In this student-friendly text, Strayer presents all of the topics necessary for a first course in number theory. Additionally, chapters on primitive roots, Diophantine equations, and continued fractions allow instructors the flexibility to tailor the material to meet their own classroom needs. Each chapter concludes with seven Student Projects, one of which always involves programming a calculator or computer. All of the projects not only engage students in solving number-theoretical problems but also help familiarize them with the relevant mathematical literature.
Author(s): James K. Strayer
Publisher: Waveland Press
Year: 1994/2002
Language: English
Pages: C, xii, 290, B
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Cover
S Title
Elementary Number Theory
Copyright © 1994 by Waveland Press, Inc.
Reissued 2002 by Waveland Press
ISBN 978-1-57766-224-2
Preface
Preface to the Student
Contents
Introduction
1 Divisibility and Factorization
1.1 Divisibility
Exercise Set 1.1
1.2 Prime Numbers
Exercise Set 1.2
1.3 Greatest Common Divisors
Exercise Set 1.3
1.4 The Euclidean Algorithm
Exercise Set 1.4
1.5 The Fundamental Theorem of Arithmetic
Exercise Set 1.5
1.6 Concluding Remarks
Student Projects
2 Congruences
2.1 Congruences
Exercise Set 2.1
2.2 Linear Congruences in One Variable
Exercise Set 2.2
2.3 The Chinese Remainder Theorem
Exercise Set 2 .3
2.4 Wilson's Theorem
Exercise Set 2.4
2.5 Fermat's Little Theorem; Pseudoprime Numbers
Exercise Set 2 .5
2.6 Euler's Theorem
Exercise Set 2.6
2.7 Concluding Remarks
Student Projects
3 Arithmetic Functions
3.1 Arithmetic Functions; Multiplicativity
Exerdse Set 3.1
3.2 The Euler Phi-Function
Exercise Set 3.2
3.3 The Number of Positive Divisors Function
Exercise Set 3.3
3.4 The Sum of Positive Divisors Function
Exercise Set 3.4
3.5 Perfect Numbers
Exercise Set 3.5
3.6 The Mobius Inversion Formula
Exercise Set 3.6
3.7 Concluding Remarks
Student Projects
4 Quadratic Residues
4.1 Quadratic Residues
Exercise Set 4.1
4.2 The Legendre Symbol
Exercise Set 4.2
4.3 The Law of Quadratic Reciprocity
Exercise Set 4.3
4.4 Concluding Remarks
Student Projects
5 Primitive Roots
5.1 The Order of an Integer; Primitive Roots
Exercise Set 5.1
5.2 Primitive Roots for Prime Numbers
Exercise Set 5.2
5.3 The Primitive Root Theorem
Exercise Set 5.3
5.4 Index Arithmetic; nth Power Residues
Exercise Set 5.4
5.5 Concluding Remarks
Student Projects
6 Diophantine Equations
6.1 Linear Diophantine Equations
Exercise Set 6.1
6.2 Nonlinear Diophantine Equations; a Congruence Method
Exercise Set 6.2
6.3 Pythagorean Triples
Exercise Set 6.3
6.4 Fermat's Last Theorem
Exercise Set 6.4
6.5 Representation of an Integer as a Sum of Squares
Exercise Set 6.5
6.6 Concluding Remarks
Student Projects
7 Continued Fractions
7.1 Rational and Irrational Numbers
Exercise Set 7.1
7.2 Finite Continued Fractions
Exercise Set 7.2
7.3 Convergents
Exercise Set 7.3
7.4 Infinite Continued Fractions
Exercise Set 7.4
7.5 Eventually Periodic Continued Fractions
Exercise Set 7.5
7.6 Periodic Continued Fractions
Exercise Set 7.6
7.7 Concluding Remarks
Student Projects
8 A Few Applications
8.1 A Recreational Application
Exercise Set 8.1
8.2 Cryptography; The RSA Encryption System
Exercise Set 8.2
8.3 Primality Testing
Exercise Set 8.3
8.4 Pell' s Equation
Exercise Set 8 .4
8.5 Concluding Remarks
Student Projects
Appendices
A Mathematical Induction
A.1 The First Principle of Mathematical Induction
A.2 The Second Principle of Mathematical Induction
Exercise Set A
B Equivalence Relations
B.1 Relations
B.2 Equivalence Relations
B.3 Partitions
Exercise Set B
C Abstract Algebra
C.1 Rings and Fields
C.2 Groups
Exercise Set C
D The Binomial Theorem
D.1 The Binomial Theorem
Exercise Set D
E Tables
Table 1 Prime Numbers
Table 2 Arithmetic Functions
Table3 Primitive Roots/Prime Numbers
Table 4 Continued Fractions
Hints and Answers to Selected Exercises
Exercise Sets 1.1, 1.2
Exercise Sets 1.3, 1.4, 1.5
Exercise. Sets 2.1, 2.2
Exercise Sets 2.3, 2.4, 2.5, 2.6
Exercise Sets 3.1, 3.2
Exercise Sets 3.3. 3.4. 3.5
Exercise Sets 3.6
Exercise Sets 4.1. 4.2
Exercise Set 4.3
Exercise Sets 5.1, 5.2, 5.3
Exercise Set 5.4
Exercise Sets 6.1, 6.2, 6.3
Exercise Sets 6.4, 6.5
Exercise Sets 7.1, 7.2
Exercise Sets 7.3, 7.4, 7.5, 7.6
Exercise Set 8.2
Exercise Sets 8.3, 8.4
Bibliography
Index
Back Cover
