Erdős–Fuchs theorem

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.

tatement

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

:$R(n)\; =\; frac\{r(1)+r(2)+cdots+r(n)\}\{n\}.$

The theorem states that

:$R(n)\; =\; C\; +\; Oleft(n^\{-3/4-varepsilon\}\; ight)$

cannot hold unless "C" = 0.

References

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