Erdős–Fuchs theorem: Difference between revisions
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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 }} | ||
Revision as of 14:22, 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.