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A360946
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Number of Pythagorean quadruples with inradius n.
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1
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1, 3, 6, 10, 9, 19, 16, 25, 29, 27, 27, 56, 31, 51, 49, 61, 42, 91, 52, 71, 89, 86, 63, 142, 64, 95, 116, 132, 83, 153, 90, 144, 149, 133, 108, 238, 108, 162, 169, 171, 122, 284, 130, 219, 200, 196, 145, 340, 174, 201, 231, 239, 164, 364, 176, 314, 278, 256, 190, 399, 195, 281, 360, 330
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OFFSET
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1,2
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COMMENTS
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A Pythagorean quadruple is a quadruple (a,b,c,d) of positive integers such that a^2 + b^2 + c^2 = d^2 with a <= b <= c. Its inradius is (a+b+c-d)/2, which is a positive integer.
For every positive integer n, there is at least one Pythagorean quadruple with inradius n.
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REFERENCES
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J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.
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LINKS
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EXAMPLE
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For n=1 the a(1)=1 solution is (1,2,2,3).
For n=2 the a(2)=3 solutions are (1,4,8,9), (2,3,6,7) and (2,4,4,6).
For n=3 the a(3)=6 solutions are (1,6,18,19), (2,5,14,15), (2,6,9,11), (3,4,12,13), (3,6,6,9) and (4,4,7,9).
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MATHEMATICA
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n=50;
div={}; suc={}; A={};
Do[A=Join[A, {Range[1, (1+1/Sqrt[3])q]}], {q, 1, n}];
Do[suc=Join[suc, {Length[div]}]; div={}; For [i=1, i<=Length[Extract[A, q]], i++, div=Join[div, Intersection[Divisors[q^2+(Extract[Extract[A, q], i]-q)^2], Range[2(Extract[Extract[A, q], i]-q), Sqrt[q^2+(Extract[Extract[A, q], i]-q)^2]]]]], {q, 1, n}]; suc=Rest[Join[suc, {Length[div]}]]; matriz={{"q", " ", "cuaternas"}}; For[j=1, j<=n, j++, matriz=Join[matriz, {{j, " ", Extract[suc, j]}}]]; MatrixForm[Transpose[matriz]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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