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Consider elliptic curves $E:= y^2=x^3+Ax+B $ (A, B are integers) which have points $P, Q$ with rational coordinates and satisfy $P=[n]Q, n>1$. Now consider the below problem:

Input: Rational coordinates = $P$.

Output: Integers $A, B, n$ (when $n>1$), and rational coordinates $Q,$ $$OR,$$

$$Return \; 0.$$

now we want to know how many points $P$ will have a curve satisfying above condition, so -

What is the percent of rational points $P$ that will not return $0$ among all rational points pair? is there anything in the literature related to the above problem?

Note: In this post/problem description, to be rational point on $E$ only $x$ coordinate has to be rational, rational y-coordinate is not mandatory.

A related problem is asked on math stackexchange.

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  • $\begingroup$ The set of points $P$ with rational $x$-coordinate is a countable collection of vertical lines. How do you propose to define percentages in such a set? $\endgroup$
    – Gerry Myerson
    Commented May 19, 2023 at 4:57
  • $\begingroup$ If only the $x$ coordinate is rational do you only input the $x$ coordinate? Also why is the rational $y$ coordinate not mandatory? $\endgroup$
    – Will Sawin
    Commented May 19, 2023 at 13:21
  • $\begingroup$ @GerryMyerson for example, in integers 50% integers are even, 50 % integers are negative... thus probability of finding an integer even or negative is... of course in above posted case, we might not be calculate exactly, but an lower/upper bound is expected. $\endgroup$
    – Consider Non-Trivial Cases
    Commented May 19, 2023 at 15:30
  • $\begingroup$ @WillSawin to relax the problem, both $x, y$ rational are more restricted than only rational $x$, I reckon, thus it might be easier, that is why. $\endgroup$
    – Consider Non-Trivial Cases
    Commented May 19, 2023 at 15:33
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    $\begingroup$ I think it's worth asking: why do you want this thing? Is there something in particular you're hoping to learn about the rational points, or the measure itself, or? This feels like one step along the path of a larger question and having a better sense of that larger question can help people with providing a meaningful answer to this one. $\endgroup$
    – Steven Stadnicki
    Commented May 20, 2023 at 20:14

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