Elliptic Curves Containing Sequences of Consecutive Cubes

TL;DR

This paper demonstrates the existence of infinitely many elliptic curves over rationals that contain a sequence of five consecutive cubes as rational points, with these points being linearly independent, implying the rank is at least five.

Contribution

It constructs an infinite family of elliptic curves with five rational points forming a consecutive cubes sequence, showing these points are linearly independent.

Findings

Existence of infinite family of elliptic curves with 5 consecutive cubes

The 5 points are linearly independent on these curves

The rank of these curves is at least 5

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 $5$.