There exists a positive integer 
such that every sufficiently
large integer is the sum of at most 
primes. It follows that there
exists a positive integer 
such that every integer 
is a sum of at most 
primes. The smallest proven
value of 
is known as the Schnirelmann constant.
Schnirelmann's theorem can be proved using Mann's theorem, although Schnirelmann used the weaker inequality
 |
where 
,
,
and 
is the Schnirelmann density. Let 
be the set of primes, together with 0 and
1, and let 
.
Using a sophisticated version of the inclusion-exclusion
principle, Schnirelmann showed that although 
, 
. By repeated applications of Mann's
theorem, the sum of 
copies of 
satisfies 
. Thus, if 
, the sum of 
copies of 
has Schnirelmann density
1, and so contains all positive integers.
