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A Universal Skolem Set of Positive Lower Density

Authors Florian Luca , Joël Ouaknine , James Worrell

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Subject Classification

ACM Subject Classification
  • Computing methodologies → Symbolic and algebraic algorithms
  • Computing methodologies → Number theory algorithms
Keywords
  • Linear Recurrence Sequences
  • Skolem Problem
  • Exponential Diophantine Equations
  • Sieve Methods

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Abstract

The Skolem Problem asks to decide whether a given integer linear recurrence sequence (LRS) has a zero term. Decidability of this problem has been open for many decades, with little progress since the 1980s. Recently, a new approach was initiated via the notion of a Skolem set - a set of positive integers relative to which the Skolem Problem is decidable. More precisely, 𝒮 is a Skolem set for a class ℒ of integer LRS if there is an effective procedure that, given an LRS in ℒ, decides whether the sequence has a zero in 𝒮. A recent work exhibited a Skolem set for the class of all LRS that, while infinite, had density zero. In the present work we construct a Skolem set of positive lower density for the class of simple LRS .

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Florian Luca, Joël Ouaknine, and James Worrell. A Universal Skolem Set of Positive Lower Density. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 73:1-73:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.MFCS.2022.73

Author Details

Florian Luca
  • School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
  • Research Group in Algebraic Structures & Applications, King Abdulaziz University, Saudi Arabia
  • Centro de Ciencias Matemáticas UNAM, Morelia, Mexico
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Germany
James Worrell
  • Department of Computer Science, University of Oxford, UK

Funding

  • Ouaknine, Joël: Also affiliated with Keble College, Oxford as http://emmy.network/ Fellow, and supported by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).

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