Preface | p. viii |

Structure of the Text | p. ix |

To the Student | p. x |

To the Instructor | p. xi |

Acknowledgements | p. xiv |

Prologue: Number Theory Through the Ages | p. xvi |

Numbers, Rational and Irrational (Historical figures: Pythagoras and Hypatia) | p. 6 |

Numbers and the Greeks | p. 6 |

Numbers You Know | p. 13 |

A First Look at Proofs | p. 17 |

Irrationality of ?2 | p. 28 |

Using Quantifiers | p. 32 |

Mathematical Induction (Historical figure: Noether) | p. 42 |

The Principle of Mathematical Induction | p. 42 |

Strong Induction and the Well-Ordering Principle | p. 55 |

The Fibonacci Sequence and the Golden Ratio | p. 67 |

The Legend of the Golden Ratio | p. 76 |

Divisibility and Primes (Historical figure: Eratosthenes) | p. 92 |

Basic Properties of Divisibility | p. 92 |

Prime and Composite Numbers | p. 98 |

Patterns in the Primes | p. 104 |

Common Divisors and Common Multiples | p. 116 |

The Division Theorem | p. 124 |

Applications of god and 1cm | p. 138 |

The Euclidean Algorithm (Historical figure: Euclid) | p. 148 |

The Euclidean Algorithm | p. 148 |

Finding the Greatest Common Divisor | p. 156 |

A Greeker Argument that ?2 Is Irrational | p. 172 |

Linear Diophantine Equations (Historical figure: Diophantus) | p. 182 |

The Equation aX + bY= 1 | p. 182 |

Using the Euclidean Algorithm to Find a Solution | p. 191 |

The Diophantine Equation aX + bY = n | p. 200 |

Finding All Solutions to a Linear Diophantine Equation | p. 205 |

The Fundamental Theorem of Arithmetic (Historical figure: Mersenne) | p. 216 |

The Fundamental Theorem | p. 216 |

Consequences of the Fundamental Theorem | p. 225 |

Modular Arithmetic (Historical figure: Gauss) | p. 241 |

Congruence Modulo n | p. 241 |

Arithmetic with Congruences | p. 254 |

Check-Digit Schemes | p. 267 |

The Chinese Remainder Theorem | p. 274 |

The Gregorian Calendar | p. 288 |

The Mayan Calendar | p. 296 |

Modular Number Systems (Historical figure: Turing) | p. 307 |

The Number System Z_{n}: An Informal View | p. 307 |

The Number System Z_{n}: Definition and Basic Properties | p. 310 |

Multiplicative Inverses in Z_{n} | p. 322 |

Elementary Cryptography | p. 338 |

Encryption Using Modular Multiplication | p. 343 |

Exponents Modulo n (Historical figure: Fermat) | p. 355 |

Fermat's Little Theorem | p. 355 |

Reduced Residues and the Euler ?-Function | p. 368 |

Euler's Theorem | p. 379 |

Exponentiation Ciphers with a Prime Modulus | p. 390 |

The RSA Encryption Algorithm | p. 399 |

Primitive Roots (Historical figure: Lagrange) | p. 415 |

The Order of an Element of Z_{n} | p. 415 |

Solving Polynomial Equations in Z_{n} | p. 429 |

Primitive Roots | p. 438 |

Applications of Primitive Roots | p. 448 |

Quadratic Residues (Historical figure: Eisenstein) | p. 466 |

Squares Modulo n | p. 466 |

Euler's Identity and the Quadratic Character of -1 | p. 478 |

The Law of Quadratic Reciprocity | p. 489 |

Gauss's Lemma | p. 495 |

Quadratic Residues and Lattice Points | p. 505 |

Proof of Quadratic Reciprocity | p. 516 |

Primality Testing (Historical figure: Erdös) | p. 529 |

Primality Testing | p. 529 |

Continued Consideration of Charmichael Numbers | p. 538 |

The Miller-Rabin Primality Test | p. 546 |

Two Special Polynomial Equations in Z_{p} | p. 556 |

Proof that Miller-Rabin Is Effective | p. 561 |

Prime Certificates | p. 573 |

The AKS Deterministic Primality Test | p. 588 |

Gaussian Integers (Historical figure: Euler) | p. 599 |

Definition of the Gaussian Integers | p. 599 |

Divisibility and Primes in Z[i] | p. 607 |

The Division Theorem for the Gaussian Integers | p. 614 |

Unique Factorization in Z[i] | p. 629 |

Gaussian Primes | p. 635 |

Fermat'sTwo Squares Theorem | p. 641 |

Continued Fractions (Historical figure: Ramanujan) | p. 653 |

Expressing Rational Numbers as Continued Fractions | p. 653 |

Expressing Irrational Numbers as Continued Fractions | p. 660 |

Approximating Irrational Numbers Using Continued Fractions | p. 673 |

Proving That Convergents are Fantastic Approximations | p. 684 |

Some Nonlinear Diophantine Equations (Historical figure: Germain) | p. 705 |

Pell's Equation | p. 705 |

Fermat's Last Theorem | p. 719 |

Proof of Fermat's Last Theorem for n = 4 | p. 726 |

Germain's Contributions to Fermat's Last Theorem | p. 735 |

A Geometric Look at the Equation x^{4} + y^{4} = z^{2} | p. 746 |

Index | p. 754 |

Appendix: Axioms to Number Theory (online) | p. A-1 |

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