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I’m working with a Diophantine equation where I express positive integers as $m^2 + n^3$ in three different ways(to show that there are infinite $c^2$ that can be expressed in this way). After finding six such points (with four distinct ones) in terms of a parameter $c$, I plotted them on the elliptic curve $y^2 + x^3 = c^2$, because why not(I just learned about what elliptic curves are and automatically assumed they applied :P)?

enter image description here

On the graph(https://www.desmos.com/calculator/vxohuwvzmy), I observed two interesting things:

  1. Four of the points(3 unique points) lie on a straight line.
  2. Two points seem to be identical and appear to be “perpendicular” to this line.

The six points on the graph are given in terms of the parameters $k_1$ and $k_2$, where $k_1 = \frac{-1+\sqrt{8c+1}}{2}$ and $k_2 = \frac{-1-\sqrt{8c+1}}{2}$, and whenever these happen to be integers, the following points are integers too:

$P_1 = \Bigl(0,\frac{k_1\bigl(k_1 + 1\bigr)}{2}\Bigr)$

$P_2 = \Bigl(0,\frac{k_2\bigl(k_2 + 1\bigr)}{2}\Bigr)$

$P_3 = \Bigl(k_1,\frac{k_1\bigl(k_1 - 1\bigr)}{2}\Bigr)$

$P_4 = \Bigl(k_2,\frac{k_2\bigl(k_2 - 1\bigr)}{2}\Bigr)$

There’s also a two points not on the original curve, that form a line with the y-intercept, perpendicular to the line formed by the points above(I'm unsure of how I got this non-solution?):

$P_5 = \Bigl(\frac{k_1^2 + k_1 - 2}{2},k_1^2+k_1-1)$

$P_6 = \Bigl(\frac{k_2^2 + k_2 - 2}{2},k_2^2+k_2-1)$

So my questions:

  1. Four of these points always seem to fall on the same straight line—why? What about the perpendicular point/line(is that even relevant?)?

  2. Is there a more general way to solve this question with elliptic curves that's staring me in the face(as person who knows basically nothing about elliptic curves)?

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  • $\begingroup$ Hello, I added in the changes you suggested(I actually drew out the line, and organized the graph more), and the solutions are in the form of the points in the graph(for every c there are three corresponding non-identical points). Thanks for your help! $\endgroup$ Commented Mar 22 at 16:36
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    $\begingroup$ What do you mean by a "point perpendicular to a line"? $\endgroup$ Commented Mar 22 at 16:47
  • $\begingroup$ The "perpendicular point" when connected to the y-intercept forms a line that is perpendicular to the other line $\endgroup$ Commented Mar 22 at 16:49
  • $\begingroup$ $c^2=x^3+y^2$ is Mordell Elliptic Curve $\endgroup$ Commented Mar 22 at 17:14
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    $\begingroup$ Okayy I just made those changes, I appreciate the feedback :)) $\endgroup$ Commented Mar 23 at 16:33

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