Journal article Open Access

Two-Parameter Generalization of the Collatz Function: Characterization of Terminal Cycles and Empirical Results

Boris B. Stefanov


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  <identifier identifierType="DOI">10.5281/zenodo.4609709</identifier>
  <creators>
    <creator>
      <creatorName>Boris B. Stefanov</creatorName>
      <nameIdentifier nameIdentifierScheme="ORCID" schemeURI="http://orcid.org/">0000-0002-1824-0980</nameIdentifier>
      <affiliation>Aisthon Ltd., Hellertown, Pennsylvania 18055, USA</affiliation>
    </creator>
  </creators>
  <titles>
    <title>Two-Parameter Generalization of the Collatz Function: Characterization of Terminal Cycles and Empirical Results</title>
  </titles>
  <publisher>Zenodo</publisher>
  <publicationYear>2021</publicationYear>
  <subjects>
    <subject>Collatz conjecture</subject>
    <subject>3x+1 problem</subject>
    <subject>iteration</subject>
    <subject>convergence</subject>
    <subject>terminal cycles</subject>
  </subjects>
  <dates>
    <date dateType="Issued">2021-03-17</date>
  </dates>
  <language>en</language>
  <resourceType resourceTypeGeneral="JournalArticle"/>
  <alternateIdentifiers>
    <alternateIdentifier alternateIdentifierType="url">https://zenodo.org/record/4609709</alternateIdentifier>
  </alternateIdentifiers>
  <relatedIdentifiers>
    <relatedIdentifier relatedIdentifierType="DOI" relationType="IsVersionOf">10.5281/zenodo.4609708</relatedIdentifier>
    <relatedIdentifier relatedIdentifierType="URL" relationType="IsPartOf">https://zenodo.org/communities/omj</relatedIdentifier>
  </relatedIdentifiers>
  <rightsList>
    <rights rightsURI="https://creativecommons.org/licenses/by/4.0/legalcode">Creative Commons Attribution 4.0 International</rights>
    <rights rightsURI="info:eu-repo/semantics/openAccess">Open Access</rights>
  </rightsList>
  <descriptions>
    <description descriptionType="Abstract">&lt;p&gt;This paper proposes a new two-parameter generalization T_{b,s}(x) of the Z -&amp;gt; Z Collatz function T(x) and restates the eponymous conjecture in terms of the proposed function. The generalization obviates some of the conditions for emergence of terminal cycles for the Collatz T(x) function over the integers. The stopping behavior of the T_{b,s}(x) is qualitatively similar to that of the T(x). &amp;nbsp;The paper presents theoretical discussion of the generalization and computational results on the terminal cycles and stopping times of T_{b,s}(x). The {1,2} cycle of T(x) is shown to be a case of coincidence of three independent cycle categories of T_{b,s}(x).&lt;/p&gt;</description>
  </descriptions>
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