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Suppose, a Mordell curve

$$y^2=x^3+k$$

has at least one integral solution. Denote $d$ to be the smallest absolute value of $x$, such that $x^3+k$ is a square, in other words, $d$ is the the smallest possible absolute $x$-coordinate of an integral solution.

Is any reasonable upper bound for $d$ depending on $k$ known ? In other words, if I want to verify that there is no solution, when can I stop the calculation ?

The upper bounds of integral solutions of an elliptic curve mentioned in wikipedia in the general case show that there are only finite many solutions, but they are completely useless in practice, because they are much too large. But the situation might be better in the case of Mordell-curves.

Peter
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1 Answers1

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Stark in 1973 obtained better bounds than Baker for the general case, on the size of integral points $(x,y)$ on the Mordell curve $$ E_k: y^2=x^3+k, $$ namely that $$ {\rm max} \{ |x|,|y|\}<e^{c_{\epsilon}|k|^{1+\epsilon}} $$ with an effectively computable constant $c_{\epsilon}> 0$ depending on a given $\epsilon >0$. Some numerical experiments led Hall in 1971 to make the following conjecture.

Hall Conjecture: For the integral points $(x,y)$ on $E_k$ we have $$ |x| < c_{\epsilon}|k|^{2+\epsilon}, $$ with a constant $c_{\epsilon}> 0$ depending only on $\epsilon >0$. For the data by Noam Elkies on this see the previous question.

Despite of this conjecture, integral points on $E_k$ can be quite large in comparison to $k$ in practice, because it seems to be hard to estimate the size of the constant $c_{\epsilon}$. For example, for $k=28024$ we have $$ (x,y)=(3790689201, 233387325399875). $$

Edit: For estimates only depending on $k$, we have Baker's astronomical result $$ {\rm max} \{ |x|,|y|\}<\exp (10^{10^5}|k|^{10^4}), $$ and some improvements, e.g., $$ {\rm max} \{ |x|,|y|\}<\exp ({\rm min_{(c,d)\in S}\{ c|k|\log(|k|)^d\}}), $$ where $$ S=\{ (10^{181}, 4), (10^{23}, 5), (10^{19}, 6)\} $$ by Juricevic in $2008$.

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Dietrich Burde
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