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arXiv · 1806.01158

Elliptic Curves Containing Sequences of Consecutive Cubes

Gamze Savaş Çelik·Gökhan Soydan
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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 $5$.

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Gamze Savaş Çelik, Gökhan Soydan. 2018-06-04. Elliptic Curves Containing Sequences of Consecutive Cubes. https://arxiv.org/abs/1806.01158

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