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Abstract

Let $E$ be an elliptic curve over the rationals ${\mathbb {Q}}$ given by $y^2=x^3-nx$ with a positive integer $n$. We consider first the case where $n=N^2$ for a square-free integer $N$. Then we show that if the Mordell–Weil group $E({\mathbb {Q}})$ has rank one, there exist at most 17 integer points on $E$. Moreover, we show that for some parameterized $N$ a certain point $P$ can be in a system of generators for $E( {\mathbb {Q}})$, and we determine the integer points in the group generated by the point $P$ and the torsion points. Secondly, we consider the case where $n=s^4+t^4$ for distinct positive integers $s$ and $t$. We then show that if $n$ is fourth-power-free, the points $P_1=(-t^2,s^2t)$ and $P_2=(-s^2,st^2)$ can be in a system of generators for $E( {\mathbb {Q}})$. Furthermore, we prove that if $n$ is square-free, then there exist at most nine integer points in the group $\varGamma $ generated by the points $P_1$, $P_2$ and the torsion point $(0,0)$. In particular, in case $n=s^4+1$ the group $\varGamma $ has exactly seven integer points.