Elliptic curves containing sequences of consecutive cubes. (English) Zbl 1405.14081

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) and, for any \(P\in E\), denote its coordinates by \((x_P,y_P)\). The paper deals with curves containing sequences of points \(P_i\) whose \(x\)-coordinates are consecutive cubes in \(\mathbb{Q}\), i.e. such that \(x_{P_0}=c^3\) and \(x_{P_i}=(c+i)^3\) for \(i\geqslant 1\). It is easy to see that such sequences have to be finite and the authors use explicit computations to write down parametric families of elliptic curves containing sequences of 5 points with \(x\)-coordinate \((c+i)^3\) with \(-2\leqslant i\leqslant 2\). Moreover, using MAGMA (and Silverman’s specialization theorem), they also show that such points are independent and of infinite order so that all curves in the family have rank \(\geqslant 5\).


14H52 Elliptic curves
11B83 Special sequences and polynomials
11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
14G05 Rational points


ecdata; Magma
Full Text: DOI arXiv Euclid


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