Vol. LXXX, 2 (2011)
p. 161 - 170

Regular tetrahedra whose vertices have integer coordinates

E. J. Ionascu

Received: December 15, 2009;   Accepted: July 1, 2011

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.

Keywords:  Diophantine equations; integers; infinite graph; fractal.  

AMS Subject classification: Primary:  11D09 

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Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic  

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