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need integer square root
In another thread I have ended up with this equation
$\displaystyle y=\dfrac{(3v_1+u_1)\pm\sqrt{u_1^2+6u_1v_1+v_1^2}}{ 2}$
I need y to be an integer <> 0, so the first requirement is that
$\displaystyle u_1^2+6u_1v_1+v_1^2$ be a perfect square
If I go at it like this
$\displaystyle x^2=u^2+6uv+v^2$
$\displaystyle x^2=(u+v)^2+4uv$
$\displaystyle x^2-(u+v)^2=4uv$
$\displaystyle (x+(u+v) )(x-(u+v) )=4uv$
$\displaystyle let\ \ \ \ \ x+(u+v)=r$
$\displaystyle and\ \ \ \ x-(u+v)=s$
$\displaystyle x=(r+s)/2$
$\displaystyle u+v=(r-s)/2$
$\displaystyle uv=rs/4$
But I have no individual values of $\displaystyle u$ and $\displaystyle v$ now so cannot calculate
$\displaystyle 3v_1+u_1$
Anyone have any other ideas how I can ensure $\displaystyle u_1^2+6u_1v_1+v_1^2$ is a perfect square?