Brun sieve

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)