Two infinite families of elliptic curves with rank greater than one

Abstract

We prove, using elementary methods, that each member of the infinite families of elliptic curves given by $E_m:y^2=x^3-x+m^6$ and $E_m^{\prime }:y^2=x^3+x-m^6$ have rank at least $2$ and 3, respectively, under mild restrictions on $m$. We also prove stronger results for $E_m$ and $E_m^{\prime }$ using more technical machinery.

To appear in Integers