In the field of number theory, the Brun sieve (also called Brun's pure sieve) is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Viggo Brun in 1915 and later generalized to the fundamental lemma of sieve theory by others.
Description.
In terms of sieve theory the Brun sieve is of "combinatorial type"; that is, it derives from a careful use of the inclusion–exclusion principle.
Let formula_1 be a finite set of positive integers. 
Let formula_2 be some set of prime numbers. 
For each prime formula_3 in formula_2, let formula_5 denote the set of elements of formula_1 that are divisible by formula_3. 
This notation can be extended to other integers formula_8 that are products of distinct primes in formula_2. In this case, define formula_10 to be the intersection of the sets formula_5 for the prime factors formula_3 of formula_8.
Finally, define formula_14 to be formula_1 itself. 
Let formula_16 be an arbitrary positive real number. 
The object of the sieve is to estimate:
formula_17
where the notation formula_18 denotes the cardinality of a set formula_19, which in this case is just its number of elements. 
Suppose in addition that formula_20 may be estimated by
formula_21
where formula_22 is some multiplicative function, and formula_23 is some error function. 
Let
formula_24
Brun's pure sieve.
This formulation is from Cojocaru &amp; Murty, Theorem 6.1.2. With the notation as above, suppose that 
Then
formula_34
where formula_19 is the cardinal of formula_1, formula_37 is any positive integer and the formula_38 invokes big O notation.
In particular, letting formula_39 denote the maximum element in formula_1, if formula_41 for a suitably small formula_42, then
formula_43
Applications.
The last two results were superseded by Chen's theorem, and the second by Goldbach's weak conjecture (formula_46).