Geodesic
A parametrization of equilateral triangles having integer coordinates
Journal of integer sequences, Tome 10 (2007) no. 6
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.
Classification : 11A67, 11D09, 11D04, 11R99, 11B99, 51N20
Keywords: Diophantine equation, equilateral triangle, quadratic reciprocity 
@article{JIS_2007__10_6_a3,
     author = {Ionascu,  Eugen J.},
     title = {A parametrization of equilateral triangles having integer coordinates},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {6},
     zbl = {1140.11322},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_6_a3/}
}
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Ionascu,  Eugen J. A parametrization of equilateral triangles having integer coordinates. Journal of integer sequences, Tome 10 (2007) no. 6. http://geodesic.mathdoc.fr/item/JIS_2007__10_6_a3/