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Thread: rational points question

  1. #1
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    rational points question

    I'm trying to find a set of all rational points for $\displaystyle {x^2}+{y^2}=3$. I know that if I can find a line with rational slope $\displaystyle m$ passing through point $\displaystyle (x_0,y_0)$, I can find the set but i'm having trouble finding the coordinates of the point such that they satisfy $\displaystyle {x^2}+{y^2}=3$ where they are rational numbers.



    Also, how can I generalize for a circle with any radius? Finding a set of all rational points for $\displaystyle {x^2}+{y^2}=t$.
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  2. #2
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    Quote Originally Posted by ordinalhigh View Post
    I'm trying to find a set of all rational points for $\displaystyle {x^2}+{y^2}=3$.
    There are no rational points on this curve. The existence of rational points on $\displaystyle {x^2}+{y^2}=3$ is equivalent to the existence of integer solutions to $\displaystyle {x^2}+{y^2}=3z^2$. But a natural number is a sum of two squares if and only if each of its prime divisors of the form 4k+3 occurs to an even power. This is evidently not the case for the number $\displaystyle 3z^2$.
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  3. #3
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    Quote Originally Posted by ordinalhigh View Post
    I'm trying to find a set of all rational points for
    ...
    for $\displaystyle {x^2}+{y^2}=t$.

    $\displaystyle \left(\dfrac{x}{u} \right )^2 + \left(\dfrac{y}{v} \right )^2 = t$

    $\displaystyle \left( xv \right )^2 + \left( yu \right )^2 = t \left( uv \right )^2$

    see Opalg's post.
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