Elementary Number Theory and Its Applications

✍ Scribed by Kenneth H. Rosen

• Year: 1984
• Tongue: English
• Leaves: 462
• Category: Library

✦ Synopsis

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 Historical content has also been enhanced to show the history for the modern material.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

Preface
Contents
Introduction
1. The Integers
The well-ordering property
Divisibility
Representation of integers
Computer operations with integers
Prime numbers
2. Greatest Common Divisors and Prime Factorization
Greatest common divisors
The Euclidean algorithm
The fundamental theorem of arithmetic
Factorization of integers and the Fermat numbers
Linear diophantine equations
3. Congruences
Introduction to congruences
Linear congruences
The Chinese remainder theorem
Systems of linear congruences
4. Applications of Congruences
Divisibility tests
The perpetual calendar
Round-robin tournaments
Computer file storage and hashing functions
5. Some Special Congruences
Wilson's theorem and Fermat's little theorem
Pseudoprimes
Euler's theorem
6. Multiplicative Functions
Euler's phi-function
The sum and number of divisors
Perfect numbers and Mersenne primes
7. Cryptology
Character ciphers
Block ciphers
Exponentiation ciphers
Public-key cryptography
Knapsack ciphers
Some applications to computer science
8. 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
Psuedo-random numbers
The splicing of telephone cables
9. Quadratic Residues and Reciprocity
Quadratic residues
Quadratic reciprocity
The Jacobi symbol
Euler pseudoprimes
10. Decimal Fractions and Continued Fractions
Decimal fractions
Finite continued fractions
Infinite continued fractions
Periodic continued fractions
11. Some Nonlinear Diophantine Equations
Pythagorean triples
Fermat's last theorem
Pell's equations
Appendix
Answers to selected problems
Bibliography
List of symbols
Index