1. The Integers. Numbers, Sequences, and Sums.
Mathematical Induction.
The Fibonacci Numbers.
Divisibility.
2. Integer Representation and Operations. Representation of Integers.
Computer Operations with Integers.
Complexity of Integer Operations.
3. Primes and Greatest Common Divisors. Prime Numbers.
Greatest Common Divisors.
The Euclidean Algorithm.
The Fundamental Theorem of Arithmetic.
Factorization Methods and the Fermat Numbers.
Linear Diophantine Equations.
4. Congruences. Introduction to Congruences.
Linear Congruences.
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. Euler's Phi-Function.
The Sum and Number of Divisors.
Perfect Numbers and Mersenne Primes.
Moebius Inversion.
8. Cryptology. Character Ciphers.
Block and Stream Ciphers.
Exponentiation Ciphers.
Public-Key Crytography.
Knapsack Ciphers.
Crytographic 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 Testing Using Orders of Integers and Primitive Roots.
Universal Exponents.
10. Applications of Primitive Roots. Pseudorandom Numbers.
The E1Gamal Cryptosystem.
An Application to the Splicing of Telephone Cables.
11. Quadratic Residues and Reciprocity. Quadratic Residues and Nonresidues.
The Law of Quadratic Reciprocity.
The Jacobi Symbol.
Euler Pseudoprimes.
Zero-Knowledge Proofs.
12. Decimal Fractions and Continued Fractions. 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 Equations.
Appendix A: Axioms for the Set of Integers. Appendix B: Binomial Coefficients. Appendix C: Using Maple (R) and Mathematica for Number Theory. Appendix D: Number Theory Web Links. Appendix E: Tables. Answers to odd-numbered exercises. Bibliography. Index of Biographies. Index. 