Let $P$ and $Q$ be two points on an elliptic curve defined over a number field $K$. For $\alpha\in \text{ End}(E)$, define $B_\alpha$ to be the $\mathcal{ O}_K$-integral ideal generated by the denominator of $x(\alpha(P)+Q)$. Let $\mathcal{ O}$ be a subring of $\text{ End}(E)$, that is a Dedekind domain. We will study the sequence $\{ B_\alpha\}_{ \alpha\in \mathcal{ O}}$. We will show that, for all but finitely many $\alpha\in \mathcal{ O}$, the ideal $B_\alpha$ has a primitive divisor when $P$ is a non-torsion point and there exist two endomorphisms $g\neq 0$ and $f$ so that $f(P)=g(Q)$. This is a generalization of previous results on elliptic divisibility sequences.
Categories: math.NT