# Elementary Number Theory

7^{th}Edition

ISBN10: 0073383147

ISBN13: 9780073383149

Copyright: 2011

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The estimated amount of time this product will be on the market is based on a number of factors, including faculty input to instructional design and the prior revision cycle and updates to academic research-which typically results in a revision cycle ranging from every two to four years for this product. Pricing subject to change at any time.

### Program Details

# 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

# 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

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