08-17-2026, 03:48 PM
Prime Numbers: A Computational Perspective
Authors: Richard Crandall & Carl Pomerance
Publication: 2005, 2nd edition
Publisher: Springer-Verlag, New York
Prime Numbers: A Computational Perspective is a substantial exploration of prime numbers that deliberately bridges classical number theory and modern computation. Rather than treating primes only through abstract theorems, Crandall and Pomerance continually ask computational questions: How can we efficiently determine whether a huge integer is prime? How can we factor large composite numbers? How quickly can arithmetic with enormous integers be performed? The book develops the mathematics behind these problems while presenting more than 100 algorithms in detailed pseudocode, making it valuable both as a theoretical text and as a practical guide to computational experimentation.
A major part of the book is devoted to primality testing and integer factorization. After establishing essential number-theoretic tools, the authors distinguish between recognizing probable primes and actually proving primality, then develop increasingly sophisticated factoring techniques. Topics include exponential and subexponential factoring algorithms, elliptic-curve methods, and fast algorithms for large-integer arithmetic. The second edition incorporates important developments such as the AKS primality test, new computational evidence concerning the Riemann hypothesis, faster GCD techniques, and updated computational records. It also discusses broader developments in prime-number theory, including arbitrarily long arithmetic progressions of primes.
One of the book's strengths is that computation is not presented as a substitute for mathematics but as another way of investigating mathematical structure. Algorithms are accompanied by explanations of why they work, while exercises range from conventional problems to questions suitable for further investigation and even research projects. Later chapters broaden the picture through elliptic-curve arithmetic, the distribution and ubiquity of primes, and efficient large-integer computation. The result is particularly suitable for advanced undergraduates, graduate students, researchers, programmers interested in computational number theory, and mathematicians interested in the connection between primes, algorithms and cryptography.
Key takeaways
Springer — Prime Numbers: A Computational Perspective
Authors: Richard Crandall & Carl Pomerance
Publication: 2005, 2nd edition
Publisher: Springer-Verlag, New York
Prime Numbers: A Computational Perspective is a substantial exploration of prime numbers that deliberately bridges classical number theory and modern computation. Rather than treating primes only through abstract theorems, Crandall and Pomerance continually ask computational questions: How can we efficiently determine whether a huge integer is prime? How can we factor large composite numbers? How quickly can arithmetic with enormous integers be performed? The book develops the mathematics behind these problems while presenting more than 100 algorithms in detailed pseudocode, making it valuable both as a theoretical text and as a practical guide to computational experimentation.
A major part of the book is devoted to primality testing and integer factorization. After establishing essential number-theoretic tools, the authors distinguish between recognizing probable primes and actually proving primality, then develop increasingly sophisticated factoring techniques. Topics include exponential and subexponential factoring algorithms, elliptic-curve methods, and fast algorithms for large-integer arithmetic. The second edition incorporates important developments such as the AKS primality test, new computational evidence concerning the Riemann hypothesis, faster GCD techniques, and updated computational records. It also discusses broader developments in prime-number theory, including arbitrarily long arithmetic progressions of primes.
One of the book's strengths is that computation is not presented as a substitute for mathematics but as another way of investigating mathematical structure. Algorithms are accompanied by explanations of why they work, while exercises range from conventional problems to questions suitable for further investigation and even research projects. Later chapters broaden the picture through elliptic-curve arithmetic, the distribution and ubiquity of primes, and efficient large-integer computation. The result is particularly suitable for advanced undergraduates, graduate students, researchers, programmers interested in computational number theory, and mathematicians interested in the connection between primes, algorithms and cryptography.
Key takeaways
- Theory meets computation: the central aim is to connect traditional prime-number theory with algorithms and computer experimentation.
- Primality and factorization are central: the book explains both how to recognize/prove primes and how sophisticated algorithms attack the factorization of large integers.
- Algorithmically rich: more than 100 algorithms, together with pseudocode, exercises and research-oriented problems, make the book useful for actual experimentation.
- Excellent for computational number theory: it is more advanced than a general introduction to prime numbers, but especially rewarding for readers who want to understand how modern mathematics turns theoretical ideas about primes into practical algorithms.
Springer — Prime Numbers: A Computational Perspective
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