A272388 - OEIS

%I #23 Aug 07 2021 08:21:03

%S 117,160,203,225,234,318,319,319,320,351,406,429,450,468,468,480,585,

%T 595,595,595,609,612,636,638,638,640,671,675,680,680,697,697,702,741,

%U 780,800,812,819,858,884,884,888,900,925,935,936,936,954,957,957,960,990,990

%N Longest side of Heronian tetrahedron.

%C A Heronian tetrahedron or perfect tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers.

%H R. H. Buchholz, <a href="https://doi.org/10.1017/S0004972700030252">Perfect Pyramids</a>, Bull. Austral. Math. Soc. 45, 353-368, 1992.

%H 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 Ivars Peterson, <a href="https://www.sciencenews.org/article/perfect-pyramids">Perfect pyramids</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTetrahedron.html">Heronian Tetrahedron</a>.

%e The following are examples of Heronian tetrahedra.

%e dAB, dAC, dBC, dCD, dBD, dAD, SABC,  SABD,  SACD,  SBCD,  Volume

%e 117, 84,  51,  52,  53,  80,  1890,  1800,  2016,  1170,  18144

%e 160, 153, 25,  39,  56,  120, 1872,  2688,  1404,  420,   8064

%e 203, 195, 148, 203, 195, 148, 13650, 13650, 13650, 13650, 611520

%e 225, 200, 65,  119, 156, 87,  6300,  4914,  2436,  3570,  35280

%e 234, 168, 102, 104, 106, 160, 7560,  7200,  8064,  4680,  145152

%e 318, 221, 203, 42,  175, 221, 22260, 18564, 4620,  2940,  206976

%e 319, 318, 175, 175, 210, 221, 26796, 23100, 18564, 14700, 1034880

%e 319, 318, 175, 203, 252, 221, 26796, 27720, 22260, 17640, 1241856

%e 320, 306, 50,  78,  112, 240, 7488,  10752, 5616,  1680,  64512

%e 351, 252, 153, 156, 159, 240, 17010, 16200, 18144, 10530, 489888

%e where

%e dPQ is the distance between vertices P and Q and

%e SPQR is the area of triangle PQR.

%t aMax=360(*WARNING:takes a long time*);

%t heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;

%t cayley=1/24Sqrt[2Det[{

%t   {0,1,1,1,1},

%t   {1,0,#1^2,#2^2,#6^2},

%t   {1,#1^2,0,#3^2,#5^2},

%t   {1,#2^2,#3^2,0,#4^2},

%t   {1,#6^2,#5^2,#4^2,0}

%t }]]&;

%t Do[

%t   S1=heron[a,b,c];

%t   If[S1//IntegerQ//Not,Continue[]];

%t   Do[

%t     S2=heron[a,e,f];

%t     If[S2//IntegerQ//Not,Continue[]];

%t     Do[

%t       If[b==e&&c>f||b==f&&c>e,Continue[]];

%t       S3=heron[b,d,f];

%t       If[S3//IntegerQ//Not,Continue[]];

%t       S4=heron[c,d,e];

%t       If[S4//IntegerQ//Not,Continue[]];

%t       V=cayley[a,b,c,d,e,f];

%t       If[V//IntegerQ//Not,Continue[]];

%t       If[V==0,Continue[]];

%t       a//Sow(*{a,b,c,d,e,f,S1,S2,S3,S4,V}//Sow*);

%t     ,{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   ,{e,a-b+1,b},{f,a-e+1,b}];

%t ,{a,117,aMax},{b,a/2//Ceiling,a},{c,a-b+1,b}]//Reap//Last//Last

%Y Cf. A120131, A120132, A120133.

%K nonn

%O 1,1

%A _Albert Lau_, May 19 2016

%E a(11)-a(53) from _Giovanni Resta_, May 20 2016