a milestone and a shining star in elementary number theory

books:


	• An Introduction to the Theory of Numbers (Oxford Science Publications) G. H. Hardy, E. M. Wright Oxford University Press, USA, 1980 - 456 pages average customer review:based on 9 reviews view larger image
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highly recommended

a milestone and a shining star in elementary number theory

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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Superb Introduction for the Mathematical Sophisticate

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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One of the greatest

First of all, let me say this about the one star review. Do not let yourself be infuenced by lesser mathematicians. Idiots in my opinion. To give this book one star, you must posses some special kind of mediocracy. Keep your stupidity to yourself Lucas.

No one writes like this anymore. Mathematicians like Hardy have passed. The subject has ballooned, and now you have to specialize within Number Theory. There are fewer and fewer that can posses knowledge of the entire subject of Number Theory. Remember what Harold M. Edwards said. You have to read the classics, and beware of secondary sources. Authors give their own spin on ideas. And who is to say they have a greater or lesser understanding of the subject. Furthermore, who can determine how well can they express themselves. How many mathematicians our days bother to study grammar and literature? The best example is Gauss' Disquisitiones Arithmeticae. Would you rather read a book written by Gauss himself, the man that established the subject? Or by some one who learned what some one learned what some one learned over a period of 200 years? Also know what Axler, author of Linear Algebra Done Right, said about reading mathematics books. For a mathematics book, if you spend less than half an hour per page you are going too fast. The last thing i will say is again attributed to Edwards. In his book on Advanced Calculus he encourages the reader to jump chapters. A book does not have to, and sometimes it should not, be read in order. It may take some practice to see how you need to jump around, but you will find that you can maximize your reading by doing so.

There are several point in which this book excels. First, in the writing style. Second, in how many ideas it introduces. Or how good an understanding the reader obtains of Number Theory. It is invaluable to have the big picture. Third, the author has in mind the future material the reader will encounter. He knows you will go beyond this book, and prepares you for what is to come. You do not enter higher courses blind.

The writting style is representative of that of Wiles and Loiville. It will show you how your mathematical writting should be. It takes a lot of practice to learn mathematical formalism and how to write proofs. This is the book to learn from. The author is not afraid to connect the ideas you are learning to other advanced ideas and to mathematical history, unlike present day authors. If you plan to be a mathematician, you must know its history. The writting is in a mathematical sense superfluos. It does not assume you are a genius, but strikes balance between what you should know and what you should be told.

The book is successful in providing you with the big picture, and how ideas you are learning reflect one ideas you will learn or have already learned. Having a big picture of the subject, which he describes in the second chapter, lets you know what you are learning now and puts the entire material in context. Gives you great perspective of the subject. Because a great deal of branches of number theory are discussed, you are not only better equiped to choose which branch might interest you, but it eases the transition to more advanced courses, such as Analytical Number Theory.

The author from the start discusses unanswered questions in Number Theory. I know alot of professors which think that the student should not be exposed to questions that surpass his mathematical knowledge. They are the weak mathematicians. Mathematics is about exploring and breaking limits. You should know what is beyond your reach, and the reach of every one else. The questions that still stand might be answered by some one that was intrigued by the challenge of answering them when they are helpless to do so. Fermat's Last Thorem is such an example. The guy learned it at the age of 10.

The last thing i will say about the book is this. Number theory has one scope. Namely, prime numbers. This book make it clear that the purpose of number theory is to determine the properties of numbers. It discusses the limitations of mathematics in attaining answers to Riemann Hypothesis, Fundamental theorem, trancedental and irrational and algebraic numbers, and so on. The book is, in my opinion, an expansion of the section on unanswered questions. And in doing so many more questions are asked and analyzed. There are prime numbers, and nothing else.

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THE BOOK on number theory---BUY IT!!!!

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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Nice intro to number theory

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."

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reviews: page 1, 2

This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford, Cambridge, Aberdeen, and other universities. It is neither a systematic treatise on the theory of numbers nor a 'popular' book for non-mathematical readers. It contains short accounts of the elements of many different sides of the theory, not usually combined in a single volume; and, although it is written for mathematicians, the range of mathematical knowledge presupposed is not greater than that of an intelligent first-year student. In this edition the main changes are in the notes at the end of each chapter; Sir Edward Wright seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present, both in the notes and in the text, a reasonably accurate account of the present state of knowledge.

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