Yesterday I read up on Diophantine equations and the property in which two rational solutions of such an equation make a line that gives a third rational solution to the equation. I was thinking about pairs of rationals where this property isn't especially obvious to someone new to the subject. I noticed that if you have the equation $y^2 = \frac{1}{3}x^3 - 2x$ and you have two rational solutions that make a vertical line (or a line with a very large slope) that you might get something like in this illustration where the red curve is the Diophantine equation and the dashed line is a line created by two rational solutions...

enter image description here

If the dashed line could represent the line created by two solutions the property in question wouldn't work so clearly it can't happen. But why? Similarly why couldn't a similar situation with a non vertical line with a large slope not happen?

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    $\begingroup$ This is almost the same question you asked yesterday (math.stackexchange.com/questions/3226195/…) about elliptic curves. It seems to me that your background in elementary algebra might be not quite good enough to master this material. At a minimum, you need to understand homogeneous coordinates in the projective plane. $\endgroup$ – Ethan Bolker May 15 '19 at 18:07
  • $\begingroup$ Ethan Bolker, ok, could you point me to some educational material on elementary algebra and the projective plane? $\endgroup$ – Algebra is Awesome May 15 '19 at 18:55
  • $\begingroup$ So I did more reading, is this Diophantine equation a smooth cubic curve? Or is it not one because it's not continuous? $\endgroup$ – Algebra is Awesome May 15 '19 at 20:00
  • $\begingroup$ I used some sources from the first question to try and answer it myself. I found a partial answer. Feel free to comment if I got it wrong or if you have a solution to the second part of my question about slopped lines for this particular diophantine equation. $\endgroup$ – Algebra is Awesome May 15 '19 at 20:38

This is a partial answer. According to https://en.wikipedia.org/wiki/Elliptic_curve a vertical line means for the points P and Q that we use to get that vertical line our P = -Q and our third homogenous coordinate that intersects the Diophantine equation is just the additive identity. But it seems like there could be some steeply slopped lines for which the line is not a tangent point to the curve for P or Q but who doesn't intersect the curve at a third homogenous coordinate R. What's going on in situations like this?enter image description here

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  • $\begingroup$ So it seems like in this case the slope of the curve (to the right of the y-axis) becomes steeper and steeper meaning any non vertical line will eventually intersect it. Very cool! It was hard to see until i just zoomed out on the graph and noticed that behavior. In retrospect I would have maybe realized that two rational homogeneous coordinates like in the graph above would have had a solution that was just very far away on the projected plane. But anyway that's the solution. $\endgroup$ – Algebra is Awesome May 15 '19 at 20:54

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