## 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 $$\geqslant 5$$.

### MSC:

 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
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### References:

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