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Mann's Theorem


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

 sigma(A direct sum B)>=min{1,sigma(A)+sigma(B)}.

Here, A direct sum B denotes the direct sum, i.e., A direct sum B={a+b:a in A,b in B}, and sigma is the Schnirelmann density.

Mann's theorem is optimal in the sense that A=B={0,1,11,12,13,...} satisfies sigma(A direct sum B)=sigma(A)+sigma(B).

Mann's theorem implies Schnirelmann's theorem as follows. Let P={0,1} union {p:p prime}, then Mann's theorem proves that sigma(P+P+P+P)>2sigma(P+P), 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 2·1/(sigma(P+P)) copies of the primes. Since the only sets with Schnirelmann density 1 are the sets containing all positive integers, Schnirelmann's theorem follows.


See also

Schnirelmann Density, Schnirelmann's Theorem

This entry contributed by Kevin O'Bryant

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References

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.

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Mann's Theorem

Cite this as:

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

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