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Harold Davenport
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The Higher Arithmetic: An Introduction to the Theory of Numbers 7th Edition

by H. Davenport (Author)
4.6 4.6 out of 5 stars 26 ratings
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Updated in a seventh edition, The Higher Arithmetic introduces concepts and theorems in a way that does not require the reader to have an in-depth knowledge of the theory of numbers, and also touches on matters of deep mathematical significance. This new edition includes state of the art material on the use of computers in number theory, as well as taking full account of the proving of Fermat's last theorem.
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  1. ISBN-10
    0521634466
  2. ISBN-13
    978-0521634465
  3. Edition
    7th
  4. Publisher
    Cambridge University Press
  5. Publication date
    January 28, 2000
  6. Language
    English
  7. Dimensions
    6.25 x 0.75 x 9.25 inches
  8. Print length
    241 pages
  9. See all details
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Editorial Reviews

Review

"Although this book is not written as a textbook but rather as a work for the general reader, it could certainly be used as a textbook for an undergraduate course in number theory and, in the reviewer's opinion, is far superior for this purpose to any other book in English." Bulletin of the American Mathematical Society

"It is a pleasant experience to see a book on Number theory in its seventh edition." Monatshefte fur Mathematik

Book Description

Seventh edition of a classic elementary number theory book.

Product details

  • Publisher ‏ : ‎ Cambridge University Press; 7th edition (January 28, 2000)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 241 pages
  • ISBN-10 ‏ : ‎ 0521634466
  • ISBN-13 ‏ : ‎ 978-0521634465
  • Item Weight ‏ : ‎ 14.3 ounces
  • Dimensions ‏ : ‎ 6.25 x 0.75 x 9.25 inches
  • Customer Reviews:
    4.6 4.6 out of 5 stars 26 ratings

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Harold Davenport

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4.6 out of 5 stars
4.6 out of 5
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26 global ratings

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5.0 out of 5 stars One of the Best
Reviewed in the United States on October 6, 2016
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Anyone familiar with Davenport knows that he was uncommonly brilliant. He is one of the few great mathematicians of the 20th century, I believe, that could take any technical idea and explain it plainly to someone at a bus stop so that at least the essential ideas are in tact. In fact, from reading his work, I think it is apparent that he truly enjoyed setting things down in a very straightforward, clear manner and he had a gift for this.

Now, some comments are in order regarding the use of this text. It is a common trend in math textbooks to present in a theorem-proof style. So naturally students get comfortable with this and it becomes a hindrance when a book does not meet such a format. Davenport's book is not written like this, so if you require the stock format, you will be disappointed. However, the clarity of this text provides an understanding beyond what a stock treatment can if you can go beyond this artificial hurdle.

Andrew Wiles has said that when he wants to review a topic he always picks up Davenport first, then goes to Hardy and Wright. Davenport's book can almost be read like a novel it is so good and clear. But you won't find symbols marking the end of proofs or telling you where they begin. He simply explains as a great teacher would in conversation. Usually, the logical flow is clear enough where you do not need to see "Proof .... QED." Furthermore, this is not an introductory text that a researcher could find nothing of interest in. Despite being elementary of character, it has innovations in development and bears a stamp of Davenport's brilliant mind. Just to provide an example, the section on continued fractions is very original and beautifully done. There is definitely plenty of meat on the bone even for an experienced reader of number theory.
18 people found this helpful
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Lainey A.
5.0 out of 5 stars Five Stars
Reviewed in the United States on July 24, 2017
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Another great addition to my library!
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Itai
3.0 out of 5 stars Disappointing
Reviewed in the United States on January 31, 2009
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I've purchased this book based on the rave reviews it's received on Amazon.com, both on this page and elsewhere. I've been greatly disappointed.

This is the eighth edition, and, as such, is low on error count, so if all you're looking for in a math textbook is that it be error-free, this may be the book for you.

If you are looking for a little more than that: say, an interesting, well-motivated and pedagogically sound lecture, you'd be better off looking for it elsewhere, for instance in Jones & Jones' superb "Elementary Number Theory".

"The Higher Arithmetic"'s style of writing is unstructured prose (as opposed to the Definition-Theorem-Proof structure), supposedly rendering the text less rigid and more "friendly", when, in fact, it accomplishes the exact opposite effect: You're never sure where a proof begins and where it ends. This compounds unnecessary intellectual and psychological strains on top of those already naturally present whenever one learns new material.

The unstructured-ness also makes this book quite useless as a work of reference.

The proofs aren't particularly elegant or insightful (in fact, they are quite difficult to follow in some cases, for no good reason).

There's very little in terms of historical background and in terms of interesting applications and recreations.

Finally, the book is uncannily devoid of that geeky sense of humor that embellishes the best of math textbooks (e.g. "in this sense, at least, the prime 2 is very odd!", Jones & Jones, 1998, p. 106).

This book can best be recommended to those who have already studied number theory, and would like a refresher of the main topics an introductory course is likely to include.

P.S.
This review is based on my impressions of the first three chapters (which constitute roughly one third of the book in terms of number of pages). I simply couldn't bear reading any further. I can't preclude the possibility that it gets better down the road.
19 people found this helpful
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Anthony Varilly
5.0 out of 5 stars This is a MUST BUY if you want to learn Number Theory!
Reviewed in the United States on June 8, 2000
This book is an AMAZING introduction to the Theory of Numbers. It assumes no previous exposure to the subject, or any technical mathematical knowledge for that matter. Its prose is lucid and the style appealing. Davenport chose NOT to write a lemma-theorem-proof kind of book, and the result is a marvelous, eminently readable introduction to the subject. Its wonderful to read a book where good prose is used to appropiately substitute a massive collection of uninviting symbols. I've also been reading other books on Number Theory, such as Hardy & Wright, but none are as clear as this one.
I found the chapter on quadratic residues (which includes the reciprocity law) to be especially well written. The section on computers and number theory is excelent as well. A concise and coherent discussion of crytography and the RSA system is included here. The organization of the book's chapters is fantastic. Each chapter builds up on results proven in the previous ones, showing well the connections between the different aspects of Number Theory. The exercises of the book range from simple to challenging, but are all accesible to someone willing to put effort into them.
This would be an excelent source for learning number theory for mathematical competition purposes, such as the ASHME, AIME, USAMO, and even for the International Mathematical Olympiad. The book contains much more than what is needed for these competitions, but the olympiad/contest reader will benefit greatly from a study of Davenport's work.
The book can certainly be used for an undergraduate course in Number Theory, though it might need supplementary materials, to cover a semester's worth of work. I know the book has been used in the past in previous editions as the main text for Math 124: Number Theory at Harvard University.
I would also recommend this book to anyone interested in acquanting themselves with Number Theory.
Awesome! There is simply no other word that describes The Higher Arithmetic.
75 people found this helpful
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Davide Berti - Cuneo
5.0 out of 5 stars A classic
Reviewed in Italy on April 27, 2014
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The book is readable without previous knowledge on number theory. Each argument is exposed in the usual way (definition - proposition - demonstration) so the reader can go on safely by himself. I particularly enjoyed: continuos fractions, almost complete; factorization of primes, introductory but well made; and the last chapter on cryptography (and the role of pseudo-random numbers). Worth the prize and the time spent but it is a book for beginners of university level.
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安部精孝
5.0 out of 5 stars 古典として書斎に置きたい本
Reviewed in Japan on November 26, 2014
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数論関係の書物に参考書として必ず推薦されているのが、本書である。
　高額でしたが、独学で読破することを目指しています。
なかなか入手できない本で、とてもラッキーでした。
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ab..c
5.0 out of 5 stars Gentle book on number theory topics
Reviewed in the United Kingdom on September 14, 2009
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Introduction

Several years ago, a tutor showed me her copy of this book and highly recommended it as a primer for this topic. I have compared the contents list (seventh edition, 1999) of mine, against this new copy contents list and both are pretty close to each other. Although this edition will be much updated.

Why is this book worth recommending?

This book encourages the reader to return to its pages again-and-again. This book i.m.h.o has a high level of initial readability, rather than featuring many equations, to create a level of understanding that is rewarding. For example, the initial topics gently explain about 'primes'. Then book clarifies this by branching into topics such as 'Congruences' and 'Quadratic residues' in a way that an author of a few complex analysis books would be proud.

Conclusion

This is a book thats not your final destination in mathematics, but a book to help you reach it.
14 people found this helpful
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Adgoodies
5.0 out of 5 stars simplified
Reviewed in the United Kingdom on May 21, 2014
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i used this book with my prof's lecture notes and it helped me grasped better what was in the lecture notes; it's also easy to read.
3 people found this helpful
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Thomas Cullen
4.0 out of 5 stars Four Stars
Reviewed in the United Kingdom on July 26, 2017
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very well written and easy to understand
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