Prime Numbers: A Computational Perspective

  title={Prime Numbers: A Computational Perspective},
  author={Richard E. Crandall and Carl Pomerance},
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… 

Survey on Prime Numbers

Primes are the building blocks of integer universe. Prime numbers plays a major role in number theory. This paper is a detailed survey on prime numbers.This describes different types of primes and

Recent Breakthrough in Primality Testing

Prime numbers are rather old objects in mathematics, however, they did not loose their fascination and importance. Invented by the ancient Greek in an alogy to theindivisible atoms in physics, primes

Primality Testing and Factorization Methods

Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers, as well as in finding prime numbers, which have no such factors.

Factoring integers

The theory of numbers is primarily concerned with the properties of the natural numbers 1,2,3,. . . . The fundamental theorem of arithmetic states that each natural number > 1 can be expressed

A collection of patterns for prime generation

This paper can be used for educational purposes, allowing students to become acquainted with the Mathematics and Algorithmics regarding the search of primes, as well as with the notion and use of patterns.

Smooth numbers: computational number theory and beyond

The analysis of many number theoretic algorithms turns on the role played by integers which have only small prime factors; such integers are known as “smooth numbers”. To be able to determine which


Three major primality proving algorithms are surveyed, which rely on disparate concepts in mathematics: Jacobi sums, elliptic curves, and exponentiation in a polynomial ring.

The Lucas-Pratt primality tree

It is shown that for all constants C ∈ R, the number of modular multiplications necessary to obtain this certificate is greater than Clog p for a set of primes p with relative asymptotic density 1.


  • Aaron Ekstrom
  • Mathematics
    Journal of the Australian Mathematical Society
  • 2012
Abstract In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex

Phillip Mates 1 Problem Summary & Motivation

Primes are used pervasively throughout modern society, and many deterministic algorithms don’t have proven upper bounds on their run time complexities, relying on heuristics.



An Introduction to the Theory of Numbers

This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,

An introduction to the theory of numbers

Divisibility congruence quadratic reciprocity and quadratic forms some functions of number theory some diophantine equations Farey fractions and irrational numbers simple continued fractions primes