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 , , 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.