## Fundamental Number Theory with ApplicationsBeginning with the arithmetic of the rational integers and proceeding to an introduction of algebraic number theory via quadratic orders, Fundamental Number Theory with Applications reveals intriguing new applications of number theory. This text details aspects of computer science related to Fundamental Number Theory with Applications also covers: Numerous exercises are included, testing the reader's knowledge of the concepts covered, introducing new and interesting topics, and providing a venue to learn background material. Written by a professor and author who is an accomplished scholar in this field, this book provides the material essential for an introduction to the fundamentals of number theory. |

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### Contents

Section 1 | 1 |

Section 2 | 15 |

Section 3 | 18 |

Section 4 | 20 |

Section 5 | 21 |

Section 6 | 22 |

Section 7 | 24 |

Section 8 | 75 |

Section 24 | 199 |

Section 25 | 206 |

Section 26 | 207 |

Section 27 | 216 |

Section 28 | 221 |

Section 29 | 227 |

Section 30 | 233 |

Section 31 | 241 |

Section 9 | 83 |

Section 10 | 89 |

Section 11 | 94 |

Section 12 | 99 |

Section 13 | 103 |

Section 14 | 118 |

Section 15 | 140 |

Section 16 | 145 |

Section 17 | 150 |

Section 18 | 172 |

Section 19 | 185 |

Section 20 | 188 |

Section 21 | 189 |

Section 22 | 194 |

Section 23 | 198 |

Section 32 | 246 |

Section 33 | 247 |

Section 34 | 272 |

Section 35 | 273 |

Section 36 | 300 |

Section 37 | 326 |

Section 38 | 347 |

Section 39 | 350 |

Section 40 | 361 |

Section 41 | 381 |

Section 42 | 416 |

Section 43 | 418 |

Section 44 | 423 |

### Common terms and phrases

arithmetic assume axioms base bit operations called Chapter cipher class number computational congruences Conjecture continued fraction expansion contradiction Conversely Corollary defined Definition denoted difference of squares Diophantine equations dividing division algorithm element elliptic curve equivalent Euler's Example Exercise exists Fermat's Little Theorem Fibonacci finite footnote function fundamental discriminant Gauss gcd(a given Hence incongruent solutions induction hypothesis instance integer squares integral polynomial irreducible Legendre symbol Lemma mathematics method multiplicative inverse namely natural numbers notation notion number theory O^-ideal odd prime perfect square pmod primality test prime divisor primitive representation primitive root modulo proof of Theorem properties Proposition 3.1.1 Prove pseudoprime quadratic irrational quadratic orders quadratic residue quadratic residue modulo radicand reader reduced residue system relatively prime residue system modulo result Section simple continued fraction solutions modulo solutions of x2 subtraction Suppose zero