Conjectural Decidability of the Skolem Problem

2026-07-16

AI Summary

This work addresses the long-standing Skolem problem—determining whether a given integer linear recurrence sequence contains a zero term—which has profound implications for program termination analysis and related fields. Introducing the notion of “large zeros,” defined as zeros occurring beyond doubly exponential indices, the paper combines tools from analytic number theory and algebraic methods to define a set of “good primes” and demonstrates their anomalous sparsity in neighborhoods of large zeros. Unconditionally, it proves that the set of large zeros has natural density zero, thereby constructing a universal Skolem set of density one. Moreover, under a strengthened form of Cramér’s conjecture, the paper establishes the nonexistence of large zeros altogether, which implies the decidability of the Skolem problem.

Abstract

The Skolem Problem asks to determine whether a given integer linear recurrence sequence (LRS) has a zero term. This problem, whose decidability has been open for many decades, arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and probabilistic model checking, amongst many others. In the present paper, we introduce a notion of "large" zeros of (non-degenerate) linear recurrence sequences, i.e., zeros occurring at an index larger than a double exponential of the magnitude of the data defining the given LRS. We establish two main results. First, we define an infinite set of prime numbers, termed "good", having density one amongst all prime numbers, with the following property: for any large zero of a given LRS, there is an interval around the large zero together with an upper bound on the number of good primes possibly present in that interval. The bound in question is much lower than one would expect if good primes were distributed similarly as ordinary prime numbers, as per the Cramér model in number theory. We therefore conclude, conditionally on a strengthening of the classical Cramér conjecture, that large zeros do not exist, which would entail decidability of the Skolem Problem. Second, we show unconditionally that large zeros are very sparse: the set of positive integers that can possibly arise as large zeros of some LRS has null density. This in turn immediately yields a Universal Skolem Set of density one, answering a question left open in the literature.

Problem

• Skolem Problem
• linear recurrence sequence
• decidability
• zero term
• number theory

Innovation

• Skolem Problem
• linear recurrence sequences
• large zeros
• good primes
• Cramér conjecture