2006/08/02 by Eugen J. Ionascu, Ionascu, Eugen J.
Mathematics · #11D09 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D09
paper · pdf · doi:10.48550/arxiv.math/0608068
3 fugures, 17 pages, submitted to Integers
arxiv created 2006/08/02 · arxiv updated 2009/12/01
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 √(2(m2-mn+n2)) for some integers m,n. We also show a similar characterization for the sides of a regular tetrahedron in \Z3: 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 \Z3 contained in a given plane is studied and the beginning analysis is presented. A more general parametrization is proven under a special assumption. Some related questions are stated in the end.