Author(s): Thomas P. Dence, Joseph B. Dence
Publisher: Harcourt Academic Press
Year: 1999
Language: English
Pages: 517+xvii
Title
Preface
Prologue to the Student
Contents
Part 1: The Fundamentals
1. Introduction: The Primes
1.1 Sets, Logic, and Proof
1.2 Real numbers and the well-ordering property
1.3 The Division Algorithm
1.4 The Primes
1.5 Infinitude of the Primes
1.6 Remarks on the Distribution of the Primes
1.7 Primes of Various Forms
1.8 Other Theorems and Conjectures about Primes
References
2. The Fundamental Theorem of Arithmetic and Its Consequences
2.1 The Fundamental Theorem of Arithmetic
2.2 A Theorem of Euclid
2.3 Groups
2.4 Greatest Common Divisor
2.5 Application to Linear Diophantine Equations
2.6 The Euclidean Algorithm
2.7 Least Common Multiple
References
3. An Introduction to Congruences
3.1 Elementary Properties of Congruences
3.2 Residue Classes
3.3 Rings
3.4 Fields
3.5 The Algebra of Residue Classes
3.6 Residue Systems
3.7 Introduction of the Euler phi-Function
3.8 Fermat’s Theorem
3.9 Pseudoprimes *
3.10 Euler’s Theorem
3.11 Solving Linear Congruences; Finite, Simple Continued Fractions
3.12 Cryptography and the RSA Method *
References
4. Polynomial Congruences
4.1 Introduction to Polynomial Congruences
4.2 A Theorem of Lagrange
4.3 Wilson’s Theorem
4.4 The Lucas-Lehmer Test
4.5 The Chinese Remainder Theorem
4.6 Polynomial Congruences with Prime-Power Moduli
References
5. Primitive Roots
5.1 Order of an Integer
5.2 Some Exploratory Computations
5.3 Primitive Roots
5.4 lndices
5.5 The Existence of Primitive Roots
References
6. Residues
6.1 Quadratic Residues and Nonresidues
6.2 Legendre’s Symbol
6.3 Jacobi's Symbol
6.4 Gauss’s Law of Quadratic Reciprocity
6.5 Cubic and Quartic Residues
6.6 A Theorem on kth-Power Residues of an Arbitrary Modulus
6.7 Other Reciprocity Laws *
References
7. Multiplicative Functions
7.1 Some Common Multiplicative Functions
7.2 A General Theorem on Multiplicative Functions
7.3 Highly Composite lntegers *
7.4 Perfect Numbers
7.5 More on Euler’s Totient; Carmichael’s Conjecture
7.6 The Möbius Inversion Formula
7.7 Convolution
7.8 The Algebraic Structure of Arithmetic Functions
References
Part 2: Special Topics
8. Representation Problems
8.1 The Equation x^2 + y^2 = z^2
8.2 The Two-Square Problem
8.3 The Four-Square Problem
8.4 The Equation x^4 + y^4 = z^4
8.5 Other Representation Problems *
References
9. An Introduction to Number Fields
9.1 Numbers Algebraic over a Field
9.2 Extensions of Number Fields
9.3 Transcendental Elements and Degree of an Extension
9.4 Algebraic Integers
9.5 Integers in a Quadratic Field
9.6 Norms
9.7 Units and Primes in Algebraic Number Fields
References
10. Partitions
10.1 Introduction to Partitions
10.2 Generating Functions
10.3 The Generating Function for Partitions
10.4 Some Partition ldentities *
10.5 Euler’s Pentagonal Number Theorem
10.6 A Recursive Formula for Unrestricted Partitions
References
11. Recurrence Relations
11.1 Introduction to Recurrence Relations
11.2 A General Theorem on Linear Recurrence Relations
11.3 Lucas Sequences
11.4 Stirling’s Numbers
11.5 Bernoulli Numbers
11.6 Connection with the Riemann Zeta Function
References
Two after-dinner desserts
Appendix I: Notation
Appendix II: Mathematical Tables
Appendix III: Sample Final Examinations
Appendix IV: Hints and Answers to Selected Problems
Name Index
Subject Index
