Divisibility sequences and powers of algebraic integers

Abstract

Let $\alpha$ be an algebraic integer and define a sequence of rational integers $d_n(\alpha )$ by the condition

$d_n(\alpha )=\max \{d\in \mathbb{Z}:{\alpha }^n\equiv 1MODd\}.$

We show that $d_n(\alpha )$ is a strong divisibility sequence and that it satisfies $\log d_n(\alpha )=o(n)$ provided that no power of $\alpha$ is in $\mathbb{Z}$ and no power of $\alpha$ is a unit in a quadratic field. We completely analyze some of the exceptional cases by showing that $d_n(\alpha )$ splits into subsequences satisfying second order linear recurrences. Finally, we provide numerical evidence for the conjecture that aside from the exceptional cases, $d_n(\alpha )=d_1(\alpha )$ for infinitely many $n$, and we ask whether the set of such $n$ has positive (lower) density.