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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:
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
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:
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