# A Guide to Elementary Number Theory

MAA, 2009 - Mathematics - 141 pages
A 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.

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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 Index 137 About the Author 141 Copyright