Divisibility sequences and powers of algebraic integers

  • Joseph H. Silverman


Let be an algebraic integer and define a sequence of rational integers by the condition

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