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I'm looking into diophantine equations and see the property that any two solutions to a diophantine equation create a line that intersects the equation at a different solution. However, what about vertical lines through the graph of $y^2 = x^3 - 2x$. I'm assuming there's a detail about this property I'm not picking up on.

Is there some reason it doesn't count to have a vertical line with our P and Q that finds a third point R?

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  • $\begingroup$ Sorry i forgot to specify that the property holds for rational solutions and I don't know if it holds for all real solutions. $\endgroup$ – Algebra is Awesome May 14 '19 at 21:24
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    $\begingroup$ It is a property of cubic equations that from two rational points you have a 3rd point (look at the 3rd root of $h(t) = f(a+ct,b+dt) = -c^3 t (t-1)(t-T)$ where $f(x,y) =y^2-x^3+2x, f(a,b)=f(a+c,b+d)=0$) $\endgroup$ – reuns May 14 '19 at 21:26
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    $\begingroup$ The theorem you are thinking about requires that you count tangency as a double root, and pay attention to solutions "at infinity". en.wikipedia.org/wiki/Elliptic_curve#The_group_law $\endgroup$ – Ethan Bolker May 14 '19 at 21:26
  • $\begingroup$ Could you elaborte on what tou mean by "count tangency as a double root"? $\endgroup$ – Algebra is Awesome May 14 '19 at 21:27
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    $\begingroup$ The quadratic curve $y=x^2$ looks as if it has just one root, at $0$. But that root is really a double root. $x$ is a factor twice. The geometry that reflects that algebra shows up because the $x$-axis is tangent to the parabola at the origin. You have to understand multiple roots of polynomials before you can make sense of what you are reading about elliptic curves. $\endgroup$ – Ethan Bolker May 14 '19 at 21:33

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