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Edition: 3RD 93

Copyright: 1993

Publisher: Addison-Wesley Longman, Inc.

Published: 1993

International: No

Copyright: 1993

Publisher: Addison-Wesley Longman, Inc.

Published: 1993

International: No

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This third edition preserves the strengths of the previous editions while enhancing the text's teachability, flexibility, and richness. It incorporates feedback from many of the more than 200 schools where this text has been used. The blending of classical theory with modern applications has always been a hallmark of the text, and this new edition builds on this strength with new examples and additional applications. Many new exercises, including more routine exercises, along with many new intermediate, challenging, and extremely challenging exercises are provided. Challenging and extremely challenging exercises are clearly marked in the text. New to this edition are answers or solutions to all odd-numbered exercises at the end of the text.

**FEATURES:**

- Sections on check digits and zero knowledge proofs are included.
- Coverage of elementary factoring methods, including the Pollard p-1 method and the Pollard rho method, were expanded.
- Information on recent developments in number theory, such as new big prime numbers and factorizations of large integers, was updated.
- Ten additional biographies of mathematicians bring the text's total to more than 25.
- Computations and Explorations allow students to use computer programs to discover new ideas.
- Applications are integrated with text material.
- Exercises range from routine to challenging.
- Coverage of number theory and cryptology is integrated and comprehensive.
- Excellent computer science applications include hashing functions, arithmetic with large integers, pseudo-primes, and probabalistic primality testing.

**Chapter 1: Introduction**

**Chapter 2: The Integers**

Basic Properties

Summations and Products

Mathematical Induction

Binomial Coefficients

Divisibility

Representations of Integers

Computer Operations with Integers

Complexity of Integer Operations

Prime Numbers

**Chapter 3: Greatest Common Divisors and Prime Factorization**

Greatest Common Divisors

The Euclidean Algorithm

The Fundamental Theorem of Arithmetic

The Fermat Numbers and Factorization Methods

Linear Diophantine Equations

**Chapter 4: Congruencies**

Introduction to Congruencies

Linear Congruencies

The Chinese Remainder Theorem

Systems of Linear Congruencies

Factoring Using the Pollard rho Method

**Chapter 5: Applications of Congruencies**

Divisibility Tests

The Perpetual Calendar

Round-Robin Tournaments

Computer File Storage and Hashing Functions

Check Digits

**Chapter 6: Some Special Congruencies**

Wilson's Theorem and Fermat's Little Theorem

Pseudoprimes

Euler's Theorem

**Chapter 7: Multiplicative Functions**

Euler's Phi-Function

The Sum and Number of Divisors

Perfect Numbers and Mersenne Primes

**Chapter 8: Cryptology**

Character Ciphers

Block Ciphers

Exponentiation Ciphers

Public-Key Cryptography

Knapsack Ciphers

Some Applications to Computer Science

**Chapter 9: Primitive Roots**

The Order of an Integer and Primitive Roots

Primitive Roots for Primes

Existence of Primitive Roots

Index Arithmetic

Primality Testing Using Primitive Roots

Universal Exponents

Pseudo-Random Numbers

An Application to the Splicing of Telephone Cables

**Chapter 10: Quadratic Residues and Reciprocity**

Quadratic Residues and Nonresidues

Quadratic Reciprocity

The Jacobi Symbol

Euler Pseudoprimes

Zero-Knowledge Proofs

**Chapter 11: Decimal Fractions and Continued Fractions**

Decimal Fractions

Finite Continued Fractions

Infinite Continued Fractions

Periodic Continued Fractions

Factoring Using Continued Fractions

**Chapter 12: Some Nonlinear Diophantine Equations**

Pythagorean Triples

Fermat's Last Theorem

Sums of Squares

Pell's Equations

Appendix

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Summary

This third edition preserves the strengths of the previous editions while enhancing the text's teachability, flexibility, and richness. It incorporates feedback from many of the more than 200 schools where this text has been used. The blending of classical theory with modern applications has always been a hallmark of the text, and this new edition builds on this strength with new examples and additional applications. Many new exercises, including more routine exercises, along with many new intermediate, challenging, and extremely challenging exercises are provided. Challenging and extremely challenging exercises are clearly marked in the text. New to this edition are answers or solutions to all odd-numbered exercises at the end of the text.

**FEATURES:**

- Sections on check digits and zero knowledge proofs are included.
- Coverage of elementary factoring methods, including the Pollard p-1 method and the Pollard rho method, were expanded.
- Information on recent developments in number theory, such as new big prime numbers and factorizations of large integers, was updated.
- Ten additional biographies of mathematicians bring the text's total to more than 25.
- Computations and Explorations allow students to use computer programs to discover new ideas.
- Applications are integrated with text material.
- Exercises range from routine to challenging.
- Coverage of number theory and cryptology is integrated and comprehensive.
- Excellent computer science applications include hashing functions, arithmetic with large integers, pseudo-primes, and probabalistic primality testing.

Table of Contents

**Chapter 1: Introduction**

**Chapter 2: The Integers**

Basic Properties

Summations and Products

Mathematical Induction

Binomial Coefficients

Divisibility

Representations of Integers

Computer Operations with Integers

Complexity of Integer Operations

Prime Numbers

**Chapter 3: Greatest Common Divisors and Prime Factorization**

Greatest Common Divisors

The Euclidean Algorithm

The Fundamental Theorem of Arithmetic

The Fermat Numbers and Factorization Methods

Linear Diophantine Equations

**Chapter 4: Congruencies**

Introduction to Congruencies

Linear Congruencies

The Chinese Remainder Theorem

Systems of Linear Congruencies

Factoring Using the Pollard rho Method

**Chapter 5: Applications of Congruencies**

Divisibility Tests

The Perpetual Calendar

Round-Robin Tournaments

Computer File Storage and Hashing Functions

Check Digits

**Chapter 6: Some Special Congruencies**

Wilson's Theorem and Fermat's Little Theorem

Pseudoprimes

Euler's Theorem

**Chapter 7: Multiplicative Functions**

Euler's Phi-Function

The Sum and Number of Divisors

Perfect Numbers and Mersenne Primes

**Chapter 8: Cryptology**

Character Ciphers

Block Ciphers

Exponentiation Ciphers

Public-Key Cryptography

Knapsack Ciphers

Some Applications to Computer Science

**Chapter 9: Primitive Roots**

The Order of an Integer and Primitive Roots

Primitive Roots for Primes

Existence of Primitive Roots

Index Arithmetic

Primality Testing Using Primitive Roots

Universal Exponents

Pseudo-Random Numbers

An Application to the Splicing of Telephone Cables

**Chapter 10: Quadratic Residues and Reciprocity**

Quadratic Residues and Nonresidues

Quadratic Reciprocity

The Jacobi Symbol

Euler Pseudoprimes

Zero-Knowledge Proofs

**Chapter 11: Decimal Fractions and Continued Fractions**

Decimal Fractions

Finite Continued Fractions

Infinite Continued Fractions

Periodic Continued Fractions

Factoring Using Continued Fractions

**Chapter 12: Some Nonlinear Diophantine Equations**

Pythagorean Triples

Fermat's Last Theorem

Sums of Squares

Pell's Equations

Appendix

Publisher Info

Publisher: Addison-Wesley Longman, Inc.

Published: 1993

International: No

Published: 1993

International: No

This third edition preserves the strengths of the previous editions while enhancing the text's teachability, flexibility, and richness. It incorporates feedback from many of the more than 200 schools where this text has been used. The blending of classical theory with modern applications has always been a hallmark of the text, and this new edition builds on this strength with new examples and additional applications. Many new exercises, including more routine exercises, along with many new intermediate, challenging, and extremely challenging exercises are provided. Challenging and extremely challenging exercises are clearly marked in the text. New to this edition are answers or solutions to all odd-numbered exercises at the end of the text.

**FEATURES:**

- Sections on check digits and zero knowledge proofs are included.
- Coverage of elementary factoring methods, including the Pollard p-1 method and the Pollard rho method, were expanded.
- Information on recent developments in number theory, such as new big prime numbers and factorizations of large integers, was updated.
- Ten additional biographies of mathematicians bring the text's total to more than 25.
- Computations and Explorations allow students to use computer programs to discover new ideas.
- Applications are integrated with text material.
- Exercises range from routine to challenging.
- Coverage of number theory and cryptology is integrated and comprehensive.
- Excellent computer science applications include hashing functions, arithmetic with large integers, pseudo-primes, and probabalistic primality testing.

**Chapter 1: Introduction**

**Chapter 2: The Integers**

Basic Properties

Summations and Products

Mathematical Induction

Binomial Coefficients

Divisibility

Representations of Integers

Computer Operations with Integers

Complexity of Integer Operations

Prime Numbers

**Chapter 3: Greatest Common Divisors and Prime Factorization**

Greatest Common Divisors

The Euclidean Algorithm

The Fundamental Theorem of Arithmetic

The Fermat Numbers and Factorization Methods

Linear Diophantine Equations

**Chapter 4: Congruencies**

Introduction to Congruencies

Linear Congruencies

The Chinese Remainder Theorem

Systems of Linear Congruencies

Factoring Using the Pollard rho Method

**Chapter 5: Applications of Congruencies**

Divisibility Tests

The Perpetual Calendar

Round-Robin Tournaments

Computer File Storage and Hashing Functions

Check Digits

**Chapter 6: Some Special Congruencies**

Wilson's Theorem and Fermat's Little Theorem

Pseudoprimes

Euler's Theorem

**Chapter 7: Multiplicative Functions**

Euler's Phi-Function

The Sum and Number of Divisors

Perfect Numbers and Mersenne Primes

**Chapter 8: Cryptology**

Character Ciphers

Block Ciphers

Exponentiation Ciphers

Public-Key Cryptography

Knapsack Ciphers

Some Applications to Computer Science

**Chapter 9: Primitive Roots**

The Order of an Integer and Primitive Roots

Primitive Roots for Primes

Existence of Primitive Roots

Index Arithmetic

Primality Testing Using Primitive Roots

Universal Exponents

Pseudo-Random Numbers

An Application to the Splicing of Telephone Cables

**Chapter 10: Quadratic Residues and Reciprocity**

Quadratic Residues and Nonresidues

Quadratic Reciprocity

The Jacobi Symbol

Euler Pseudoprimes

Zero-Knowledge Proofs

**Chapter 11: Decimal Fractions and Continued Fractions**

Decimal Fractions

Finite Continued Fractions

Infinite Continued Fractions

Periodic Continued Fractions

Factoring Using Continued Fractions

**Chapter 12: Some Nonlinear Diophantine Equations**

Pythagorean Triples

Fermat's Last Theorem

Sums of Squares

Pell's Equations

Appendix