A characterization of regular tetrahedra in Z3

JOURNAL OF NUMBER THEORY

卷:129

Article

Ionascu, Eugen J.

关键词: Diophantine equations;    Regular tetrahedra;    Equilateral triangles;    Integers;    Parametrization;    Characterization;   

DOI  :  10.1016/j.jnt.2009.01.003

来源: Elsevier

【 摘 要 】

Text. In this note we characterize all regular tetrahedra whose vertices in R-3 have integer coordinates. The main result is a consequence of the characterization of all equilateral triangles having integer coordinates [R. Chandler, E.J. Ionascu. A characterization of all equilateral triangles in Z(3), Integers 8 (2008), Article A19]. Previous work on this topic begun in [E.J. Ionascu, A parametrization of equilateral triangles having integer coordinates, J. Integer Seq. 10 (2007), Article 07.6.7]. We will use this characterization to point out some corollaries. The number of such tetrahedra whose vertices are in the finite set {0, 1, 2, . . . , n}(3), n is an element of N, is related to the sequence A103158 in the Online Encyclopedia of Integer Sequences [Neil J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at: http://www.research.att.com/-njas/sequences/, 2005]. Video. For a video summary of this paper, please visit http://www.youtube.com/watch?v=LT3aAUUFMFk. (C) Published by Elsevier Inc.

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