Elementary Number Theory [Hardcover]

David Burton (Author)

Book Description

ISBN-10: 0073051888 | ISBN-13: 978-0073051888 | Publication Date: September 27, 2005 | Edition: 6

Elementary Number Theory, Sixth Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton’s engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.

Product Details

• Hardcover: 448 pages
• Publisher: McGraw-Hill Science/Engineering/Math; 6 edition (September 27, 2005)
• Language: English
• ISBN-10: 0073051888
• ISBN-13: 978-0073051888
• Product Dimensions: 9.3 x 6.5 x 0.8 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (13 customer reviews)
• Amazon Best Sellers Rank: #193,274 in Books (See Top 100 in Books)

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26 of 27 people found the following review helpful:

4.0 out of 5 stars A Worthy Number Theory Text, May 12, 2006

By 

Justin Bost (Salisbury, NC) - See all my reviews

This review is from: Elementary Number Theory (Hardcover)

We used this book in an number theory course I took recently. Burton is a skilled writer, and his book is extremely easy to read even for those devoid of "mathematical maturity". There is a student solutions manual, but I recommend that you abstain from buying it. Many of the exercises have generous hints provided. In fact, Burton probably overdoes it in the hint department. Some of the exercises are ruined that way. Nonetheless, Burton provides excellent exercise sets. Some of the problems are trivial, some aren't. He is careful to point out certain themes that recur in number theory in the text and the exercises.

As previous reviewers have noted, there are brief biographical sketches of certain mathematicians that were integral to the development of number theory. It is interesting to read about the lives and personalities of the men (and women!) that worked on the subject that Gauss coined as "the queen of mathematics".

Chapters 1-9 are the core of an undergraduate course in number theory. I was not that impressed by Burton's introduction to cryptography in Chapter 10. Chapters 11-13 are a nice read though. I do question the wisdom of wasting an entire chapter (Chapter 14) on Fibonacci numbers. Continued fractions and Pell's equation (or "Fermat's equation", as Pell was a mathematical fraud, according to E.T. Bell) are covered in Chapter 15. Chapter 16 is a delightful (but necessarily brief) introduction to twentieth century innovations in number theory. The reader will definitely be left wanting more after the final pages on the Prime Number Theorem.

All in all, not a bad effort. Burton could raise the level of his work from 4 stars to 5 stars with a couple of modifications. Chapter 14 should probably be condensed to an appendix or inserted in another chapter. Also, Burton goes out of his way not to discuss algebraic concepts (groups, rings, fields). Presumably, this is to make the text more friendly to math education majors. Still, there is a whole other side to the subject that the reader is not exposed to by this regrettable omission. Algebraic number theory is not covered.

For a second number theory read, I recommend one, or several of the following:

(1) "Introduction to Analytic Number Theory" by Tom Apostol. An excellent book. Apostol develops the theory necessary to prove Dirichlet's theorem on primes in arithmetic progressions and of course the Prime Number Theorem (an analytic proof). Apostol's book is noteworthy for its treatment of arithmetical functions, which is extensively developed throughout the text.

(2) "An Introduction to the Theory of Numbers" by Niven, Zuckerman, and Montgomery. This book gives a nice coverage of the algebraic aspects of number theory. It has an entire chapter on algebraic numbers that is well worth the read. Also, the more recent edition with Montgomery delves into the geometric results in number theory. This is a well rounded book written by mathematicians preeminent in their field.

(3) "An Introduction to the Theory of Numbers" by Dence and Dence. Quite reader friendly, and surprisingly complete. They promote a deep understanding of the relevant algebra, which is covered at a comfortable pace. They provide an easier read than say Niven, Zuckerman and Montgomery with approximately the same coverage of material.

(4) "An Introduction to the Theory of Numbers" by Hardy and Wright. Written by a legendary number theorist, this book is like a history lesson of 20th century number theory (up through Selberg's "elementary" proof of the Prime Number Theorem). Not so fun to read, but worthwhile as a reference.

(5) "An Introduction to Number Theory" by L.K. Hua. Regrettably, this book is out of print. Nevertheless, you should take a look at it. You can read it with no prior knowledge of number theory and go quite far. Has a comprehensive treatment of (elementary) algebraic number theory. Best appreciated after reading Niven, Zuckerman, and Montgomery.

(6) "Number Theory" by George Andrews is recommended for a combinatorial approach to number theory. The Dover publication is very cheap. Also has some nice introductory material to the theory of partitions.

Of course, there are many others. You can probably find all of the above (except maybe #3) in your local university library.

Recommended.

14 of 15 people found the following review helpful:

3.0 out of 5 stars Adequate introductory text at an outrageous price., February 13, 2005

By 

From Detroit (Michigan, USA) - See all my reviews

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This review is from: Elementary Number Theory (Hardcover)

This text has served me through my first course in number theory. It follows the traditional "definition - theorem - proof - example - exercises" format throughout it's sections. For some flavor, it even throws in a little history behind the mathematics it presents. This book, however, IS NOT worth the ridiculous price that McGraw Hill has retailers charging; nothing in it is that spectacular (well, not spectacular at all, really).

16 of 18 people found the following review helpful:

5.0 out of 5 stars Elementary Number Theory by David. M. Burton, April 5, 2000

By 

Pantelis Damianou (University of Cyprus, Nicosia) - See all my reviews

This review is from: Elementary Number Theory: Fourth Edition (Hardcover)

This is an excellent textbook for an introductory course in Number Theory. I have used it a number of times for my own courses and I believe it is the most popular book for elementary Number theory courses in the United States. It covers all the standard topics in Number Theory including congruences, properties of prime numbers and their distribution, the theorems of Fermat and Wilson, quadratic residues, quadratic reciprocity, perfect numbers, pythagorian triples, representation of integers as sums of squares and a chapter on continued fractions and Pell's equation. The book includes historical notes, useful tables and a great number of interesting exercises. I recommend this book for begginers in Number Theory but I believe that even the advanced reader may find something interesting.