Divisibility sequences and powers of algebraic integers

  • Joseph H. Silverman


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

dn(α)=max{dZ:αn1MODd}.d_n(\alpha) = \max\{d\in\mathbb{Z} : \alpha^n \equiv 1 {MOD}{d} \}.

We show that dn(α)d_n(\alpha) is a strong divisibility sequence and that it satisfies logdn(α)=o(n)\log d_n(\alpha)=o(n) provided that no power of α\alpha is in Z\mathbb{Z} and no power of α\alpha is a unit in a quadratic field. We completely analyze some of the exceptional cases by showing that dn(α)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, dn(α)=d1(α)d_n(\alpha)=d_1(\alpha) for infinitely many nn, and we ask whether the set of such nn has positive (lower) density.