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.

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.