  Let $P$ be a non-torsion point on the elliptic curve $E_{a}: y^{2}=x^{3}+ax$.
We show that if $a$ is fourth-power-free and either $n>2$ is even or $n>1$ is
odd with $x(P)<0$ or $x(P)$ a perfect square, then the $n$-th element of the
elliptic divisibility sequence generated by $P$ always has a primitive divisor.
