Introduction to Number Theory covers the essential content of an introductory number theory course including divisibility and prime factorization, congruences, and quadratic reciprocity. The instructor may also choose from a collection of additional topics.
Aligning with the trend toward smaller, essential texts in mathematics, the author strives for clarity of exposition. Proof techniques and proofs are presented slowly and clearly.
The book employs a versatile approach to the use of algebraic ideas. Instructors who wish to put this material into a broader context may do so, though the author introduces these concepts in a non-essential way.
A final chapter discusses algebraic systems (like the Gaussian integers) presuming no previous exposure to abstract algebra. Studying general systems helps students to realize unique factorization into primes is a more subtle idea than may at first appear; students will find this chapter interesting, fun and quite accessible.
Applications of number theory include several sections on cryptography and other applications to further interest instructors and students alike.
Author(s): Mark Hunacek
Series: Textbooks in Mathematics
Edition: 1
Publisher: Chapman and Hall/CRC
Year: 2023
Language: English
Pages: 164
City: Boca Raton, FL
Tags: Number Theory; Divisibility; Prime Factorization; Congruences; Quadratic Reciprocity
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Author
Introduction: What is Number Theory?
0.1 Exercises
1 Divisibility
1.1 The Principles of Well-Ordering and Mathematical Induction
Exercises
1.2 Basic Properties of Divisibility
Exercises
1.3 The Greatest Common Divisor
Exercises
1.4 The Euclidean Algorithm
Exercises
1.5 Primes
Exercises
1.6 Numbers to Different Bases
Exercises
Challenge Problems for Chapter 1
2 Congruences and Modular Arithmetic
2.1 Basic Definitions and Principles
Exercises
2.2 Arithmetic in Z[sub(n)]
Exercises
2.3 Linear Equations in Z[sub(n)]
Exercises
2.4 The Euler Phi Function
Exercises
2.5 Theorems of Wilson, Fermat and Euler
Exercises
2.6 Pythagorean Triples
Exercises
Challenge Problems for Chapter 2
3 Cryptography: An Introduction
3.1 Basic Definitions
3.2 Classical Cryptography
Exercises
3.3 Public Key Cryptography: RSA
Exercises
Challenge Problems for Chapter 3
4 Perfect Numbers
4.1 Basic Definitions and Principles: The Sigma Function
Exercises
4.2 Even Perfect Numbers
Exercises
Challenge Problems for Chapter 4
5 Primitive Roots
5.1 Order of an Integer
Exercises
5.2 Primitive Roots
Exercises
5.3 Polynomials in Z[sub(p)]
Exercises
5.4 Primitive Roots Modulo a Prime
Exercises
5.5 An Application: Diffie-Hellman Key Exchange
5.6 Another Application: ElGamal Cryptosystem
Challenge Problems for Chapter 5
6 Quadratic Reciprocity
6.1 Squares Modulo a Prime
Exercises
6.2 Euler's Criterion and Legendre Symbols
Exercises
6.3 The Law of Quadratic Reciprocity
Exercises
6.4 The Supplemental Relations
Exercises
6.5 The Jacobi Symbol
Exercises
Challenge Problems for Chapter 6
7 Arithmetic Beyond the Integers
7.1 Gaussian Integers: Introduction and Basic Facts
Exercises
7.2 A Geometric Interlude
Exercises
7.3 Divisibility and Primes in the Gaussian Integers
Exercises
7.4 The Division Algorithm and the Greatest Common Divisor in Z[i]
Exercises
7.5 An Application: Sums of Two Squares
Exercises
7.6 Another Application: Diophantine Equations
Exercises
7.7 A Third Application: Pythagorean Triples
Exercises
7.8 Irreducible Gaussian Integers
Exercises
7.9 Other Quadratic Extensions
Exercises
7.10 Algebraic Numbers and Integers
Exercises
7.11 The Quaternions
Exercises
7.12 Sums of Four Squares
Challenge Problems for Chapter 7
Appendix A: A Proof Primer
Appendix B: Axioms for the Integers
Appendix C: Basic Algebraic Terminology
Bibliography
Index
