Preface | p. viii |

New to This Edition | p. x |

Preliminaries | p. 01 |

Mathematical Induction | p. 01 |

The Binomial Theorem | p. 08 |

Divisibility Theory in the Integers | p. 13 |

Early Number Theory | p. 13 |

The Division Algorithm | p. 17 |

The Greatest Common Divisor | p. 19 |

The Euclidean Algorithm | p. 26 |

The Diophantine Equation ax + by = c | p. 32 |

Primes and Their Distribution | p. 39 |

The Fundamental Theorem of Arithmetic | p. 39 |

The Sieve of Eratosthenes | p. 44 |

The Goldbach Conjecture | p. 50 |

The Theory of Congruences | p. 61 |

Carl Friedrich Gauss | p. 61 |

Basic Properties of Congruence | p. 63 |

Binary and Decimal Representations of Integers | p. 69 |

Linear Congruences and the Chinese Remainder Theorem | p. 76 |

Fermat's Theorem | p. 85 |

Pierre de Fermat | p. 85 |

Fermat's Little Theorem and Pseudoprimes | p. 87 |

Wilson's Theorem | p. 93 |

The Fermat-Kraitchik Factorization Method | p. 97 |

Number-Theoretic Functions | p. 103 |

The Sum and Number of Divisors | p. 103 |

The Möbius Inversion Formula | p. 112 |

The Greatest Integer Function | p. 117 |

An Application to the Calendar | p. 122 |

Euler's Generalization of Fermat's Theorem | p. 129 |

Leonhard Euler | p. 129 |

Euler's Phi-Function | p. 131 |

Euler's Theorem | p. 136 |

Some Properties of the Phi-Function | p. 141 |

Primitive Roots and Indices | p. 147 |

The Order of an Integer Modulo n | p. 147 |

Primitive Roots for Primes | p. 152 |

Composite Numbers Having Primitive Roots | p. 158 |

The Theory of Indices | p. 163 |

The Quadratic Reciprocity Law | p. 169 |

Euler's Criterion | p. 169 |

The Legendre Symbol and Its Properties | p. 175 |

Quadratic Reciprocity | p. 185 |

Quadratic Congruences with Composite Moduli | p. 192 |

Introduction to Cryptography | p. 197 |

From Caesar Cipher to Public Key Cryptography | p. 197 |

The Knapsack Cryptosystem | p. 209 |

An Application of Primitive Roots to Cryptography | p. 214 |

Numbers of Special Form | p. 219 |

Marin Mersenne | p. 219 |

Perfect Numbers | p. 221 |

Mersenne Primes and Amicable Numbers | p. 227 |

Fermat Numbers | p. 237 |

Certain Nonlinear Diophantine Equations | p. 245 |

The Equation x^{2} + y^{2} = z^{2} | p. 245 |

Fermat's Last Theorem | p. 252 |

Representation of Integers as Sums of Squares | p. 261 |

Joseph Louis Lagrange | p. 261 |

Sums of Two Squares | p. 263 |

Sums of More Than Two Squares | p. 272 |

Fibonacci Numbers | p. 283 |

Fibonacci | p. 283 |

The Fibonacci Sequence | p. 285 |

Certain Identities Involving Fibonacci Numbers | p. 292 |

Continued Fractions | p. 303 |

Srinivasa Ramanujan | p. 303 |

Finite Continued Fractions | p. 306 |

Infinite Continued Fractions | p. 319 |

Farey Fractions | p. 334 |

Pell's Equation | p. 337 |

Some Modern Developments | p. 353 |

Hardy, Dickson, and Erdös | p. 353 |

Primality Testing and Factorization | p. 358 |

An Application to Factoring: Remote Coin Flipping | p. 371 |

The Prime Number Theorem and Zeta Function | p. 375 |

Miscellaneous Problems | p. 384 |

Appendixes | p. 387 |

General References | p. 387 |

Suggested Further Reading | p. 390 |

Tables | p. 393 |

Answers to Selected Problems | p. 410 |

Index | p. 421 |

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