(In reply to re: A ffine problem (spoiler)
Any point on the curve will give us a solution provided its coordinates are rational. For a cubic curve like this, if a line passes through three points on the curve and two of them have rational coordinates then so will the third (since no surds are involved in the algebra).
Similarly, as here, if we use a tangent at a rational point (called doubling) then still no surds are involved and the 'third' point (i.e. the second distinct point) will always also have rational coordinates.
Whether a line meets the curve in three points of course depends on its shape, and whether the number of solutions is finite is a more interesting problem in group theory - but worth exploring.
Posted by Harry
on 2016-05-02 18:57:18