Friendly Introduction to Number Theory, A (3rd Edition)

Starting with nothing more than basic high school algebra, this volume leads readers gradually from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. Features an informal writing style and includes many numerical examples. Emphasizes the methods used for proving theorems rather than specific results. Includes a new chapter on big-Oh notation and how it is used to describe the growth rate of number theoretic functions and to describe the complexity of algorithms. Provides a new chapter that introduces the theory of continued fractions. Includes a new chapter on “Continued Fractions, Square Roots and Pell’s Equation.” Contains additional historical material, including material on Pell’s equation and the Chinese Remainder Theorem. A useful reference for mathematics teachers.

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.


A Valuable Book!   November 11, 2003
Joonsang Baek (Wollongong, NSW Australia)
6 out of 6 found this review helpful

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.


What's up with the bad reviews?   December 5, 2007
Michael R. Steele (Salt Lake City, Utah United States)
3 out of 3 found this review helpful

I'm a bit shocked by the bad reviews of this book. I guess if a book is clear, understandable, and interesting it doesn't qualify as a worthy math book. I've noticed this in other math book reviews, there seems to be a real element of machismo among alot of mathematicians, if the book is very formal usually theorem proof theorem it gets high marks. If the book is like this, one where the author is clear and engaging the book is discounted. I personally found this book to be one of the great intellectual adventures of my life, but here again in college I only minored in math not majored.


This book was NOT written for math majors   January 19, 2001
Darin Brown (Goleta, CA United States)
50 out of 52 found this review helpful

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.