**Main Description**

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.

# 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

## Answers to Selected Problems

## Index