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EXAMPLE
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Below shows some example: (might contains gap)
a, b, c, S, V
203, 195, 148, 13650, 611520
888, 875, 533, 223860, 37608480
1804, 1479, 1183, 870870, 214582368
2431, 2296, 2175, 2277660, 1403038560
2873, 2748, 1825, 2419950, 1355172000
5512, 5215, 1887, 4919460, 1377448800
8484, 6625, 6409, 20980050, 30546952800
11275, 10136, 8619, 41861820, 103147524480
19695, 16448, 13073, 106675680, 323290060800
32708, 31493, 24525, 363332970, 2685757314240
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MATHEMATICA
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heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;
cayley=1/24Sqrt[2Det[{
{0, 1, 1, 1, 1},
{1, 0, #1^2, #2^2, #6^2},
{1, #1^2, 0, #3^2, #5^2},
{1, #2^2, #3^2, 0, #4^2},
{1, #6^2, #5^2, #4^2, 0}
}]]&;
aMin=203;
aMax=2000(*WARNING:runs very slow*);
Do[
If[GCD[a, b, c]>1, Continue[]];
S=heron[a, b, c];
If[S//IntegerQ//Not, Continue[]];
V=cayley[a, b, c, a, b, c];
If[V//IntegerQ//Not, Continue[]];
a(*{a, b, c, S, V}*)//Sow;
, {a, aMin, aMax}
, {b, a/Sqrt[2]//Ceiling, a-1}
, {c, Mod[a+b, 2, Floor[Sqrt[a^2-b^2]]+1], b-1, 2}
]//Reap//Last//Last(*//TableForm*)
{S, V}=.;
(*
(*this piece of code runs much faster but might contains gap*)
mMax=100;
Do[
{a, b, c}={n(m^2+k^2), m(n^2+k^2), (m+n)(m n-k^2)};
{a, b, c}={a, b, c}/GCD[a, b, c];
V=cayley[a, b, c, a, b, c];
If[V//IntegerQ//Not, Continue[]];
a(*{a, b, c, heron[a, b, c], V}*)//Sow
, {m, mMax}
, {n, m-1}
, {k, Floor[Sqrt[(m^2 n)/(2m+n)]+1], n-1}
]//Reap//Last//Last//Union(*TableForm*)
{a, b, c, V}=.;
*)
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