Erdős–Fuchs theorem

From Citizendium
Revision as of 20:22, 29 October 2008 by Richard Pinch (Talk | contribs) (remove WPmarkup; subpages)

Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
This editable Main Article is under development and subject to a disclaimer.

In mathematics, in the area of combinatorial number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function.

The theorem is named after Paul Erdős and Wolfgang Heinrich Johannes Fuchs.


Let A be a subset of the natural numbers and r(n) denote the number of ways that a natural number n can be expressed as the sum of two elements of A (taking order into account). We consider the average

The theorem states that

cannot hold unless C=0.


  • P. Erdős; W.H.J. Fuchs (1956). "On a Problem of Additive Number Theory". Journal of the London Mathematical Society 31 (1): 67-73.