5204 Explicit machine computation and programs (not the theory of computation or programming)
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Maximal integral point sets over Z^2
(2008)

Sascha Kurz
Andrey Radoslavov Antonov
 Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P={p(1),...,p(n)} in Z^2 a maximal integral point set over Z^2 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.

There are integral heptagons, no three points on a line, no four on a circle
(2007)

Tobias Kreisel
Sasch Kurz
 We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul ErdÃ¶s.