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In mathematics, the Skolem problem is the problem of determining whether the values of a linear recurrence sequence include the number zero. The problem can be formulated for recurrences over different types of numbers, including integers, rational numbers, and algebraic numbers. It is an open problem whether there exists an algorithm that can solve this problem, with only special cases and conditional results known. The Skolem problem is named after Thoralf Skolem, due to his 1933 paper proving the Skolem–Mahler–Lech theorem for rational sequences, which describes the structure of the set of zeroes of a linear recurrence sequence.

There does exist an algorithm to test whether a linear recurrence 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. The remaining difficult part of the Skolem problem is determining whether the finite set of non-repeating zeros is empty or not.

Partial solutions to the Skolem problem are known, covering the special case of the problem for recurrences of degree at most four. However, these solutions do not apply to recurrences of degree five or more. For integer recurrences, the Skolem problem is known to be NP-hard.

The Skolem Problem can be seen as a linear version of the Halting problem, which is known to be undecidable. It has been said by Terrence Tao that it is "faintly outrageous that this problem is still open", as we do not yet know how to decide the Halting problem for even "linear" automata.

Motivation
A linear recurrence sequence (often shortened to LRS) is a sequence $$ \mathbf{u} = \langle u_n \rangle_{n=0}^\infty $$ satisfying a recurrence relation $$ u_{n+d} = a_{d-1} u_{n+d-1} + \dots + a_0 u_n $$and defined by initial values $$ u_0, \dots , u_{d-1} $$. If the initial values and the coefficients $$ a_0, \dots, a_{d-1} $$ lie in a ring $$ R $$ then we call $$ \mathbf u $$ an $$ R $$-LRS (for example, an integer-valued linear recurrence sequence is called a $$ \mathbb Z $$-LRS). The Skolem problem asks whether a given LRS $$ \mathbf u $$ has a zero, that is, whether there is an integer $$ n $$ such that $$ u_n = 0 $$. As an example,

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on the zeros of a sequence satisfying a linear recurrence with constant coefficients. This theorem states that, if such a sequence has zeros, then with finitely many exceptions the positions of the zeros repeat regularly. Skolem proved this for recurrences over the rational numbers, and Mahler and Lech extended it to other systems of numbers. However, the proofs of the theorem do not show how to test whether there exist any zeros (i.e. if any of these finite exceptions exist).

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