# Difference between revisions of "Erdős–Fuchs theorem"

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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]]. | ||

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* {{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 }} | ||

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## Revision as of 20:22, 29 October 2008

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.

- Donald J. Newman (1998).
*Analytic number theory*, 31-38. ISBN 0-387-98308-2.