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Elementary Number Theory (Springer Undergraduate Mathematics Series), by Gareth A. Jones, Josephine M. Jones
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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.
- Sales Rank: #325182 in eBooks
- Published on: 2012-12-06
- Released on: 2012-12-06
- Format: Kindle eBook
Review
From the reviews:
BULLETIN OF MATHEMATICS BOOKS
"?as a nice concluding chapter on Fermat? Last Theorem, with a brief discussion on the coup de grace."
G.A. Jones and J.M. Jones
Elementary Number Theory
"A welcome addition . . . a carefully and well-written book."―THE MATHEMATICAL GAZETTE
"This book would make an excellent text for an undergraduate course on number theory."
―MATHEMATICAL REVIEWS
Most helpful customer reviews
103 of 105 people found the following review helpful.
Pure Joy
By Katherine M. Lawrence
I picked this up when I hit a small snag as I reviewed my undergraduate mathematics in order to return to graduate school for my master's in mathematics.
What started off as a small aside while reviewing another text (to recall some fundamentals, but in a rigorous way), turned into pure joy as I began a delightful excursion into "Elementary Number Theory," for its own sake, under the guidance of Jones & Jones.
Although many find Gallian and a host of others, Rudin included, to be the way to go, Jones & Jones [parallel to these authors] have a way of setting out proofs that appealed to me - for whatever that's worth.
ALL exercises have answers at the back, practically a sine qua non for all people who self-study and have to "grade" their own homework. The authors tie the relevance of the theories together without the sometimes heavy handed pop references to the Beatles, or to arcane things such as "yellow pigs." This is not to say the authors did not pay attention to the history and dates which they sprinkle in as they spin the development of the theories. Yet, they are always mindful of the mathematics which they teach and never get too cute.
It is the beauty of the number theory that is center stage, here, and like Zen, is achieved on the basis of its own elegant simplicity. But simplicity does not mean simple minded nor so brief that the authors lose the student. I felt in lock step with the authors page after page, proof after proof.
Perhaps I never understood Abstract Algebra quite well enough because I did not have as strong as grasp in elementary number theory as I should have had, but Jones & Jones certainly present the subject matter in a way that a somewhat rusty college grad could quickly sink her teeth into and enjoy. In short, this helped me close ground, but fast, while at the same time it opened my eyes to other proofs in other courses that I had committed to memory yet never full appreciated.
In any case this book was money VERY well spent and worth its modest price of admission.
59 of 61 people found the following review helpful.
Good starting point!
By Lex van de Pol
A few weeks ago I ordered three books about Number Theory, and this is the one I like most. I am not a beginner in maths, but I am a beginner in Number Theory. This book start with the basics, and it has exercises with answers! I think this is a good book for self-study, it is easier to read than the books from Leveque.
33 of 33 people found the following review helpful.
Excellent intro book on number theory
By Sumkindamathgeek
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.
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