Erdős–Anning theorem

Erdős–Anning theorem

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. A set of points to which no more can be added, with all points on the integer grid, forms an Erdős-Diophantine graph.

Proof

Let "A", "B" and "C" be non-collinear 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" hyperbolas 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.

References

*citation
first1 = Norman H.
last1 = Anning
first2 = Paul
last2 = Erdős
authorlink2 = Paul Erdős
title = Integral distances
journal = Bulletin of the American Mathematical Society
volume = 51
pages = 598–600
year = 1945
url = http://www.ams.org/bull/1945-51-08/S0002-9904-1945-08407-9/

External links

*mathworld | urlname = Erdos-AnningTheorem | title = Erdos-Anning Theorem


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Erdős-Diophantine graph — In Diophantine geometry, an Erdős Diophantine graph, named after Paul Erdős and Diophantus of Alexandria, is a complete graph with vertices located on the integer square grid scriptstylemathbb{Z}^2 such that all mutual distances between the… …   Wikipedia

  • List of things named after Paul Erdős — The following were named after Paul Erdős:* Erdős number * Erdős cardinal * Erdős conjecture a list of numerous conjectures named after Erdős ** Erdős conjecture on arithmetic progressions ** Cameron–Erdős conjecture ** Erdős–Burr conjecture **… …   Wikipedia

  • List of mathematics articles (E) — NOTOC E E₇ E (mathematical constant) E function E₈ lattice E₈ manifold E∞ operad E7½ E8 investigation tool Earley parser Early stopping Earnshaw s theorem Earth mover s distance East Journal on Approximations Eastern Arabic numerals Easton s… …   Wikipedia

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”