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Sieve of Eratosthenes
Ancient algorithm for generating prime numbers / From Wikipedia, the free encyclopedia
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In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
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It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime.[1] This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.[2] Once all the multiples of each discovered prime have been marked as composites, the remaining unmarked numbers are primes.
The earliest known reference to the sieve (Ancient Greek: κόσκινον Ἐρατοσθένους, kóskinon Eratosthénous) is in Nicomachus of Gerasa's Introduction to Arithmetic,[3] an early 2nd cent. CE book which attributes it to Eratosthenes of Cyrene, a 3rd cent. BCE Greek mathematician, though describing the sieving by odd numbers instead of by primes.[4]
One of a number of prime number sieves, it is one of the most efficient ways to find all of the smaller primes. It may be used to find primes in arithmetic progressions.[5]