The problem is decidable when the linear recurrence equation has order at most four, but the general case remains open.
Order five is the first unresolved order (Lipton et al. 2022). The Skolem-Mahler-Lech
theorem shows that the indices at which it vanishes consist of finitely many
exceptional indices together with finitely many full arithmetic
progressions, but it does not provide a general effective method for finding
the exceptional indices.
Lipton, R. J.; Luca, F.; Nieuwveld, J.; Ouaknine, J.; Purser, D.; and Worrell, J. "On the Skolem Problem and the Skolem Conjecture."
In LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer
Science. (Ed. C. Baier and D. Fisman). New York: ACM, pp. 5:1-5:9,
2022. https://doi.org/10.1145/3531130.3533328.Ouaknine,
J. and Worrell, J. "Decision Problems for Linear Recurrence Sequences."
In Reachability Problems: 6th International Workshop, RP 2012 (Ed. A. Finkel,
J. Leroux, and I. Potapov). Berlin, Germany: Springer, pp. 21-28,
2012. https://doi.org/10.1007/978-3-642-33512-9_3.
with 
in the linear
recurrence equation
for some nonnegative integer 
.
