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Elements of Number Theory (Dover Phoenix Editions)

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This book has two parts: a very elementary introduction to number theory concentrating on congruences presented in a number of chapters. Each chapter ends with a set of straightforward numerical exercises. The second part consists of a set of problems presented at the end of each chapter but having no relation to the content of the chapter. Each problem is provided with a solution or outline of a solution. It is this set of problems that makes the book really interesting. The problems are challenging, in fact many of them appear as theorems in Hardy and Wrights' An Introduction to the Theory of Numbers (Oxford Science Publications).

I bought this book some thirty years ago when I studied mathematics at the university. Working through these problems greatly improved my problem solving abilities. To be able to work through the problems the only prerequisites are arithmetic and some elementary calculus. But be sure to have the right mind set, only for budding mathematicians!

The best way to get a feeling for the subject is by doing problems, and this back has great ones, which emphasize elementary methods, but very quickly get you to the research frontier (one of the many charms of the subject is that some very simple and natural questions are still wide open). After working through this book, you will really understand the subject. The only negative is that the areas covered are a little narrow (since they follow Vinogradov's personal interests), so essentially no algebraic number theory is introduced, for example.