Mann's theorem is a theorem widely circulated as the " conjecture" that was subsequently proven by
Mann (1942). It states that if and are sets of integers each containing 0, then

Mann's theorem is optimal in the sense that satisfies .

Mann's theorem implies Schnirelmann's theorem as follows. Let ,
then Mann's theorem proves that , so as more and more copies of
the primes are included, the Schnirelmann density
increases at least linearly, and so reaches 1 with at most copies of the primes. Since the only sets
with Schnirelmann density 1 are the sets
containing all positive integers, Schnirelmann's
theorem follows.

Garrison, B. K. "A Nontransformation Proof of Mann's Density Theorem." J. reine angew. Math.245, 41-46, 1970.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.Mann,
H. B. "A Proof of the Fundamental Theorem on the Density of Sets of Positive
Integers." Ann. Math.43, 523-527, 1942.