An Introduction to the Theory of Numbers

Rating:
Summary: A classic introduction to a wide range of topics.
Review: Every serious student of number theory should have this classic book on their shelf. Even though only "elementary" calculus and abstract algebra are used, a certain mathematical maturity is required. I feel the book is strongest in the area of elementary --not necessarily easy though -- analytic number theory (Hardy was a world class expert in analytic number theory). An elementary, but difficult proof of the Prime number Theorem using Selberg's Theorem is thoroughly covered in chapter 22.

While modern results in the area of algorithmic number theory are not presented nor is a systematic presentation of number theory given (it is not a textbook), it contains a flavor, inspiration and feel that is completely unique. It covers more disparate topics in number theory than any other n.t. book I know of. The fundamental results in classical, algebraic, additive, geometric, and analytic number theory are all covered. A beautifully written book.

Other recommended books on number theory in increasing order of difficulty:

1) Elementary Number Theory, By David Burton, Third Edition. Covers classical number theory. Suitable for an upper level undergraduate course. Primarily intended as a textbook for a one semester number theory course. No abstract algebra required for this book. Not a gem of a book like Davenport's The Higher Arithmetic, but a great book to seriously start learning number theory.

2) The Queen of Mathematics, by Jay Goldman. A historically motivated guide to number theory. A very clearly written book that covers number theory at a graduate or advanced undergraduate level. Covers much of the material in Gauss's Disquisitiones, but without all the detail. The book covers elementary number theory, binary quadratic forms, cyclotomy, Gaussian integers, quadratic fields, ideals, algebraic curves, rational points on elliptic curves, geometry of numbers, and introduces p-adic numbers. Only a slight bit of analytic number theory is covered. The best book in my opinion to start learning algebraic number theory. Wonderfully fills the otherwise troublesome gap between undergraduate and graduate level number theory.

Full of historical information hard to find elsewhere, very well researched. To cover all the material in this book would likely take two semesters, though most of the important material could be covered in one semester. Requires a background in abstract algebra (undergraduate level), and a little advanced calculus. Some complex analysis for sections 19.7 and 19.8 would be helpful, but not at all a requirement. The author recommends Harold Davenport's ' The Higher Arithmetic as a companion volume for the first 12 chapters ' according to Goldman a gem of a book.

3) Additive Number Theory, by Melvyn Nathanson. Graduate level text in additive number theory, covers the classical bases. This book is the first comprehensive treatment of the subject in 40 years. Some highlights: 1) Chen's theorem that every sufficiently large even integer is the sum of a prime and a number that is either prime or the product of two primes. 2) Brun's sieve for upper bound on the number of twin primes. 3) Vinogradov's simplification of the Hardy, Littlewood, and Ramanujan's circle method.

Rating:
Summary: Didn't think it could be this good...
Review: I've got Algebra (I & II) by Bartel van der Waerden and I didn't think that any book, math or otherwise, could bring me to tears like that one did. Everything in this (Hardy & Wright) is absolutely crystal clear, from the basics like divisibility, to much more advanced topics like Erdos's and Selberg's proof of the Prime Number Theorem. The writing style is very similar to van der Waerden's and Rudin's (only says what needs to be said-just the way I like it). I just can't think of a more comprehensive and densely written number theory book.

Rating:
Summary: THE BOOK on number theory---BUY IT!!!!
Review: It was always claimed that of all the mathematicians who ever lived, Hardy was one of the greatest writers. This book certainly confirms that view. From the very beginning, one thinks, "Wow, this guy REALLY knows what he's talking about." Hardy was, in fact, one of the greatest number theorists of the twentieth century. Hardy gives actual intuitive motivation for almost all of the theorems in the book (intuition is often overlooked by mathematical authors who use the confusing traditional "theorem-proof" approach), and his proofs are elegant and easy to follow. Once, I spoke to the chair of the math department at a major University (Wash U. in St. Louis) and he told me that he reads Hardy and Wright at least once a year to refresh himself on the basics. I would recommend this book to anyone who is learning about number theory for the first time, and wishes to pursue the subject through self-study.