A123673 Smaller side of right triangles with integer sides and integer side inscribed squares with two vertices on the hypotenuse.

111, 222, 333, 444, 555, 666, 777, 888, 999, 1110, 1145, 1221, 1332, 1443, 1554, 1665, 1776, 1887, 1998, 2109, 2220, 2290, 2331, 2442, 2553, 2664, 2775, 2886, 2997, 3108, 3219, 3272, 3330, 3435, 3441, 3552, 3663, 3774, 3885, 3996, 4107, 4218, 4329, 4440

Offset: 1

Author

Cino Hilliard, Nov 17 2006

Keywords

• nonn

Comments

The side of the inscribed square having two vertices on the hypotenuse of a right triangle, sides x= 4*z^4*s^2/(z-s)^2. So it follows for a solution to exist, z >= 3s. For s=60 and z = 185 we have a = 185^2, b = 185^4*60^2/125^2=269879184 then x1 = sqrt((185^2 - sqrt(185^4-4*269879184))/2) = 111 x2 = sqrt((185^2 - sqrt(185^4+4*269879184))/2) = 148 So x= 111 and y = 148. It is interesting to note that 37 almost always divides these numbers. Some exceptions are 1145, 2290, 3272, and 3435.

Programs

PARI

g(n)= { for(x=1,n, for(y=x,n, z=sqrt(x^2+y^2); s=x*y*z/(z^2+x*y); if(s==floor(s), print1(floor(x)",") ) ) ) }