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Sieve of Pritchard: algorithm steps for primes up to 150. In mathematics, the sieve of Pritchard is an algorithm for finding all prime numbers up to a specified bound. Like the ancient sieve of Eratosthenes, it has a simple conceptual basis in number theory. [1] It is especially suited to quick hand computation for small bounds.
A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially, let p equal 2, the smallest prime number.
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.
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit X.
The following is pseudocode which combines Atkin's algorithms 3.1, 3.2, and 3.3 [1] by using a combined set s of all the numbers modulo 60 excluding those which are multiples of the prime numbers 2, 3, and 5, as per the algorithms, for a straightforward version of the algorithm that supports optional bit-packing of the wheel; although not specifically mentioned in the referenced paper, this ...
REM Eratosthenes Sieve Prime Number Program in BASIC 1 SIZE = 8190 2 DIM FLAGS (8191) 3 PRINT "Only 1 iteration" 5 COUNT = 0 6 FOR I = 0 TO SIZE 7 FLAGS (I) = 1 8 NEXT I 9 FOR I = 0 TO SIZE 10 IF FLAGS (I) = 0 THEN 18 11 PRIME = I + I + 3 12 K = I + PRIME 13 IF K > SIZE THEN 17 14 FLAGS (K) = 0 15 K = K + PRIME 16 GOTO 13 17 COUNT = COUNT + 1 ...
The sieve starts with a list of the integers from 1 to n. From this list, all numbers of the form i + j + 2ij are removed, where i and j are positive integers such that 1 ≤ i ≤ j and i + j + 2ij ≤ n. The remaining numbers are doubled and incremented by one, giving a list of the odd prime numbers (that is, all primes except 2) below 2n + 2.
In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general algorithm, it is conceptually simpler. It serves as a helpful first step in understanding how the general number field sieve works.