On the Skolem Problem and the Skolem Conjecture

Abstract

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. Our main contributions in this paper are as follows: First, we show that the Skolem Problem is decidable for reversible LRS of order 7 or less. (An integer LRS is reversible if its unique extension to a bi-infinite LRS also takes exclusively integer values; a typical example is the classical Fibonacci sequence, whose bi-infinite extension is ⟨…, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, …⟩.) Second, assuming the Skolem Conjecture (a central hypothesis in Diophantine analysis, also known as the Exponential Local-Global Principle), we show that the Skolem Problem for LRS of order 5 is decidable, and exhibit a concrete procedure for solving it.