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| An Introduction to the Theory of Numbers | 
 enlarge | Authors: G. H. Hardy, Edward M. Wright, Andrew Wiles
Creators: Roger Heath-brown, Joseph Silverman
Publisher: Oxford University Press, USA
Category: Book
List Price: $60.00
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Avg. Customer Rating:  11 reviews
Sales Rank: 377602
Media: Paperback
Edition: 6
Number Of Items: 1
Pages: 500
Shipping Weight (lbs): 2.2
Dimensions (in): 9.1 x 6.1 x 1.5
ISBN: 0199219869
Dewey Decimal Number: 512
EAN: 9780199219865
ASIN: 0199219869
Publication Date: September 15, 2008
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Product Description
An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D. R. Heath-Brown, this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones and developments in number theory.
Updates include a chapter by J. H. Silverman on one of the most important developments in number theory - modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader.
The text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists.
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| Customer Reviews: Read 6 more reviews...
 Sixth edition is modest upgrade December 24, 2008
The 2008 Sixth Edition adds a new chapter on Elliptic Curves and expands the chapter endnotes, but is otherwise little changed from the Fifth Edition. The best new feature is a comprehensive index in addition to the name index that was in earlier editions.
This book is a wide-ranging survey of elementary number theory. It has no exercises, and is written more for mathematicians than students (though many bright students love the book). It's somewhat dated today, and if you want something modern you might instead look at Niven & Zuckerman & Montgomery's An Introduction to the Theory of Numbers. That book has a more uniform coverage of elementary number theory (but does not go as deep), and includes some newer topics such as factorization algorithms, public key cryptography, and Schnirelmann density. They're both very good books.
The Sixth Edition of Hardy & Wright has been re-typeset and has a more open, less dense look. Unfortunately this process introduced a lot of typographical errors, at least in the first printing. Hopefully these can be corrected in a second printing.
 Syntax and lack of background December 16, 2008
0 out of 1 found this review helpful
I have yet to write a review on any of the textbooks that I have purchased from amazon, but I felt the need to give my insight into this book. I'm currently a student studying engineering in Australia. I purchased the Dover publication on number theory a while back with an interest in getting a deeper understanding of the mathematics I'm learning. I don't claim to be an expert in the field of maths I'm merely interested. I came around to getting this book after I was stuck on the Basis theorem explanation in the Dover publication. I stopped on that book when the author explained that the fundamental theorem of arithmetic required a good understanding of the Basis theorem. I tried to follow the proof outlined in the book but was unable to grasp it. I followed some proofs outlined online like on Wikipedia which gave me a better understanding of it, but they used methods of abstract algebra (Noetherian rings and fields) which I hadn't studied.
This book starts out by outlining the logic operations it will be using throughout chapter 1. Section 1.1 explains the divisibility of integers followed by statements of fundamental theorem of arithmetic and some notes on prime number series.
Section is 1.6 called "some notations". This is where this book starts to fall apart for me. It starts out by introducing some of the Landau Symbols ( though nowhere does it tell you the name of these symbols with any reference) The back of the book does have a symbols index but it never names these symbols just page numbers.
Here the author starts to explain the definitions of Big-O, little-o and asymptotically equivalent, with some examples.
Future down he explains that o(1) = 0(1) but usually not the inverse. Symmetry doesn't hold. I fail to see why is the case, the author gives no explanation other then " to be observed...."
The next section 1.7 " The logarithmic function". Here he outlines the Taylor series of the e^x function. He then states that e^x tends to infinity more rapidly then any power of x
he shows this by rearranging the Taylor series and making a reference to the e^x.x^-n > x/(n+1)! ( does x/(n+1)! have something to do with a formula for the product of powers of x ?? I don't know )I don't see how this proves that e^x is increases more rapidly to infinity then the powers of x. Maybe my maths isn't up to scratch...
After graphing these functions it would seem to be not to be true when x^y ( y > 3 ) in earlier integers. But this might not hold for larger numbers thus it would follow from the x > x0 as defined in the wikipedia definition of Big-O but it's not defined that way in this book.
Following this the author states that log x being the inverse rises slower then any power of x. He then gives an example of
x = 10^9 then log x = 20.72.. ( it should be noted that the author puts a tiny footnote on a previous page implying that this is in fact a Napierian logarithm, which he means natural logarithm, but he doesn't state this. )
Chapter one didn't impress me in the slightest. The following chapters seems to give a much better development and clearer understanding.
a milestone and a shining star in elementary number theory March 8, 2008
1 out of 1 found this review helpful
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.
Nice intro to number theory March 13, 2007
3 out of 3 found this review helpful
This is an unusual number theory book in that it covers topics of interest to the authors which are not often found in the "standard" introductory treatment. My only mild complaints are: no subject index and some ambiguous and unusual notation here and there.
I agree that this book should be in the library of anyone serious about the topic, however, if you are beginning your study of number theory from scratch there are other books that may provide a better start. I would recommend Joe Roberts "Elementary Number Theory: A Problem Oriented Approach" and/or "An Introduction to the Theory of Numbers" by Niven, Zuckerman, and Montgomery.
Roberts offers a wide spectrum of problems, with detailed solutions, written along the lines of Polya & Szego's "Problems and Theorems in Analysis I & II". Nivens book is a solid traditional introduction.
It is fun to read Hardy and Wright though, it exhibits a style that is sadly missing today.
I have to say in closing that it would be good to ignore some of the previous reviews, specifically ones making reference to "idiots". They're unproductive, miss the point of reviewing, and exhibit a level of ignorance which Mark Twain identified years ago: "It is better to keep your mouth shut and appear stupid than to open it and remove all doubt."
Superb Introduction for the Mathematical Sophisticate August 8, 2006
3 out of 3 found this review helpful
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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