OFFSET
1,1
COMMENTS
For every primitive trirectangular tetrahedron (0, a, b, c) with coprime integer sides, (b*c)^2 + (a*b)^2 + (c*a)^2 is divisible by 144.
The square of the basis is related by De Gua's theorem on the square of the main diagonal of a (different, not necessarily primitive) Euler brick (a*b/12=A031173(k), a*c/12=A031174(k), b*c/12=A031175(k)) also having integer sides and integer face diagonals including a trirectangular tetrahedron (0, a*b/12, a*c/12, b*c/12), such as a(1) = 6434041 = A023185(8) = A031173(8)^2 + A031174(8)^2 + A031175(8)^2.
By this process a cycle of primitive trirectangular tetrahedrons is defined, such as with indices k: (1 8), (2 6), (3 5), (4 7), (9 19), ...
LINKS
Eric Weisstein's World of Mathematics, Trirectangular Tetrahedron
Wikipedia, De Gua's theorem
CROSSREFS
KEYWORD
nonn
AUTHOR
Ralf Steiner, Dec 28 2017
STATUS
approved