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