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Elementary Number Theory, 7e, by David M. Burton Table of ContentsPreface New to this Edition 1 Preliminaries 1.1 MathematicalInduction 1.2 The Binomial Theorem 2 Divisibility Theory in theIntegers 2.1 Early Number Theory 2.2 The Division Algorithm 2.3 TheGreatest Common Divisor 2.4 The Euclidean Algorithm 2.5 The DiophantineEquation 3 Primes and Their Distribution 3.1 The Fundamental Theorem ofArithmetic 3.2 The Sieve of Eratosthenes 3.3 The Goldbach Conjecture4 The Theory of Congruences 4.1 Carl Friedrich Gauss 4.2 BasicProperties of Congruence 4.3 Binary and Decimal Representations ofIntegers 4.4 Linear Congruences and the Chinese Remainder Theorem 5Fermat's Theorem 5.1 Pierre de Fermat 5.2 Fermat's Little Theorem andPseudoprimes 5.3 Wilson's Theorem 5.4 The Fermat-KraitchikFactorization Method 6 Number-Theoretic Functions 6.1 The Sum andNumber of Divisors 6.2 The Mobius Inversion Formula 6.3 The GreatestInteger Function 6.4 An Application to the Calendar 7 Euler'sGeneralization of Fermat's Theorem 7.1 Leonhard Euler 7.2 Euler'sPhi-Function 7.3 Euler's Theorem 7.4 Some Properties of thePhi-Function 8 Primitive Roots and Indices 8.1 The Order of an IntegerModulo n 8.2 Primitive Roots for Primes 8.3 Composite Numbers HavingPrimitive Roots 8.4 The Theory of Indices 9 The Quadratic ReciprocityLaw 9.1 Euler's Criterion 9.2 The Legendre Symbol and Its Properties9.3 Quadratic Reciprocity 9.4 Quadratic Congruences with Composite Moduli10 Introduction to Cryptography 10.1 From Caesar Cipher to Public KeyCryptography 10.2 The Knapsack Cryptosystem 10.3 An Application ofPrimitive Roots to Cryptography 11 Numbers of Special Form 11.1 MarinMersenne 11.2 Perfect Numbers 11.3 Mersenne Primes and Amicable Numbers11.4 Fermat Numbers 12 Certain Nonlinear Diophantine Equations 12.1The Equation 12.2 Fermat's Last Theorem 13 Representation of Integersas Sums of Squares 13.1 Joseph Louis Lagrange 13.2 Sums of Two Squares13.3 Sums of More Than Two Squares 14 Fibonacci Numbers 14.1 Fibonacci14.2 The Fibonacci Sequence 14.3 Certain Identities Involving FibonacciNumbers 15 Continued Fractions 15.1 Srinivasa Ramanujan 15.2 FiniteContinued Fractions 15.3 Infinite Continued Fractions 15.4 FareyFractions 15.5 Pell's Equation 16 Some Recent Developments 16.1Hardy, Dickson, and Erdos 16.2 Primality Testing and Factorization 16.3An Application to Factoring: Remote Coin Flipping 16.4 The Prime NumberTheorem and Zeta Function Miscellaneous Problems Appendixes GeneralReferences Suggested Further Reading Tables