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Sieve of Eratosthenes: algorithm steps for primes below 121 (including optimization of starting from prime's square). In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
The sieve methods discussed in this article are not closely related to the integer factorization sieve methods such as the quadratic sieve and the general number field sieve. Those factorization methods use the idea of the sieve of Eratosthenes to determine efficiently which members of a list of numbers can be completely factored into small primes.
A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin [1] (2003), sieve of Pritchard (1979), and various wheel sieves [2] are most common.
Eratosthenes' sieve in Javascript Archived 2001-03-01 at the Wayback Machine; About Eratosthenes' methods, including a Java applet; How the Greeks estimated the distances to the Moon and Sun; Measuring the Earth with Eratosthenes' method; List of ancient Greek mathematicians and contemporaries of Eratosthenes
Sieve method, or the method of sieves, can mean: in mathematics and computer science, the sieve of Eratosthenes, a simple method for finding prime numbers in number theory, any of a variety of methods studied in sieve theory; in combinatorics, the set of methods dealt with in sieve theory or more specifically, the inclusion–exclusion principle
In mathematics, the Legendre sieve, named after Adrien-Marie Legendre, is the simplest method in modern sieve theory.It applies the concept of the Sieve of Eratosthenes to find upper or lower bounds on the number of primes within a given set of integers.
Observations analogous to the preceding can be applied recursively, giving the Sieve of Eratosthenes. One way to speed up these methods (and all the others mentioned below) is to pre-compute and store a list of all primes up to a certain bound, such as all primes up to 200.
Sieve of Eratosthenes; Probabilistic algorithm; Fermat primality test. Pseudoprime; Carmichael number; Euler pseudoprime; Euler–Jacobi pseudoprime; Fibonacci pseudoprime; Probable prime; Baillie–PSW primality test; Miller–Rabin primality test; Lucas–Lehmer primality test; Lucas–Lehmer test for Mersenne numbers; AKS primality test