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.

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

R(n) = (r(1)+r(2)+\cdots+r(n) ) / n .

The theorem states that

R(n) = C + O\left(n^{-3/4-\epsilon}\right)

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.
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