Product Description

Numbers are part of our everyday experience and their properties have fascinated mankind since ancient times. Deciding whether a number is prime and if not, what its factors are, are both fundamental problems. In recent years analysis and solution of these problems have assumed commercial significance since large primes are an essential feature of secure methods of information transmission. The purely mathematical fascination that led to the development of methods for primality testing has been supplemented by the need to test within reasonable timescales, and computational methods have entered at all levels of number theory. In this book, Peter Giblin describes, in the context of an introduction to the theory of numbers, some of the more elementary methods for factorization and primality testing; that is, methods independent of a knowledge of other areas of mathematics. Indeed everything is developed from scratch so the mathematical prerequisites are minimal. An essential feature of the book is the large number of computer programs (written in Pascal) and a wealth of computational exercises and projects (in addition to more usual theory exercises). The theoretical development includes continued fractions and quadratic residues, directed always towards the two fundamental problems of primality testing and factorization. There is time, all the same, to include a number of topics and projects of a purely ‘recreational’ nature.

Product Details

• Amazon Sales Rank: #646081 in Books
• Published on: 1993-09-02
• Original language: English
• Number of items: 1
• Binding: Paperback
• 252 pages

Review
"An interesting, sophisticated introduction to number theory..." American Mathematical Monthly

"Of the many volumes I have seen about `number theory and computing', this delightful, if unorthodox, introductory text is probably the finest...a great strength of this book is its emphasis on computing and on computing examples. There are several programs included in the text, often different algorithms for achieving the same computational result, and both theoretical and practical reasons for preferring one method over another are discussed. The programming language is Pascal, which is perfectly appropriate...[and] there are a great many numerial exercises and examples...only the deadest of students could possibly consider this dry; the author has brought life and energy to the subject by his presentation." Duncan Buell, Mathematical Reviews

Customer Reviews

Excellent, clear, writing style ...
This book deals with elementary number theory, the main topics are (1) primality tests (Lucas-Lehmer test for Mersenne primes, Lucas n-1 over q test, Rabin-Miller test and pseudo-primes, Pepin's tests for Fermat numbers, Proth's theorem ...) and (2) results to find actual factors (trial division, Pollard's rho method, continued fraction of square root of n method).

However, this "theme" of factoring and testing for primality is spread over 11 short chapters that deal with traditional topics such as the greatest common divisor, congruences, chinese remainder theorem, Fermat's theorem, the divisor function, the Euler phi function, continued fractions, and the Legendre symbol (quadratic reciprocity).

Most of the results related to factoring or primality tests, are actually developed as (solved) exercises.

The author has a clear writing style and a habit of following every theorem with a list of examples and projects (exercises), so the reader is truly invited to solve exercises.

There's no emphasis on group theory or abstract algebraic interpration, all results are "basic", and the clear writing style of the author never lets you wonder why a certain result is true or false ...

There's also some other topics covered such as Shanks' algorithm for solving x^2=a mod p and the Pell equation x^2-ny^2=1 solving by continued fractions.

And there is also material such as perfect numbers, amicable numbers, lucky numbers ... In short, this book covers the theory of elementary number theory (up to quadratic reciprocity) very well and may also appeal to people who like "mathematical puzzles" ...