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