Difference between revisions of "Eratosthenes, sieve of"
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| − | A method worked out by Eratosthenes (3rd century B.C.) for eliminating composite numbers from the sequence of natural numbers. The essence of the sieve of Eratosthenes consists in the following. First the number 1 is crossed out. Now 2 is a prime number. Next all numbers divisible by 2 are crossed out. Now 3, the first number not crossed out, is a prime number. Then all the numbers divisible by 3 are crossed out. Now 5, the first number not crossed out, is a prime number. Continuing in this way one can find an arbitrary large segment of the sequence of prime numbers. The sieve of Eratosthenes has been developed into other stronger [[sieve method]]s, such as, for example, the [[Brun theorem|Brun sieve]]. | + | A method worked out by Eratosthenes (3rd century B.C.) for eliminating composite numbers from the sequence of natural numbers. The essence of the sieve of Eratosthenes consists in the following. First the number 1 is crossed out. Now 2 is a prime number. Next all numbers divisible by 2 are crossed out. Now 3, the first number not crossed out, is a prime number. Then all the numbers divisible by 3 are crossed out. Now 5, the first number not crossed out, is a prime number. Continuing in this way one can find an arbitrary large segment of the sequence of prime numbers. |
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| + | In 1808 A.M. Legendre used the ideas of this sieve to derive an estimate for the [[distribution of prime numbers]] of the form $\pi(x) < x/\log\log x$. | ||
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| + | The sieve of Eratosthenes has been developed into other stronger [[sieve method]]s, such as, for example, the [[Brun theorem|Brun sieve]]. | ||
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[[Category:Number theory]] | [[Category:Number theory]] | ||
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Revision as of 18:04, 1 November 2014
A method worked out by Eratosthenes (3rd century B.C.) for eliminating composite numbers from the sequence of natural numbers. The essence of the sieve of Eratosthenes consists in the following. First the number 1 is crossed out. Now 2 is a prime number. Next all numbers divisible by 2 are crossed out. Now 3, the first number not crossed out, is a prime number. Then all the numbers divisible by 3 are crossed out. Now 5, the first number not crossed out, is a prime number. Continuing in this way one can find an arbitrary large segment of the sequence of prime numbers.
In 1808 A.M. Legendre used the ideas of this sieve to derive an estimate for the distribution of prime numbers of the form $\pi(x) < x/\log\log x$.
The sieve of Eratosthenes has been developed into other stronger sieve methods, such as, for example, the Brun sieve.
Comments
For references see Sieve method.
Eratosthenes, sieve of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eratosthenes,_sieve_of&oldid=34158