Elementary Number Theory (Springer Undergraduate Mathematics Series)

Some reviewers have said it assumes almost no previous knowledge, but that's not entirely true. It does use an elementary approach, and assumes little knowledge of advanced mathematics from the reader, but if you are a neophyte in mathematics this is not the book, it is better to use a more elementary book to have more benefits. For the small size it has there is a lot inside it, and the writing style is pleasant, although, sometimes, important details are, in an undesirable way, left out. The fact the book has answers is indeed a very good thing, and good teachers should make their own set of exercises, so I don't think the fact that the exercises are answered at the end of the book is a drawback. Quite the contrary, it is a perfect suitable book for being self-taught and independent. All the exercises have answers, or give you directions. They are spread around the text, appearing every time a new concept is been given, or a theorem is been shown. All theorems have proofs easy to follow. Someone might find that the book should have addressed this or that, instead of the topics chosen, but this is how far any criticism of this textbook can go. In few words: buy it!

This is a reasonably complete overview of number theory that does not require any understanding of either algebra or analysis. As such, it can be used for an introductory undergraduate level course. It would be less useful for a graduate level class in which students have a better background in abstract mathematics. The proofs are straightforward and complete. The exercises are very useful. Hints and solutions are provided if you get stuck. Students with limited background in abstract mathematics will find this book accessible. Although it is not a book on how to do proofs, working through this book can give a good and relatively painless introduction to some of the proofs that will be useful in algebra other advanced classes. Students already versed in algebra will likely find this book to be a too low a level, and should look to more advanced texts.

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)).

The pacing in this book is phenomenal. Every topic is explained with more than enough detail and without tangents. The one thing this book does better than any other Number Theory book are the in-chapter questions. The questions are done so well that WANT to do them. You feel as if you are missing vital information by not doing them. This is one of the few books I can study for hours on end because you are constantly being engaged as you are working through the chapter, instead of mindlessly reading and hoping you remember everything you've just read. They questions aren't very difficult (full worked solutions for every problem if needed), but they are just hard enough to make you have to think. That is also it's only downside. You cannot jump between sections sometimes. The book assumes you follow it from the beginning of a section to the end as it sometimes teaches you vital information via the in-chapter questions. If you are looking to self-study then you will not find a better book. I've tried every Number Theory book with decent ratings on Amazon, and this is by far the best.

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).

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 love the combined introduction to number theory, practice of full proof writing, and exercises and solutions. Maybe not the meatiest text if you are looking for a stripped introduction to the concepts of number theory without doing work yourself. Practice is important and proof writing an important skill in mathematics.

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.

I bought several other classics to start in number theory. They were recommended by the pros. I don't doubt these other books (like Hardy ...) are really good but the people recommending them don't have a clue what it means to be a novice in anything or they have long since forgotten. The "elementary number theory" on the other hand is really what it claims to be. The explanations are really good. So are the proofs. I like this book because it never claims something to be too obvious not to give an example of it. I only start reading chapter 4 myself now but I am already convinced that this pace and level is what I need. With Riemann and Fermat in the next chapters, I can't wait.

Buena introducción sin perder el rigor matemático. Una buena lectura antes de empezar con otros más avanzados como el Hardy&Wright o el Rosen.

Which level is the book aimed? This book (i.m.h.o) is a carefully composed introduction to underpinning knowledge of Number Theory topics. Its true -as its stated in the book- that introductory parts may be 'A'-level, but this builds beyond this without becoming worrying or intimidating! After a several weeks of effort on my part, I have found this book both helpful and clarifying and worth the time studying it. I found this book layout enjoyable, progressive and interesting. Also its questions and answers (in the back) kept me going. Later on,after several chapters, if you have algebra insight from other sources than this book, you may sail through these areas with some ease. Final thoughts Personally, I found this book as a whole as stimulating as a maths book could be. Overall I really enjoyed this book, and I have seen the 'Springer' library is sometimes recommended in academic study book lists. It contains a simple proof of the Pythagorean theorem which is beautiful.

I would just echo everything that the previous reviewers have said. I have used this for self study and have found it wonderfully clear. I also bought some Open University texts for a few more exercises on Diophantine equations and congruences, but really all of the key ideas are in the book.

This book is an excellent introduction to the subject