Preface
　　List of Symbols
　　What Is Number Theory?
　　1 The Integers
　　 1.1 Numbers and Sequences
　　 1.2 Sums and Products
　　 1.3 Mathematical Induction
　　 1.4 The Fibonacci Numbers
　　 1.5 Divisibility
　　2 Integer Representations and Operations
　　 2.1 Representations of Integers
　　 2.2 Computer Operations with Integers
　　 2.3 Complexity of Integer Operations
　　3 Primes and Greatest Common Divisors
　　 3.1 Prime Numbers
　　 3.2 The Distribution of Primes
　　 3.3 Greatest Common Divisors and their Properties
　　 3.4 The Euclidean Algorithm
　　 3.5 The Fundamental Theorem of Arithmetic
　　 3.6 Factorization Methods and the Fermat Numbers
　　 3.7 Linear Diophantine Equations
　　4 Congruences
　　 4.1 Introduction to Congruences
　　 4.2 Linear Congruences
　　 4.3 The Chinese Remainder Theorem
　　 4.4 Solving Polynomial Congruences
　　 4.5 Systems of Linear Congruences
　　 4.6 Factoring Using the Pollard Rho Method
　　5 Applications of Congruences
　　 5.1 Divisibility Tests
　　 5.2 The Perpetual Calendar
　　 5.3 Round-Robin Tournaments
　　 5.4 Hashing Functions
　　 5.5 Check Dieits
　　6 Some Special Congruences
　　 6.1 Wilson's Theorem and Fermat's Little Theorem
　　 6.2 Pseudoprimes
　　 6.3 Euler's Theorem
　　7 Multiplicative Functions
　　 7.1 The Euler Phi-Function
　　 7.2 The Sum and Number of Divisors
　　 7.3 Perfect Numbers and Mersenne Primes
　　 7.4 M6bius Inversion
　　 7.5 Partitions
　　8 Cryptology
　　 8.1 Character Ciphers
　　 8.2 Block and Stream Ciphers
　　 8.3 Exponentiation Ciphers
　　 8.4 Public Key Cryptography
　　 8.5 Knapsack Ciphers
　　 8.6 Cryptographic Protocols and Applications
　　9 Primitive Roots
　　 9.1 The Order of an Integer and Primitive Roots
　　 9.2 Primitive Roots for Primes
　　 9.3 The Existence of Primitive Roots
　　 9.4 Discrete Logarithms and Index Arithmetic
　　 9.5 Primality Tests Using Orders of Integers and Primitive Roots
　　 9.6 Universal Exponents
　　10 Applications of Primitive Roots and the
　　 Order of an Integer
　　 10.1 Pseudorandom Numbers
　　 10.2 The E1Gamal Cryptosystem
　　 10.3 An Application to the Splicing of Telephone Cables
　　11 Quadratic Residues
　　 11.1 Quadratic Residues and Nonresidues
　　 11.2 The Law of Quadratic Reciprocity
　　 11.3 The Jacobi Symbol
　　 11.4 Euler Pseudoprimes
　　 11.5 Zero-Knowledge Proofs
　　12 Decimal Fractions and Continued Fractions
　　 12.1 Decimal Fractions
　　 12.2 Finite Continued Fractions
　　 12.3 Infinite Continued Fractions
　　 12.4 Periodic Continued Fractions
　　 12.5 Factoring Using Continued Fractions
　　13 Some Nonlinear Diophantine Equations
　　 13.1 Pythagorean Triples
　　 13.2 Fermat's Last Theorem
　　 13.3 Sums of Squares
　　 13.4 Pell's Equation
　　 13.5 Congruent Numbers
　　14 The Gaussian Integers
　　 14.1 Gaussian Integers and Gaussian Primes
　　 14.2 Greatest Common Divisors and Unique Factorization
　　 14.3 Gaussian Integers and Sums of Squares
　　Appendix A Axioms for the Set of Integers
　　Appendix B Binomial Coefficients
　　Appendix C Using Maple and Mathematica for Number Theory
　　 C.1 Using Maple for Number Theory
　　 C.2 Using 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