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A272389 Longest side of primitive Heronian tetrahedron.

%I A272389 #7 May 01 2022 19:13:22
%S A272389 117,160,203,225,318,319,319
%N A272389 Longest side of primitive Heronian tetrahedron.
%C A272389 A Heronian tetrahedron or perfect tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers.
%C A272389 Primitive tetrahedron means 6 sides don't share a common factor.
%e A272389 see A272388
%t A272389 aMax=360(*WARNING:takes a long time*);
%t A272389 heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;
%t A272389 cayley=1/24Sqrt[2Det[{
%t A272389   {0,1,1,1,1},
%t A272389   {1,0,#1^2,#2^2,#6^2},
%t A272389   {1,#1^2,0,#3^2,#5^2},
%t A272389   {1,#2^2,#3^2,0,#4^2},
%t A272389   {1,#6^2,#5^2,#4^2,0}
%t A272389 }]]&;
%t A272389 Do[
%t A272389   S1=heron[a,b,c];
%t A272389   If[S1//IntegerQ//Not,Continue[]];
%t A272389   Do[
%t A272389     S2=heron[a,e,f];
%t A272389     If[S2//IntegerQ//Not,Continue[]];
%t A272389     Do[
%t A272389       If[GCD[a, b, c, d, e, f] > 1, Continue[]];
%t A272389       If[b==e&&c>f||b==f&&c>e,Continue[]];
%t A272389       S3=heron[b,d,f];
%t A272389       If[S3//IntegerQ//Not,Continue[]];
%t A272389       S4=heron[c,d,e];
%t A272389       If[S4//IntegerQ//Not,Continue[]];
%t A272389       V=cayley[a,b,c,d,e,f];
%t A272389       If[V//IntegerQ//Not,Continue[]];
%t A272389       If[V==0,Continue[]];
%t A272389       a//Sow(*{a,b,c,d,e,f,S1,S2,S3,S4,V}//Sow*);
%t A272389     ,{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 A272389   ,{e,a-b+1,b},{f,a-e+1,b}];
%t A272389 ,{a,117,aMax},{b,a/2//Ceiling,a},{c,a-b+1,b}]//Reap//Last//Last
%Y A272389 Cf. A272388
%K A272389 nonn,more
%O A272389 1,1
%A A272389 _Albert Lau_, May 21 2016