Which Mordell-curves have exactly $2$ integral points with large coordinates?

Question

I found the list of the integral solutions of the Mordell-equation

$$y^2=x^3+n$$ with non-zero integer $n$ for $-10^7\le n\le 10^7$

I am interested in integers $n$ such that the Mordell-equation has exactly $2$ integral points with large coordinates. For example, for $n=25895$ the $x$-coordinate of the points is $103289609$

I would like to search larger examples (if existent), but I do not know whether and how the pdf-reader allows to search a string like $[******]$ , where $*$ stands for an arbitary digit. Therefore my question

Are there easy to verify conditions for $n$ to satisfy this property ? Can I construct $n$ for which we will have exactly two integral points with extremely large coordinates ?

Answer 1

The record-holder in the range $[-10^6,10^6]$ of the numbers listed in the file is $$n=706\ 394$$ The $x$-coordinate of the two integral points is $30\ 050\ 613\ 311$

As Mike Bennett reported in the linked question, there is a problem with the data's. So, there might be an even more spectacular example. I verified that the $x$-coordinate is in fact the smallest, but I do now know how to prove that the Mordell-curve only contains $2$ integral points.