Introduction

    Divisibility and Primes

    Modular Arithmetic
      Congruences

      Inverses Modulo m and Linear Congruences

      Chinese Remainder Theorem

      Polynomial Congruences

      Magic Squares

    Fundamental Theorems of Modular Arithmetic
      Fermat's Theorem

      Euler's Phi Function

      Euler's Theorem

      Lagrange's Theorem

    Cryptography
      Classical Cryptosystems

      Public-Key Cryptography

      The RSA Scheme

    Primality Testing and Factoring
      Pseudoprimes and Carmichael Numbers

      Strong Pseudoprimes and Probabilistic Primality Testing

      Pollard's (p-1)-method

      Pollard's rho-method

    Primitive Roots
      The Concept of Order

      The Primitive Root Theorem

      The Discrete Logarithm

      Primality Testing

    Applications
      The El Gamal System

      Signature Schemes

      Psuedo-Random Number Generators

    Quadratic Congruences
      Quadratic Residues

      Computing Square Roots mod p

      Complete Solution of Quadratic Congruences

    Applications
      Identification Schemes

      Psuedo-Random Number Generators-II

    Continued Fractions
      Introduction

      Convergents

      Infinite Continued Fractions

      Quadratic Irrationals

      Purely Periodic Continued Fractions

      Classical Continued Fraction Expansions

    Factoring Methods
      Continued Fraction Factoring Method

      Quadratic Sieve

    Diophantine Approximations
      Best Approximations

      Intermediate Convergents and Good Approximations

      }Algebraic and Transcendental Numbers

    Diophantine Equations
      Introduction

      Congruence Methods

      Pythagorean Triples

      Sums of Two Squares

      The Brahmagupta-Bhaskara Equation

      }Fermat's Last Theorem

    Arithmetical Functions and Dirichlet Series
      Arithmetical Functions

      Dirichlet Series

      Euler Products

      Mobius Inversion Formula

      Dirichlet's Theorem

    Distribution of Primes
      Counting Primes

      Chebyshev's Functions

      Chebyshev's Theorem

      The Riemann Zeta Function

    Quadratic Reciprocity Law
      Quadratic Reciprocity

      Proof of the Quadratic Reciprocity Law

      The Jacobi Symbol

    Quadratic Forms
      Basic Notions

      Reduction of Definite Forms-I

      Reduction of Definite Forms-II

      Genus Theory

      Generic Characters and Genera

      The Number of Representations by Forms

    Elliptic Curves
      Introduction

      Projective Spaces

      The Group Law on Elliptic Curves

      Elliptic Curves Modulo Primes

      Rational Points on Elliptic Curves

      Elliptic Curve Factorization Method