Theory of Numbers: A Text and Source Book of Problems

421 Pages·1995·90.61 MB·English

by  Andrew Adler

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
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Copyright © 1995 by Jones and Bartlett Publishers, Inc.
All rights reserved. No part of the material protected by this copyright notice
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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