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2018 Elliptic curves containing sequences of consecutive cubes
Gamze Savas Celik, Gokhan Soydan
Rocky Mountain J. Math. 48(7): 2163-2174 (2018). DOI: 10.1216/RMJ-2018-48-7-2163

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 $5$.

Citation

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Gamze Savas Celik. Gokhan Soydan. "Elliptic curves containing sequences of consecutive cubes." Rocky Mountain J. Math. 48 (7) 2163 - 2174, 2018. https://doi.org/10.1216/RMJ-2018-48-7-2163

Information

Published: 2018
First available in Project Euclid: 14 December 2018

zbMATH: 06999258
MathSciNet: MR3892128
Digital Object Identifier: 10.1216/RMJ-2018-48-7-2163

Subjects:
Primary: 14G05
Secondary: 11B83

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 7 • 2018
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