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 $i=1,2,\cdots ,k$, is said to be a sequence of consecutive cubes on $E$ if the $x-$coordinates of the points $x_i$'s for $i=1,2,\cdots$ 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. Morever, these five rational points in $E(\mathbb{Q})$ are linearly independent and the rank $r$ of $E(\mathbb{Q})$ is at least $5$.

10 pages, to appear, Rocky Mountain Journal of Mathematics (2018)