Fundamental Number Theory with Applications (Discrete Mathematics and Its Applications) - Hardcover

Mollin, Richard A.

Hardcover

ISBN 10: 0849339871 ISBN 13: 9780849339875

Publisher: CRC Press, 1997

Synopsis

Beginning with the arithmetic of the rational integers and proceeding to an introduction of algebraic number theory via quadratic orders, Fundamental Number Theory with Applications reveals intriguing new applications of number theory. This text details aspects of computer science related to

cryptography

factoring

primality testing

complexity analysis

computer arithmetic

computational number theory
Fundamental Number Theory with Applications also covers:

Carmichael numbers

Dirichlet products

Jacobsthal sums

Mersenne primes

perfect numbers

powerful numbers

self-contained numbers
Numerous exercises are included, testing the reader's knowledge of the concepts covered, introducing new and interesting topics, and providing a venue to learn background material.
Written by a professor and author who is an accomplished scholar in this field, this book provides the material essential for an introduction to the fundamentals of number theory.

About this title

Review

"The book's strengths lie in its currency, its many worked examples, historical footnotes, and references to the literature." - D.V. Feldman, University of New Hampshire, in CHOICE, April 1998 "a very useful addition to the many books on number theory with applications, and it is meant to be accessible to anyone from the novice to the research scientistprovides an excellent supplementary source of information for the reader, not least in the many biographical footnotes on the mathematicians involved in the subject matter, and there are also more than a thousand exercises and examples in the text. Besides various tables for results in computational number theory, the nine appendices cover material from set theory, which includes discussions on the axiom of choice and Zorn's Lemma, to the ABC conjecture." --P. Shiu, Zentralblatt MATH, Vol. 943

From the Author

A book rich in applications of number theory to cryptography
If you want to give an introductory number theory course at any level, then this text is for you. The applications to computer science, especially cryptography, are rich and extensive, ranging from computer arithmetic to an elementary presentation of the elliptic curve factoring method. There are also optional applications to algebraic number theory. A human touch is given by the more than 70 biographies of mathematicians, which are woven through the text as footnotes.

The text is exercise-rich, with over 740 problems, ranging from the routine to the more difficult star problems, and all odd numbered exercises are solved in detail at the end of the text. To aid the instructor who adopts the text for a course, a solutions manual for the even numbered exercises is available free of charge.

There are over 270 examples illustrating and motivating all topics covered, as well as 9 appendices chock-full of information ranging from set theory to Cunningham factorizations, and new records. The index has over one thousand entries to ensure maximum ease in extracting information.

Steve Galovich says of the text: ``The book is engaging, and I am sure that students will learn a great deal from it."

Andrew Granville says: ``I like the chatty style and the human touch that are evident throughout the text. I especially enjoyed reading the frequent historical notes, bringing back to life many of the great mathematicians by amusing and pertinent anecdotes."

Nikos Tzanakis says: "This is a new book written by an active number-theorist, who has very refreshing views about how basic number theory should be presented to beginners and advanced students. Actually, it is so rich that the instructor can find many interesting paths through its pages in order to teach not only the basic theory, but also more advanced topics and applications. The author dares to include topics, which are rarely found in "elementary" books, and he does this successfully. I wouldn't hesitate to say that it may be a breakthrough for "elementary" number theory books, starting with the title, which is very well chosen."

Detailed Description from the PREFACE:

This text is structured for an undergraduate course in number theory at any level. The last two sections of each chapter are marked by the pointing hand symbol to denote that they are the optional sections on applications. The penultimate section of each chapter is devoted to applications in computer science, especially to cryptography. The last section of each chapter develops applications to algebraic number theory via quadratics, meaning applications to quadratic fields and quadratic orders in general. The student who works through these last interconnected sections of each chapter will have a solid foundation and introduction to an upper level algebraic number theory course. The remaining sections of each chapter constitute the core material (plus some added gems, and more challenging material marked by the pencil symbol to distinguish them from the essential core material), for what is usually termed a course in Elementary Number Theory. The interrelated penultimate sections of each chapter provide a mini-course in aspects of computer science related to cryptography, factoring, primality testing, complexity analysis, computer arithmetic, and computational number theory in general. No background in computer science is assumed, so the elementary approach will appeal to anyone, however uninitiated, who is interested in learning about these areas. Computer scientists, engineers, mathematicians, and students in science will also find that these sections provide a quick introduction to these applications of number theory.

The instructor may gear the text for an upper-division course in number theory by covering the fundamentals plus any choice of the applications in the last two sections of each chapter. On the other hand, the text is also intended for a lower-division introductory number theory course. In this case, the instructor may cover only the core material with perhaps an application or two for extra interest from the sections marked by the pointing hand, or some of the more challenging non-core material markedby the pencil symbol.

For the beginner, which may include enlightened high school students, we have made no assumptions as to the background of the reader starting in the Introduction in Section One of Chapter One. We begin there with a history of number theory going back to the ancient Sumerians in the fourth millennium B.C. We continue this historical thread throughout the text. We explore, for example, the very notions of number and addition, among other primitive notions in this introduction to set the stage for the fundamental laws of arithmetic that follow in the subsequent section, and we build from there. Hence, the complete novice can use the text as a self-contained entity. The text is flexible enough to satisfy disparate needs in the presentation of number theory at any level.

Edition details

• Publisher: CRC Press
• Publication date: 1997
• Language: English
• ISBN 10: 0849339871
• ISBN 13: 9780849339875
• Binding: Hardcover
• Edition number: 1
• Number of pages: 464