# Erdős–Fuchs theorem

From Citizendium, the Citizens' Compendium

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.