Difference between revisions of "Erdős–Fuchs theorem"

From Citizendium
Jump to: navigation, search
(New article, my own wording from Wikipedia)
 
m (Erdos-Fuchs theorem moved to Erdős–Fuchs theorem: correct spelling (with accent))
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
{{subpages}}
 
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]].
 
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]].
  
Line 18: Line 19:
 
* {{cite journal | title=On a Problem of Additive Number Theory | author=P. Erdős | authorlink=Paul Erdős | coauthors=W.H.J. Fuchs | journal=Journal of the London Mathematical Society | year=1956 | volume=31 | issue=1 | pages=67-73 }}
 
* {{cite journal | title=On a Problem of Additive Number Theory | author=P. Erdős | authorlink=Paul Erdős | coauthors=W.H.J. Fuchs | journal=Journal of the London Mathematical Society | year=1956 | volume=31 | issue=1 | pages=67-73 }}
 
* {{cite book | author=Donald J. Newman | title=Analytic number theory | series=[[Graduate Texts in Mathematics|GTM]] | volume=177 | year=1998 | isbn=0-387-98308-2 | pages=31-38 }}
 
* {{cite book | author=Donald J. Newman | title=Analytic number theory | series=[[Graduate Texts in Mathematics|GTM]] | volume=177 | year=1998 | isbn=0-387-98308-2 | pages=31-38 }}
 
[[Category:Combinatorics]]
 
[[Category:Number theory]]
 
 
{{numtheory-stub}}
 

Latest revision as of 15:50, 18 June 2009

This article is a stub and thus not approved.
Main Article
Discussion
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.

Statement

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.


References

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