Divisibility Sequences and Powers of Algebraic Integers

Let α be an algebraic integer and define a sequence of rational integers dn(α) by the condition dn(α) = max{d ∈ Z : α ≡ 1 (mod d)}. We show that dn(α) is a strong divisibility sequence and that it satisfies log dn(α) = o(n) provided that no power of α is in Z and no power of α is a unit in a quadratic field. We completely analyze some of the exceptional cases by showing that dn(α) splits into subsequences satisfying second order linear recurrences. Finally, we provide numerical evidence for the conjecture that aside from the exceptional cases, dn(α) = d1(α) 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