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Thread: need integer square root

  1. #1
    Sep 2011

    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?
    Last edited by moriman; Sep 29th 2011 at 11:39 AM. Reason: submitted early by mistake
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