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Journal of Integer Sequences


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 p 2(m2 − mn + n2) for some integers m, n. We also show a similar characterization for the sides of a regular tetrahedron in Z 3 : such a tetrahedron exists if and only if the sides are of the form k √ 2, for some k ∈ N. The classification of all the equilateral triangles in 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.

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