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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.
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.
Meissel already found that for k ≥ 3, P k (x, a) = 0 if a = π(x 1/3).He used the resulting equation for calculations of π(x) for big values of x. [1]Meissel calculated π(x) for values of x up to 10 9, but he narrowly missed the correct result for the biggest value of 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 ...
The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. [1] [2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π 0 (x), which is equal to π(x) − 1 ⁄ 2 if x is exactly a prime number, and equal to π(x) otherwise.
trial divisions, where () denotes the prime-counting function, the number of primes less than x. This does not take into account the overhead of primality testing to obtain the prime numbers as candidate factors. A useful table need not be large: P(3512) = 32749, the last prime that fits into a sixteen-bit signed integer and P(6542) = 65521 for ...
⎕CR 'PrimeNumbers' ⍝ Show APL user-function PrimeNumbers Primes ← PrimeNumbers N ⍝ Function takes one right arg N (e.g., show prime numbers for 1 ... int N) Primes ← (2 =+ ⌿ 0 = (⍳ N) ∘. |⍳ N) / ⍳ N ⍝ The Ken Iverson one-liner PrimeNumbers 100 ⍝ Show all prime numbers from 1 to 100 2 3 5 7 11 13 17 19 23 29 31 37 41 43 ...