A031173 Longest edge a of smallest (measured by the longest edge) primitive Euler bricks (a, b, c, sqrt(a^2 + b^2), sqrt(b^2 + c^2), sqrt(a^2 + c^2) are integers).

240, 275, 693, 720, 792, 1155, 1584, 2340, 2640, 2992, 3120, 5984, 6325, 6336, 6688, 6732, 8160, 9120, 9405, 10725, 11220, 12075, 13860, 14560, 16800, 17472, 17748, 18560, 19305, 21476, 23760, 23760, 24684, 25704, 26649, 29920, 30780

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

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.
• Index entries for sequences related to bricks

Crossrefs

Cf. A031174, A031175.