Families of cubic elliptic curves containing sequences of consecutive powers

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ defined by $y^2=a{x}^3+b{x}^2+cx+d$. We say a sequence of rational points $\left(x_i,y_i\right)\in E\left(\mathbb{Q}\right)$, $i=0,1,\dots ,\ell$, forms a sequence of consecutive $n$-th powers on $E$ of length $\ell$ whenever the sequence of $x$-coordinates, $x_i$, $i=0,1,\dots ,\ell$, consists of consecutive powers of degree $n$ in the form $x_i={\left(g+i\right)}^n$, for some rational $g$. Applying the known Mestre’s theorem, for an arbitrary natural number $n\ge 2$, we produce a one-parameter family of elliptic curves over $\mathbb{Q}$ which contains an $8$-term sequence of consecutive $n$-th powers. Furthermore, we show that for $n=2$ and $3$ the associated families of elliptic curves are of generic rank $\ge 6$ and $7$, respectively. We also provide an explicit set of linearly independent points for those families. Finally, according to our limited trial conducted for $4\le n\le 50$, we discovered that the generic rank of the corresponding families is $\ge 7$. We guess that this holds for all $n\ge 4$; however, we are not able to prove it at this time.