Divisibility Sequences and Powers of Algebraic Integers

Let $\a$ be an algebraic integer and define a sequence of rational integers $d_n(\a)$ by the condition \[ d_n(\a) = \max\{d\in\ZZ : \a^n \equiv 1 \MOD{d} \}. \] We show that $d_n(\a)$ is a strong divisibility sequence and that it satisfies $\log d_n(\a)=o(n)$ provided that no power of $\a$ is in $\ZZ$ and no power of $\a$ is a unit in a quadratic field. We completely analyze some of the exceptional cases by showing that $d_n(\a)$ splits into subsequences satisfying second order linear recurrences. Finally, we provide numerical evidence for the conjecture that aside from the exceptional cases, $d_n(\a)=d_1(\a)$ for infinitely many $n$, and we ask whether the set of such $n$ has postive (lower) density.

2000 Mathematics Subject Classification: Primary: 11R04; Secondary: 11A05, 11D61

Keywords and Phrases: divisibility sequence, multiplicative group

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