Prime Numbers

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." ^ L'Enseignement Mathématique

"...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

Autorentext

Richard Crandall currently holds the title of Apple Distinguished Scientist, having previously been Apples Chief Cryptographer, the Chief Scientist at NeXT, Inc., and recipient of the Vollum Chair of Science at Reed College. His primary interest is interdisciplinary scientific computation, though he has authored numerous theoretical papers in quantum physics, biology, mathematics, and chemistry, as well as various patents across engineering fields.

Carl Pomerance received his Ph.D. in mathematics from Harvard University in 1972. Currently he is a professor at Dartmouth College. A popular lecturer and winner of the Chauvenet and Conant Prizes for expository mathematical writing, Pomerance is well known for his research in computational number theory, his efforts having produced important algorithms now in wide use.

Zusammenfassung

Prime numbers beckon to the beginner, the basic notion of primality being accessible to a child. Yet, some of the simplest questions about primes have stumped humankind for millennia. In this book, the authors concentrate on the computational aspects of prime numbers, such as recognizing primes and discovering the fundamental prime factors of a given number. Over 100 explicit algorithms cast in detailed pseudocode are included in the book. Applications and theoretical digressions serve to illuminate, justify, and underscore the practical power of these algorithms. The 2nd edition adds new material on primality and algorithms and updates all the numerical records, such as the largest prime, etc. It has been revised throughout.

From the reviews of the first edition:

"The exercises are a gold mine of interesting examples, pointers to the literature and potential research projects. Prime Numbers is a welcome addition to the literature of number theorycomprehensive, 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." American Scientist

"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. It is also a timely book, since primes and factoring have reached a certain vogue, partly because of cryptography. " L'Enseignement Mathématique

"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." Bulletin of the AMS

Inhalt
Preface.- Primes!.- Number-Theoretical Tools.- Recognizing Primes and Composites.-Primality Proving.- Exponential Factoring Algorithms.- Subexponential Factoring Algorithms.- Elliptic Curve Arithmetic.- The Ubiquity of Prime Numbers.- Fast Algorithms for Large-Integer Arithmetic.- Book Pseudocode.- References.- Index.