Self Study of number theory

There are lots of great number theory texts for beginners and it really depends on your tastes,level of preparation and interests.

One of the most commonly used introductory texts and one of the most user-friendly is Kenneth Rosen's Elementary Number Theory And Its Applications.It starts from jump and covers basically the entire range of what can be learned in number theory using just high school algebra and without abstract algebra or real analysis- which a surprising amount. It has many solved examples,many wonderful side historical notes as well as applications to computer science. If you have very minimal background and are looking to learn strictly by self-study, I don't think you can do better. The one problem is the book is rather expensive. If you can get a used copy or an international edition, then by all means,do so.

If you're short on time and want to learn the subject actively,there's the old classic,Number Theory For Beginners by Andre Weil,which I love. It was recommended to me as a graduate student by Melvyn Nathanson, who first learned number theory from Weil himself using this text. It's a crystal clear problem course with a lot of very straightforward exercises-it's perfect for those trying to learn it quickly.

If you're looking for a historical presentation of number theory, John Stillwell's Elements of Number Theory can't be beat. It's a bit more difficult then the other 2 books as it functions as a companion to his abstract algebra text and presents the integers in the context of the development of basic ring theory from the Ancient Greek to today. Like all Stillwell's texts,it's scholarly, well written and provides a narrative for the basics of number theory, such as the development of Diophantine equations in antiquity and number theoretic codes in computer science.A wonderful read, as all Stillwell's books are.

That should get you started. Good luck!