Prime Numbers: A Computational Perspective [Hardcover]

Richard Crandall (Author), Carl B. Pomerance (Author)

Book Description

ISBN-10: 0387252827 | ISBN-13: 978-0387252827 | Publication Date: August 4, 2005 | Edition: 2nd

Bridges the gap between theoretical and computational aspects of prime numbers Exercise sections are a goldmine of interesting examples, pointers to the literature and potential research projects Authors are well-known and highly-regarded in the field

Editorial Reviews

Review

From the reviews: MATHEMATICAL REVIEWS "There are many books about the theory of prime numbers and a few about computations concerning primes. This book bridges the gap between theoretical and computational aspects of prime numbers. It considers such matters as how to recognize primes, how to compute them, how to count them, and how to test conjectures about them¿The book is clearly written and is a pleasure to read. It is largely self-contained. A first course in number theory and some knowledge of computer algorithms should be sufficient background for reading it…Each chapter concludes with a long list of interesting exercises and research problems." BULLETIN OF THE AMS "The book is an excellent resource for anyone who wants to understand these algorithms, learn how to implement them, and make them go fast. It's also a lot of fun to read! It's rare to say this of a math book, but open Prime Numbers to a random page and it's hard to put down. Crandall and Pomerance have written a terrific book." AMERICAN SCIENTIST "…a welcome addition to the literature of number theory – comprehensive, up-to-date and written with style. It will be useful to anyone interested in algorithms dealing with the arithmetic of the integers and related computational issues." SIAM REVIEW "Overall, this book by Crandall and Pomerance fills a unique niche a deserves a place on the bookshelf of anyone with more than a passing interest in prime numbers. It would provide a gold mine of information and problems for a graduate class on computationl number theory." From the reviews of the second edition: "This book is a very successful attempt of the authors to describe the current state-of-the-art of computational number theory … . One of the many attractive features of this book is the rich and beautiful set of exercises and research problems … . the authors have managed to lay down their broad and deep insight in primes into this book in a very lucid and vivid way. … The book provides excellent material for graduate and undergraduate courses on computational theory. Warmly recommended … ." (H.J.J. te Riele, Nieuw Archief voor Wiskunde, Vol. 7 (3), 2006) "An absolutely wonderful book! Written in a readable and enthusiastic style the authors try to share the elegance of the prime numbers with the readers … . Weaving together a wealth of ideas and experience from theory and practice they enable the reader to have more than a glimpse into the current state of the knowledge … . any chapter or section can be singled out for high praise. … Indeed it is destined to become a definitive text on … prime numbers and factoring." (Peter Shiu, Zentralblatt MATH, Vol. 1088 (14), 2006) "This impressive book represents a comprehensive collection of the properties of prime numbers. … in the exercises at the end of each chapter valuable hints are given how the theorems have been attained. The chapters end with research exercises. The book is up to date and carefully written. … The volume is very vividly and even entertainingly written and is best suited for students and for teachers as well." (J. Schoissengeier, Monatshefte für Mathematik, Vol. 150 (1), 2007) "The aim of this book is to bridge the gap between prime-number theory covered in many books and the relatively new area of computer experimentation and algorithms. The aim is admirably met. … There is a comprehensive and useful list of almost 500 references including many to websites. … This is an interesting, well-written and informative book neatly covering both the theoretical as well as the practical computational implementation of prime numbers and many related topics at first-year undergraduate level." (Ron Knott, The Mathematical Gazette, Vol. 92 (523), 2008)

From the Back Cover

Prime numbers beckon to the beginner, as the basic notion of primality is accessible even to children. Yet, some of the simplest questions about primes have confounded humankind for millennia. In the new edition of this highly successful book, Richard Crandall and Carl Pomerance have provided updated material on theoretical, computational, and algorithmic fronts. New results discussed include the AKS test for recognizing primes, computational evidence for the Riemann hypothesis, a fast binary algorithm for the greatest common divisor, nonuniform fast Fourier transforms, and more. The authors also list new computational records and survey new developments in the theory of prime numbers, including the magnificent proof that there are arbitrarily long arithmetic progressions of primes, and the final resolution of the Catalan problem. Numerous exercises have been added. Richard Crandall currently holds the title of Apple Distinguished Scientist, having previously been Apple's Chief Cryptographer, the Chief Scientist at NeXT, Inc., and recipient of the Vollum Chair of Science at Reed College. Though he publishes in quantum physics, biology, mathematics, and chemistry, and holds various engineering patents, his primary interest is interdisciplinary scientific computation. Carl Pomerance is the recipient of the Chauvenet and Conant Prizes for expository mathematical writing. He is currently a mathematics professor at Dartmouth College, having previously been at the University of Georgia and Bell Labs. A popular lecturer, he is well known for his research in computational number theory, his efforts having produced important algorithms now in use. From the reviews of the first edition: "Destined to become a definitive textbook conveying the most modern computational ideas about prime numbers and factoring, this book will stand as an excellent reference for this kind of computation, and thus be of interest to both educators and researchers." "...Prime Numbers is a welcome addition to the literature of number theory---comprehensive, up-to-date and written with style." - American Scientist "It's rare to say this of a math book, but open Prime Numbers to a random page and it's hard to put down. Crandall and Pomerance have written a terrific book." - Bulletin of the AMS

See all Editorial Reviews

Product Details

• Hardcover: 613 pages
• Publisher: Springer; 2nd edition (August 4, 2005)
• Language: English
• ISBN-10: 0387252827
• ISBN-13: 978-0387252827
• Product Dimensions: 9.3 x 6.3 x 1.4 inches
• Shipping Weight: 2.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #604,330 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

27 of 27 people found the following review helpful:

5.0 out of 5 stars Standard reference on the subject., April 21, 2003

By 

Decio Luiz Gazzoni Filho (Londrina, PR Brazil) - See all my reviews

This review is from: Prime Numbers (Hardcover)

While recently published, this book is shaping to become the standard reference on the theory that surrounds prime numbers in a computational setting, drawing from all branches of number theory, as well as abstract algebra, analysis, combinatorics, statistics, complexity theory and elliptic curves. Surely a multidisciplinary treatise if there ever was one.

The authors' writing style, while not conversational, never gets in the way, and allows reading at many levels (from light reading to deep research). Theorems are proved only when it makes sense to do so, i.e. when the proof adds insight into the matter. The exercises are interesting and challenging, and closing each chapter are avenues of further research, referencing open problems in the literature and the authors' own opinion on interesting subjects for research.

The first chapter is an overview of theoretical and computational developments, with anything from Euclid's proof of the infinitude of primes, Riemann's study of the zeta function, down to the latest huge computation of the twin prime constant and zeros of the zeta function in the critical line. Some famous open problems are displayed as well.

The necessary number theory background is covered on Chapter 2, though the interested reader should seek a more complete treatise on the subject.

Trial division, sieving and pseudo-primality tests are fully covered in Chapter 3. There is really nothing to complain about this chapter of the book.

Chapter 4 concerns proving the primality of integers. Many results are presented from the classical (meaning not involving elliptic curves) primality tests, and again there is nothing to complain.

Many people, such as myself, are drawn to the book for the integer factoring algorithms, and they're not going to be disappointed. Unfortunately, modern factoing algorithms deserve a book on its own, and it's impossible to cover all the ground in the space alloted to them in this book. The authors do a pretty good job of introducing them, even if the explanation is unclear and a bit shallow at times, and they always reference other works on the field for further information they were unable to cover.

Chapter 7, ``Elliptic Curve Arithmetic,'' is a great starting point for elliptic curve studies, with a no-nonsense introduction to the subject that is certainly enough for the algorithms that follow. These include Lenstra's Elliptic Curve Method of factorization; Shanks-Mestre's, Schoof's and Atkin-Morain's algorithms for assessing curve order; and Goldwasser-Kilian's and Atkin-Morain's primality proving algorithms.

Almost as valuable as the rest of the book itself (at least for implementers) is the ninth and last chapter, ``Fast algorithms for large-integer arithmetic.'' Many of these can be carried over without effort to floating point, so the scope of the material is even broader than the authors claim. Having read parts of Knuth's ``The Art of Computer Programming: Seminumerical Algorithms,'' I can attest to the superb exposition of Crandall and Pomerance being a breath of fresh air in this field. This book belongs on the shelf of every programmer implementing multiprecision arithmetic for this chapter alone.

22 of 22 people found the following review helpful:

5.0 out of 5 stars A Factoring "Bible", March 6, 2002

By 

Ed Prothro (Stillwater, OK United States) - See all my reviews

This review is from: Prime Numbers (Hardcover)

This book has all the recent (2001) developments in factoring algorithms and related number theory. It has chapters on algorithms for large numbers. While graduate-level, much of it should be accessible by an undergraduate. It has exercises and research questions after each chapter.

To find all the info in it, you would have to scour a research library for all the papers that have been published on factoring and primality testing -- they are scattered thru many math journals. It also covers things like quantum computing and cryptography.

It's a good reference - no need to read the whole thing. It would also make an excellent graduate-level textbook.

7 of 8 people found the following review helpful:

4.0 out of 5 stars advanced coverage, September 22, 2005

By 

W Boudville (Terra, Sol 3) - See all my reviews
(VINE VOICE)    (TOP 1000 REVIEWER)    (HALL OF FAME REVIEWER)    (REAL NAME)   

This review is from: Prime Numbers (Hardcover)

This is an advanced treatment of prime numbers. But it is not all abstract number theory. The recurrent theme is how to compute these and how to use primes in other computationally intensive tasks.

The book summarises centuries of effort. Notably with Goldbach's Conjecture about every even number>2 being the sum of two primes. But intriguing issues like the density of primes along the number line are gone into. Along with the Mersenne primes and prime producing formulae.

An entire chapter discusses cryptography and related matters. Primes are at the heart of PKI and its RSA implementation. There is even a section briefly covering quantum computing and a quantum Turing Machine. Rather sparse detail because, well, the experimental results are still very new. Only baby steps have been forthcoming. The phase coherence difficulties are formidable. But it is a potentially vast area of future work.