Maximal integral point sets over ℤ²

2008/04/08 by Andrey Antonov, Andrey Radoslavov Antonov, Antonov, Andrey Radoslavov +2 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Analytic Number Theory Research #Mathematical Approximation and Integration #math.CO #math.NT #msc:05D99 #msc:11D99 #msc:52-04 #msc:52C10 #msc:52C45

paper · pdf · doi:10.48550/arxiv.0804.1280

23 pages, 10 figures, 4 tables

arxiv created 2008/04/08 · arxiv updated 2009/12/01

Abstract

Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P=\p₁,...,p_n\⊂ℤ² a maximal integral point set over ℤ² if all pairwise distances are integral and every additional point p_n+1 destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle.

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