Elementary Number Theory

407 Pages·2015·3.665 MB·English

by  James S. Kraft, Lawrence C. Washington

Mathematics
TEXTBOOKS in MATHEMATICS TEXTBOOKS in MATHEMATICS
Elementary Number Theory takes an accessible approach to teaching E
readers about the role of number theory in pure mathematics and its im- l Elementary
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portant applications to cryptography and other areas. The first chapter
m
of the book explains how to do proofs and includes a brief discussion
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of lemmas, propositions, theorems, and corollaries. The core of the text Number
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covers linear Diophantine equations; unique factorization; congruences;
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Fermat’s, Euler’s, and Wilson’s theorems; order and primitive roots; and
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quadratic reciprocity. The authors also discuss numerous cryptographic
Theory
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topics, such as RSA and discrete logarithms, along with recent develop-
y
ments.

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• Describes exciting applications of number theory to various
problems in cryptography u
• Discusses recent developments m
• Includes “check your understanding” problems that test readers on
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fundamental ideas, with solutions at the end of each chapter
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• Provides numerous examples as well as standard and
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computational exercises, with selected answers or hints in an

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appendix
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• Reviews the prerequisites in an appendix
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About the Authors o
James S. Kraft teaches mathematics at the Gilman School. He has pre-
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viously taught at the University of Rochester, St. Mary’s College of Cali-
y
fornia, and Ithaca College. He has also worked in communications secu-
rity. Dr. Kraft has published several research papers in algebraic number
theory. He earned his Ph.D. from the University of Maryland.
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Lawrence C. Washington is a professor of mathematics and Distin-
a
guished Scholar-Teacher at the University of Maryland. Dr. Washington sK
h
has published extensively in number theory, including books on cryp- ir
na James S. Kraft
tography (with Wade Trappe), cyclotomic fields, and elliptic curves. He gf
t
t
earned his Ph.D. from Princeton University. o
Lawrence C. Washington
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K24624
www.crcpress.com

TEXTBOOKS in MATHEMATICS
Elementary
Number
Theory
James S. Kraft
Gilman School
Baltimore, Maryland, USA
Lawrence C. Washington
University of Maryland
College Park, Maryland, USA
K24624_FM.indd 1 10/22/14 12:45 PM
TEXTBOOKS in MATHEMATICS
Series Editors: Al Boggess and Ken Rosen
PUBLISHED TITLES
ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH
Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom
ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH
William Paulsen
ADVANCED CALCULUS: THEORY AND PRACTICE
John Srdjan Petrovic
ADVANCED LINEAR ALGEBRA
Nicholas Loehr
ANALYSIS WITH ULTRASMALL NUMBERS
Karel Hrbacek, Olivier Lessmann, and Richard O’Donovan
APPLYING ANALYTICS: A PRACTICAL APPROACH
Evan S. Levine
COMPUTATIONS OF IMPROPER REIMANN INTEGRALS
Ioannis Roussos
CONVEX ANALYSIS
Steven G. Krantz
COUNTEREXAMPLES: FROM ELEMENTARY CALCULUS TO THE BEGINNINGS OF ANALYSIS
Andrei Bourchtein and Ludmila Bourchtein
DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION
Steven G. Krantz
DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY
Mark A. McKibben and Micah D. Webster
ELEMENTARY NUMBER THEORY
James S. Kraft and Lawrence C. Washington
ELEMENTS OF ADVANCED MATHEMATICS, THIRD EDITION
Steven G. Krantz
EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA®
Crista Arangala
K24624_FM.indd 2 10/22/14 12:45 PM
PUBLISHED TITLES CONTINUED
AN INTRODUCTION TO NUMBER THEORY WITH CRYPTOGRAPHY
James Kraft and Larry Washington
AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH MATLAB®, SECOND EDITION
Mathew Coleman
INTRODUCTION TO THE CALCULUS OF VARIATIONS AND CONTROL WITH MODERN APPLICATIONS
John T. Burns
LINEAR ALGEBRA, GEOMETRY AND TRANSFORMATION
Bruce Solomon
THE MATHEMATICS OF GAMES: AN INTRODUCTION TO PROBABILITY
David G. Taylor
QUADRACTIC IRRATIONALS: AN INTRODUCTION TO CLASSICAL NUMBER THEORY
Franz Holter-Koch
REAL ANALYSIS AND FOUNDATIONS, THIRD EDITION
Steven G. Krantz
RISK ANALYSIS IN ENGINEERING AND ECONOMICS, SECOND EDITION
Bilal M. Ayyub
RISK MANAGEMENT AND SIMULATION
Aparna Gupta
TRANSFORMATIONAL PLANE GEOMETRY
Ronald N. Umble and Zhigang Han
K24624_FM.indd 3 10/22/14 12:45 PM
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To Kristi, Danny, and Aaron,
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To Susan and Patrick,
and to Ida H. Washington and Lawrence M. Washington

Contents
Preface xiii
Introduction 1
1 Divisibility 9
1.1 What Is a Proof? . . . . . . . . . . . . . . . . . . . . . 9
1.1.1 Proof by Contradiction . . . . . . . . . . . . . . 15
1.2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Euclid’s Theorem . . . . . . . . . . . . . . . . . . . . . 19
1.4 Euclid’s Original Proof . . . . . . . . . . . . . . . . . . 21
1.5 The Sieve of Eratosthenes . . . . . . . . . . . . . . . . . 23
1.6 The Division Algorithm . . . . . . . . . . . . . . . . . . 25
1.6.1 A Cryptographic Application . . . . . . . . . . . 26
1.7 The Greatest Common Divisor . . . . . . . . . . . . . . 27
1.8 The Euclidean Algorithm . . . . . . . . . . . . . . . . . 30
1.8.1 The Extended Euclidean Algorithm . . . . . . . 32
1.9 Other Bases . . . . . . . . . . . . . . . . . . . . . . . . 37
1.10 Fermat and Mersenne Numbers . . . . . . . . . . . . . 40
1.11 Chapter Highlights . . . . . . . . . . . . . . . . . . . . 43
1.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.12.1 Exercises . . . . . . . . . . . . . . . . . . . . . . 44
1.12.2 Computer Explorations . . . . . . . . . . . . . . 50
1.12.3 Answers to “Check Your Understanding” . . . . 51
2 Linear Diophantine Equations 55
2.1 ax+by = c . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 The Postage Stamp Problem . . . . . . . . . . . . . . . 60
vii
viii Contents
2.3 Chapter Highlights . . . . . . . . . . . . . . . . . . . . 64
2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . 64
2.4.2 Answers to “Check Your Understanding” . . . . 66
3 Unique Factorization 67
3.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . 67
3.2 The Fundamental Theorem of Arithmetic . . . . . . . . 69
3.3 Euclid and the Fundamental Theorem of Arithmetic . . 74
3.4 Chapter Highlights . . . . . . . . . . . . . . . . . . . . 75
3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . 75
3.5.2 Answers to “Check Your Understanding” . . . . 76
4 Applications of Unique Factorization 77
4.1 A Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Irrationality Proofs . . . . . . . . . . . . . . . . . . . . 79
4.3 The Rational Root Theorem . . . . . . . . . . . . . . . 81
4.4 Pythagorean Triples . . . . . . . . . . . . . . . . . . . . 83
4.5 Differences of Squares . . . . . . . . . . . . . . . . . . . 87
4.6 Chapter Highlights . . . . . . . . . . . . . . . . . . . . 89
4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . 90
4.7.2 Computer Explorations . . . . . . . . . . . . . . 93
4.7.3 Answers to “Check Your Understanding” . . . . 93
5 Congruences 95
5.1 Definitions and Examples . . . . . . . . . . . . . . . . . 95
5.2 Modular Exponentiation . . . . . . . . . . . . . . . . . 102
5.3 Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Linear Congruences . . . . . . . . . . . . . . . . . . . . 108
5.5 The Chinese Remainder Theorem . . . . . . . . . . . . 115
5.6 Fractions Mod m . . . . . . . . . . . . . . . . . . . . . 119