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Edition: 4TH 00

Copyright: 2000

Publisher: Addison-Wesley Longman, Inc.

Published: 2000

International: No

Copyright: 2000

Publisher: Addison-Wesley Longman, Inc.

Published: 2000

International: No

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This latest edition of Kenneth Rosen's widely used *Elementary Number Theory and Its Applications* enhances the flexibility and depth of previous editions while preserving their strengths. Rosen effortlessly blends classic theory with contemporary applications. New examples, additional applications and increased cryptology coverage are also included. The book has also been accuracy-checked to ensure the quality of the content. A diverse group of exercises are presented to help develop skills. Also included are computer projects. The book contains updated and increased coverage of Cryptography and new sections on Mvbius Inversion and solving Polynomial Congruences. Historical content has also been enhanced to show the history for the modern material. For those interested in number theory.

**What is Number Theory?**

**1. The Integers. **

Numbers, Sequences, and Sums.

Mathematical Induction.

Fibonacci Numbers.

Divisibility.

**2. Integer Representation and Operations. **

Representation of Integers.

Computer Operations with Integers.

Complexity of Integer Operations.

**3. Greatest Common Divisors and Prime Factorization. **

Prime Numbers.

Greatest Common Divisors.

The Euclidean Algorithm.

The Fundamental Theorem of Arithmetic.

Factorization Methods and the Fermat Numbers.

Linear Diophantine Equations.

**4. Congruences. **

Introduction to Congruences.

Linear Congruences.

The Chinese Remainder Theorem.

Solving Polynomial Congruences.

Systems of Linear Congruences.

Factoring Using the Pollard rho Method.

**5. Applications of Congruences. **

Divisibility Tests.

The Perpetual Calendar.

Round-Robin Tournaments.

Hashing Functions.

Check Digits.

**6. Some Special Congruences. **

Wilson's Theorem and Fermat's Little Theorem.

Pseudoprimes.

Euler's Theorem.

**7. Multiplicative Functions. **

Euler's Phi-Function.

The Sum and Number of Divisors.

Perfect Numbers and Mersenne Primes.

Möbius Inversion.

**8. Cryptology. **

Character Ciphers.

Block and Stream Ciphers.

Exponentiation Ciphers.

Public-Key Crytography.

Knapsack Ciphers.

Crytographic Protocols and Applications.

**9. Primitive Roots. **

The Order of an Integer and Primitive Roots.

Primitive Roots for Primes.

Existence of Primitive Roots.

Discrete Logarithms and Index Arithmetic.

Primality Testing Using Orders of Integers and Primitive Roots.

Universal Exponents.

**10. Applications of Primitive Roots. **

Pseudorandom Numbers.

The E1Gamal Cryptosystem.

An Application to the Splicing of Telephone Cables.

**11. Quadratic Residues and Reciprocity. **

Quadratic Residues and Nonresidues.

The Law of Quadratic Reciprocity.

The Jacobi Symbol.

Euler Pseudoprimes.

Zero-Knowledge Proofs.

**12. Decimal Fractions and Continued Fractions. **

Decimal Fractions.

Finite Continued Fractions.

Infinite Continued Fractions.

Periodic Continued Fractions.

Factoring Using Continued Fractions.

**13. Some Nonlinear Diophantine Equations. **

Pythagorean Triples.

Fermat's Last Theorem.

Sums of Squares.

Pell's Equations.

Appendix A: Axioms for the Set of Integers.

Appendix B: Binomial Coefficients.

Appendix C: Using Maple® and *Mathematica* for Number Theory.

Appendix D: Number Theory Web Links.

Appendix E: Tables.

Answers to odd-numbered exercises.

Bibliography.

Index of Biographies.

Index.

Photo Credits.

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Summary

This latest edition of Kenneth Rosen's widely used *Elementary Number Theory and Its Applications* enhances the flexibility and depth of previous editions while preserving their strengths. Rosen effortlessly blends classic theory with contemporary applications. New examples, additional applications and increased cryptology coverage are also included. The book has also been accuracy-checked to ensure the quality of the content. A diverse group of exercises are presented to help develop skills. Also included are computer projects. The book contains updated and increased coverage of Cryptography and new sections on Mvbius Inversion and solving Polynomial Congruences. Historical content has also been enhanced to show the history for the modern material. For those interested in number theory.

Table of Contents

**What is Number Theory?**

**1. The Integers. **

Numbers, Sequences, and Sums.

Mathematical Induction.

Fibonacci Numbers.

Divisibility.

**2. Integer Representation and Operations. **

Representation of Integers.

Computer Operations with Integers.

Complexity of Integer Operations.

**3. Greatest Common Divisors and Prime Factorization. **

Prime Numbers.

Greatest Common Divisors.

The Euclidean Algorithm.

The Fundamental Theorem of Arithmetic.

Factorization Methods and the Fermat Numbers.

Linear Diophantine Equations.

**4. Congruences. **

Introduction to Congruences.

Linear Congruences.

The Chinese Remainder Theorem.

Solving Polynomial Congruences.

Systems of Linear Congruences.

Factoring Using the Pollard rho Method.

**5. Applications of Congruences. **

Divisibility Tests.

The Perpetual Calendar.

Round-Robin Tournaments.

Hashing Functions.

Check Digits.

**6. Some Special Congruences. **

Wilson's Theorem and Fermat's Little Theorem.

Pseudoprimes.

Euler's Theorem.

**7. Multiplicative Functions. **

Euler's Phi-Function.

The Sum and Number of Divisors.

Perfect Numbers and Mersenne Primes.

Möbius Inversion.

**8. Cryptology. **

Character Ciphers.

Block and Stream Ciphers.

Exponentiation Ciphers.

Public-Key Crytography.

Knapsack Ciphers.

Crytographic Protocols and Applications.

**9. Primitive Roots. **

The Order of an Integer and Primitive Roots.

Primitive Roots for Primes.

Existence of Primitive Roots.

Discrete Logarithms and Index Arithmetic.

Primality Testing Using Orders of Integers and Primitive Roots.

Universal Exponents.

**10. Applications of Primitive Roots. **

Pseudorandom Numbers.

The E1Gamal Cryptosystem.

An Application to the Splicing of Telephone Cables.

**11. Quadratic Residues and Reciprocity. **

Quadratic Residues and Nonresidues.

The Law of Quadratic Reciprocity.

The Jacobi Symbol.

Euler Pseudoprimes.

Zero-Knowledge Proofs.

**12. Decimal Fractions and Continued Fractions. **

Decimal Fractions.

Finite Continued Fractions.

Infinite Continued Fractions.

Periodic Continued Fractions.

Factoring Using Continued Fractions.

**13. Some Nonlinear Diophantine Equations. **

Pythagorean Triples.

Fermat's Last Theorem.

Sums of Squares.

Pell's Equations.

Appendix A: Axioms for the Set of Integers.

Appendix B: Binomial Coefficients.

Appendix C: Using Maple® and *Mathematica* for Number Theory.

Appendix D: Number Theory Web Links.

Appendix E: Tables.

Answers to odd-numbered exercises.

Bibliography.

Index of Biographies.

Index.

Photo Credits.

Publisher Info

Publisher: Addison-Wesley Longman, Inc.

Published: 2000

International: No

Published: 2000

International: No

*Elementary Number Theory and Its Applications* enhances the flexibility and depth of previous editions while preserving their strengths. Rosen effortlessly blends classic theory with contemporary applications. New examples, additional applications and increased cryptology coverage are also included. The book has also been accuracy-checked to ensure the quality of the content. A diverse group of exercises are presented to help develop skills. Also included are computer projects. The book contains updated and increased coverage of Cryptography and new sections on Mvbius Inversion and solving Polynomial Congruences. Historical content has also been enhanced to show the history for the modern material. For those interested in number theory.

**What is Number Theory?**

**1. The Integers. **

Numbers, Sequences, and Sums.

Mathematical Induction.

Fibonacci Numbers.

Divisibility.

**2. Integer Representation and Operations. **

Representation of Integers.

Computer Operations with Integers.

Complexity of Integer Operations.

**3. Greatest Common Divisors and Prime Factorization. **

Prime Numbers.

Greatest Common Divisors.

The Euclidean Algorithm.

The Fundamental Theorem of Arithmetic.

Factorization Methods and the Fermat Numbers.

Linear Diophantine Equations.

**4. Congruences. **

Introduction to Congruences.

Linear Congruences.

The Chinese Remainder Theorem.

Solving Polynomial Congruences.

Systems of Linear Congruences.

Factoring Using the Pollard rho Method.

**5. Applications of Congruences. **

Divisibility Tests.

The Perpetual Calendar.

Round-Robin Tournaments.

Hashing Functions.

Check Digits.

**6. Some Special Congruences. **

Wilson's Theorem and Fermat's Little Theorem.

Pseudoprimes.

Euler's Theorem.

**7. Multiplicative Functions. **

Euler's Phi-Function.

The Sum and Number of Divisors.

Perfect Numbers and Mersenne Primes.

Möbius Inversion.

**8. Cryptology. **

Character Ciphers.

Block and Stream Ciphers.

Exponentiation Ciphers.

Public-Key Crytography.

Knapsack Ciphers.

Crytographic Protocols and Applications.

**9. Primitive Roots. **

The Order of an Integer and Primitive Roots.

Primitive Roots for Primes.

Existence of Primitive Roots.

Discrete Logarithms and Index Arithmetic.

Primality Testing Using Orders of Integers and Primitive Roots.

Universal Exponents.

**10. Applications of Primitive Roots. **

Pseudorandom Numbers.

The E1Gamal Cryptosystem.

An Application to the Splicing of Telephone Cables.

**11. Quadratic Residues and Reciprocity. **

Quadratic Residues and Nonresidues.

The Law of Quadratic Reciprocity.

The Jacobi Symbol.

Euler Pseudoprimes.

Zero-Knowledge Proofs.

**12. Decimal Fractions and Continued Fractions. **

Decimal Fractions.

Finite Continued Fractions.

Infinite Continued Fractions.

Periodic Continued Fractions.

Factoring Using Continued Fractions.

**13. Some Nonlinear Diophantine Equations. **

Pythagorean Triples.

Fermat's Last Theorem.

Sums of Squares.

Pell's Equations.

Appendix A: Axioms for the Set of Integers.

Appendix B: Binomial Coefficients.

Appendix C: Using Maple® and *Mathematica* for Number Theory.

Appendix D: Number Theory Web Links.

Appendix E: Tables.

Answers to odd-numbered exercises.

Bibliography.

Index of Biographies.

Index.

Photo Credits.