A Friendly Introduction to Number Theory

Rating:
Summary: A Valuable Book!
Review: I admit that this book might not be suitable especially for pure mathematicians. But I very much liked Silverman's way of writing: He cast questions and encourage readers to tackle them! Indeed, this is a unique number thoery book written in that way.

Rating:
Summary: This book was NOT written for math majors
Review: I just wanted to make clear the point that each textbook or math book written is written for an INTENDED audience, and it's not fair to negatively criticize a book by using the reviewer's own personal background, rather than the INTENDED audience, as the guide for criticism.

This book was not written for math majors. So, I find it kind of distressing to hear that many math majors are saying this was textbook for a beginning number theory class for math majors. Silverman makes effort to point out that the book was written as the textbook for a general liberal arts math class, which is actually taken by non-science and non-math majors at the university where Silverman teaches. It requires nothing beyond basic calculus (if that), and I don't see anywhere where Silverman gives the impression that the book is meant to be used as a strong introduction to writing proofs or becoming fluent in rigorous mathematical arguments which math majors will later see.

So, of course, math majors will find fault...but the book wasn't written for them. It was written primarily to get people who have little interest in math or little exposure to math, some opportunity to see something more interesting beyond high school algebra and calculus. The emphasis on computation is warranted in any case, because although number theory is mathematics and has rigorous proofs, intuition and working familiarity with the concepts and constructions of number theory only come through hours and hours of simple computations with the positive integers. Computation is a legitimate and necessary part of number theory.

As for rational points on the circle (and Fermat's Last Theorem) being unusual or out of the ordinary material, this is farthest from the truth. The example of rational points on the circle is one of the oldest (2,000 years or so???) and most basic constructions of number theory, revealing how geometric number theory is, and the example directly leads to more general ideas and concepts which are central to current research (Diophantine equations, elliptic curves, projective geometry, for example) and pick up many of the standard graduate references on elliptic curves and the first 5-10 pages are a detailed examination of this very example.

I'm a graduate student studying number theory, so I'm pretty far away from the intended audience. But I can see that the book does a pretty good job at what it sets out to do, namely present an exposition of certain problems in mathematics, accessible to non-science and liberal arts majors, in a leisurely and engaging fashion, and to get the students to do their own basic pattern-searching, computation, data collection and conjecturing (ALL important facets of mathematics...proof is the polished product, but lots of time is spent by mathematicians before even GETTING to the point of proving things.)

This sounds like a fairly "friendly" introduction to me. If you want more, check out Niven, Hardy/Wright, Ireland/Rosen, Apostol, Gauss, etc.

Rating:
Summary: A really friendly, enjoyable introduction to number theory
Review: I very much enjoyed this book. The book is indeed an excellent and "friendly" introduction to number theory. Dr. Silverman writes in a conversation style. I felt like I had a friendly tutor standing over my shoulder explaining not only how the mathematics worked, but, more importantly, why the topics he described or was about to describe are important and their relevancy in either the world of mathematics or in the "real" world. While he has very few "formal" proofs compared to most number theory texts, Dr. Silverman thoroughly works through numerous numerical examples to give the reader a "feel" for what is going on.

I was particularly pleased with Dr. Silverman's chapter explanation of why quadratic residues are important and how they are used.

Dr. Silverman presents introductory explanations of a number of frequently mentioned number theory topics such as Mersenne Primes, number sieves, RSA cryptography, elliptic curves. He ties together lucid explanations of Pythagorean triples, x2 + y2 = z2, x4 + y4 = z4, and elliptic curves to build to an explanation of Wiles proof of Fermat's Last Theorem.

Rating:
Summary: I have kind of mixed feelings on this one
Review: This book was used in an introductory number theory class at my University last spring. Then (and still now), I have mixed feelings about it.

Good points: The material that is covered is explained well, with the author focusing more on understanding and application than mindless theorem/proof theorem/proof, etc. Also, some of the topics covered in this book are a bit out of the ordinary, (including a chapter at the beginning about rational points on circles and a series of chapters at the end leading to a (very) rough sketch of the proof of Fermat's Last Theorem.

Bad points: Having worked through (virtually) the entire book, I am left with the feeling that there is plenty left out. The book is THIN, and it would be interesting to see some of the theory behind quadratic reciprocity, etc. fleshed out more instead of just learning enough to do the basic computation. Also, the problems are HEAVILY focused on computation, so it does not serve as a very good introduction to proof-based mathematics (I think many people would tend to skim over the proofs in the book and forget them, focusing instead only on the computational methods).

In all, this might be a good book for self-study for the beginning number-theorist. If you're looking for a rigorous introduction to number theory though, this book might be far too "friendly" for your tastes.