Elementary Number Theory (Mcgraw-Hill International Editions) Review

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(More customer reviews)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.

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This text provides a simple account of classical number theory, as well as some of the historical background in which the subject evolved. It is intended for use in a one-semester, undergraduate number theory course taken primarily by mathematics majors and students preparing to be secondary school teachers. Although the text was written with this readership in mind, very few formal prerequisites are required. Much of the text can be read by students with a sound background in high school mathematics.

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