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\begin{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 $\sqrt{2(m^2-mn+n^2)}$ for some integers $m,n$. We also
show a similar characterization for the sides of a regular
tetrahedron in $\mathbb Z^3$: such a tetrahedron exists if and only
if the sides are of the form $k\sqrt{2}$, for some $k\in\mathbb N$.
The classification of all the equilateral triangles in $\mathbb Z^3$
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.
\end{abstract}

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