Occurrence Indices of Elements in Linear Recurrence Sequences over Primary Residue Rings

We study distances to the first occurrence (occurrence indices) of a given element in a linear recurrence sequence over a primary residue ring $\mathbb{Z}{p^n}$. We give conditions on the characteristic polynomial $F(x)$ of a linear recurrence sequence $u$ which guarantee that all elements of the ring occur in $u$. For the case where $F(x)$ is a reversible Galois polynomial over $\mathbb{Z}{p^n}$, we give upper bounds for occurrence indices of elements in a linear recurrence sequence $u$. A situation where the characteristic polynomial $F(x)$ of a linear recurrence sequence $u$ is a trinomial of a special form over $\mathbb{Z}_4$ is considered separately. In this case we give tight upper bounds for occurrence indices of elements of $u$.