**P. What is Number Theory?**

**1. The Integers.**

Numbers and Sequences.

Sums and Products.

Mathematical Induction.

The Fibonacci Numbers.

**2. Integer Representations and Operations.**

Representations of Integers.

Computer Operations with Integers.

Complexity of Integer Operations.

**3. Primes and Greatest Common Divisors.**

Prime Numbers.

The Distribution of Primes.

Greatest Common Divisors.

The Euclidean Algorithm.

The Fundemental Theorem of Arithmetic.

Factorization Methods and Fermat Numbers.

Linear Diophantine Equations.

**4. Congruences.**

Introduction to Congruences.

Linear Congrences.

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.**

The Euler Phi-Function.

The Sum and Number of Divisors.

Perfect Numbers and Mersenne Primes.

Mobius Inversion.

Partitions.

**8. Cryptology.**

Character Ciphers.

Block and Stream Ciphers.

Exponentiation Ciphers.

Knapsack Ciphers.

Cryptographic 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 Tests Using Orders of Integers and Primitive Roots.

Universal Exponents.

**10. Applications of Primitive Roots and the Order of an Integer.**

Pseudorandom Numbers.

The EIGamal Cryptosystem.

An Application to the Splicing of Telephone Cables.

**11. Quadratic Residues.**

Quadratic Residues and nonresidues.

The Law of Quadratic Reciprocity.

The Jacobi Symbol.

Euler Pseudoprimes.

Zero-Knowledge Proofs.

**12. Decimal Fractions and Continued.**

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 Equation.

Congruent Numbers.

**14. The Gaussian Integers.**

Gaussian Primes.

Unique Factorization of Gaussian Integers.

Gaussian Integers and Sums of Squares.