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%I #101 Feb 25 2020 21:26:27
%S 240,275,693,720,792,1155,1584,2340,2640,2992,3120,5984,6325,6336,
%T 6688,6732,8160,9120,9405,10725,11220,12075,13860,14560,16800,17472,
%U 17748,18560,19305,21476,23760,23760,24684,25704,26649,29920,30780
%N 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).
%C Primitive means that gcd(a,b,c) = 1.
%C 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
%C There are some longest edges a which occur multiple times, such as a(31) = a(32)) = 23760. - _Ralf Steiner_, Jan 07 2018
%C 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
%D Calculated by F. Helenius (fredh(AT)ix.netcom.com).
%H Giovanni Resta, <a href="/A031173/b031173.txt">Table of n, a(n) for n = 1..3556</a>
%H R. R. Gallyamov, I. R. Kadyrov, D. D. Kashelevskiy, <a href="https://arxiv.org/abs/1601.00636">A fast modulo primes algorithm for searching perfect cuboids and its implementation</a>, arXiv preprint arXiv:1601.00636 [math.NT], 2016.
%H A. A. Masharov and R. A. Sharipov, <a href="http://arxiv.org/abs/1504.07161">A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture</a>, arXiv preprint arXiv:1504.07161 [math.NT], 2015.
%H Dustin Moody, Mohammad Sadek, Arman Shamsi Zargar, <a href="https://doi.org/10.1216/RMJ-2019-49-7-2253">Families of elliptic curves of rank >= 5 over Q(t)</a>, Rocky Mountain Journal of Mathematics (2019) Vol. 49, No. 7, 2253-2266.
%H Giovanni Resta, <a href="/A031173/a031173.txt">The 3556 primitive bricks with c < b < a < 5*10^8</a>
%H J. Ramsden and H. Sharipov, <a href="http://arxiv.org/abs/1207.6764">Inverse problems associated with perfect cuboids</a>, arXiv preprint arXiv:1207.6764 [math.NT], 2012.
%H J. Ramsden and H. Sharipov, <a href="http://arxiv.org/abs/1208.1859">On singularities of the inverse problems associated with perfect cuboids</a>, arXiv preprint arXiv:1208.1859 [math.NT], 2012.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1108.5348">Perfect cuboids and irreducible polynomials</a>, arXiv:1108.5348 [math.NT], 2011.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1109.2534">A note on the first cuboid conjecture</a>, arXiv:1109.2534 [math.NT], 2011.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1201.1229">A note on the second cuboid conjecture. Part I</a>, arXiv:1201.1229 [math.NT], 2012.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1205.3135">Perfect cuboids and multisymmetric polynomials</a>, arXiv preprint arXiv:1205.3135 [math.NT], 2012.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1206.6769">On an ideal of multisymmetric polynomials associated with perfect cuboids</a>, arXiv preprint arXiv:1206.6769 [math.NT], 2012.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1207.2102">On the equivalence of cuboid equations and their factor equations</a>, arXiv preprint arXiv:1207.2102 [math.NT], 2012.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1207.4081">A biquadratic Diophantine equation associated with perfect cuboids</a>, arXiv preprint arXiv:1207.4081 [math.NT], 2012.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1208.0308">On a pair of cubic equations associated with perfect cuboids</a>, arXiv preprint arXiv:1208.0308 [math.NT], 2012.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1208.1227">On two elliptic curves associated with perfect cuboids</a>, arXiv preprint arXiv:1208.1227 [math.NT], 2012.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1505.02745">Asymptotic estimates for roots of the cuboid characteristic equation in the linear region</a>, arXiv preprint arXiv:1505.02745 [math.NT], 2015.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1505.00724">Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture</a>, arXiv preprint arXiv:1505.00724 [math.NT], 2015.
%H Ruslan Sharipov, <a href="http://arxiv.org/abs/1507.01861">A note on invertible quadratic transformations of the real plane</a>, arXiv preprint arXiv:1507.01861 [math.AG], 2015.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerBrick.html">Euler Brick</a>
%H Walter Wyss, <a href="https://arxiv.org/abs/1506.02215">No Perfect Cuboid</a>, arXiv:1506.02215 [math.NT], 2015-2017.
%H <a href="/index/Br#bricks">Index entries for sequences related to bricks</a>
%Y Cf. A031174, A031175.
%K nonn
%O 1,1
%A _Eric W. Weisstein_
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