A lower bound for the number of integral solutions of Mordell equation

Abstract

For a nonzero integer $d$, a celebrated Siegel Theorem says that the number $N(d)$ of integral solutions of Mordell equation $y^2+x^3=d$ is finite. We find a lower bound for $N(d)$, showing that the number of solutions of Mordell equation increases dramatically. We also prove that for any positive integer $n$, there is an integer square multiply represented by Mordell equations, i.e., $k^2=y_1^2+x_1^3=y_2^2+x_2^3=\cdots =y_n^2+x_n^3$.