A Guide to Elementary Number Theory

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MAA, 2009 - Mathematics - 141 pages
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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

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About the author (2009)

Underwood Dudley received the Ph.D. degree (number theory) from the University of Michigan in 1965. He taught at the Ohio State University and at DePauw University, from which he retired in 2004. He is the author of three books on mathematical oddities, The Trisectors, Mathematical Cranks, and Numerology all published by the Mathematical Association of America. He has also served as editor of the College Mathematics Journal, the Pi Mu Epsilon Journal, and two of the Mathematical Association of America's book series.

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