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!

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.

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 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.

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).