Erdős–Anning theorem

The Erdős–Anning theorem states that, whenever an infinite number of points in the plane all have integer distances, the points lie on a straight line. The same result holds in higher dimensional Euclidean spaces.

The theorem cannot be strengthened to give a finite bound on the number of points: there exist arbitrarily large finite sets of points that are not on a line and have integer distances.

The theorem is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1945. Erdős later supplied a simpler proof, which can also be used to check whether a point set forms an Erdős–Diophantine graph, an inextensible system of integer points with integer distances. The Erdős–Anning theorem inspired the Erdős–Ulam problem on the existence of dense point sets with rational distances. Some later publications credit the same result to Gustave Choquet and Germain Kreweras. It was independently rediscovered, with an application to the design of phased arrays, by B. Gleijeses in 1981.

Rationality versus integrality

Although there can be no infinite non-collinear set of points with integer distances, there are infinite non-collinear sets of points whose distances are rational numbers. For instance, the subset of points on a unit circle obtained as the even multiples of one of the acute angles of an integer-sided right triangle (such as the triangle with side lengths 3, 4, and 5) has this property.