Elementary Number Theory

Preface New To This Edition 1 Preliminaries 1.1 Mathematical Induction 1.2 The Binomial Theorem 2 Divisibility Theory in the Integers 2.1 Early Number Theory 2.1 The Division Algorithm 2.2 The Greatest Common Divisor 2.3 The Euclidean Algorithm 2.4 The Diophantine Equation ax + by = c 3 Primes and Their Distribution 3.1 The Fundamental Theorem of Arithmetic 3.2 The Sieve of Eratosthenes 3.3 The Goldbach Conjecture 4 The Theory of Congruences 4.1 Carl Friedrich Gauss 4.2 Basic Properties of Congruence 4.3 Binary and Decimal Representations of Integers 4.4 Linear Congruences and the Chinese Remainder Theorem 5 Fermat's Theorem 5.1 Pierre de Fermat 5.2 Fermat's Little Theorem and Pseudoprimes 5.3 Wilson's Theorem 5.4 The Fermat-Kraitchik Factorization Method 6 Number-Theoretic Functions 6.1 The Sum and Number of Divisors 6.2 The Mobius Inversion Formula 6.3 The Greatest Integer Function 6.4 An Application to the Calendar 7 Euler's Generalization of Fermat's Theorem 7.1 Leonhard Euler 7.2 Euler's Phi-Function 7.3 Euler's Theorem 7.4 Some Properties of the Phi-Function 8 Primitive Roots and Indices 8.1 The Order of an Integer Modulo n 8.2 Primitive Roots for Primes 8.3 Composite Numbers Having Primitive Roots 8.4 The Theory of Indices 9 The Quadratic Reciprocity Law 9.1 Euler's Criterion 9.2 The Legendre Symbol and Its Properties 9.3 Quadratic Reciprocity 9.4 Quadratic Congruences with Composite Moduli 10 Introduction to Cryptography 10.1 From Caesar Cipher to Public Key Cryptography 10.2 The Knapsack Cryptosystem 10.3 An Application of Primitive Roots to Cryptography 11 Numbers of Special Form 11.1 Marin Mersenne 11.2 Perfect Numbers 11.3 Mersenne Primes and Amicable Numbers 11.4 Fermat Numbers 12 Certain Nonlinear Diophantine Equations 12.1 The Equation x2 + y2 = z2 12.2 Fermat's Last Theorem 13 Representation of Integers as Sums of Squares 13.1 Joseph Louis Lagrange 13.2 Sums of Two Squares 13.3 Sums of More Than Two Squares 14 Fibonacci Numbers 14.1 Fibonacci 14.2 The Fibonacci Sequence 14.3 Certain Identities Involving Fibonacci Numbers 15 Continued Fractions 15.1 Srinivasa Ramanujan 15.2 Finite Continued Fractions 15.3 Infinite Continued Fractions 15.4 Pell's Equation 16 Some Twentieth-Century Developments 16.1 Hardy, Dickson, and Erdos 16.2 Primality Testing and Factorization 16.3 An Application to Factoring: Remote Coin Flipping 16.4 The Prime Number Theorem and Zeta Function Miscellaneous Problems Appendixes General References Suggested Further Reading Tables Answers to Selected Problems Index