An Introduction to the Theory of Numbers

This book (5th edition) cover the topics of undergraduate number theory well. The chapters are - (1)divisibility (2)congruences (3)quadratic reciprocity and quadratic forms (4)some funtions of number theory (5)some diophantine equations (6)farey fractions and irrational numbers (7)simple continued fractions (8)prime estimates and multiplicative number theory (9)algebraic numbers (10)partition funtion (11)density of sequences of integers.It also contains basic cryptography, basic group theory and basic elliptical curves in some of the chapters. The authors give notes on the end of each chapter about some research results, which I enjoy reading.However, the author give too much hints spoling the fun of solving the problems. Eg 32-36, 40-3, 59-53, 108-36, 136-17, 312-8, and most of the problems in chapter 8. The author should put these hints at the back of the book. I suggest you look up IMO (imo.math.ca) for problems suitable for chapter 1-7 because IMO is well-knowned for its excellent number theory problems (especially 1990-3).Overall this is an excellent book. I give it a rating of 4.5/5, I don't give it 5 because of the author give too much hints to problems instead of putting hints at back of the book.

This is a fantastic book on number theory. It covers far more ground than most introductory text (comparable to Hardy and Wright in depth with much less concern for the big O). It covers material usually only available in separate texts: Rational points on elliptic curves, the partition function, and Dirchlet series. Quite readable chapters, well motivated theoretically, although the historic motivation for the subject matter comes largely in the end-of-the-chapter notes. It's an excellent refresher and reference for non-specialist who find themselves using an algorithm or formula they've forgotten (number theory now playing a role in physics and CS, like never before). It is well cross-referenced with regards to methods of proofs the can be accomplished in different section by different methods - this again making it an excellent reference.Alas, it is pre-FLT. So you'll have to look elsewhere for that.

I have started my studies in Number Theory reading this book from the preface to the last word. It is amazing! I think it is a better introduction to the subject than the classical Hardy and Wright...it is "more objective" and almost 100% elementary...a good high school reader could do well with it. The chapter of diophantine equations has some divine proofs, very clever and very beautiful. And there is an easy proof of the irracionality of Pi. The only negative point is the existence of some points where the authors could be less concise and a bit clearer, stating the theorems before giving the demonstrations, instead of saying at the end of the paragraph "we then have proved the theorem of..." Its a good book for self-study. It has many exercises.

It's a excellent book. Guide you through the simplest proofs until the great ones. If you can follow the book since start until end you'll be prepared for beginning research in this incredible world.