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An Introduction to the Theory of Numbers (Oxford Science Publications) 5th Edition

4.9 out of 5 stars 14 customer reviews
ISBN-13: 978-0198531715
ISBN-10: 0198531710
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Editorial Reviews

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"A really good book!"--Fernando Gouvea, Colby College


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Product Details

  • Series: Oxford Science Publications
  • Paperback: 456 pages
  • Publisher: Oxford University Press; 5 edition (April 17, 1980)
  • Language: English
  • ISBN-10: 0198531710
  • ISBN-13: 978-0198531715
  • Product Dimensions: 9.2 x 1 x 6.1 inches
  • Shipping Weight: 1.4 pounds
  • Average Customer Review: 4.9 out of 5 stars  See all reviews (14 customer reviews)
  • Amazon Best Sellers Rank: #476,414 in Books (See Top 100 in Books)

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Format: Paperback
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.
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Format: Paperback
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.
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Format: Paperback
This classic deserves its reputation but be warned that it is not an introduction for mathematical neophytes. The authors take detours (which sometimes are looks ahead) from the main path of development that the sophisticate will enjoy but the novice may not be able to recognize as detours. Examples are the geometry of numbers (introduced in chapter 3), the Farey dissection of the continuum, and trigonometric sums.

The authors also present deeper material than is usually considered an introduction. Their presentations are excellent but require sophistication for the following topics among others: quadratic fields, generating functions of arithmetical functions, Selberg's proof of the Prime Number Theorem, and Kronecker's theorem.

This is a book to buy and keep provided you have the necessary mathematical sophistication.

Final note: this book nicely complements Apostol's Introduction to Analytic Number Theory.
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Format: Paperback
it is surprising to find that so few people have anything to say about this book; Hardy was a giant among mathematicians and at last this book is translated in french...Although it is an old book, the younger author saw that it was updated through 5 editions in the 20th century; this book cannot truly become obsolete because it is about number theory from an elementary viewpoint; so no complex analysis, no modular forms and no proof of Fermat's last theorem either but a wealth of results that could keep you busy quite for a while. Moreover, most of the proofs are still up to date and usable in secondary school or college; most of the proofs about arithmetical functions given in this work have found a new life and home in more recent books such as Natanson's: Elementary methods in number theory (another fine book by the way in which Hardy and Littlewood tauberian theorem is proven via Karamata's method to ensure a density theorem on partitions). The main parts of the book I went through are those on arithmetical functions and series of prime and especially mertens's theorem but there is a lot to learn from it on such subjects as gaussian integers (chapter 12), diophantine equations (chapter 13), Rogers-Ramanujan identities, Jacobi and Euler theorems in the chapter about partitions (numbered 19...), Kronecker's theorem on irrational numbers and on a smaller scale e and pi's irrationality (easy) and transcendence (not so easy) in chapter 11 and of course congruences including a famous theorem on Bernoulli numbers of Von Staudt which gives the fractional part of those enigmatic numbers as a sum of picked inverse of prime numbers . Let say it again: a wonderful book.
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