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A272388 Longest side of Heronian tetrahedron.

%I A272388 #24 Feb 16 2025 08:33:34
%S A272388 117,160,203,225,234,318,319,319,320,351,406,429,450,468,468,480,585,
%T A272388 595,595,595,609,612,636,638,638,640,671,675,680,680,697,697,702,741,
%U A272388 780,800,812,819,858,884,884,888,900,925,935,936,936,954,957,957,960,990,990
%N A272388 Longest side of Heronian tetrahedron.
%C A272388 A Heronian tetrahedron or perfect tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers.
%H A272388 R. H. Buchholz, <a href="https://doi.org/10.1017/S0004972700030252">Perfect Pyramids</a>, Bull. Austral. Math. Soc. 45, 353-368, 1992.
%H A272388 Susan H. Marshall and Alexander R. Perlis, <a href="http://www.maa.org/sites/default/files/pdf/upload_library/2/Marshall2-Monthly-2014.pdf">Heronian Tetrahedra Are Lattice Tetrahedra</a>, American Mathematical Monthly 120:2 (2013), 140-149.
%H A272388 Ivars Peterson, <a href="https://www.sciencenews.org/article/perfect-pyramids">Perfect pyramids</a>.
%H A272388 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTetrahedron.html">Heronian Tetrahedron</a>.
%e A272388 The following are examples of Heronian tetrahedra.
%e A272388 dAB, dAC, dBC, dCD, dBD, dAD, SABC,  SABD,  SACD,  SBCD,  Volume
%e A272388 117, 84,  51,  52,  53,  80,  1890,  1800,  2016,  1170,  18144
%e A272388 160, 153, 25,  39,  56,  120, 1872,  2688,  1404,  420,   8064
%e A272388 203, 195, 148, 203, 195, 148, 13650, 13650, 13650, 13650, 611520
%e A272388 225, 200, 65,  119, 156, 87,  6300,  4914,  2436,  3570,  35280
%e A272388 234, 168, 102, 104, 106, 160, 7560,  7200,  8064,  4680,  145152
%e A272388 318, 221, 203, 42,  175, 221, 22260, 18564, 4620,  2940,  206976
%e A272388 319, 318, 175, 175, 210, 221, 26796, 23100, 18564, 14700, 1034880
%e A272388 319, 318, 175, 203, 252, 221, 26796, 27720, 22260, 17640, 1241856
%e A272388 320, 306, 50,  78,  112, 240, 7488,  10752, 5616,  1680,  64512
%e A272388 351, 252, 153, 156, 159, 240, 17010, 16200, 18144, 10530, 489888
%e A272388 where
%e A272388 dPQ is the distance between vertices P and Q and
%e A272388 SPQR is the area of triangle PQR.
%t A272388 aMax=360(*WARNING:takes a long time*);
%t A272388 heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;
%t A272388 cayley=1/24Sqrt[2Det[{
%t A272388   {0,1,1,1,1},
%t A272388   {1,0,#1^2,#2^2,#6^2},
%t A272388   {1,#1^2,0,#3^2,#5^2},
%t A272388   {1,#2^2,#3^2,0,#4^2},
%t A272388   {1,#6^2,#5^2,#4^2,0}
%t A272388 }]]&;
%t A272388 Do[
%t A272388   S1=heron[a,b,c];
%t A272388   If[S1//IntegerQ//Not,Continue[]];
%t A272388   Do[
%t A272388     S2=heron[a,e,f];
%t A272388     If[S2//IntegerQ//Not,Continue[]];
%t A272388     Do[
%t A272388       If[b==e&&c>f||b==f&&c>e,Continue[]];
%t A272388       S3=heron[b,d,f];
%t A272388       If[S3//IntegerQ//Not,Continue[]];
%t A272388       S4=heron[c,d,e];
%t A272388       If[S4//IntegerQ//Not,Continue[]];
%t A272388       V=cayley[a,b,c,d,e,f];
%t A272388       If[V//IntegerQ//Not,Continue[]];
%t A272388       If[V==0,Continue[]];
%t A272388       a//Sow(*{a,b,c,d,e,f,S1,S2,S3,S4,V}//Sow*);
%t A272388     ,{d,Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1-S2)/a)^2]//Ceiling,Min[a,Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1+S2)/a)^2]]}];
%t A272388   ,{e,a-b+1,b},{f,a-e+1,b}];
%t A272388 ,{a,117,aMax},{b,a/2//Ceiling,a},{c,a-b+1,b}]//Reap//Last//Last
%Y A272388 Cf. A120131, A120132, A120133.
%K A272388 nonn
%O A272388 1,1
%A A272388 _Albert Lau_, May 19 2016
%E A272388 a(11)-a(53) from _Giovanni Resta_, May 20 2016