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 a^{2} + b^{2}
+ c^{2} = 3d^{2}. 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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