Elementary Number Theory, 5/E Kenneth H. Rosen, AT&T Laboratories ISBN: 0321237072 Publisher: Addison-Wesley Copyright: 2005 Format: Cloth; 744 pages Published: 10/19/2004 Status: Published

Elementary Number Theory and Its Applications is noted for its outstanding exercise sets, including basic exercises, exercises designed to help students explore key concepts, and challenging exercises. Computational exercises and computer projects are also provided. In addition to years of use and professor feedback, the fifth edition of this text has been thoroughly checked to ensure the quality and accuracy of the mathematical content and the exercises.

The blending of classical theory with modern applications is a hallmark feature of the text. The Fifth Edition builds on this strength with new examples and exercises, additional applications and increased cryptology coverage. The author devotes a great deal of attention to making this new edition up-to-date, incorporating new results and discoveries in number theory made in the past few years.

• Exercises: Extensive and diverse exercise sets, with routine exercises to develop basic skills, exercises devoted to developing new concepts, and challenging exercises. Answers/solutions are provided to all odd-numbered exercises in the text.

• Applications: Lots and lots of applications of number theory well integrated into the text, illustrating the usefulness of the theory.

• Cryptography: The first number theory book to integrate coverage of cryptography—even more a strength of the text as cryptography and cryptographic protocols are covered in depth, not as an afterthought. Moreover, results important for cryptography are developed with the theory in early chapters.

• Flexibility: Instructors can design their own courses choosing from a wealth of topics.

• Proofs: Carefully motivated and fully explained proofs of the key results in number theory.

• Computer Projects: Each section of the text presents computation exercise and projects focused on concepts or algorithms from that section. Students can tie together mathematics in the text with their computer skills using these projects.

• Accuracy: Extra attention has been devoted to ensure that accuracy of the mathematical content and exercises in the text.

• Website: A website will accompany the text, including a variety of resources that can enrich the use of the book.

• Historical Content and Biographies: Illustrate the human side, both ancient and modern, of number theory.

• Ancillaries: A student solutions guide provides complete solutions to the odd-numbered exercises and other resources for students. An extensive Instructor's Solution Manual provides complete solutions to all exerices, material on programming projects, and an extensive test bank.

• A more teachable organization has been achieved by splitting the beginning sections in Chapters 1 and 3 into two sections each.

• Advanced topics, including the Riemann zeta function and its connection with the primes, Primes=P, results about gaps between consecutive primes, and the abc conjecture are discussed.

• A chapter on the Gaussian integers has been added.

• Cryptanalysis of the Vigenere cipher is covered in depth.

• Euclid's proof of the infinitude of primes is now in the text; many other proofs of this result are developed in the exercises or text.

• The pigeonhole principle and its application to Diophantine approximation are now introduced.

P. What is Number Theory?


1. The Integers.
Numbers and Sequences.
Sums and Products.
Mathematical Induction.
The Fibonacci Numbers.

2. Integer Representations and Operations.
Representations of Integers.
Computer Operations with Integers.
Complexity of Integer Operations.

3. Primes and Greatest Common Divisors.
Prime Numbers.
The Distribution of Primes.
Greatest Common Divisors.
The Euclidean Algorithm.
The Fundemental Theorem of Arithmetic.
Factorization Methods and Fermat Numbers.
Linear Diophantine Equations.

4. Congruences.
Introduction to Congruences.
Linear Congrences.
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.
The Euler Phi-Function.
The Sum and Number of Divisors.
Perfect Numbers and Mersenne Primes.
Mobius Inversion.

8. Cryptology.
Character Ciphers.
Block and Stream Ciphers.
Exponentiation Ciphers.
Knapsack Ciphers.
Cryptographic Protocols and Applications.

9. Primitive Roots.
The Order of an Integer and Primitive Roots.
Primitive Roots for Primes.
The Existence of Primitive Roots.
Index Arithmetic.
Primality Tests Using Orders of Integers and Primitive Roots.
Universal Exponents.

10. Applications of Primitive Roots and the Order of an Integer.
Pseudorandom Numbers.
The EIGamal Cryptosystem.
An Application to the Splicing of Telephone Cables.

11. Quadratic Residues.
Quadratic Residues and nonresidues.
The Law of Quadratic Reciprocity.
The Jacobi Symbol.
Euler Pseudoprimes.
Zero-Knowledge Proofs.

12. Decimal Fractions and Continued.
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 Equation.

14. The Gaussian Integers.
Gaussian Primes.
Unique Factorization of Gaussian Integers.
Gaussian Integers and Sums of Squares.

MM0611
Number Theory (Mathematics)

 

Instructor's Solutions Manual
by Kenneth H. Rosen
© 2005 | 0-321-26842-3 | Paper; 400 pages

Companion Website
by Kenneth H. Rosen
© 2005 | 0-321-26841-5 | On-line Supplement


 

Student Solutions Manual
by Kenneth H. Rosen
© 2005 | 0-321-26840-7 | Paper; 250 pages

Companion Website
by Kenneth H. Rosen
© 2005 | 0-321-26841-5 | On-line Supplement