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

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

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.