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from math import isqrt def sieve_of_Sundaram (n): """The sieve of Sundaram is a simple deterministic algorithm for finding all the prime numbers up to a specified integer.""" if n < 3: if n < 2: return 0 else: return 1 k = (n-3) // 2 + 1 integers_list = [True for i in range (k)] ops = 0 for i in range ((isqrt (n)-3) // 2 + 1): # if integers ...
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers and a target-sum T {\displaystyle T} , and the question is to decide whether any subset of the integers sum to precisely T {\displaystyle T} . [ 1 ]
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.
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.
Let {q 1, q 2, …} be successive prime numbers in the interval (B 1, B 2] and d n = q n − q n−1 the difference between consecutive prime numbers. Since typically B 1 > 2, d n are even numbers. The distribution of prime numbers is such that the d n will all be relatively small. It is suggested that d n ≤ ln 2 B 2.
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 ...
Given an integer n, choose some integer a coprime to n and calculate a n − 1 modulo n. If the result is different from 1, then n is composite. If it is 1, then n may be prime. If a n −1 (modulo n) is 1 but n is not prime, then n is called a pseudoprime to base a. In practice, if a n −1 (modulo n) is 1, then n is usually prime.
Chen's theorem, which shows that there are infinitely many primes p such that p + 2 is either a prime or a semiprime (the product of two primes); a closely related theorem of Chen Jingrun asserts that every sufficiently large even number is the sum of a prime and another number which is either a prime or a semiprime.