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
