On the Skolem problem and the Skolem conjecture - Lipton, RJ; Luca, L; Nieuwveld, J; Ouaknine, J; Purser, D; Worrell, J (Association for Computing Machinery, 2022)
It is a longstanding open problem whether there is an algorithm to decide the Skolem Problem for linear recurrence sequences (LRS) over the integers, namely whether a given such sequence has a zero term (i.e., whether un = 0 for some n). A major breakthrough in the early 1980s established decidability for LRS of order 4 or less, i.e., for LRS in which every new term depends linearly on the previous four (or fewer) terms. The Skolem Problem for LRS of order 5 or more, in particular, remains a major open challenge to this day.