Journal of Integer Sequences, Vol. 10 (2007), Article 07.6.7 |

Department of Mathematics

Columbus State University

Columbus, GA 31907

USA

**Abstract:**

We study the existence of equilateral triangles of given side
lengths and with integer coordinates in dimension three. We show
that such a triangle exists if and only if their side lengths are of
the form
for some integers . We also
show a similar characterization for the sides of a regular
tetrahedron in : such a tetrahedron exists if and only
if the sides are of the form , for some .
The classification of all the equilateral triangles in
contained in a given plane is studied and the beginning analysis for
small side lengths is included. A more general parametrization is
proven under special assumptions. Some related questions about the
exceptional situation are formulated in the end.

(Concerned with sequence A102698 .)

Received August 28 2006;
revised version received June 16 2007.
Published in *Journal of Integer Sequences*, June 18 2007.

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