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Regular tetrahedra whose vertices have integer coordinates

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Author(s): E. J. Ionascu

Journal: Acta Mathematica Universitatis Comenianae
ISSN 0862-9544

Volume: LXXX;
Issue: 2;
Start page: 161;
Date: 2011;
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Keywords: Diophantine equations | integers | infinite graph | fractal

ABSTRACT
In this paper we introduce theoretical arguments for constructing a procedure that allows one to find the number of all regular tetrahedra that have coordinates in the set {0,1, . . . , n}. The terms of this sequence are twice the values of the sequence A103158 in the Online Encyclopedia of Integer Sequences.These results lead to the consideration of an infinite graph having a fractal nature which is tightly connected to the set of orthogonal 3-by-3 matrices with rational coefficients. The vertices of this graph are the primitive integer solutions of the Diophantine equation a2 + b2 + c2 = 3d2. Our aim here is to laid down the basis of finding good estimates, if not exact formulae, for the sequence A103158.
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