Theory of Numbers: A Text and Source Book of Problems PDF
Preview Theory of Numbers: A Text and Source Book of Problems
IMPORTANT FIGURES IN THE HISTORY OF NUMBER THEORY (Alphabetical) al-Karaji c.1090 Archimedes 287 - 212 B.C. Aryabhata c.475 - 550 Claude Bachet 1587 - 1638 Bhaskhara 1114 - c.1185 Rafael Bombelli 1526 - 1572 Brahmagupta c.625 William Brouncker 1620 - 1684 Pavnuty Chebyshev 1821 - 1894 Ch'in Chiu-shao c.1202 - 1261 Diophantus c.250 Peter Lejeune Dirichlet 1805 - 1859 Ferdinand Eisenstein 1823 - 1852 Eratosthenes c.230 B.C. Euclid 323 - 285 B.C. Leonhard Euler 1707 - 1783 Pierre de Fermat 1601 - 1665 Fibonacci c.1175 - 1250 Bernard Frenicle de Bessy c.1602 - 1675 Carl Friedrich Gauss 1777 - 1855 Christian Goldbach 1690 - 1764 Jacques Hadamard 1865 - 1963 G. H. Hardy 1877 - 1947 David Hilbert 1862 - 1943 Carl Gustav Jacobi 1804 - 1851 Ernst Kummer 1810 - 1893 Joseph Louis Lagrange 1736 - 1813 Gabriel Lame 1795 - 1870 Adrien-Marie Legendre 1752 - 1833 Edouard Lucas 1842 - 1891 Marin Meranne 1588 - 1648 John Pell 1610 - 1685 Pythagoras 585 - 501 B.C. Sun-Tzu c.250 Axel Thue 1863 - 1922 Charles de la Vallee-Poussin 1866 - 1962 I. M. Vinogradov 1891 1983 - John Wallis 1610 1703 - Edward Waring 1734 1793 - John Wilson 1741 1793 - IMPORTANT FIGURES IN THE HISTORY OF NUMBER THEORY (Chronological) Pythagoras 585 - 501 B.C. Euclid 323 - 285 B.C. Archimedes 287 - 212 B.C. Eratosthenes c.230 B.C. Diophantus c.250 A.D. Sun-Tzu c.250 Aryabhata c.475 - 550 Brahmagupta c.625 al-Karaji c.1090 Bhaskhara 1114 - c.1185 Fibonacci c.1175 - 1250 Ch'in Chiu-shao c.1202 - 1261 Rafael Bombelli 1526 - 1572 Claude Bachet 1587 - 1638 Marin Mersenne 1588 - 1648 Pierre de Fermat 1601 - 1665 Bernard Frenicle de Bessy c.1602 - 1675 John Pell 1610 - 1685 John Wallis 1610 - 1703 William Brouncker 1620 - 1684 Christian Goldbach 1690 - 1764 Leonhard Euler 1707 - 1783 Edward Waring 1734 - 1793 Joseph Louis Lagrange 1736 - 1813 John Wilson 1741 - 1793 Adrien-Marie Legendre 1752 - 1833 Carl Friedrich Gauss 1777 - 1855 Gabriel Lame 1795 - 1870 Carl Gustav Jacobi 1804 - 1851 Peter Lejeune Dirichlet 1805 - 1859 Ernst Kummer 1810 - 1893 Pavnuty Chebyshev 1821 - 1894 Ferdinand Eisenstein 1823 - 1852 Edouard Lucas 1842 - 1891 David Hilbert 1862 - 1943 Axel Thue 1863 - 1922 Jacques Hadamard 1865 - 1963 Charles de la Vallee-Poussin 1866 - 1962 G. H. Hardy 1877 - 1947 I. M. Vinogradov 1891 - 1983 The Theory of Numbers A TEXT AND SOURCE BOOK OF PROBLEMS Andrew Adler (cid:9) John E. Coury The University of British Columbia Jones and Bartlett Publishers Sudbury, Massachusetts Boston(cid:9) London Singapore (cid:9)(cid:9) Editorial, Sales, and Customer Service Offices Jones and Bartlett Publishers 40 Tall Pine Drive Sudbury, MA 01776 1-508-443-5000 1-800-832-0034 [email protected] http://www.jbpub.com Jones and Bartlett Publishers International Barb House, Barb Mews London W6 7PA UK Copyright © 1995 by Jones and Bartlett Publishers, Inc. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Adler, Andrew. The theory of numbers: a text and source book of problems / Andrew Adler, John E. Coury. p. cm. Includes bibliographical references and index. ISBN 0-86720-472-9 1. Number theory. I. Coury, John E. II. Title. QA241.A244(cid:9) 1995 5121.7—dc20(cid:9) 94-41865 CIP Printed in the United States of America 98 97 96(cid:9) 10 9 8 7 6 5 4 3 2 For Marie, Bob, and Michael J.C. For Alan, Lissett, Rachel, and Sara A.A. Contents Preface(cid:9) ix Introduction(cid:9) 1 Chapter One: Divisibility, Primes, and the Euclidean Algorithm(cid:9) 6 Results 7 Divisibility 7 Primes 10 The Euclidean Algorithm 13 The Equation ax + by = c 15 Problems and Solutions 17 Exercises 32 Notes, Biographical Sketches, References 35 Chapter Two: Congruences(cid:9) 39 Results 40 Divisibility Tests 42 Linear Congruences 43 Techniques for Solving ax =- b (mod m) 44 - The Chinese Remainder Theorem 46 An Application: Finding the Day of the Week 47 Problems and Solutions 48 Exercises 64 Notes, Biographical Sketches, References 66 Chapter Three: The Theorems of Fermat, Euler, and Wilson(cid:9) 71 Results 72 Fermat's Theorem and Wilson's Theorem 72 Euler's Theorem and the Euler 0 -function 75 Problems and Solutions 78 v vi(cid:9) CONTENTS Exercises 94 Notes, Biographical Sketches, References 96 (cid:9) Chapter Four: Polynomial Congruences 101 Results 101 General Polynomial Congruences 101 Solutions of f (x) 0 (mod pk) 106 The Congruence x2 a( mod plc) 109 Problems and Solutions 110 Exercises 122 Notes, References 124 Chapter Five: Quadratic Congruences and the Law (cid:9) of Quadratic Reciprocity 125 Results 126 General Quadratic Congruences 126 The Congruence x2 a( mod m) 127 Quadratic Residues 128 The Law of Quadratic Reciprocity 131 Problems and Solutions 137 Exercises 153 Notes, Biographical Sketches, References 155 (cid:9) Chapter Six: Primitive Roots and Indices 158 Results 158 The Order of an Integer 158 Primitive Roots 160 Power Residues and Indices 164 The Existence of Primitive Roots 167 Problems and Solutions 169 Exercises 189 Notes, Biographical Sketches, References 191 (cid:9) Chapter Seven: Prime Numbers 194 Results 195 The Sieve of Eratosthenes 195 Perfect Numbers 196 Mersenne Primes 197 Fermat Numbers 198 The Prime Number Theorem 200 Dirichlet's Theorem 202 Goldbach's Conjecture 203 Other Open Problems 204
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