Preface | p. v |

Flowchart of Chapter Dependencies | p. ix |

Introduction | p. 1 |

What Is Number Theory? | p. 6 |

Pythagorean Triples | p. 13 |

Pythagorean Triples and the Unit Circle | p. 21 |

Sums of Higher Powers and Fermat's Last Theorem | p. 26 |

Divisibility and the Greatest Common Divisor | p. 30 |

Linear Equations and the Greatest Common Divisor | p. 37 |

Factorization and the Fundamental Theorem of Arithmetic | p. 46 |

Congruences | p. 55 |

Congruences, Powers, and Fermat's Little Theorem | p. 65 |

Congruences, Powers, and Euler's Formula | p. 71 |

Euler's Phi Function and the Chinese Remainder Theorem | p. 75 |

Prime Numbers | p. 83 |

Counting Primes | p. 90 |

Mersenne Primes | p. 96 |

Mersenne Primes and Perfect Numbers | p. 101 |

Powers Modulo m and Successive Squaring | p. 111 |

Computing k^{th} Roots Modulo m | p. 118 |

Powers, Roots, and "Unbreakable" Codes | p. 123 |

Primality Testing and Carmichael Numbers | p. 129 |

Squares Modulo p | p. 141 |

Is-1 a Square Modulo p? Is 2? | p. 148 |

Quadratic Reciprocity | p. 159 |

Proof of Quadratic Reciprocity | p. 171 |

Which Primes Are Sums of Two Squares? | p. 181 |

Which Numbers Are Sums of Two Squares? | p. 193 |

As Easy as One, Two, Three | p. 199 |

Euler's Phi Function and Sums of Divisors | p. 206 |

Powers Modulo p and Primitive Roots | p. 211 |

Primitive Roots and Indices | p. 224 |

The Equation X^{4}+Y^{4}=Z^{4} | p. 231 |

Square-Triangular Numbers Revisited | p. 236 |

Pell's Equation | p. 245 |

Diophantine Approximation | p. 251 |

Diophantine Approximation and Pell's Equation | p. 260 |

Number Theory and Imaginary Numbers | p. 267 |

The Gaussian Integers and Unique Factorization | p. 281 |

Irrational Numbers and Transcendental Numbers | p. 297 |

Binomial Coefficients and Pascal's Triangle | p. 313 |

Fibonacci's Rabbits and Linear Recurrence Sequences | p. 324 |

Oh, What a Beautiful Function | p. 339 |

Cubic Curves and Elliptic Curves | p. 353 |

Elliptic Curves with Few Rational Points | p. 366 |

Points on Elliptic Curves Modulo p | p. 373 |

Torsion Collections Modulo p and Bad Primes | p. 384 |

Defect Bounds and Modularity Patterns | p. 388 |

Elliptic Curves and Fermat's Last Theorem | p. 394 |

Further Reading | p. 396 |

Index | p. 397 |

The Topsy-Turvy World of Continued Fractions [online] | p. 410 |

Continued Fractions and Pell's Equation [online] | p. 426 |

Generating Functions [online] | p. 442 |

Sums of Powers [online] | p. 452 |

Factorization of Small Composite Integers [online] | p. 464 |

A List of Primes [online] | p. 466 |

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