EBOOK: Elementary Number Theory

Elementary Number Theory, 7e, by David M. Burton

Table of Contents

Preface

New to this Edition

1 Preliminaries

1.1 Mathematical Induction
1.2 The Binomial Theorem

2 Divisibility Theory in the Integers

2.1 Early Number Theory 2.2 The Division Algorithm
2.3 The Greatest Common Divisor
2.4 The Euclidean Algorithm
2.5 The Diophantine Equation

3 Primes and Their Distribution

3.1 The Fundamental Theorem of Arithmetic
3.2 The Sieve of Eratosthenes
3.3 The Goldbach Conjecture

4 The Theory of Congruences

4.1 Carl Friedrich Gauss
4.2 Basic Properties of Congruence
4.3 Binary and Decimal Representations of Integers
4.4 Linear Congruences and the Chinese Remainder Theorem

5 Fermat’s Theorem

5.1 Pierre de Fermat
5.2 Fermat’s Little Theorem and Pseudoprimes
5.3 Wilson’s Theorem
5.4 The Fermat-Kraitchik Factorization Method

6 Number-Theoretic Functions

6.1 The Sum and Number of Divisors
6.2 The Möbius Inversion Formula
6.3 The Greatest Integer Function
6.4 An Application to the Calendar

7 Euler’s Generalization of Fermat’s Theorem

7.1 Leonhard Euler
7.2 Euler’s Phi-Function
7.3 Euler’s Theorem
7.4 Some Properties of the Phi-Function

8 Primitive Roots and Indices

8.1 The Order of an Integer Modulo n
8.2 Primitive Roots for Primes 8.3 Composite Numbers Having Primitive Roots
8.4 The Theory of Indices

9 The Quadratic Reciprocity Law

9.1 Euler’s Criterion
9.2 The Legendre Symbol and Its Properties
9.3 Quadratic Reciprocity
9.4 Quadratic Congruences with Composite Moduli

10 Introduction to Cryptography

10.1 From Caesar Cipher to Public Key Cryptography
10.2 The Knapsack Cryptosystem
10.3 An Application of Primitive Roots to Cryptography

11 Numbers of Special Form

11.1 Marin Mersenne
11.2 Perfect Numbers
11.3 Mersenne Primes and Amicable Numbers
11.4 Fermat Numbers

12 Certain Nonlinear Diophantine Equations

12.1 The Equation
12.2 Fermat’s Last Theorem

13 Representation of Integers as Sums of Squares

13.1 Joseph Louis Lagrange
13.2 Sums of Two Squares
13.3 Sums of More Than Two Squares

14 Fibonacci Numbers

14.1 Fibonacci
14.2 The Fibonacci Sequence
14.3 Certain Identities Involving Fibonacci Numbers

15 Continued Fractions

15.1 Srinivasa Ramanujan
15.2 Finite Continued Fractions
15.3 Infinite Continued Fractions
15.4 Farey Fractions
15.5 Pell’s Equation

16 Some Recent Developments

16.1 Hardy, Dickson, and Erdös
16.2 Primality Testing and Factorization
16.3 An Application to Factoring: Remote Coin Flipping
16.4 The Prime Number Theorem and Zeta Function

Miscellaneous Problems

Appendixes

General References

Suggested Further Reading

Tables

Elementary Number Theory, Seventh Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton’s engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.