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Since it is not easy to determine the integral points of a Mordell curve $$y^2=x^3+n$$ with integer $n\ne 0$, I came to the following questions :

$1)$ What is the smallest (in absolute value) integer $n$ , such that it is unknown whether the Mordell-curve $y^2=y^3+n$ has an integral point ?

$2)$ What is the smallest (in absolute value) integer $n$, such that the complete list of the integral points is not known ?

Since I found a list for $-10^7\le n\le 10^7$, the absolute value should be greater than $10^7$.

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Peter
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Mike Bennet has mentioned the example on MSE, due to Noam Elkies with $$n = 509142596247656696242225,$$ where there are at least $125$ pairs of integral solutions, but where not all integral solutions are known. But of course this may not be the smallest $n$. Noam Elkies has more information concerning the Mordell equation $x^3 – y^2 = k$ on his homepage, e.g., here, concerning Hall's conjecture, but you will certainly know this already (as well as Cremona's database etc.).

Dietrich Burde
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