Customer Reviews

Elementary Number Theory

5 star:		(6)
4 star:		(5)
3 star:		(1)
2 star:		(0)
1 star:		(1)

 

 

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.

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).

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.

This is a textbook about Elementary Number Theory, where "elementary" does not
mean "simple" or "beginning", but rather those portions of the mathematics of
integers that do not rely on analysis (infinitesmal calculus).

Number theory allows many different orderings of topics, without omitting
proofs. I found Burton's order to be easy to follow. Many results in number
theory follow easily from results in abstract algebra or linear algebra.
The author does not depend on results beyond elementary algebra, but some
degree of mathematical maturity is required. Readers with a math degree will
still have to work to absorb the material.

There are many problems. Those with numerical answers are answered in the back
of the book. About half of the others are answered in an answer guide,
available separately. Almost everything is proved. I only recall two cases
of "left to the reader" except for the problems, of course. None of the
problems are used for future developments in the main text.

The author has a separate text about the history of mathematics. Most of the
chapters in this book start with a section about the history of the material
in the chapter and about the people that developed it. This is interesting
extra material, or padding that makes the book even more expensive than it
should be, depending on you.

This is the 6th edition. The only error I encountered was a consistent misspelling
of one name in chapter 10. I could not find any reported errors on the WWW.

I've used several other number theory books over the years. This one seems the best
for me. Perhaps that is due to Burton's skill, or perhaps it is because I finally
worked through one from front to back, instead of searching for the information
I needed just then.

Got the book for free in electronic format. The prices for these books are criminal. Intellectual property is the fraud of frauds.

The book itself is excellent. I had not had any formal study of proofs when I started reading, but this book guides you very gently and clearly through the process. Number theory is NOT boring, and going through this book will make you feel enlightened. It's not easy, but very doable for anyone of average intelligence.

Most of this stuff I found in different books,
but usually not as well presented as this.
This book is a perfect one for someone starting in on number theory.
It even has some neat tables at the end.
I wish I had had this book ten years ago!

before finally selecting this book for reading i 've spent a few hours in the library browsing through some number theory books. Coming from a a different background electical & computer engineer, I had no notion at all of number theory. I like his way of writing with the embedded historical notes and furthermore the proofs of the theorem and their chronological order particularly in the second chapter. I have to admit that I comment on the previous version but new versions are supposed to improve. It is not time consuming to go through the proofs while you understand the theorems and the techniques used behind. The flow is very coherent and solidly written. Overall an excellent introductory book as cited in a previous review.

I'm a first-year Ph.D. student, taking a graduate-level number theory course, and I still use this book from my undergrad years as a reference. Just about any basic number theory topic you're looking for is in here. I can't recommend it highly enough!

I bought this book to study number theory on my own. (but let me say I had great knowledge about the material b4 I got into it). I studied the first three chapters on my own, and it was great experience, but then I had to stop cuz I did not have any free time to continue. From the first three chapters, I rank this book 5 stars!

This book is awesome, written very rigorously!! Its the right way to write any book in mathematics, and I love it.

I must admit, the exposition can get a little hairy at (very few) spots, but the problems are good, and it has served me well as a reference (for certain limited topics). Great introduction.