 TOPICS # 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.

Chen's Theorem, Goldbach Conjecture, Mann's Theorem, Prime Number, Prime Partition, Schnirelmann Constant, Schnirelmann Density, Waring's Prime Number Conjecture, Waring's Problem

This entry contributed by Kevin O'Bryant

## Explore with Wolfram|Alpha More things to try:

## 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.

## Referenced on Wolfram|Alpha

Schnirelmann's Theorem

## Cite this as:

O'Bryant, Kevin. "Schnirelmann's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SchnirelmannsTheorem.html