OFFSET
3,1
COMMENTS
Given an ordered primitive Pythagorean triple (a,b,c) with a<b, we call (b+c)/a the median radial proportion. This number gives us the proportion (r_b)/r, where r_b is the exinradius associated to the greater cathetus of the Pythagorean triangle (a,b,c) and r is the inradius of the same triangle.
For n>=3 there exists a unique ordered primitive Pythagorean triple such that (b+c)/a = n.
For n odd, the triple is {n, (n^2-1)/2, (n^2+1)/2}.
For n even, the triple is { 2*n, n^2-1, n^2+1 }.
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, Radial proportions (in Spanish).
FORMULA
EXAMPLE
Irregular array begins:
n=3: 3, 4, 5;
n=4: 8, 15, 17;
n=5: 5, 12, 13;
n=6: 12, 35, 37;
n=7: 7, 24, 25;
...
Row n=3 is (3,4,5) and has (b+c)/a = (4+5)/3 = 3.
Row n=4 is (8,15,17) and has (b+c)/a = (15+17)/8 = 4.
MATHEMATICA
k=50;
ternas={{n, " ", a, b, c, " ", r, " ", " γ2 ", " ", s, " ", rb}}; Do[If[Mod[t, 2]==0, ternas=Join[ternas, {{t, " ", 2t, t^2-1, t^2+1, " ", t-1, " ", t, " ", t(t+1), " ", t(t-1)}}], ternas=Join[ternas, {{t, " ", t, (t^2-1)/2, (t^2+1)/2, " ", (t-1)/2, " ", t, " ", (t(t+1))/2, " ", (t(t-1))/2}}]], {t, 3, k+2}]
MatrixForm[Transpose[ternas]]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Miguel-Ángel Pérez García-Ortega, Apr 11 2023
EXTENSIONS
Edited by N. J. A. Sloane, Apr 30 2023
STATUS
approved