In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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

Linear function

In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....

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The theorem is named after Paul Erdős

Paul Erdos

Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...

 and Wolfgang Heinrich Johannes Fuchs

Wolfgang Heinrich Johannes Fuchs

Wolfgang Heinrich Johannes Fuchs was a mathematician specializing in complex analysis. His main area of research was Nevanlinna theory....

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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.