Decidability of the Positivity Problem for Linear Recurrence Sequences

Determine whether there exists an algorithm that, given an integer linear recurrence sequence of arbitrary order, decides whether all terms are nonnegative (i.e., u_n \geqslant 0 for all n \in \mathbb{N}); presently, decidability is established only for orders s \leqslant 5.

Background

The positivity problem asks, for a given LRS, whether all its terms are nonnegative. It is a natural modification of Skolem's problem and has important implications for algorithmic questions in number theory and computer science.

The paper notes that while the problem is decidable for small orders (up to 5), the general case has resisted resolution and remains an explicit uncertainty.