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.
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.
- P. Erdős; W.H.J. Fuchs (1956). "On a Problem of Additive Number Theory". Journal of the London Mathematical Society 31 (1): 67-73.