core course in the subject. Such a course should stick pretty close to the naive questions, which in number theory concern prime numbers and Diophantine equations. The emphasis in this book is on Diophantine equations, especially quadratic equations in two variables. My own conscious interest in Diophantine equations goes back to a long winter’s night in a St. Louis basement in 1962 when my father and I tried to solve the notorious problem of the monkey and the coconuts as presented by Martin Gardner. No one told me then that Diophantine equations belong to a subject called “ number theory,” and I found little help in the public library. I needed a teacher trained in number theory. It pleases me that several of my students of Gauss are now teaching in the schools. I might particularly mention Mr. Lee Ah Huat with whom I discovered Gauss’s first proof of the law of quadratic reciprocity. This book is closely based on lectures I gave to able groups of students during three consecutive years at the National University of Singapore. I thank the students for constantly demanding “ the notes,” which was how the text began. I tried during the writing always to keep my students in mind, always to remember that I was writing a textbook. I have sought to avoid the twin traps of doing algebra to the exclusion of number theory and of doing only trivial number theory. I take it for granted that the material I have chosen is interesting. My supreme stylistic goal is clarity. Viii PREFACE By the time this book is published I shall have gone on from Singapore. Singapore has been part of my life for three and a half years, and I shall miss it. I have many friends here. I wish them all well. Singapore Dan Flath August 1987 Acknowledgment I am very grateful to Jean-Pierre Serre and the Singapore Mathematical Society for permission to reprint the article “ A = b2 - Aac” which was first published in the Mathematical Medley. Daniel E. Flath ix Contents 1. Prime Numbers and Unique Factorization 1 1. Introduction, 1 2. Prime Numbers, 2 3. Unique Factorization, 6 4. 4- # 2 ^ 2 * *' +QrXr = n, 11 5. The Distribution of the Primes, 17 2. Sums of Two Squares 24 1. Introduction, 24 2. Integers mod m, 25 3. Applications of Lemma 1.4, 32 4. Gaussian Integers, 34 5. Farey Sequences, 39 6. Minkowski’s Theorem, 47 7. Method of Descent, 50 8. Reduction of Positive Definite Binary Quadratic Forms, 55 3. Quadratic Reciprocity 63 1. Introduction, 63 2. Composite Moduli, 65 3. The Legendre Symbol, 70 4. The First Proof, 77 5. The Gauss Lemma, 80 xi xii CONTENTS 6. Gauss Sums, 83 7. The Ring Z[e2”i/n], 89 8. The Jacobi Symbol, 92 9. The Kronecker Symbol, 95 10. Binary Quadratic Forms, 99 4. Indefinite Forms 104 1. Introduction, 104 2. The Square Root of 2, 105 3. The Pell Equation, 108 4. aX2 + bXY + cY2 = m, 118 5. Automorphisms, 124 6. Reduction of Indefinite Forms, 128 7. Continued Fractions, 135 8. Reduction (II), 139 9. Automorphisms (II), 142 5. The Oass Group and Genera 147 1. Introduction, 147 2. The Class Group, 148 3. The Genus Group, 156 4. What Gauss Did, 161 5. Counting Ambiguous Classes, 164 6. The Ternary Form Y 2 - XZ, 168 7. The Duplication Theorem, 172 8. Sums of Three Squares, 175 Appendix A. A = b1 — 4 ac by Jean-Pierre Serre 180 Appendix B. Tables 190 Bibliography 204 Subject Index 207 Notation Index 211 Bibliography Number theory has been blessed with many excellent books. This bibliography is a list of those that were available to me during the writing of this book. I have drawn most heavily from [5], [12], [13], [14], [31], and [33]. The reader who would learn more about rational and integral quadratic forms must begin by studying the fields of />-adic numbers. There are fine introductions in [4] and [27]. His ultimate goal must be a close study of [5], which is magnificent. A proof of Dirichlet’s Theorem on Primes in Arithmetic Progressions can be found in [27]. For a superb introduction to the distribution of prime numbers, see [28]. [1] Auslander, L. and Tolimieri, R., Ring structure and the Fourier transform, The Mathematical Intelligencer 7, 49-52 (1985). [2] Bachmann, P., Niedere Zahlentheorie y Chelsea, New York, 1968. [3] Baker, A., A Concise Introduction to the Theory of Numbers, Cambridge University Press, Cambridge, 1984. [4] Borevich, Z. I. and Shafarevich, I. R., Number Theory, Academic, New York, 1966. [5] Cassels, J. W. S., Rational Quadratic Forms, Academic, London, 1978. [6] Cohn, H., A Second Course in Number Theory, Wiley, New York, 1962. [7] Davenport, H., The Higher Arithmetic, 5th ed., Cambridge University Press, Cambridge, 1982. [8] Dickson, L. E., Introduction to the Theory of Numbers, The University of Chicago Press, Chicago, 1929. [9] Dirichlet, P. G. L., Vorlesungen uber Zahlentheorie, herausgegeben von R. Dedekind, 4th ed., Chelsea, New York, 1968. [10] Edwards, H. M., Fermat's Last Theorem, Springer, New York, 1977. 204 BIBLIOGRAPHY 205 [11] Frei, G., Leonhard Euler’s Convenient Numbers, The Mathematical Intelligencer, 7, 55-58, 64 (1985). [12] Gauss, C. F., Disquisitiones Arithmeticae, Springer, New York, 1986. [13] Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 5th ed., Oxford University Press, Oxford, 1979. [14] Hua, L. K., Introduction to Number Theory, Springer, New York, 1982. [15] Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Theory, Springer, New York, 1982. [16] Jones, B. W., The Arithmetic Theory of Quadratic Forms, The Mathematical Association of America, Providence, R.I., 1950. [17] Knuth, D. E., The Art of Computer Programming, Vol. 2, 2nd ed., Addison-Wesley, Reading, Mass., 1981. [18] Landau, E., Elementary Number Theory, Chelsea, New York, 1958. [19] Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea, New York, 1953. [20] LeVeque, W. J., Fundamentals of Number Theory, Addison-Wesley, Reading, Mass., 1977. [21] LeVeque, W. J., Topics in Number Theory, Vol. II, Addison-Wesley, Reading, Mass., 1956. [22] Mathews, G. B., Theory of Numbers, Part I, Deighton, Bell and Co., Cambridge, 1892. [23] Mordell, L. J., Diophantine Equations, Academic, London, 1969. [24] Nagell, T., Introduction to Number Theory, 2nd ed., Chelsea, New York, 1981. [25] Narkiewicz, W., Number Theory, World Scientific, Singapore, 1983. [26] Niven, I. and Zuckerman, H. S., An Introduction to the Theory of Numbers, 4th ed., Wiley, New York, 1980. [27] Serre, J.-P., A Course in Arithmetic, Springer, New York, 1973. [28] Shapiro, H. N., Introduction to the Theory of Numbers, Wiley, New York, 1983. [29] Sierpinski, W., Elementary Theory of Numbers, Panstwowe Wydawnictwo Naukowe, Warsaw, 1964. [30] Stark, H. M., An Introduction to Number Theory, MIT Press, Cambridge, Mass., 1978. [31] Venkov, B. A., Elementary Number Theory, Wolters-Noordhoff Publishing, Groningen, 1970. [32] Vinogradov, I. M., Elements of Number Theory, Dover, London, 1954. [33] Weil, A., Number Theory: An Approach through History, Birkhauser, Boston, 1983. [34] Weil, A., Number Theory for Beginners, Springer, New York, 1979. Subject Index Algebraic number theory, 64, 89 linear, 66, 69, 127, 155 Ambiguous class, see Class group quadratic, see also Legendre symbol Ambiguous form, see Quadratic forms, binary x 1 a -1 , 25, 27-28, 30, 178 Automorphism, see Quadratic forms x2 a ±2, 31, 53-54, 76 x2 a -3 , 31, 38 Baker, A., 187 X 2 S 5, 31 Binomial coefficients, 19, 20, 30 x2 a a, 28, 63-64, 67-69, 70, 95, 97-98, Birch and Swinnerton-Dyer conjecture, 188 99, 102 Box principle, 48.
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