[1] A linear recurrence relation expresses the values of a sequence of numbers as a linear combination of earlier values; for instance, the Fibonacci numbers may be defined from the recurrence relation together with the initial values F(0) = 0 and F(1) = 1.
The Skolem problem is named after Thoralf Skolem, because of his 1933 paper proving the Skolem–Mahler–Lech theorem on the zeros of a sequence satisfying a linear recurrence with constant coefficients.
However, the proofs of the theorem do not show how to test whether there exist any zeros.
There does exist an algorithm to test whether a constant-recursive sequence has infinitely many zeros, and if so to construct a decomposition of the positions of those zeros into periodic subsequences, based on the algebraic properties of the roots of the characteristic polynomial of the given recurrence.
[3] The remaining difficult part of the Skolem problem is determining whether the finite set of non-repeating zeros is empty or not.