On the generation of Heronian triangles

We describe several algorithms for the generation of integer Heronian triangles with diameter at most $n$. Two of them have running time $\mathcal{O}\left(n^{2+\varepsilon}\right)$. We enumerate all integer Heronian triangles for $n\le 600000$ and apply the complete list on some related problems.Comment: 10 pages, 2 figures, 2 tables. http://sci-gems.math.bas.bg/jspui/handle/10525/38

Constructing 77-clusters

A set of $n$-lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and all coordinates are integral is called an $n$-cluster (in $\mathbb{R}^2$). We determine the smallest existent $7$-cluster with respect to its diameter. Additionally we provide a toolbox of algorithms which allowed us to computationally locate over 1000 different $7$-clusters, some of them having huge integer edge lengths. On the way, we exhaustively determined all Heronian triangles with largest edge length up to $6\cdot 10^6$.Comment: 18 pages, 2 figures, 2 table

Heron triangles with two fixed sides

In this paper, we study the function $H(a,b)$, which associates to every pair of positive integers $a$ and $b$ the number of positive integers $c$ such that the triangle of sides $a,b$ and $c$ is Heron, i.e., has integral area. In particular, we prove that $H(p,q)\le 5$ if $p$ and $q$ are primes, and that $H(a,b)=0$ for a random choice of positive integers $a$ and $b$.Comment: 20 pges, no figures, submitted in 200

On the Generation of Heronian Triangles

We describe several algorithms for the generation of integer Heronian triangles with diameter at most n. Two of them have running time O(n^(2+ε)). We enumerate all integer Heronian triangles for n ≤ 600000 and apply the complete list on some related problems

Heron triangles with two fixed sides

In this paper, we study the function H(a, b), which associates to every pair of positive integers a and b the number of positive integers c such that the triangle of sides a, b and c is Heron, i.e., has integral area. In particular, we prove that H(p, q) ≤ 5 if p and q are primes, and that H(a, b) = 0 for a random choice of positive integers a and b

Maximal integral point sets over Z^2

Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P = {p1,..., pn} ⊂ Z² a maximal integral point set over Z 2 if all pairwise distances are integral and every additional point pn+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