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© 1994

Prime Numbers and Computer Methods for Factorization

Book

Part of the Progress in Mathematics book series (PM, volume 126)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Hans Riesel
    Pages 37-59
  3. Hans Riesel
    Pages 84-140
  4. Hans Riesel
    Pages 141-172
  5. Hans Riesel
    Pages 173-225
  6. Hans Riesel
    Pages 226-238
  7. Back Matter
    Pages 239-464

About this book

Introduction

In the modern age of almost universal computer usage, practically every individual in a technologically developed society has routine access to the most up-to-date cryptographic technology that exists, the so-called RSA public-key cryptosystem. A major component of this system is the factorization of large numbers into their primes. Thus an ancient number-theory concept now plays a crucial role in communication among millions of people who may have little or no knowledge of even elementary mathematics.

Hans Riesel’s highly successful first edition of this book has now been enlarged and updated with the goal of satisfying the needs of researchers, students, practitioners of cryptography, and non-scientific readers with a mathematical inclination. It includes important advances in computational prime number theory and in factorization as well as re-computed and enlarged tables, accompanied by new tables reflecting current research by both the author and his coworkers and by independent researchers.

The book treats four fundamental problems: the number of primes below a given limit, the approximate number of primes, the recognition of primes and the factorization of large numbers. The author provides explicit algorithms and computer programs, and has attempted to discuss as many of the classically important results as possible, as well as the most recent discoveries. The programs include are written in PASCAL to allow readers to translate the programs into the language of their own computers.

The independent structure of each chapter of the book makes it highly readable for a wide variety of mathematicians, students of applied number theory, and others interested in both study and research in number theory and cryptography.

Keywords

Computer Factorization Maxima Mersenne prime Prime Prime Numbers Prime number Sage algorithms continued fraction cryptography number theory

Authors and affiliations

  1. 1.Department of MathematicsThe Royal Institute of TechnologyStockholmSweden

Bibliographic information

  • Book Title Prime Numbers and Computer Methods for Factorization
  • Authors Hans Riesel
  • Series Title Progress in Mathematics
  • Series Abbreviated Title Progress in Mathematics(Birkhäuser)
  • DOI https://doi.org/10.1007/978-1-4612-0251-6
  • Copyright Information Birkhäuser Boston 1994
  • Publisher Name Birkhäuser, Boston, MA
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-8176-3743-9
  • Softcover ISBN 978-1-4612-6681-5
  • eBook ISBN 978-1-4612-0251-6
  • Series ISSN 0743-1643
  • Series E-ISSN 2296-505X
  • Edition Number 2
  • Number of Pages XVI, 464
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Number Theory
    Computational Mathematics and Numerical Analysis
  • Buy this book on publisher's site
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Reviews

"Here is an outstanding technical monograph on recursive number theory and its numerous automated techniques. It successfully passes a critical milestone not allowed to many books, viz., a second edition. Many good things have happened to computational number theory during the ten years since the first edition appeared and the author includes their highlights in great depth. Several major sections have been rewritten and totally new sections have been added. The new material includes advances on applications of the elliptic curve method, uses of the number field sieve, and two new appendices on the basics of higher algebraic number fields and elliptic curves. Further, the table of prime factors of Fermat numbers has been significantly up-dated. ...Several other tables have been added so as to provide data to look for large prime factors of certain 'generalized' Fermat numbers, while several other tables on special numbers were simply deleted in the second edition. Still one can make several perplexing assertions or challenges: (1) prove that F\sb 5, F\sb 6, F\sb 7, F\sb 8 are the only four consecutive Fermat numbers which are bi-composite; (2) Show that F\sb{14} is bi- composite. (This accounts for the difficulty in finding a prime factor for it.) (3) What is the smallest Fermat quadri-composite?; and (4) Does there exist a Fermat number with an arbitrarily prescribed number of prime factors? All in all, this handy volume continues to be an attractive combination of number-theoretic precision, practicality, and theory with a rich blend of computer science."

–Zentralblatt Math