Erdos–Anning theorem
Encyclopedia
The Erdős–Anning theorem states that an infinite number of points in the plane can have mutual integer
distances only if all the points lie on a straight line. It is named after Paul Erdős
and Norman H. Anning, who proved it in 1945.
An alternative way of stating the theorem is that a non-collinear set of points in the plane with integer distances can only be extended by adding finitely many additional points, before no more points can be added. More specifically, if a set of three or more non-collinear points have integer distances, all at most some number d, then at most 4(d + 1)2 points at integer distances can be added to the set.
A set of points on the integer grid with integer distances, to which no more can be added, forms an Erdős–Diophantine graph.
points with mutual distances D(AB), D(BC) and D(AC) not exceeding d, and X a point at integer distance from A, B and C. From the triangle inequality
it follows that |D(AX) − D(BX)| is a non-negative integer not exceeding d. So X is on one of the d + 1 hyperbola
s through A and B. Similarly, X is situated on one of the d + 1 hyperbolas through B and C. As two distinct hyperbolas can not intersect in more than four points, there are at most 4(d + 1)2 points X.
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
distances only if all the points lie on a straight line. It 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 Norman H. Anning, who proved it in 1945.
An alternative way of stating the theorem is that a non-collinear set of points in the plane with integer distances can only be extended by adding finitely many additional points, before no more points can be added. More specifically, if a set of three or more non-collinear points have integer distances, all at most some number d, then at most 4(d + 1)2 points at integer distances can be added to the set.
A set of points on the integer grid with integer distances, to which no more can be added, forms an Erdős–Diophantine graph.
Proof
Let A, B and C be non-collinearLine (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
points with mutual distances D(AB), D(BC) and D(AC) not exceeding d, and X a point at integer distance from A, B and C. From the triangle inequality
Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....
it follows that |D(AX) − D(BX)| is a non-negative integer not exceeding d. So X is on one of the d + 1 hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
s through A and B. Similarly, X is situated on one of the d + 1 hyperbolas through B and C. As two distinct hyperbolas can not intersect in more than four points, there are at most 4(d + 1)2 points X.