## A Guide to Elementary Number TheoryA Guide to Elementary Number Theory is a short exposition of the topics considered in a first course in number theory. It is intended for those who have had some exposure to the material before but have half-forgotten it, and also for those who may have never taken a course in number theory but who want to understand it without having to engage with the more traditional texts which are often extensive, and dense. Number theory has an impressive history, which this guide investigates. Rather than being a textbook with exercises and solutions, this guide is an exploration of this interesting and exciting field. Its important results are all included, usually with accompanying proofs: the Quadratic Reciprocity Theorem is proved as Gauss did it. The material has been chosen to be maximally broad whilst remaining concise and accessible. |

### What people are saying - Write a review

This book is likely the most enlightening book I have read in quite a while. It clearly explains all the theorems needed and between the layout and very descriptive definitions the book really helps bring a certain comprehension to the information being given. This books text initially helps the reader connect to what Mr. Underwood Dudley was trying to explain. Based on the layout of the chapters themselves the book really eases the reader into the new theorems and equations needed to understand the elementary number theory. If I was a teacher I would no doubt use this book to help my students better understand the subject and maybe even just use the text itself to help the students learn how to correctly display mathematical information to their peers and to themselves.

### Contents

Introduction V11 1 Greatest Common Divisors | 1 |

Unique Factorization | 7 |

Linear Diophantine Equations | 11 |

Congruences | 13 |

Linear Congruences | 17 |

The Chinese Remainder Theorem | 21 |

Fermats Theorem | 25 |

Wilsons Theorem | 27 |

Sums of Two Squares | 79 |

Sums of Three Squares | 83 |

Sums of Four Squares | 85 |

Warings Problem | 89 |

Pells Equation | 91 |

Continued Fractions | 95 |

Multigrades | 101 |

Carmichael Numbers | 103 |

The Number of Divisors of an Integer | 29 |

The Sum of the Divisors of an Integer | 31 |

Amicable Numbers | 33 |

Perfect Numbers | 35 |

Eulers Theorem and Function | 37 |

Primitive Roots and Orders | 41 |

Decimals | 49 |

Quadratic Congruences | 51 |

Gausss Lemma | 57 |

The Quadratic Reciprocity Theorem | 61 |

The Jacobi Symbol | 67 |

Pythagorean Triangles | 71 |

Sophie Germain Primes | 105 |

The Group of Multiplicative Functions | 107 |

Bounds for jtjc Ill 33 The Sum of the Reciprocals of the Primes | 117 |

The Riemann Hypothesis | 121 |

The Prime Number Theorem | 123 |

The abc Conjecture | 125 |

Factorization and Testing for Primes | 127 |

Algebraic and Transcendental Numbers | 131 |

Unsolved Problems | 135 |

137 | |

About the Author | 141 |