View source for Brun sieve

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017700/b0177001.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017700/b0177002.png" /> be the amount of these numbers. It can be shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017700/b0177003.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017700/b0177004.png" /> can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017700/b0177005.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017700/b0177006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017700/b0177007.png" /> contain at most 9 prime factors.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.O. Gel'fond,   Yu.V. Linnik,   "Elementary methods in the analytic theory of numbers" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Trost,   "Primzahlen" , Birkhäuser  (1953)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Halberstam,   H.-E. Richert,   "Sieve methods" , Acad. Press  (1974)</TD></TR></table>