proof of Erdös-Anning Theorem
Let A,B and C be three non-collinear points. For an additional point P consider the triangle ABP. By using the triangle inequality for the sides PB and PA we find -|AB|≤|PB|-|PA|≤|AB|. Likewise, for triangle BCP we get -|BC|≤|PB|-|PC|≤|BC|. Geometrically, this means the point P lies on two hyperbola
with A and B or B and C respectively as foci. Since all the lengths involved here are by assumption
integer there are only 2|AB|+1 possibilities for |PB|-|PA| and 2|BC|+1 possibilities for |PB|-|PC|. These hyperbola are distinct since they don’t have the same major axis. So for each pair of hyperbola we can have at most 4 points of intersection
and there can be no more than 4(2|AB|+1)(2|BC|+1) points satisfying the conditions.
Title | proof of Erdös-Anning Theorem |
---|---|
Canonical name | ProofOfErdosAnningTheorem |
Date of creation | 2013-03-22 13:19:11 |
Last modified on | 2013-03-22 13:19:11 |
Owner | lieven (1075) |
Last modified by | lieven (1075) |
Numerical id | 4 |
Author | lieven (1075) |
Entry type | Proof |
Classification | msc 51-00 |