The multiplicity of binary recurrences

Abstract

of the recurrence relation (0.1). It follows from the Skolem–Mahler–Lech Theorem that if U =∞ then one at least of ; and = is a root of unity. Consequently we call {um} nondegenerate if none of these quantities is a root of unity. In the thirties M. Ward conjectured that a nondegenerate binary recurrence sequence of integers um has multiplicity U 5 5. This conjecture was proved by Kubota [4]. Kubota also proved that any nondegenerate binary recurrence {um} with elements um in a number eld K of degree d has multiplicity U 5 c(d) (cf. [5]). Beukers and Tijdeman [2] established for this case the bound