OFFSET
1,1
COMMENTS
Primitive means that gcd(a,b,c) = 1.
The trirectangular tetrahedron (0, a=a(n), b=A031174(n), c=A031175(n)) has three right triangles with area divisible by 6 = 2*3 each and a volume divisible by 15840 = 2^5*3^2*5*11. The biquadratic term b^2*c^2 + a^2*(b^2 + c^2) is divisible by 144 = 2^4*3^2. Also gcd(b + c, c + a, a + b) = 1. - Ralf Steiner, Nov 22 2017
There are some longest edges a which occur multiple times, such as a(31) = a(32) = 23760. - Ralf Steiner, Jan 07 2018
A trirectangular tetrahedron is never a perfect body (in the sense of Wyss) because it always has an irrational area of the base (a,b,c) whose value is half of the length of the space-diagonal of the related cuboid (b*c, c*a, a*b). The trirectangular bipyramid (6 faces, 9 edges, 5 vertices) built from these trirectangular tetrahedrons and the related left-handed ones connected on their bases have rational numbers for volume, face areas and edge lengths, but again an irrational value for the length of the space-diagonal which is a rational part of the length of the space-diagonal of the related cuboid (b*c, c*a, a*b). - Ralf Steiner, Jan 14 2018
REFERENCES
Calculated by F. Helenius (fredh(AT)ix.netcom.com).
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..3556
R. R. Gallyamov, I. R. Kadyrov, D. D. Kashelevskiy, A fast modulo primes algorithm for searching perfect cuboids and its implementation, arXiv preprint arXiv:1601.00636 [math.NT], 2016.
A. A. Masharov and R. A. Sharipov, A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture, arXiv preprint arXiv:1504.07161 [math.NT], 2015.
Dustin Moody, Mohammad Sadek, Arman Shamsi Zargar, Families of elliptic curves of rank >= 5 over Q(t), Rocky Mountain Journal of Mathematics (2019) Vol. 49, No. 7, 2253-2266.
Giovanni Resta, The 3556 primitive bricks with c < b < a < 5*10^8
J. Ramsden and H. Sharipov, Inverse problems associated with perfect cuboids, arXiv preprint arXiv:1207.6764 [math.NT], 2012.
J. Ramsden and H. Sharipov, On singularities of the inverse problems associated with perfect cuboids, arXiv preprint arXiv:1208.1859 [math.NT], 2012.
Ruslan Sharipov, Perfect cuboids and irreducible polynomials, arXiv:1108.5348 [math.NT], 2011.
Ruslan Sharipov, A note on the first cuboid conjecture, arXiv:1109.2534 [math.NT], 2011.
Ruslan Sharipov, A note on the second cuboid conjecture. Part I, arXiv:1201.1229 [math.NT], 2012.
Ruslan Sharipov, Perfect cuboids and multisymmetric polynomials, arXiv preprint arXiv:1205.3135 [math.NT], 2012.
Ruslan Sharipov, On an ideal of multisymmetric polynomials associated with perfect cuboids, arXiv preprint arXiv:1206.6769 [math.NT], 2012.
Ruslan Sharipov, On the equivalence of cuboid equations and their factor equations, arXiv preprint arXiv:1207.2102 [math.NT], 2012.
Ruslan Sharipov, A biquadratic Diophantine equation associated with perfect cuboids, arXiv preprint arXiv:1207.4081 [math.NT], 2012.
Ruslan Sharipov, On a pair of cubic equations associated with perfect cuboids, arXiv preprint arXiv:1208.0308 [math.NT], 2012.
Ruslan Sharipov, On two elliptic curves associated with perfect cuboids, arXiv preprint arXiv:1208.1227 [math.NT], 2012.
Ruslan Sharipov, Asymptotic estimates for roots of the cuboid characteristic equation in the linear region, arXiv preprint arXiv:1505.02745 [math.NT], 2015.
Ruslan Sharipov, Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture, arXiv preprint arXiv:1505.00724 [math.NT], 2015.
Ruslan Sharipov, A note on invertible quadratic transformations of the real plane, arXiv preprint arXiv:1507.01861 [math.AG], 2015.
Eric Weisstein's World of Mathematics, Euler Brick
Walter Wyss, No Perfect Cuboid, arXiv:1506.02215 [math.NT], 2015-2017.
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved