Rocky Mountain Journal of Mathematics

Elliptic curves containing sequences of consecutive cubes

Gamze Savas Celik and Gokhan 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 $i=1, 2, \ldots , k$, 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 $5$.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2163-2174.

Dates
First available in Project Euclid: 14 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1544756805

Digital Object Identifier
doi:10.1216/RMJ-2018-48-7-2163

Mathematical Reviews number (MathSciNet)
MR3892128

Zentralblatt MATH identifier
06999258

Subjects
Primary: 14G05: Rational points
Secondary: 11B83: Special sequences and polynomials

Keywords
Elliptic curves rational points sequences of consecutive cubes

Citation

Celik, Gamze Savas; Soydan, Gokhan. Elliptic curves containing sequences of consecutive cubes. Rocky Mountain J. Math. 48 (2018), no. 7, 2163--2174. doi:10.1216/RMJ-2018-48-7-2163. https://projecteuclid.org/euclid.rmjm/1544756805


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