Schnirelmann's Theorem
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 
, ![A direct sum B=<span style=]()
{a+b:a in A,b in B}" src="https://mathworld.wolfram.com/images/equations/SchnirelmannsTheorem/Inline8.gif" style="height:15px; width:169px" />, and 
is the Schnirelmann density. Let ![P=<span style=]()
{0,1,2,3,5,...}" src="https://mathworld.wolfram.com/images/equations/SchnirelmannsTheorem/Inline10.gif" style="height:15px; width:125px" /> 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 ![sigma(Q+Q+...+Q)>=min<span style=]()
{1,ksigma(Q)}" src="https://mathworld.wolfram.com/images/equations/SchnirelmannsTheorem/Inline16.gif" style="height:15px; width:216px" />. Thus, if 
, the sum of 
copies of 
has Schnirelmann density 1, and so contains all positive integers.
REFERENCES:
Khinchin, A. Y. "The Landau-Schnirelmann Hypothesis and Mann's Theorem." Ch. 2 in Three Pearls of Number Theory. New York: Dover, pp. 18-36, 1998.
