Erathostenes Of Cyrene

Eratosthenes of Cyrene

Eratosthenes (276-194 B.C.) was the third librarian of the famous library in Alexandria and an outstanding scholar all around. He is remembered by his measurement of thecircumference of the Earth, estimates of the distances to the sun and the moon, and, in mathematics, for the invention of an algorithm for collecting prime numbers. The algorithm is known as the Sieveof Eratosthenes. This sieve was described and attributed to Eratosthenes in the Introduction to Arithmetic by Nicomachus

Concept

In mathematics, the sieve of Eratosthenes , one of a number ofprime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite the multiples of each prime, starting with themultiples of 2.
The multiples of a given prime are generated starting from that prime, as a sequence of numbers with the same difference, equal to that prime, between consecutive numbers. This is thesieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.
The sieve of Eratosthenes is one of the most efficient ways to find all ofthe smaller primes (below 10 million or so).

Construction

A prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself.
* To find all the primenumbers less than or equal to a given integer n by Eratosthenes' method:
1.- Create a list of consecutive integers from 2 to n: (2, 3, 4, ..., n).
2.- Initially, let p equal 2, the first primenumber.
3.- Starting from p, count up in increments of p and mark each of these numbers greater than p itself in the list. These numbers will be 2p, 3p, 4p, etc.; note that some of them may have alreadybeen marked. In other words, remove all multiples of p from p to n.
4.- Find the first number greater than p in the list that is not marked. If there was no such number, stop. Otherwise, let p now...