Elliptic curves containing sequences of consecutive cubes

Abstract

Let $E$ be an elliptic curve over $\mathbb {Q}$ described by $y^2= x^3+ Kx+ L$, where $K, L \in \mathbb {Q}$. A set of rational points $(x_i,y_i) \in E(\mathbb {Q})$ for $i=1, 2, \ldots , k$, is said to be a sequence of consecutive cubes on $E$ if the $x$-coordinates of the points $x_i$'s for $i=1, 2,\ldots $, form consecutive cubes. In this note, we show the existence of an infinite family of elliptic curves containing a length-$5$-term sequence of consecutive cubes. Moreover, these five rational points in $E (\mathbb {Q})$ are linearly independent, and the rank $r$ of $E(\mathbb {Q})$ is at least $5$.