Superb introduction for readers with limited formal mathematical training. Topics include Euclidean algorithm and its consequences, congruences, powers of an integer modulo m, continued fractions, Gaussian integers, Diophantine equations, more. Carefully selected problems included throughout, with answers. Only high school math needed. Bibliography.
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Most Helpful Customer Reviews
19 of 21 people found the following review helpful:
4.0 out of 5 stars
Good Introduction to Key Topics and Proofs of Number Theory,
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This review is from: Elementary Theory of Numbers (Dover Books on Mathematics) (Paperback)
William J. LeVeque's short book (120 pages), Elementary Theory of Numbers, is quite satisfactory as a self-tutorial text. It should appeal to math majors new to number theory as well as others that enjoy studying mathematics.
Chapter 1 introduces proofs by induction (in various forms), proofs by contradiction, and the radix representation of integers that often proves more useful than the familiar decimal system for theoretical purposes. Chapter 2 derives the Euclidian algorithm, the cornerstone of multiplicative number theory, as well as the unique factorization theorem and the theorem of the least common multiple. Speaking from experience, I recommend that you take the time necessary to master Chapter 2, not just because these basic proofs are important, but more critically to reinforce the skills and discipline necessary for the subsequent chapters. Two integers a and b are congruent for the modulus m when their difference a-b is divisible by the integer m. In chapter 3 this seemingly simple concept, introduced by Gauss, leads to topics like residue classes and arithmetic (mod m), linear congruences, polynomial congruences, and quadratic congruences with prime modulus. The short chapter 4 was devoted to the powers of an integer, modulo m. Continued fractions, the subject of chapter 5, was not unfamiliar and yet, as with congruences, I quickly found myself enmeshed in complexity, wrestling with basic identities, the continued fraction expansion of a rational number, the expansion of an irrational number, the expansion of quadratic identities, and approximation theorems. I have yet to tackle the last two chapters, the Gaussian integers and Diophantine equations, but my expectation is that both topics will also require substantial effort and time. LeVeque's Elementary Theory of Numbers is not an elementary text, nor a basic introduction to number theory. Nonetheless, it is not out of reach of non-mathematics majors, but it will require a degree of dedication and persistence. For a reader new to number theory, LeVeque may be too much too soon. I suggest first reading Excursions in Number Theory by C. Stanley Ogilvy and John T. Anderson, another Dover reprint. It is quite good. Some caution: LeVeque emphasizes that many theorems are easy to understand, and yet this very simplicity is a two-edged sword. Simple theorems often provide no clues, no hints, on how to proceed. Discovering a short and elegant proof is often far from easy. LeVeque also stresses that a technique ceases to be a trick and becomes a method only when it has been encountered enough times to seem natural. A reader new to number theory may initially be overwhelmed by the variety of techniques used. A nit: The Dover edition of LeVeque's Elementary Theory of Numbers would benefit from a larger font size. I occasionally found myself squinting to read tiny subscripts and superscripts.
8 of 9 people found the following review helpful:
5.0 out of 5 stars
Very good,
This review is from: Elementary Theory of Numbers (Dover Books on Mathematics) (Paperback)
Very good book. First to comment on the fact that LeVeque has 2 dover books that cover basically the same topics (this one, and Fundamentals of Number Theory). I have looked at both, and this one is the better of the two. The other one uses slightly different definitions that have an Abstract Algebra twist to it. But the other book still doesn't use the power of abstract algebra so the different/akward definitions and explanations just make it hard to read. An elementary number theory book should use elementary definitions and concepts (abstract algebra is meant for ALGEBRAIC number theory books). So avoid his other book, which is good, but not as easy to read as this one. This book is very easy to read and concepts are introdced very clearly. Things come in small chunks which are easily digested. The thing about this book is, you can go through it faster than normal textbooks but you still end up learning everything you would by going slowing through hard-to-read texts (not like The Higher Arithmetic by Davenport, that book can lull you into reading it like a story book, but you end up learning nothing).
3 of 3 people found the following review helpful:
4.0 out of 5 stars
Readable, clear, but needs an errata page,
By
This review is from: Elementary Theory of Numbers (Dover Books on Mathematics) (Paperback)
As others have said, this is a fairly easy read. For me it's actually fun and I'm working through it for that reason. But:
- I don't normally use a highlighter, but found it necessary to highlight symbols where they were defined, because some of them come up only once in a while and it's easy to forget where the definition is. Symbols are not indexed. I have started my own symbol index in the back of the book. - There are some annoying errors. The theorem to be proven in section 1-3, problem 2 is false for n=1. The decimal expansions in the chapter on continued fractions (page 75) are wrong (for example 1.273820... should actually be 1.273239...). It seems to me if you're going to give 7 digits they should be the right 7 digits. On the other hand, these errors don't affect the overall flow of the text, and I'm having a great time working through this book on my own. I've read through the whole thing over the summer, and I'm going back through doing problems and writing programs. I was a math major 40 years ago, and haven't done much with it since, to give a context for that remark.
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