A291591 Numbers k such that there exist exactly five distinct Pythagorean triangles, at least one of them primitive, with area k.

71831760, 73513440, 1675212000, 6913932480, 4323749790360, 2678930100000, 175434192299520, 503151375767040

Offset: 1

Author

Sture Sjöstedt, Aug 27 2017

Keywords

• nonn
• more

Comments

I solve x^2 + 3*y^2 = (2*r)^2 over the positive integers. q, r, q-p and p are the y-coordinates in the first quadrant. Area = q*r*(q-p)*p. There are three Pythagorean triangles with this area. j, x, y with x > y and Area = j^2*x*y*(x-y)*(x+y) gives the area of an Pythagorean triangle.

Example: r = 169 in x^2 + 3*y^2 = (2*169)^2 gives q = 176, r = 169, q-p = 161 and p = 15;

k = q*r*(q-p)*p = 176*169*161*15 = 71831760.

j = 26, x = 23, y = 12 and j = 26, x = 28, y = 5 gives two Pythagorean triangles with k = 71831760;

k = 676*23*12*11*35 = 71831760 and k = 676*28*5*23*33 = 71831760.

Examples

			p^2 - p*q + q^2 = r^2;
p = 115, q = 448, q-p = 333, r = 403;
k = p*q*(q-p)*r = 115*448*333*403 = 6913932480.
x = 414, y = 104 and x = 558, y = 40 gives the same area.
k = x*y*(x-y)*(x+y) = 414*104*310*518 = 6913932480.
k = x*y*(x-y)*(x+y) = 558*40*518*598 = 6913932480.
		

Crossrefs

Cf. A055193.

Extensions

a(2), a(4)-a(7) from Giovanni Resta, Aug 28 2017

Missing term 73513440 inserted by Miguel-Ángel Pérez García-Ortega, Jul 19 2021