2006
DOI: 10.1016/j.jnt.2006.02.009
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Rational and Heron tetrahedra

Abstract: Buchholz [R.H. Buchholz, Perfect pyramids, Bull. Austral. Math. Soc. 45 (1991) 353-368] began a systematic search for tetrahedra having integer edges and volume by restricting his attention to those with two or three different edge lengths. Of the fifteen configurations identified for such tetrahedra, Buchholz leaves six unsolved. In this paper we examine these remaining cases for integer volume, completely solving all but one of them. Buchholz also considered Heron tetrahedra, which are tetrahedra with integ… Show more

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Cited by 7 publications
(13 citation statements)
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“…Due to the formula for A∆(a, b, c) for an integer sided triangle with characteristic 1, the area is rational and may in principle be non-integral. Nevertheless one may consider the cases of the side lengths modulo 8 (see [6]) and conclude that such triangles have to be integral. We summarize these findings in: Corollary 2.6.…”
Section: Basic Results and Notationmentioning
confidence: 99%
“…Due to the formula for A∆(a, b, c) for an integer sided triangle with characteristic 1, the area is rational and may in principle be non-integral. Nevertheless one may consider the cases of the side lengths modulo 8 (see [6]) and conclude that such triangles have to be integral. We summarize these findings in: Corollary 2.6.…”
Section: Basic Results and Notationmentioning
confidence: 99%
“…It is a conjecture of [8] that no non-degenerated solution exists in this case. For the cases 4(iv), 4(vii), 5(ii), and 6(i) sporadic solutions are known.…”
Section: Perfect Pyramidsmentioning
confidence: 96%
“…Here it is possible that the edge lengths of a tetrahedron are integral and that the volume is genuinely rational. If the edge lengths are integral and the face areas and the volume are rational, then all values must be integral, see [8]. In [3] it was also shown that a perfect pyramid with at most two different edge lengths cannot exist.…”
Section: Perfect Pyramidsmentioning
confidence: 99%
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