A circle is centred at $(\pi,e)$. What is the maximum no. of rational points it can have? (A rational point is one with both coordinates rational). 1 rational point is definitely possible, just choose any rational point, and alter the radius to get it through. My book says that only one rational point is possible, as $\pi\neq qe\quad q\in Q$. That's their whole explanation. I don't understand how that's enough. Edit: It has been pointed out that the problem is equivalent to showing $q_1\pi+q_2e=q_3$ has no non trivial solutions. Is this known to be true? Can someone prove it in an elementary way?
Okay I got it, suppose it passes through (a,b). Then the equation of circle is $x^2-a^2+y^2-b^2-2\pi( x-a)-2e(y-b)=0$ If x and y are both rational then $q_1\pi+q_2e=q_3$ with not everything 0 .I still have to prove this impossible. I don't think it's equivalent to $\pi \neq qe$. Edit: As has been pointed out to me, two rational points are not possible if $\pi $ and $ e$ are linearly independent over the rationals, and this is still an open problem