Schnirelmann's Theorem

There exists a positive integer s such that every sufficiently large integer is the sum of at most s primes. It follows that there exists a positive integer s_0>=s such that every integer >1 is a sum of at most s_0 primes. The smallest proven value of s_0 is known as the Schnirelmann constant.

Schnirelmann's theorem can be proved using Mann's theorem, although Schnirelmann used the weaker inequality

 sigma(A direct sum B)>=sigma(A)+sigma(B)-sigma(A)sigma(B),

where 0 in A intersection B, A direct sum B={a+b:a in A,b in B}, and sigma is the Schnirelmann density. Let P={0,1,2,3,5,...} be the set of primes, together with 0 and 1, and let Q=P direct sum P. Using a sophisticated version of the inclusion-exclusion principle, Schnirelmann showed that although sigma(P)=0, sigma(Q)>0. By repeated applications of Mann's theorem, the sum of k copies of Q satisfies sigma(Q+Q+...+Q)>=min{1,ksigma(Q)}. Thus, if k>1/sigma(Q), the sum of k copies of Q has Schnirelmann density 1, and so contains all positive integers.

See also

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

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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.

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Schnirelmann's Theorem

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O'Bryant, Kevin. "Schnirelmann's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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