Customer Reviews

Elementary Number Theory

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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.

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.

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.

I bought this book while studying cryptography, a field that relies heavily on Number Theory for inspiration and from which it draws many, if not most, of its constructions. Most books on Cryptography summarily relegate the relevant number-theoretic aspects to short appendices that fail to build any intuition about what is going on. This book delivers precisely what is missing: a very readable, easily accessible introduction to the main topics of number theory that leaves the reader with a much better idea of how everything fits together. The book is very well suited for self-study, and includes answers to all exercises.

It should be noted, though, that the book does not address any of the computational aspects of Number Theory that are so dear to Cryptography (e.g it's easy to take square roots mod p if p is prime, hard to take square roots mod pq unless you know p,q). This, however, does not reduce its usefulness, since such results become very easy to absorb once one has a decent understanding of number theory and its workings. To fill the computational gaps, I would suggest Dana Angluin's "Lecture Notes on the Complexity of Some Problems in Number Theory" which are freely available on the web (the 2001 LaTeX'ed version)

This book presumes so little of the reader that anyone can start learning number theory using this book. There are plenty of exercises and all of them have solutions. All the major topics are covered, and in a fashion and pace that allows you to grasp the underlying concepts. This book maintains accessibility and quality throughout. Highly recommended, particularly for beginners.

I used this book as a reference book to review the basic number theory that I tend to foreget while I was in graduate school. I hope I had that book while I was undergraduate, because the all the proofs are clearly written. The exercises are all meaningful, but it will be nice to include some challenging problems. I highly recommend this book to any undergraduate or even bright high school students who want a quick introduction to number theory.

That the book's almost square is easily gathered from the photo. That it's almost perfect must be verified by reading it. And what an enjoyable verification indeed awaits those who take on the challenge! Not that reading it is a challenge - on the contrary, the Joneses take every effort to ensure your learning experience be as painless as possible. Every proof is complete, all exercises are solved. The proofs are always selected for their instructional merit, rather than for their mathematical "elegance" (read: brevity and algebraic gimmickry). As one Amazonian reviewer put it: you could read it through, lying in a bubble bath.

Another Amazonian reviewer commented that "Number theory is like the cement on your driveway. Real and Complex analysis are the Porsche and Ferrari you drive home every night." I disagree. In any case, in my opinion the book's weak spots are those sections where the discussion forays into the realm of real and complex analysis, namely 9.4-6 ("Random Integers", "Evaluating Zeta(2)", "Evaluating Zeta(2k)"), 9.9 ("Complex variables"), 10.2 ("The Gaussian Integers"), a part of 10.6 ("Minkowsky's Theorem") and 11.9 ("Lame and Kummer"). "Sums of two squares" (Section 10.1) could also use improvement, but this is compensated by the excellent, independent treatment this topic receives in the "Minkowsky's Theorem" chapter.

On several occasions, from the very beginning, the book assumes familiarity with single-variable polynomials (particularly the division algorithm and the x^n-y^n expansion). Be prepared.

If it weren't for the forays mentioned above, the book would have been a straight fiver. But even as it stands, it's a tour-de-force of pedagogy and expository mathematical writing.

One last quibble. The book doesn't have a homepage, nor is there any indication of a way to contact the authors. Textbook publishers should learn from their colleagues in the applied computer science publishing industry (such as O'Reilly, Wrox, Apress, etc.) and always make a homepage available for every book, with, at the minimum, a link to an errata page, and a forum where readers of the book can discuss it, (preferably with the involvement of the author(s)).

This is a nice little book (290 pages), which can be used as course litterature for an introductory course in number theory or a by-side reading for somebody taking a first course. It's exposition is so pedagogical and clear that I could study the book from the beginning to the end on my own without help. This is pretty rare for a mathematical book. It covers not only the basic subjects likes divisibility, primes and congruences but more advanced subjects like Euler's functions, quadratic residues, Riemann zeta function as well. there is even a final chapter on Fermat's Last Theorem, which is quite accessible. I would not hesitate to recommend this book to anybody starting to study number theory. Finally it contains complete solutions to all exercises.

This is a great little book thats packed full of great number theory results. It is well written.

I'm a real fan of the SUMS books (I've bought 4 of the titles in the series), because all of the books I've bought are well written, they're jammed full of useful information and they're relatively cheap!

The book strikes a good balance between keeping focused on number theory (there are chapters requirng a knowledge of rings and groups, but these structures only support the numbers, not abstract them away) and not being trivial (I've read too many number theory books that are 'bitty', in the sense that there is too much breadth and not enough depth).

I've recently received my copy of Elementary Number Theory by Jones and Jones, and I'm (thus far) satisfied with the textbook. Although I'm not a professional mathematician, I have worked toward a degree in math and still love to study it. For me, the current textbook for number theory is a challenge to master, but with all the solutions to problems provided, I find it quite palatable to work toward an understanding of number theory, using this text.
In view of my current experiences with this textbook, I would recommend it to a mathematical hobbyist like myself, or to a professional student of mathematics -- or anyone wishing to tackle number theory.