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Math Help - rational points question

  1. #1
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    rational points question

    I'm trying to find a set of all rational points for {x^2}+{y^2}=3. I know that if I can find a line with rational slope m passing through point (x_0,y_0), I can find the set but i'm having trouble finding the coordinates of the point such that they satisfy {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 {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 {x^2}+{y^2}=3.
    There are no rational points on this curve. The existence of rational points on {x^2}+{y^2}=3 is equivalent to the existence of integer solutions to {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 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 {x^2}+{y^2}=t.

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

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

    see Opalg's post.
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