Elementary Number Theory
by Gareth A. Jones, Josephine M. Jones
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Book Summary Information
Author: Gareth A. Jones, Josephine M. JonesEdition: Paperback
Audio: English (Unknown); English (Original Language); English (Published)
Published: 1998-07-31
ISBN: 3540761977
Number of pages: 200
Publisher: Springer
Accessories:
- The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae
- Binary Quadratic Forms: An Algorithmic Approach (Algorithms and Computation in Mathematics)
- 104 Number Theory Problems: From the Training of the USA IMO Team
Book Reviews of Elementary Number Theory
Book Review: Excellent intro book on number theory
Summary: 5 Stars
Ever since my undergraduate days aeons ago, I have always had an aversion to any number theory, but Jones and Jones have changed my mind completely. In the last year, I came across a few articles that made me want to learn more about the topic, but wasn't sure where to start, as I wanted a book that had proofs that I could follow, and yet also gave me some motivation to dive into more complicated mathematics such as elliptic curves. Elementary Number Theory fit the bill perfectly and has served as a wonderful introduction to the subject that I could follow and enjoy.
This book is the perfect blend of text and formulae for me, and seems an excellent combination of rigour and looseness, always trying to keep a steady pace for the reader without bogging down in pedantic details that are irrelevant to any but the most fastidious of readers. At the same time, the authors also ensure that the reader gains an appreciation of actually proving theorems about numbers, instead of relying on mere intuition or hunches.
As mentioned by other reviews here, the authors have included complete solutions to all of the exercises, which are sprinkled throughout each chapter, as well as at the end of each chapter. This is a welcome change to so many math texts that have "exercises left to the reader," and has been a requirement for me when reading a text in an unfamiliar subject. The exercises are selected appropriately to the content of the chapters and I found them to be a welcome complement to the rest of the book.
In addition, the book discusses applications of number theory to cryptography in a very readable fashion, with any additional mathematics required for the book (in this case some simple group theory and analysis) in two appendices. A book on number theory would also be incomplete without at least a brief discussion of Andrew Wiles and Fermat's Last Theorem. Of course, Elementary Number Theory steps up to the plate appropriately and gives an overview of the history of the theorem and a (necessarily) thin overview of Wiles' proof.
I think, however, one of the best features of the book is that Jones and Jones have attempted to make the text very readable, in the sense that you could sit in a bath and enjoy part of a chapter without any trouble. I have always enjoyed reading mathematics without pen and paper handy, mainly because it improves my memory and visualization when working through problems, and this text helps greatly in that regard. They do not go for the obscure, and realize that the people who are reading this text are doing so for the first time (hence the title) and will not be overly impressed if the authors had chosen to blind us with their brilliance. The authors understand that we are mere mortals with busy lives, and appreciate a smoothly flowing textbook without having to stumble through unique and cryptic notation or a difficult proof without any explanation.
Summary of Elementary Number Theory
An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples. Exercises, together with their solutions are integrated into the text, and the first few chapters assume only basic school algebra. Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares. In particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.Mathematics Books
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