1. ## 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 $\displaystyle {x^2}+{y^2}=t$.

2. Originally Posted by ordinalhigh
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 $\displaystyle 3z^2$.

3. Originally Posted by ordinalhigh
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.