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Longest side of primitive Heronian tetrahedron.
0

%I #7 May 01 2022 19:13:22

%S 117,160,203,225,318,319,319

%N Longest side of primitive Heronian tetrahedron.

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

%C Primitive tetrahedron means 6 sides don't share a common factor.

%e see A272388

%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[GCD[a, b, c, d, e, f] > 1, Continue[]];

%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. A272388

%K nonn,more

%O 1,1

%A _Albert Lau_, May 21 2016