OFFSET
1,2
COMMENTS
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.
REFERENCES
J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.
LINKS
Miguel-Ángel Pérez García-Ortega, Pythagorean Quadruples (in Spanish).
Wikipedia, Pythagorean quadruple.
EXAMPLE
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).
MATHEMATICA
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]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Miguel-Ángel Pérez García-Ortega, Feb 26 2023
STATUS
approved