Difference between revisions of "Brun sieve"
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| − | A [[Sieve method|sieve method]] in elementary number theory, proposed by V. Brun [[#References|[1]]]; it is an extension of the sieve of Eratosthenes (cf. [[Eratosthenes, sieve of|Eratosthenes, sieve of]]). The method of Brun's sieve may be described as follows. Out of a sequence of natural numbers | + | {{TEX|done}} |
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| + | A [[Sieve method|sieve method]] in elementary number theory, proposed by V. Brun [[#References|[1]]]; it is an extension of the sieve of Eratosthenes (cf. [[Eratosthenes, sieve of|Eratosthenes, sieve of]]). The method of Brun's sieve may be described as follows. Out of a sequence of natural numbers $a_n\leq x$ the numbers with small prime divisors are eliminated ( "sieved out" ) leaving behind prime and almost-prime numbers (cf. [[Almost-prime number|Almost-prime number]]) with only large prime divisors. Let $P(x)$ be the amount of these numbers. It can be shown that $P(x)$ is included between two sums with a relatively-small number of summands, which may be estimated from above and from below. Thus, it is possible to evaluate from above the number of [[Twins|twins]] in a given interval. Brun's sieve is employed in additive number theory. Brun used his sieve to prove that all large even numbers $N$ can be represented in the form $N=P_1+P_2$, where $P_1$ and $P_2$ contain at most 9 prime factors. | ||
====References==== | ====References==== | ||
Revision as of 07:18, 11 December 2012
A sieve method in elementary number theory, proposed by V. Brun [1]; it is an extension of the sieve of Eratosthenes (cf. Eratosthenes, sieve of). The method of Brun's sieve may be described as follows. Out of a sequence of natural numbers $a_n\leq x$ the numbers with small prime divisors are eliminated ( "sieved out" ) leaving behind prime and almost-prime numbers (cf. Almost-prime number) with only large prime divisors. Let $P(x)$ be the amount of these numbers. It can be shown that $P(x)$ is included between two sums with a relatively-small number of summands, which may be estimated from above and from below. Thus, it is possible to evaluate from above the number of twins in a given interval. Brun's sieve is employed in additive number theory. Brun used his sieve to prove that all large even numbers $N$ can be represented in the form $N=P_1+P_2$, where $P_1$ and $P_2$ contain at most 9 prime factors.
References
| [1] | V. Brun, "Le crible d'Eratosthène et le théorème de Goldbach" C.R. Acad. Sci. Paris Sér. I Math. , 168 : 11 (1919) pp. 544–546 |
| [2] | A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian) |
| [3] | E. Trost, "Primzahlen" , Birkhäuser (1953) |
Comments
References
| [a1] | H. Halberstam, H.-E. Richert, "Sieve methods" , Acad. Press (1974) |
How to Cite This Entry:
Brun sieve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brun_sieve&oldid=29166
Brun sieve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brun_sieve&oldid=29166
This article was adapted from an original article by N.I. Klimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article