Collatz conjecture

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href="#Visualizations"> <div class="vector-toc-text"> 3 Visualizations </div> </a> <ul id="toc-Visualizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Supporting_arguments" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Supporting_arguments"> <div class="vector-toc-text"> 4 Supporting arguments </div> </a> <button aria-controls="toc-Supporting_arguments-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Supporting arguments subsection </button> <ul id="toc-Supporting_arguments-sublist" class="vector-toc-list"> <li id="toc-Experimental_evidence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Experimental_evidence"> <div class="vector-toc-text"> 4.1 Experimental evidence </div> </a> <ul id="toc-Experimental_evidence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_probabilistic_heuristic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_probabilistic_heuristic"> <div class="vector-toc-text"> 4.2 A probabilistic heuristic </div> </a> <ul id="toc-A_probabilistic_heuristic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stopping_times" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stopping_times"> <div class="vector-toc-text"> 4.3 Stopping times </div> </a> <ul id="toc-Stopping_times-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lower_bounds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lower_bounds"> <div class="vector-toc-text"> 4.4 Lower bounds </div> </a> <ul id="toc-Lower_bounds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cycles" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Cycles"> <div class="vector-toc-text"> 5 Cycles </div> </a> <button aria-controls="toc-Cycles-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Cycles subsection </button> <ul id="toc-Cycles-sublist" class="vector-toc-list"> <li id="toc-Cycle_length" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cycle_length"> <div class="vector-toc-text"> 5.1 Cycle length </div> </a> <ul id="toc-Cycle_length-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-k-cycles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#k-cycles"> <div class="vector-toc-text"> 5.2 k-cycles </div> </a> <ul id="toc-k-cycles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_formulations_of_the_conjecture" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_formulations_of_the_conjecture"> <div class="vector-toc-text"> 6 Other formulations of the conjecture </div> </a> <button aria-controls="toc-Other_formulations_of_the_conjecture-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Other formulations of the conjecture subsection </button> <ul id="toc-Other_formulations_of_the_conjecture-sublist" class="vector-toc-list"> <li id="toc-In_reverse" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_reverse"> <div class="vector-toc-text"> 6.1 In reverse </div> </a> <ul id="toc-In_reverse-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_an_abstract_machine_that_computes_in_base_two" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_an_abstract_machine_that_computes_in_base_two"> <div class="vector-toc-text"> 6.2 As an abstract machine that computes in base two </div> </a> <ul id="toc-As_an_abstract_machine_that_computes_in_base_two-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> 6.2.1 Example </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-As_a_parity_sequence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_parity_sequence"> <div class="vector-toc-text"> 6.3 As a parity sequence </div> </a> <ul id="toc-As_a_parity_sequence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_tag_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_tag_system"> <div class="vector-toc-text"> 6.4 As a tag system </div> </a> <ul id="toc-As_a_tag_system-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions_to_larger_domains" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extensions_to_larger_domains"> <div class="vector-toc-text"> 7 Extensions to larger domains </div> </a> <button aria-controls="toc-Extensions_to_larger_domains-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Extensions to larger domains subsection </button> <ul id="toc-Extensions_to_larger_domains-sublist" class="vector-toc-list"> <li id="toc-Iterating_on_all_integers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Iterating_on_all_integers"> <div class="vector-toc-text"> 7.1 Iterating on all integers </div> </a> <ul id="toc-Iterating_on_all_integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Iterating_on_rationals_with_odd_denominators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Iterating_on_rationals_with_odd_denominators"> <div class="vector-toc-text"> 7.2 Iterating on rationals with odd denominators </div> </a> <ul id="toc-Iterating_on_rationals_with_odd_denominators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-2-adic_extension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#2-adic_extension"> <div class="vector-toc-text"> 7.3 2-adic extension </div> </a> <ul id="toc-2-adic_extension-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Iterating_on_real_or_complex_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Iterating_on_real_or_complex_numbers"> <div class="vector-toc-text"> 7.4 Iterating on real or complex numbers </div> </a> <ul id="toc-Iterating_on_real_or_complex_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Optimizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Optimizations"> <div class="vector-toc-text"> 8 Optimizations </div> </a> <button aria-controls="toc-Optimizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Optimizations subsection </button> <ul id="toc-Optimizations-sublist" class="vector-toc-list"> <li id="toc-Time–space_tradeoff" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time–space_tradeoff"> <div class="vector-toc-text"> 8.1 Time–space tradeoff </div> </a> <ul id="toc-Time–space_tradeoff-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modular_restrictions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modular_restrictions"> <div class="vector-toc-text"> 8.2 Modular restrictions </div> </a> <ul id="toc-Modular_restrictions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Syracuse_function" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Syracuse_function"> <div class="vector-toc-text"> 9 Syracuse function </div> </a> <ul id="toc-Syracuse_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Undecidable_generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Undecidable_generalizations"> <div class="vector-toc-text"> 10 Undecidable generalizations </div> </a> <ul id="toc-Undecidable_generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_computational_complexity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_computational_complexity"> <div class="vector-toc-text"> 11 In computational complexity </div> </a> <ul id="toc-In_computational_complexity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 12 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 13 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 14 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 15 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Collatz conjecture</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 38 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-38" aria-hidden="true" > 38 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%AF%D8%B3%D9%8A%D8%A9_%D9%83%D9%88%D9%84%D8%A7%D8%AA%D8%B2" title="حدسية كولاتز – Arabic" lang="ar" hreflang="ar" data-title="حدسية كولاتز" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%B2%E0%A6%BE%E0%A6%9F%E0%A6%B8_%E0%A6%85%E0%A6%A8%E0%A7%81%E0%A6%AE%E0%A6%BE%E0%A6%A8" title="কোলাটস অনুমান – Bangla" lang="bn" hreflang="bn" data-title="কোলাটস অনুমান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Conjectura_de_Collatz" title="Conjectura de Collatz – Catalan" lang="ca" hreflang="ca" data-title="Conjectura de Collatz" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Collatz%C5%AFv_probl%C3%A9m" title="Collatzův problém – Czech" lang="cs" hreflang="cs" data-title="Collatzův problém" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Collatz-formodningen" title="Collatz-formodningen – Danish" lang="da" hreflang="da" data-title="Collatz-formodningen" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Collatz-Problem" title="Collatz-Problem – German" lang="de" hreflang="de" data-title="Collatz-Problem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%B9%CE%BA%CE%B1%CF%83%CE%AF%CE%B1_%CF%84%CE%BF%CF%85_%CE%9A%CF%8C%CE%BB%CE%B1%CF%84%CE%B6" title="Εικασία του Κόλατζ – Greek" lang="el" hreflang="el" data-title="Εικασία του Κόλατζ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Conjetura_de_Collatz" title="Conjetura de Collatz – Spanish" lang="es" hreflang="es" data-title="Conjetura de Collatz" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Konjekto_de_Collatz" title="Konjekto de Collatz – Esperanto" lang="eo" hreflang="eo" data-title="Konjekto de Collatz" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Collatzen_aierua" title="Collatzen aierua – Basque" lang="eu" hreflang="eu" data-title="Collatzen aierua" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%AF%D8%B3_%DA%A9%D9%88%D9%84%D8%A7%D8%AA%D8%B2" title="حدس کولاتز – Persian" lang="fa" hreflang="fa" data-title="حدس کولاتز" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Conjecture_de_Syracuse" title="Conjecture de Syracuse – French" lang="fr" hreflang="fr" data-title="Conjecture de Syracuse" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%BD%9C%EB%9D%BC%EC%B8%A0_%EC%B6%94%EC%B8%A1" title="콜라츠 추측 – Korean" lang="ko" hreflang="ko" data-title="콜라츠 추측" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%B2%E0%A4%BE%E0%A4%9C_%E0%A4%85%E0%A4%9F%E0%A4%95%E0%A4%B2" title="कोलाज अटकल – Hindi" lang="hi" hreflang="hi" data-title="कोलाज अटकल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Konjektur_Collatz" title="Konjektur Collatz – Indonesian" lang="id" hreflang="id" data-title="Konjektur Collatz" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Tilg%C3%A1ta_Collatz" title="Tilgáta Collatz – Icelandic" lang="is" hreflang="is" data-title="Tilgáta Collatz" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Congettura_di_Collatz" title="Congettura di Collatz – Italian" lang="it" hreflang="it" data-title="Congettura di Collatz" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A9%D7%A2%D7%A8%D7%AA_%D7%A7%D7%95%D7%9C%D7%A5" title="השערת קולץ – Hebrew" lang="he" hreflang="he" data-title="השערת קולץ" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Konjekti_Sirakiz" title="Konjekti Sirakiz – Haitian Creole" lang="ht" hreflang="ht" data-title="Konjekti Sirakiz" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Congettura_de_Collatz" title="Congettura de Collatz – Lombard" lang="lmo" hreflang="lmo" data-title="Congettura de Collatz" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Collatz-sejt%C3%A9s" title="Collatz-sejtés – Hungarian" lang="hu" hreflang="hu" data-title="Collatz-sejtés" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vermoeden_van_Collatz" title="Vermoeden van Collatz – Dutch" lang="nl" hreflang="nl" data-title="Vermoeden van Collatz" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B3%E3%83%A9%E3%83%83%E3%83%84%E3%81%AE%E5%95%8F%E9%A1%8C" title="コラッツの問題 – Japanese" lang="ja" hreflang="ja" data-title="コラッツの問題" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Collatz%E2%80%99_formodning" title="Collatz’ formodning – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Collatz’ formodning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Problem_Collatza" title="Problem Collatza – Polish" lang="pl" hreflang="pl" data-title="Problem Collatza" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Conjectura_de_Collatz" title="Conjectura de Collatz – Portuguese" lang="pt" hreflang="pt" data-title="Conjectura de Collatz" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%BE%D1%82%D0%B5%D0%B7%D0%B0_%D0%9A%D0%BE%D0%BB%D0%BB%D0%B0%D1%82%D1%86%D0%B0" title="Гипотеза Коллатца – Russian" lang="ru" hreflang="ru" data-title="Гипотеза Коллатца" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Collatz_conjecture" title="Collatz conjecture – Simple English" lang="en-simple" hreflang="en-simple" data-title="Collatz conjecture" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Collatzeva_domneva" title="Collatzeva domneva – Slovenian" lang="sl" hreflang="sl" data-title="Collatzeva domneva" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D0%B0%D1%86%D0%BE%D0%B2%D0%B0_%D1%85%D0%B8%D0%BF%D0%BE%D1%82%D0%B5%D0%B7%D0%B0" title="Колацова хипотеза – Serbian" lang="sr" hreflang="sr" data-title="Колацова хипотеза" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Collatzin_konjektuuri" title="Collatzin konjektuuri – Finnish" lang="fi" hreflang="fi" data-title="Collatzin konjektuuri" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Collatz_problem" title="Collatz problem – Swedish" lang="sv" hreflang="sv" data-title="Collatz problem" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Collatz_san%C4%B1s%C4%B1" title="Collatz sanısı – Turkish" lang="tr" hreflang="tr" data-title="Collatz sanısı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%96%D0%BF%D0%BE%D1%82%D0%B5%D0%B7%D0%B0_%D0%9A%D0%BE%D0%BB%D0%BB%D0%B0%D1%82%D1%86%D0%B0" title="Гіпотеза Коллатца – Ukrainian" lang="uk" hreflang="uk" data-title="Гіпотеза Коллатца" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gi%E1%BA%A3_thuy%E1%BA%BFt_Collatz" title="Giả thuyết Collatz – Vietnamese" lang="vi" hreflang="vi" data-title="Giả thuyết Collatz" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%80%83%E6%8B%89%E8%8C%B2%E7%8C%9C%E6%83%B3" title="考拉茲猜想 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="考拉茲猜想" data-language-autonym="文言" 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2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div>Unsolved problem in mathematics:</div> <div class="unsolved-body"><div><ul><li>For even numbers, divide by 2;</li><li>For odd numbers, multiply by 3 and add 1.</li></ul></div>With enough repetition, do all positive integers converge to 1?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Collatz-graph-50-no27.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Collatz-graph-50-no27.svg/130px-Collatz-graph-50-no27.svg.png" decoding="async" width="130" height="346" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Collatz-graph-50-no27.svg/195px-Collatz-graph-50-no27.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Collatz-graph-50-no27.svg/260px-Collatz-graph-50-no27.svg.png 2x" data-file-width="393" data-file-height="1045" /></a><figcaption><a href="/wiki/Directed_graph" title="Directed graph">Directed graph</a> showing the <a href="/wiki/Orbit_(dynamics)" title="Orbit (dynamics)">orbits</a> of small numbers under the Collatz map, skipping even numbers. The Collatz conjecture states that all paths eventually lead to 1.</figcaption></figure> The Collatz conjecture<a href="#cite_note-4">[a]</a> is one of the most famous <a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">unsolved problems in mathematics</a>. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every <a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a> into 1. It concerns <a href="/wiki/Integer_sequence" title="Integer sequence">sequences of integers</a> in which each term is obtained from the previous term as follows: if a term is <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a>, the next term is one half of it. If a term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. The conjecture has been shown to hold for all positive integers up to 2.95×1020, but no general proof has been found. It is named after the mathematician <a href="/wiki/Lothar_Collatz" title="Lothar Collatz">Lothar Collatz</a>, who introduced the idea in 1937, two years after receiving his doctorate.<a href="#cite_note-5">[4]</a> The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like <a href="/wiki/Hailstones" class="mw-redirect" title="Hailstones">hailstones</a> in a cloud),<a href="#cite_note-6">[5]</a> or as wondrous numbers.<a href="#cite_note-7">[6]</a> <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> said about the Collatz conjecture: "Mathematics may not be ready for such problems."<a href="#cite_note-Guy_(2004)-8">[7]</a> <a href="/wiki/Jeffrey_Lagarias" title="Jeffrey Lagarias">Jeffrey Lagarias</a> stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics".<a href="#cite_note-Lagarias_(2010)-9">[8]</a> However, though the Collatz conjecture itself remains open, efforts to solve the problem have led to new techniques and many partial results.<a href="#cite_note-Lagarias_(2010)-9">[8]</a><a href="#cite_note-Tao-10">[9]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Statement_of_the_problem">Statement of the problem</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Collatz-stopping-time.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Collatz-stopping-time.svg/220px-Collatz-stopping-time.svg.png" decoding="async" width="220" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Collatz-stopping-time.svg/330px-Collatz-stopping-time.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Collatz-stopping-time.svg/440px-Collatz-stopping-time.svg.png 2x" data-file-width="570" data-file-height="590" /></a><figcaption>Numbers from 1 to 9999 and their corresponding total stopping time</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CollatzStatistic100million.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/CollatzStatistic100million.png/220px-CollatzStatistic100million.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/CollatzStatistic100million.png/330px-CollatzStatistic100million.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/CollatzStatistic100million.png/440px-CollatzStatistic100million.png 2x" data-file-width="1921" data-file-height="964" /></a><figcaption>Histogram of total stopping times for the numbers 1 to 108. Total stopping time is on the x axis, frequency on the y axis.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CollatzStatistic1billion.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/CollatzStatistic1billion.png/220px-CollatzStatistic1billion.png" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/CollatzStatistic1billion.png/330px-CollatzStatistic1billion.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/CollatzStatistic1billion.png/440px-CollatzStatistic1billion.png 2x" data-file-width="1504" data-file-height="940" /></a><figcaption>Histogram of total stopping times for the numbers 1 to 109. Total stopping time is on the x axis, frequency on the y axis.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Collatz-10Million.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Collatz-10Million.png/220px-Collatz-10Million.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Collatz-10Million.png/330px-Collatz-10Million.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Collatz-10Million.png/440px-Collatz-10Million.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption>Iteration time for inputs of 2 to 107.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Collatz_Gif.gif" class="mw-file-description"><img alt="Total Stopping Time: numbers up to 250, 1000, 4000, 20000, 100000, 500000" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Collatz_Gif.gif/220px-Collatz_Gif.gif" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Collatz_Gif.gif/330px-Collatz_Gif.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Collatz_Gif.gif/440px-Collatz_Gif.gif 2x" data-file-width="982" data-file-height="715" /></a><figcaption>Total stopping time of numbers up to 250, 1000, 4000, 20000, 100000, 500000</figcaption></figure> Consider the following operation on an arbitrary <a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a>: <ul><li>If the number is even, divide it by two.</li> <li>If the number is odd, triple it and add one.</li></ul> In <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> notation, define the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> f as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)={\begin{cases}n/2&{\text{if }}n\equiv 0{\pmod {2}},\\[4px]3n+1&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)={\begin{cases}n/2&{\text{if }}n\equiv 0{\pmod {2}},\\[4px]3n+1&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b2a03faf9d31a8de0abb3c4a3d318490105da12" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.886ex; height:7.509ex;" alt="{\displaystyle f(n)={\begin{cases}n/2&{\text{if }}n\equiv 0{\pmod {2}},\\[4px]3n+1&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}"> Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next. In notation: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}={\begin{cases}n&{\text{for }}i=0,\\f(a_{i-1})&{\text{for }}i>0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>n</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>i</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}={\begin{cases}n&{\text{for }}i=0,\\f(a_{i-1})&{\text{for }}i>0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f220f8b8d9aaa456552e64310e8fbe65e356718" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.24ex; height:6.176ex;" alt="{\displaystyle a_{i}={\begin{cases}n&{\text{for }}i=0,\\f(a_{i-1})&{\text{for }}i>0\end{cases}}}"> (that is: ai is the value of f applied to n recursively i times; ai = f i(n)). The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially. That is, for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, there is some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6240a76a2092902c5f04b31c60b8ff185eec3310" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}=1}">. If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. The smallest i such that ai < a0 is called the stopping time of n. Similarly, the smallest k such that ak = 1 is called the total stopping time of n.<a href="#cite_note-Lagarias_(1985)-2">[2]</a> If one of the indexes i or k doesn't exist, we say that the stopping time or the total stopping time, respectively, is infinite. The Collatz conjecture asserts that the total stopping time of every n is finite. It is also equivalent to saying that every n ≥ 2 has a finite stopping time. Since 3n + 1 is even whenever n is odd, one may instead use the "shortcut" form of the Collatz function: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae238aa62598cce67c57371012b818b65d1ad6e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.146ex; margin-bottom: -0.192ex; width:35.071ex; height:7.843ex;" alt="{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}"> This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process. <div class="mw-heading mw-heading2"><h2 id="Empirical_data">Empirical data</h2></div> For instance, starting with n = 12 and applying the function f without "shortcut", one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1 . The number n = 19 takes longer to reach 1: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 . The sequence for n = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1. <dl><dd><big>27</big>, 82, <big>41</big>, 124, 62, <big>31</big>, 94, <big>47</big>, 142, <big>71</big>, 214, <big>107</big>, 322, <big>161</big>, 484, 242, <big>121</big>, 364, 182, <big>91</big>, 274, <big>137</big>, 412, 206, <big>103</big>, 310, <big>155</big>, 466, <big>233</big>, 700, 350, <big>175</big>, 526, <big>263</big>, 790, <big>395</big>, 1186, <big>593</big>, 1780, 890, <big>445</big>, 1336, 668, 334, <big>167</big>, 502, <big>251</big>, 754, <big>377</big>, 1132, 566, <big>283</big>, 850, <big>425</big>, 1276, 638, <big>319</big>, 958, <big>479</big>, 1438, <big>719</big>, 2158, <big>1079</big>, 3238, <big>1619</big>, 4858, <big>2429</big>, 7288, 3644, 1822, <big>911</big>, 2734, <big>1367</big>, 4102, <big>2051</big>, 6154, <big>3077</big>, 9232, 4616, 2308, 1154, <big>577</big>, 1732, 866, <big>433</big>, 1300, 650, <big>325</big>, 976, 488, 244, 122, <big>61</big>, 184, 92, 46, <big>23</big>, 70, <big>35</big>, 106, <big>53</big>, 160, 80, 40, 20, 10, <big>5</big>, 16, 8, 4, 2, <big>1</big></dd></dl> (sequence <a href="//oeis.org/A008884" class="extiw" title="oeis:A008884">A008884</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) <figure class="mw-default-size mw-halign-center" typeof="mw:File/Frameless"><a href="/wiki/File:Collatz5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Collatz5.svg/440px-Collatz5.svg.png" decoding="async" width="440" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Collatz5.svg/660px-Collatz5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Collatz5.svg/880px-Collatz5.svg.png 2x" data-file-width="360" data-file-height="180" /></a><figcaption></figcaption></figure> Numbers with a total stopping time longer than that of any smaller starting value form a sequence beginning with: <dl><dd>1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, ... (sequence <a href="//oeis.org/A006877" class="extiw" title="oeis:A006877">A006877</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> The starting values whose <a href="/wiki/Maximum" class="mw-redirect" title="Maximum">maximum</a> trajectory point is greater than that of any smaller starting value are as follows: <dl><dd>1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, ... (sequence <a href="//oeis.org/A006884" class="extiw" title="oeis:A006884">A006884</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> Number of steps for n to reach 1 are <dl><dd>0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106, 5, 26, 13, 13, 21, 21, 21, 34, 8, 109, 8, 29, 16, 16, 16, 104, 11, 24, 24, ... (sequence <a href="//oeis.org/A006577" class="extiw" title="oeis:A006577">A006577</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> The starting value having the largest total stopping time while being <dl><dd>less than 10 is 9, which has 19 steps,</dd> <dd>less than 100 is 97, which has 118 steps,</dd> <dd>less than 1000 is 871, which has 178 steps,</dd> <dd>less than 104 is 6171, which has 261 steps,</dd> <dd>less than 105 is 77031, which has 350 steps,</dd> <dd>less than 106 is 837799, which has 524 steps,</dd> <dd>less than 107 is 8400511, which has 685 steps,</dd> <dd>less than 108 is 63728127, which has 949 steps,</dd> <dd>less than 109 is 670617279, which has 986 steps,</dd> <dd>less than 1010 is 9780657630, which has 1132 steps,<a href="#cite_note-11">[10]</a></dd> <dd>less than 1011 is 75128138247, which has 1228 steps,</dd> <dd>less than 1012 is 989345275647, which has 1348 steps.<a href="#cite_note-Roosendaal-12">[11]</a> (sequence <a href="//oeis.org/A284668" class="extiw" title="oeis:A284668">A284668</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example, 9780657631 has 1132 steps, as does 9780657630. The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the <a href="/wiki/Power_of_two" title="Power of two">powers of two</a> since 2n is halved n times to reach 1, and is never increased. <div class="mw-heading mw-heading2"><h2 id="Visualizations">Visualizations</h2></div> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 280px;"><a href="/wiki/File:Collatz_orbits_of_the_all_integers_up_to_1000.svg" class="mw-file-description" title="Directed graph showing the orbits of the first 1000 numbers."><img alt="Directed graph showing the orbits of the first 1000 numbers." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Collatz_orbits_of_the_all_integers_up_to_1000.svg/63px-Collatz_orbits_of_the_all_integers_up_to_1000.svg.png" decoding="async" width="63" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Collatz_orbits_of_the_all_integers_up_to_1000.svg/95px-Collatz_orbits_of_the_all_integers_up_to_1000.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Collatz_orbits_of_the_all_integers_up_to_1000.svg/126px-Collatz_orbits_of_the_all_integers_up_to_1000.svg.png 2x" data-file-width="4725" data-file-height="18634" /></a></div> <div class="gallerytext">Directed graph showing the orbits of the first 1000 numbers.</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 280px;"><a href="/wiki/File:CollatzConjectureGraphMaxValues.jpg" class="mw-file-description" title="The x axis represents starting number, the y axis represents the highest number reached during the chain to 1. This plot shows a restricted y axis: some x values produce intermediates as high as 2.7×107 (for x = 9663)"><img alt="The x axis represents starting number, the y axis represents the highest number reached during the chain to 1. This plot shows a restricted y axis: some x values produce intermediates as high as 2.7×107 (for x = 9663)" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/CollatzConjectureGraphMaxValues.jpg/133px-CollatzConjectureGraphMaxValues.jpg" decoding="async" width="133" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/CollatzConjectureGraphMaxValues.jpg/200px-CollatzConjectureGraphMaxValues.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/CollatzConjectureGraphMaxValues.jpg/267px-CollatzConjectureGraphMaxValues.jpg 2x" data-file-width="437" data-file-height="818" /></a></div> <div class="gallerytext">The x axis represents starting number, the y axis represents the highest number reached during the chain to 1. This plot shows a restricted y axis: some x values produce intermediates as high as 2.7×107 (for x = 9663)</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 280px;"><a href="/wiki/File:Collatz-max.png" class="mw-file-description" title="The same plot as the previous one but on log scale, so all y values are shown. The first thick line towards the middle of the plot corresponds to the tip at 27, which reaches a maximum at 9232."><img alt="The same plot as the previous one but on log scale, so all y values are shown. The first thick line towards the middle of the plot corresponds to the tip at 27, which reaches a maximum at 9232." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Collatz-max.png/133px-Collatz-max.png" decoding="async" width="133" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Collatz-max.png/200px-Collatz-max.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Collatz-max.png/267px-Collatz-max.png 2x" data-file-width="437" data-file-height="818" /></a></div> <div class="gallerytext">The same plot as the previous one but on log scale, so all y values are shown. The first thick line towards the middle of the plot corresponds to the tip at 27, which reaches a maximum at 9232.</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 280px;"><a href="/wiki/File:All_Collatz_sequences_of_a_length_inferior_to_20.svg" class="mw-file-description" title="The tree of all the numbers having fewer than 20 steps."><img alt="The tree of all the numbers having fewer than 20 steps." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/All_Collatz_sequences_of_a_length_inferior_to_20.svg/180px-All_Collatz_sequences_of_a_length_inferior_to_20.svg.png" decoding="async" width="180" height="178" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/All_Collatz_sequences_of_a_length_inferior_to_20.svg/270px-All_Collatz_sequences_of_a_length_inferior_to_20.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/All_Collatz_sequences_of_a_length_inferior_to_20.svg/360px-All_Collatz_sequences_of_a_length_inferior_to_20.svg.png 2x" data-file-width="3696" data-file-height="3653" /></a></div> <div class="gallerytext">The tree of all the numbers having fewer than 20 steps.</div> </li> <li class="gallerybox" style="width: 215px"> <div class="thumb" style="width: 210px; height: 280px;"><a href="/wiki/File:Collatz_Conjecture_100M.jpg" class="mw-file-description" title="The number of iterations it takes to get to one for the first 100 million numbers."><img alt="Collatz Conjecture 100M" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Collatz_Conjecture_100M.jpg/180px-Collatz_Conjecture_100M.jpg" decoding="async" width="180" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Collatz_Conjecture_100M.jpg/270px-Collatz_Conjecture_100M.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Collatz_Conjecture_100M.jpg/360px-Collatz_Conjecture_100M.jpg 2x" data-file-width="1600" data-file-height="1200" /></a></div> <div class="gallerytext">The number of iterations it takes to get to one for the first 100 million numbers.</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Supporting_arguments">Supporting arguments</h2></div> Although the conjecture has not been proven, most mathematicians who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it. <div class="mw-heading mw-heading3"><h3 id="Experimental_evidence">Experimental evidence</h3></div> The conjecture has been checked by computer for all starting values up to 268 ≈ 2.95×1020. All values tested so far converge to 1.<a href="#cite_note-Barina-13">[12]</a> This computer evidence is still not rigorous proof that the conjecture is true for all starting values, as <a href="/wiki/Counterexamples" class="mw-redirect" title="Counterexamples">counterexamples</a> may be found when considering very large (or possibly immense) positive integers, as in the case of the disproven <a href="/wiki/P%C3%B3lya_conjecture" title="Pólya conjecture">Pólya conjecture</a> and <a href="/wiki/Mertens_conjecture" title="Mertens conjecture">Mertens conjecture</a>. However, such verifications may have other implications. Certain constraints on any non-trivial cycle, such as <a href="/wiki/Lower_bound" class="mw-redirect" title="Lower bound">lower bounds</a> on the length of the cycle, can be proven based on the value of the lowest term in the cycle. Therefore, computer searches to rule out cycles that have a small lowest term can strengthen these constraints.<a href="#cite_note-Garner_(1981)-14">[13]</a><a href="#cite_note-Eliahou_(1993)-15">[14]</a><a href="#cite_note-Simons_&_de_Weger_(2005)-16">[15]</a> <div class="mw-heading mw-heading3"><h3 id="A_probabilistic_heuristic">A probabilistic heuristic</h3></div> If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠3/4⁠ of the previous one.<a href="#cite_note-17">[16]</a> (More precisely, the geometric mean of the ratios of outcomes is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence. The argument is not a proof because it assumes that Hailstone sequences are assembled from uncorrelated probabilistic events. (It does rigorously establish that the <a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">2-adic</a> extension of the Collatz process has two division steps for every multiplication step for <a href="/wiki/Almost_all" title="Almost all">almost all</a> 2-adic starting values.) <div class="mw-heading mw-heading3"><h3 id="Stopping_times">Stopping times</h3></div> As proven by <a href="/wiki/Riho_Terras_(mathematician)" title="Riho Terras (mathematician)">Riho Terras</a>, almost every positive integer has a finite stopping time.<a href="#cite_note-Terras_(1976)-18">[17]</a> In other words, almost every Collatz sequence reaches a point that is strictly below its initial value. The proof is based on the distribution of <a href="#As_a_parity_sequence">parity vectors</a> and uses the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>. In 2019, <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a> improved this result by showing, using logarithmic <a href="/wiki/Probability_density_function" title="Probability density function">density</a>, that <a href="/wiki/Almost_all" title="Almost all">almost all</a> (in the sense of logarithmic density) Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. Responding to this work, <a href="/wiki/Quanta_Magazine" title="Quanta Magazine">Quanta Magazine</a> wrote that Tao "came away with one of the most significant results on the Collatz conjecture in decades".<a href="#cite_note-Tao-10">[9]</a><a href="#cite_note-19">[18]</a> <div class="mw-heading mw-heading3"><h3 id="Lower_bounds">Lower bounds</h3></div> In a <a href="/wiki/Computer-aided_proof" class="mw-redirect" title="Computer-aided proof">computer-aided proof</a>, Krasikov and Lagarias showed that the number of integers in the interval [1,x] that eventually reach 1 is at least equal to x0.84 for all sufficiently large x.<a href="#cite_note-20">[19]</a> <div class="mw-heading mw-heading2"><h2 id="Cycles">Cycles</h2></div> In this part, consider the shortcut form of the Collatz function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b00f4f000899ad1f6844df7b58c7efdfe130023" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.071ex; height:7.509ex;" alt="{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}"> A <a href="/wiki/Periodic_sequence" title="Periodic sequence">cycle</a> is a sequence (a0, a1, ..., aq) of distinct positive integers where f(a0) = a1, f(a1) = a2, ..., and f(aq) = a0. The only known cycle is (1,2) of period 2, called the trivial cycle. <div class="mw-heading mw-heading3"><h3 id="Cycle_length">Cycle length</h3></div> The length of a non-trivial cycle is known to be at least 114208327604 (or 186265759595 without shortcut). If it can be shown that for all positive integers less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 2^{69}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>69</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 2^{69}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a7a1a5ff26bec26826cf4ee6543c4d88c24261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.042ex; height:2.676ex;" alt="{\displaystyle 3\times 2^{69}}"> the Collatz sequences reach 1, then this bound would raise to 217976794617 (355504839929 without shortcut).<a href="#cite_note-Hercher_(2023)-21">[20]</a><a href="#cite_note-Eliahou_(1993)-15">[14]</a> In fact, Eliahou (1993) proved that the period p of any non-trivial cycle is of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=301994a+17087915b+85137581c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>301994</mn> <mi>a</mi> <mo>+</mo> <mn>17087915</mn> <mi>b</mi> <mo>+</mo> <mn>85137581</mn> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=301994a+17087915b+85137581c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7fa38c40c5092cb2a1dc81e55024a67565aa2a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:38.846ex; height:2.509ex;" alt="{\displaystyle p=301994a+17087915b+85137581c}"> where a, b and c are non-negative integers, b ≥ 1 and ac = 0. This result is based on the <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">simple continued fraction</a> expansion of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ln 3/ln 2⁠.<a href="#cite_note-Eliahou_(1993)-15">[14]</a> <div class="mw-heading mw-heading3"><h3 id="k-cycles">k-cycles</h3></div> A k-cycle is a cycle that can be partitioned into k contiguous subsequences, each consisting of an increasing sequence of odd numbers, followed by a decreasing sequence of even numbers.<a href="#cite_note-Simons_&_de_Weger_(2005)-16">[15]</a> For instance, if the cycle consists of a single increasing sequence of odd numbers followed by a decreasing sequence of even numbers, it is called a 1-cycle. Steiner (1977) proved that there is no 1-cycle other than the trivial (1; 2).<a href="#cite_note-Steiner_(1977)-22">[21]</a> Simons (2005) used Steiner's method to prove that there is no 2-cycle.<a href="#cite_note-23">[22]</a> Simons and de Weger (2005) extended this proof up to 68-cycles; there is no k-cycle up to k = 68.<a href="#cite_note-Simons_&_de_Weger_(2005)-16">[15]</a> Hercher extended the method further and proved that there exists no k-cycle with k ≤ 91.<a href="#cite_note-Hercher_(2023)-21">[20]</a> As exhaustive computer searches continue, larger k values may be ruled out. To state the argument more intuitively; we do not have to search for cycles that have less than 92 subsequences, where each subsequence consists of consecutive ups followed by consecutive downs.[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] <div class="mw-heading mw-heading2"><h2 id="Other_formulations_of_the_conjecture">Other formulations of the conjecture</h2></div> <div class="mw-heading mw-heading3"><h3 id="In_reverse">In reverse</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Collatz-tree,_depth%3D20.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Collatz-tree%2C_depth%3D20.svg/440px-Collatz-tree%2C_depth%3D20.svg.png" decoding="async" width="440" height="59" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Collatz-tree%2C_depth%3D20.svg/660px-Collatz-tree%2C_depth%3D20.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Collatz-tree%2C_depth%3D20.svg/880px-Collatz-tree%2C_depth%3D20.svg.png 2x" data-file-width="14843" data-file-height="2000" /></a><figcaption>The first 21 levels of the Collatz <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> generated in bottom-up fashion. The graph includes all numbers with an orbit length of 21 or less.</figcaption></figure> There is another approach to prove the conjecture, which considers the bottom-up method of growing the so-called Collatz graph. The Collatz graph is a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> defined by the inverse <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1,2,3,5\\[4px]\left\{2n,{\frac {n-1}{3}}\right\}&{\text{if }}n\equiv 4\end{cases}}{\pmod {6}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>{</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>4</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1,2,3,5\\[4px]\left\{2n,{\frac {n-1}{3}}\right\}&{\text{if }}n\equiv 4\end{cases}}{\pmod {6}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41a322eb17a1ca4fa107beb57cac4867b0c89c07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:51.695ex; height:8.509ex;" alt="{\displaystyle R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1,2,3,5\\[4px]\left\{2n,{\frac {n-1}{3}}\right\}&{\text{if }}n\equiv 4\end{cases}}{\pmod {6}}.}"> So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads backwards to all positive integers. For any integer n, n ≡ 1 (mod 2) <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> 3n + 1 ≡ 4 (mod 6). Equivalently, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n − 1/3⁠ ≡ 1 (mod 2) if and only if n ≡ 4 (mod 6). Conjecturally, this inverse relation forms a <a href="/wiki/Tree_(graph_theory)" title="Tree (graph theory)">tree</a> except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function f defined in the <a href="#Statement_of_the_problem">Statement of the problem</a> section of this article). When the relation 3n + 1 of the function f is replaced by the common substitute "shortcut" relation <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3n + 1/2⁠, the Collatz graph is defined by the inverse relation, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1\\[4px]\left\{2n,{\frac {2n-1}{3}}\right\}&{\text{if }}n\equiv 2\end{cases}}{\pmod {3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>{</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1\\[4px]\left\{2n,{\frac {2n-1}{3}}\right\}&{\text{if }}n\equiv 2\end{cases}}{\pmod {3}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/443d6acdf06087446392a7e48e733c009d60c4fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:45.928ex; height:8.509ex;" alt="{\displaystyle R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1\\[4px]\left\{2n,{\frac {2n-1}{3}}\right\}&{\text{if }}n\equiv 2\end{cases}}{\pmod {3}}.}"> For any integer n, n ≡ 1 (mod 2) if and only if <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3n + 1/2⁠ ≡ 2 (mod 3). Equivalently, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2n − 1/3⁠ ≡ 1 (mod 2) if and only if n ≡ 2 (mod 3). Conjecturally, this inverse relation forms a tree except for a 1–2 loop (the inverse of the 1–2 loop of the function f(n) revised as indicated above). Alternatively, replace the 3n + 1 with <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n′/H(n′)⁠ where n′ = 3n + 1 and H(n′) is the highest <a href="/wiki/Power_of_2" class="mw-redirect" title="Power of 2">power of 2</a> that divides n′ (with no <a href="/wiki/Remainder" title="Remainder">remainder</a>). The resulting function f maps from <a href="/wiki/Odd_number" class="mw-redirect" title="Odd number">odd numbers</a> to odd numbers. Now suppose that for some odd number n, applying this operation k times yields the number 1 (that is, fk(n) = 1). Then in <a href="/wiki/Binary_number" title="Binary number">binary</a>, the number n can be written as the concatenation of <a href="/wiki/String_(computer_science)" title="String (computer science)">strings</a> wk wk−1 ... w1 where each wh is a finite and contiguous extract from the representation of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3h⁠.<a href="#cite_note-Colussi2011-24">[23]</a> The representation of n therefore holds the <a href="/wiki/Repeating_decimal" title="Repeating decimal">repetends</a> of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3h⁠, where each repetend is optionally rotated and then replicated up to a finite number of bits. It is only in binary that this occurs.<a href="#cite_note-Hew2016-25">[24]</a> Conjecturally, every binary string s that ends with a '1' can be reached by a representation of this form (where we may add or delete leading '0's to s). <div class="mw-heading mw-heading3"><h3 id="As_an_abstract_machine_that_computes_in_base_two">As an abstract machine that computes in base two</h3></div> Repeated applications of the Collatz function can be represented as an <a href="/wiki/Abstract_machine" title="Abstract machine">abstract machine</a> that handles <a href="/wiki/String_(computer_science)" title="String (computer science)">strings</a> of <a href="/wiki/Bit" title="Bit">bits</a>. The machine will perform the following three steps on any odd number until only one <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style>1 remains: <ol><li>Append <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">1 to the (right) end of the number in binary (giving 2n + 1);</li> <li>Add this to the original number by binary addition (giving 2n + 1 + n = 3n + 1);</li> <li>Remove all trailing <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">0s (that is, repeatedly divide by 2 until the result is odd).</li></ol> <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4></div> The starting number 7 is written in base two as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">111. The resulting Collatz sequence is: <div style="font-family:monospace"> <pre> 111 1111 1011<s>0</s> 10111 10001<s>0</s> 100011 1101<s>00</s> 11011 101<s>000</s> 1011 1<s>0000</s> </pre> </div> <div class="mw-heading mw-heading3"><h3 id="As_a_parity_sequence">As a parity sequence</h3></div> For this section, consider the shortcut form of the Collatz function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1\end{cases}}{\pmod {2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1\end{cases}}{\pmod {2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a240a4b8a847f8d81e77a3e609b865b8a998c487" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.146ex; margin-bottom: -0.192ex; width:36.361ex; height:7.843ex;" alt="{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1\end{cases}}{\pmod {2}}.}"> If P(...) is the parity of a number, that is P(2n) = 0 and P(2n + 1) = 1, then we can define the Collatz parity sequence (or parity vector) for a number n as pi = P(ai), where a0 = n, and ai+1 = f(ai). Which operation is performed, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3n + 1/2⁠ or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n/2⁠, depends on the parity. The parity sequence is the same as the sequence of operations. Using this form for f(n), it can be shown that the parity sequences for two numbers m and n will agree in the first k terms if and only if m and n are equivalent modulo 2k. This implies that every number is uniquely identified by its parity sequence, and moreover that if there are multiple Hailstone cycles, then their corresponding parity cycles must be different.<a href="#cite_note-Lagarias_(1985)-2">[2]</a><a href="#cite_note-Terras_(1976)-18">[17]</a> Applying the f function k times to the number n = 2ka + b will give the result 3ca + d, where d is the result of applying the f function k times to b, and c is how many increases were encountered during that sequence. For example, for 25a + 1 there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is 33a + 2; for 22a + 1 there is only 1 increase as 1 rises to 2 and falls to 1 so the result is 3a + 1. When b is 2k − 1 then there will be k rises and the result will be 3ka + 3k − 1. The power of 3 multiplying a is independent of the value of a; it depends only on the behavior of b. This allows one to predict that certain forms of numbers will always lead to a smaller number after a certain number of iterations: for example, 4a + 1 becomes 3a + 1 after two applications of f and 16a + 3 becomes 9a + 2 after four applications of f. Whether those smaller numbers continue to 1, however, depends on the value of a. <div class="mw-heading mw-heading3"><h3 id="As_a_tag_system">As a tag system</h3></div> For the Collatz function in the shortcut form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1.\end{cases}}{\pmod {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>≡</mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1.\end{cases}}{\pmod {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbad5e3e25f45b2f51c7ef1d36bfc3da5c3e5e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.146ex; margin-bottom: -0.192ex; width:36.361ex; height:7.843ex;" alt="{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1.\end{cases}}{\pmod {2}}}"> Hailstone sequences can be computed by the <a href="/wiki/Tag_system#Example:_Computation_of_Collatz_sequences" title="Tag system">2-tag system</a> with production rules <dl><dd>a → bc, b → a, c → aaa.</dd></dl> In this system, the positive integer n is represented by a string of n copies of a, and iteration of the tag operation halts on any word of length less than 2. (Adapted from De Mol.) The Collatz conjecture equivalently states that this tag system, with an arbitrary finite string of a as the initial word, eventually halts (see <a href="/wiki/Tag_system#Example:_Computation_of_Collatz_sequences" title="Tag system">Tag system</a> for a worked example). <div class="mw-heading mw-heading2"><h2 id="Extensions_to_larger_domains">Extensions to larger domains</h2></div> <div class="mw-heading mw-heading3"><h3 id="Iterating_on_all_integers">Iterating on all integers</h3></div> An extension to the Collatz conjecture is to include all integers, not just positive integers. Leaving aside the cycle 0 → 0 which cannot be entered from outside, there are a total of four known cycles, which all nonzero integers seem to eventually fall into under iteration of f. These cycles are listed here, starting with the well-known cycle for positive n: Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first. <table class="wikitable" style="text-align: center;"> <tbody><tr> <th>Cycle</th> <th>Odd-value cycle length</th> <th>Full cycle length </th></tr> <tr> <td style="text-align: left;"><big>1</big> → 4 → 2 → <big>1</big> ...</td> <td>1</td> <td>3 </td></tr> <tr> <td style="text-align: left;"><big>−1</big> → −2 → <big>−1</big> ...</td> <td>1</td> <td>2 </td></tr> <tr> <td style="text-align: left;"><big>−5</big> → −14 → <big>−7</big> → −20 → −10 → <big>−5</big> ...</td> <td>2</td> <td>5 </td></tr> <tr> <td style="text-align: left;"><big>−17</big> → −50 → <big>−25</big> → −74 → <big>−37</big> → −110 → <big>−55</big> → −164 → −82 → <big>−41</big> → −122 → <big>−61</big> → −182 → <big>−91</big> → −272 → −136 → −68 → −34 → <big>−17</big> ...</td> <td>7</td> <td>18 </td></tr></tbody></table> The generalized Collatz conjecture is the assertion that every integer, under iteration by f, eventually falls into one of the four cycles above or the cycle 0 → 0. <div class="mw-heading mw-heading3"><h3 id="Iterating_on_rationals_with_odd_denominators">Iterating on rationals with odd denominators</h3></div> The Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms. The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added. A closely related fact is that the Collatz map extends to the ring of <a href="/wiki/2-adic_integers" class="mw-redirect" title="2-adic integers">2-adic integers</a>, which contains the ring of rationals with odd denominators as a subring. When using the "shortcut" definition of the Collatz map, it is known that any periodic <a href="#As_a_parity_sequence">parity sequence</a> is generated by exactly one rational.<a href="#cite_note-26">[25]</a> Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture<a href="#cite_note-Lagarias_(1985)-2">[2]</a>). If a parity cycle has length n and includes odd numbers exactly m times at indices k0 < ⋯ < km−1, then the unique rational which generates immediately and periodically this parity cycle is <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3^{m-1}2^{k_{0}}+\cdots +3^{0}2^{k_{m-1}}}{2^{n}-3^{m}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3^{m-1}2^{k_{0}}+\cdots +3^{0}2^{k_{m-1}}}{2^{n}-3^{m}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b45c3d33f5fb4529b9230201af63c7249017e21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.417ex; height:6.009ex;" alt="{\displaystyle {\frac {3^{m-1}2^{k_{0}}+\cdots +3^{0}2^{k_{m-1}}}{2^{n}-3^{m}}}.}"></td> <td></td> <td class="nowrap">1</td></tr></tbody></table> For example, the parity cycle (1 0 1 1 0 0 1) has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3^{3}2^{0}+3^{2}2^{2}+3^{1}2^{3}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {151}{47}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>−</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>151</mn> <mn>47</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3^{3}2^{0}+3^{2}2^{2}+3^{1}2^{3}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {151}{47}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d09ca430bc5e3d95586279992c77ab7125521af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.513ex; height:6.343ex;" alt="{\displaystyle {\frac {3^{3}2^{0}+3^{2}2^{2}+3^{1}2^{3}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {151}{47}}}"> as the latter leads to the rational cycle <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {151}{47}}\rightarrow {\frac {250}{47}}\rightarrow {\frac {125}{47}}\rightarrow {\frac {211}{47}}\rightarrow {\frac {340}{47}}\rightarrow {\frac {170}{47}}\rightarrow {\frac {85}{47}}\rightarrow {\frac {151}{47}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>151</mn> <mn>47</mn> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>250</mn> <mn>47</mn> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>125</mn> <mn>47</mn> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>211</mn> <mn>47</mn> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>340</mn> <mn>47</mn> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>170</mn> <mn>47</mn> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>85</mn> <mn>47</mn> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>151</mn> <mn>47</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {151}{47}}\rightarrow {\frac {250}{47}}\rightarrow {\frac {125}{47}}\rightarrow {\frac {211}{47}}\rightarrow {\frac {340}{47}}\rightarrow {\frac {170}{47}}\rightarrow {\frac {85}{47}}\rightarrow {\frac {151}{47}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348c423b97d24af32fe868f70aafa7edb990e97a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:59.371ex; height:5.343ex;" alt="{\displaystyle {\frac {151}{47}}\rightarrow {\frac {250}{47}}\rightarrow {\frac {125}{47}}\rightarrow {\frac {211}{47}}\rightarrow {\frac {340}{47}}\rightarrow {\frac {170}{47}}\rightarrow {\frac {85}{47}}\rightarrow {\frac {151}{47}}.}"> Any cyclic permutation of (1 0 1 1 0 0 1) is associated to one of the above fractions. For instance, the cycle (0 1 1 0 0 1 1) is produced by the fraction <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3^{3}2^{1}+3^{2}2^{2}+3^{1}2^{5}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {250}{47}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>−</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>250</mn> <mn>47</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3^{3}2^{1}+3^{2}2^{2}+3^{1}2^{5}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {250}{47}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f852b5c2f3acd335545acdd7926cebe5c072b24" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.16ex; height:6.343ex;" alt="{\displaystyle {\frac {3^{3}2^{1}+3^{2}2^{2}+3^{1}2^{5}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {250}{47}}.}"> For a one-to-one correspondence, a parity cycle should be irreducible, that is, not partitionable into identical sub-cycles. As an illustration of this, the parity cycle (1 1 0 0 1 1 0 0) and its sub-cycle (1 1 0 0) are associated to the same fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/7⁠ when reduced to lowest terms. In this context, assuming the validity of the Collatz conjecture implies that (1 0) and (0 1) are the only parity cycles generated by positive whole numbers (1 and 2, respectively). If the odd denominator d of a rational is not a multiple of 3, then all the iterates have the same denominator and the sequence of numerators can be obtained by applying the "3n + d " generalization<a href="#cite_note-Belaga_(1998a)-27">[26]</a> of the Collatz function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{d}(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}},\\[4px]{\frac {3x+d}{2}}&{\text{if }}x\equiv 1{\pmod {2}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mi>d</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{d}(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}},\\[4px]{\frac {3x+d}{2}}&{\text{if }}x\equiv 1{\pmod {2}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160f092b9c61909d7176b6ab67678ec85bc73a20" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:36.104ex; height:8.009ex;" alt="{\displaystyle T_{d}(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}},\\[4px]{\frac {3x+d}{2}}&{\text{if }}x\equiv 1{\pmod {2}}.\end{cases}}}"> <div class="mw-heading mw-heading3"><h3 id="2-adic_extension">2-adic extension</h3></div> The function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}}\\[4px]{\frac {3x+1}{2}}&{\text{if }}x\equiv 1{\pmod {2}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}}\\[4px]{\frac {3x+1}{2}}&{\text{if }}x\equiv 1{\pmod {2}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87de26c7247f378d6f1a08f32578751a4a1ee3a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.146ex; margin-bottom: -0.192ex; width:34.606ex; height:7.843ex;" alt="{\displaystyle T(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}}\\[4px]{\frac {3x+1}{2}}&{\text{if }}x\equiv 1{\pmod {2}}\end{cases}}}"> is well-defined on the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aedfb5c02eff978ab963421ce930f46801657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.605ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{2}}"> of <a href="/wiki/2-adic_integers" class="mw-redirect" title="2-adic integers">2-adic integers</a>, where it is continuous and <a href="/wiki/Measure-preserving_transformation" class="mw-redirect" title="Measure-preserving transformation">measure-preserving</a> with respect to the 2-adic measure. Moreover, its dynamics is known to be <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic</a>.<a href="#cite_note-Lagarias_(1985)-2">[2]</a> Define the parity vector function Q acting on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aedfb5c02eff978ab963421ce930f46801657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.605ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{2}}"> as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=\sum _{k=0}^{\infty }\left(T^{k}(x)\mod 2\right)2^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=\sum _{k=0}^{\infty }\left(T^{k}(x)\mod 2\right)2^{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc35c2ffeb8b3e818b2b90e054011db410ac6f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.605ex; height:7.009ex;" alt="{\displaystyle Q(x)=\sum _{k=0}^{\infty }\left(T^{k}(x)\mod 2\right)2^{k}.}"> The function Q is a 2-adic <a href="/wiki/Isometry" title="Isometry">isometry</a>.<a href="#cite_note-28">[27]</a> Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that <a href="/wiki/Almost_all" title="Almost all">almost all</a> trajectories are acyclic in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aedfb5c02eff978ab963421ce930f46801657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.605ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{2}}">. An equivalent formulation of the Collatz conjecture is that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\left(\mathbb {Z} ^{+}\right)\subset {\tfrac {1}{3}}\mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\left(\mathbb {Z} ^{+}\right)\subset {\tfrac {1}{3}}\mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd37358b94be339c15a461cb787b8df16ed7d60d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.37ex; height:3.676ex;" alt="{\displaystyle Q\left(\mathbb {Z} ^{+}\right)\subset {\tfrac {1}{3}}\mathbb {Z} .}"> <div class="mw-heading mw-heading3"><h3 id="Iterating_on_real_or_complex_numbers">Iterating on real or complex numbers</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Collatz_Cobweb.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Collatz_Cobweb.svg/220px-Collatz_Cobweb.svg.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Collatz_Cobweb.svg/330px-Collatz_Cobweb.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Collatz_Cobweb.svg/440px-Collatz_Cobweb.svg.png 2x" data-file-width="422" data-file-height="421" /></a><figcaption><a href="/wiki/Cobweb_plot" title="Cobweb plot">Cobweb plot</a> of the orbit 10 → 5 → 8 → 4 → 2 → 1 → ... in an extension of the Collatz map to the real line.</figcaption></figure> The Collatz map can be extended to the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> by choosing any function which evaluates to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c68203be699cce32f38b291115d5bc2c6220dc9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle x/2}"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is an even integer, and to either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d61bf1338ba5b30d23b750ad76009792dc1dbcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.495ex; height:2.343ex;" alt="{\displaystyle 3x+1}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3x+1)/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3x+1)/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c097a29f0910d4b32a361ab2fdadeac6fb76c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.629ex; height:2.843ex;" alt="{\displaystyle (3x+1)/2}"> (for the "shortcut" version) when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is an odd integer. This is called an <a href="/wiki/Interpolating" class="mw-redirect" title="Interpolating">interpolating</a> function. A simple way to do this is to pick two functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3755e3e04ec295992b2b5331655ef83a500a05c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}">, where: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}(n)={\begin{cases}1,&n{\text{ is even,}}\\0,&n{\text{ is odd,}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even,</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is odd,</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}(n)={\begin{cases}1,&n{\text{ is even,}}\\0,&n{\text{ is odd,}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a5dabee67b8728e0d91f8eab95c68e917b8fd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.443ex; height:6.176ex;" alt="{\displaystyle g_{1}(n)={\begin{cases}1,&n{\text{ is even,}}\\0,&n{\text{ is odd,}}\end{cases}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}(n)={\begin{cases}0,&n{\text{ is even,}}\\1,&n{\text{ is odd,}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even,</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is odd,</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}(n)={\begin{cases}0,&n{\text{ is even,}}\\1,&n{\text{ is odd,}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d3d596f1d61e22003f013e33ab6e853f68bfda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.443ex; height:6.176ex;" alt="{\displaystyle g_{2}(n)={\begin{cases}0,&n{\text{ is even,}}\\1,&n{\text{ is odd,}}\end{cases}}}"></dd></dl> and use them as switches for our desired values: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\triangleq {\frac {x}{2}}\cdot g_{1}(x)\,+\,{\frac {3x+1}{2}}\cdot g_{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\triangleq {\frac {x}{2}}\cdot g_{1}(x)\,+\,{\frac {3x+1}{2}}\cdot g_{2}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4f04eb20bae8921d4bd6bfabd3b39453955221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.59ex; height:5.176ex;" alt="{\displaystyle f(x)\triangleq {\frac {x}{2}}\cdot g_{1}(x)\,+\,{\frac {3x+1}{2}}\cdot g_{2}(x)}">.</dd></dl> One such choice is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}(x)\triangleq \cos ^{2}\left({\tfrac {\pi }{2}}x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}(x)\triangleq \cos ^{2}\left({\tfrac {\pi }{2}}x\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c50db0be5b91abb8f11b4d8d5f905ac8e715d1bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.804ex; height:3.509ex;" alt="{\displaystyle g_{1}(x)\triangleq \cos ^{2}\left({\tfrac {\pi }{2}}x\right)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}(x)\triangleq \sin ^{2}\left({\tfrac {\pi }{2}}x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}(x)\triangleq \sin ^{2}\left({\tfrac {\pi }{2}}x\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/819d25604592f548685d4e6d9a103144e11802b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.548ex; height:3.509ex;" alt="{\displaystyle g_{2}(x)\triangleq \sin ^{2}\left({\tfrac {\pi }{2}}x\right)}">. The <a href="/wiki/Iterations" class="mw-redirect" title="Iterations">iterations</a> of this map lead to a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a>, further investigated by Marc Chamberland.<a href="#cite_note-Chamberland_(1996)-29">[28]</a> He showed that the conjecture does not hold for positive real numbers since there are infinitely many <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a>, as well as <a href="/wiki/Orbit_(dynamics)" title="Orbit (dynamics)">orbits</a> escaping <a href="/wiki/Monotonic_function" title="Monotonic function">monotonically</a> to infinity. The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> has two <a href="/wiki/Attractor" title="Attractor">attracting</a> cycles of period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1;\,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>;</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1;\,2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a27ea8fa8e1e876c74f6e5bddcd4143340b4639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.555ex; height:2.843ex;" alt="{\displaystyle (1;\,2)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1.1925...;\,2.1386...)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1.1925...</mn> <mo>;</mo> <mspace width="thinmathspace" /> <mn>2.1386...</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1.1925...;\,2.1386...)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2d4ccffbaf80146608db1b5d8dc65105c2c53f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.03ex; height:2.843ex;" alt="{\displaystyle (1.1925...;\,2.1386...)}">. Moreover, the set of unbounded orbits is conjectured to be of <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">measure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">. Letherman, Schleicher, and Wood extended the study to the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>.<a href="#cite_note-Letherman,_Schleicher,_and_Wood_(1999)-30">[29]</a> They used Chamberland's function for <a href="/wiki/Trigonometric_functions#In_the_complex_plane" title="Trigonometric functions">complex sine and cosine</a> and added the extra term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\pi }}\left({\tfrac {1}{2}}-\cos(\pi z)\right)\sin(\pi z)\,+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\pi }}\left({\tfrac {1}{2}}-\cos(\pi z)\right)\sin(\pi z)\,+}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5da2597df2f962f89fe42415349517d523af5188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.801ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{\pi }}\left({\tfrac {1}{2}}-\cos(\pi z)\right)\sin(\pi z)\,+}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(z)\sin ^{2}(\pi z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(z)\sin ^{2}(\pi z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc85eb4796817374160119564d9f6b55f6a6443b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.763ex; height:3.176ex;" alt="{\displaystyle h(z)\sin ^{2}(\pi z)}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb5239f31e57a3120e928486857a5ea8fb8eee80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.236ex; height:2.843ex;" alt="{\displaystyle h(z)}"> is any <a href="/wiki/Entire_function" title="Entire function">entire function</a>. Since this expression evaluates to zero for real integers, the extended function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(z)\triangleq \;&{\frac {z}{2}}\cos ^{2}\left({\frac {\pi }{2}}z\right)+{\frac {3z+1}{2}}\sin ^{2}\left({\frac {\pi }{2}}z\right)\,+\\&{\frac {1}{\pi }}\left({\frac {1}{2}}-\cos(\pi z)\right)\sin(\pi z)+h(z)\sin ^{2}(\pi z)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(z)\triangleq \;&{\frac {z}{2}}\cos ^{2}\left({\frac {\pi }{2}}z\right)+{\frac {3z+1}{2}}\sin ^{2}\left({\frac {\pi }{2}}z\right)\,+\\&{\frac {1}{\pi }}\left({\frac {1}{2}}-\cos(\pi z)\right)\sin(\pi z)+h(z)\sin ^{2}(\pi z)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/149caa0c7bbc886e6a876b3c276f238eccf6bde5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.914ex; margin-bottom: -0.257ex; width:49.257ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}f(z)\triangleq \;&{\frac {z}{2}}\cos ^{2}\left({\frac {\pi }{2}}z\right)+{\frac {3z+1}{2}}\sin ^{2}\left({\frac {\pi }{2}}z\right)\,+\\&{\frac {1}{\pi }}\left({\frac {1}{2}}-\cos(\pi z)\right)\sin(\pi z)+h(z)\sin ^{2}(\pi z)\end{aligned}}}"></dd></dl> is an interpolation of the Collatz map to the complex plane. The reason for adding the extra term is to make all integers <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">. With this, they show that no integer is in a <a href="/wiki/Classification_of_Fatou_components#Baker_domain" title="Classification of Fatou components">Baker domain</a>, which implies that any integer is either eventually periodic or belongs to a <a href="/wiki/Wandering_set" title="Wandering set">wandering domain</a>. They conjectured that the latter is not the case, which would make all integer orbits finite. <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Collatz_Fractal.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Collatz_Fractal.jpg/220px-Collatz_Fractal.jpg" decoding="async" width="220" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Collatz_Fractal.jpg/330px-Collatz_Fractal.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Collatz_Fractal.jpg/440px-Collatz_Fractal.jpg 2x" data-file-width="40320" data-file-height="17280" /></a><figcaption>A Collatz <a href="/wiki/Fractal" title="Fractal">fractal</a> centered at the origin, with real parts from -5 to 5.</figcaption></figure> Most of the points have orbits that diverge to infinity. Coloring these points based on how fast they diverge produces the image on the left, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(z)\triangleq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(z)\triangleq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe86082a7be07c0a29e9ae2c69b5baff04f201c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.497ex; height:3.009ex;" alt="{\displaystyle h(z)\triangleq 0}">. The inner black regions and the outer region are the <a href="/wiki/Classification_of_Fatou_components" title="Classification of Fatou components">Fatou components</a>, and the boundary between them is the <a href="/wiki/Julia_set" title="Julia set">Julia set</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">, which forms a <a href="/wiki/Fractal" title="Fractal">fractal</a> pattern, sometimes called a "Collatz fractal". <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Exponential_Collatz_Fractal.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Exponential_Collatz_Fractal.jpg/220px-Exponential_Collatz_Fractal.jpg" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Exponential_Collatz_Fractal.jpg/330px-Exponential_Collatz_Fractal.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Exponential_Collatz_Fractal.jpg/440px-Exponential_Collatz_Fractal.jpg 2x" data-file-width="30720" data-file-height="17280" /></a><figcaption>Julia set of the exponential interpolation.</figcaption></figure> There are many other ways to define a complex interpolating function, such as using the <a href="/wiki/Exponential_function#Complex_plane" title="Exponential function">complex exponential</a> instead of sine and cosine: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)\triangleq {\frac {z}{2}}+{\frac {1}{4}}(2z+1)\left(1-e^{i\pi z}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>z</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>π</mi> <mi>z</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)\triangleq {\frac {z}{2}}+{\frac {1}{4}}(2z+1)\left(1-e^{i\pi z}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a10304a8361731ba1b7d13372217996a0aa78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.289ex; height:5.176ex;" alt="{\displaystyle f(z)\triangleq {\frac {z}{2}}+{\frac {1}{4}}(2z+1)\left(1-e^{i\pi z}\right)}">,</dd></dl> which exhibit different dynamics. In this case, for instance, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} (z)\gg 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≫</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (z)\gg 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7161fe8d617d5307328bdef3ef8dd7f89e5903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.449ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (z)\gg 1}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)\approx z+{\tfrac {1}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)\approx z+{\tfrac {1}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d87e2d585743dd2e992874834321b377c5eb49c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.861ex; height:3.509ex;" alt="{\displaystyle f(z)\approx z+{\tfrac {1}{4}}}">. The corresponding Julia set, shown on the right, consists of uncountably many curves, called hairs, or rays. <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Optimizations">Optimizations</h2></div> <div class="mw-heading mw-heading3"><h3 id="Time–space_tradeoff">Time–space tradeoff</h3></div> The section <a href="#As_a_parity_sequence">As a parity sequence</a> above gives a way to speed up simulation of the sequence. To jump ahead k steps on each iteration (using the f function from that section), break up the current number into two parts, b (the k least significant bits, interpreted as an integer), and a (the rest of the bits as an integer). The result of jumping ahead k is given by <dl><dd>fk(2ka + b) = 3c(b, k)a + d(b, k).</dd></dl> The values of c (or better 3c) and d can be precalculated for all possible k-bit numbers b, where d(b, k) is the result of applying the f function k times to b, and c(b, k) is the number of odd numbers encountered on the way.<a href="#cite_note-31">[30]</a> For example, if k = 5, one can jump ahead 5 steps on each iteration by separating out the 5 least significant bits of a number and using <dl><dd>c(0...31, 5) = { 0, 3, 2, 2, 2, 2, 2, 4, 1, 4, 1, 3, 2, 2, 3, 4, 1, 2, 3, 3, 1, 1, 3, 3, 2, 3, 2, 4, 3, 3, 4, 5 },</dd> <dd>d(0...31, 5) = { 0, 2, 1, 1, 2, 2, 2, 20, 1, 26, 1, 10, 4, 4, 13, 40, 2, 5, 17, 17, 2, 2, 20, 20, 8, 22, 8, 71, 26, 26, 80, 242 }.</dd></dl> This requires 2k <a href="/wiki/Precomputation" title="Precomputation">precomputation</a> and storage to speed up the resulting calculation by a factor of k, a <a href="/wiki/Space%E2%80%93time_tradeoff" title="Space–time tradeoff">space–time tradeoff</a>. <div class="mw-heading mw-heading3"><h3 id="Modular_restrictions">Modular restrictions</h3></div> For the special purpose of searching for a counterexample to the Collatz conjecture, this precomputation leads to an even more important acceleration, used by Tomás Oliveira e Silva in his computational confirmations of the Collatz conjecture up to large values of n. If, for some given b and k, the inequality <dl><dd>fk(2ka + b) = 3c(b)a + d(b) < 2ka + b</dd></dl> holds for all a, then the first counterexample, if it exists, cannot be b modulo 2k.<a href="#cite_note-Garner_(1981)-14">[13]</a> For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f2(4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting value is as good as checking an entire congruence class. As k increases, the search only needs to check those residues b that are not eliminated by lower values of k. Only an exponentially small fraction of the residues survive.<a href="#cite_note-32">[31]</a> For example, the only surviving residues mod 32 are 7, 15, 27, and 31. Integers divisible by 3 cannot form a cycle, so these integers do not need to be checked as counter examples.<a href="#cite_note-Clay-33">[32]</a> <div class="mw-heading mw-heading2"><h2 id="Syracuse_function">Syracuse function</h2></div> If k is an odd integer, then 3k + 1 is even, so 3k + 1 = 2ak′ with k′ odd and a ≥ 1. The Syracuse function is the function f from the set I of positive odd integers into itself, for which f(k) = k′ (sequence <a href="//oeis.org/A075677" class="extiw" title="oeis:A075677">A075677</a> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). Some properties of the Syracuse function are: <ul><li>For all k ∈ I, f(4k + 1) = f(k). (Because 3(4k + 1) + 1 = 12k + 4 = 4(3k + 1).)</li> <li>In more generality: For all p ≥ 1 and odd h, fp − 1(2ph − 1) = 2 × 3p − 1h − 1. (Here fp − 1 is <a href="/wiki/Functional_power" class="mw-redirect" title="Functional power">function iteration notation</a>.)</li> <li>For all odd h, f(2h − 1) ≤ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3h − 1/2⁠</li></ul> The Collatz conjecture is equivalent to the statement that, for all k in I, there exists an integer n ≥ 1 such that fn(k) = 1. <div class="mw-heading mw-heading2"><h2 id="Undecidable_generalizations">Undecidable generalizations</h2></div> In 1972, <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Horton Conway</a> proved that a natural generalization of the Collatz problem is algorithmically <a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a>.<a href="#cite_note-34">[33]</a> Specifically, he considered functions of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {g(n)=a_{i}n+b_{i}}{\text{ when }}{n\equiv i{\pmod {P}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>n</mi> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> when </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≡</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {g(n)=a_{i}n+b_{i}}{\text{ when }}{n\equiv i{\pmod {P}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500cf2fb657fc847e50a076af09a460e5b3c147e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.309ex; height:2.843ex;" alt="{\displaystyle {g(n)=a_{i}n+b_{i}}{\text{ when }}{n\equiv i{\pmod {P}}},}"> where a0, b0, ..., aP − 1, bP − 1 are rational numbers which are so chosen that g(n) is always an integer. The standard Collatz function is given by P = 2, a0 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, b0 = 0, a1 = 3, b1 = 1. Conway proved that the problem <dl><dd>Given g and n, does the sequence of iterates gk(n) reach 1?</dd></dl> is undecidable, by representing the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a> in this way. Closer to the Collatz problem is the following universally quantified problem: <dl><dd>Given g, does the sequence of iterates gk(n) reach 1, for all n > 0?</dd></dl> Modifying the condition in this way can make a problem either harder or easier to solve (intuitively, it is harder to justify a positive answer but might be easier to justify a negative one). Kurtz and Simon<a href="#cite_note-KurtzSimon-35">[34]</a> proved that the universally quantified problem is, in fact, undecidable and even higher in the <a href="/wiki/Arithmetical_hierarchy" title="Arithmetical hierarchy">arithmetical hierarchy</a>; specifically, it is Π0 2-complete. This hardness result holds even if one restricts the class of functions g by fixing the modulus P to 6480.<a href="#cite_note-36">[35]</a> Iterations of g in a simplified version of this form, with all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a8c2db2990a53c683e75961826167c5adac7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.797ex; height:2.509ex;" alt="{\displaystyle b_{i}}"> equal to zero, are formalized in an <a href="/wiki/Esoteric_programming_language" title="Esoteric programming language">esoteric programming language</a> called <a href="/wiki/FRACTRAN" title="FRACTRAN">FRACTRAN</a>. <div class="mw-heading mw-heading2"><h2 id="In_computational_complexity">In computational complexity</h2></div> Collatz and related conjectures are often used when studying computation complexity.<a href="#cite_note-37">[36]</a><a href="#cite_note-38">[37]</a> The connection is made through the <a href="/wiki/Busy_Beaver" class="mw-redirect" title="Busy Beaver">Busy Beaver</a> function, where BB(n) is the maximum number of steps taken by any n state <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> that halts. There is a 15 state Turing machine that halts if and only if a conjecture by <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> (closely related to the Collatz conjecture) is false. Hence if BB(15) was known, and this machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples exist (which proves the conjecture true). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(15) will be very hard to compute, at least as difficult as settling this Collatz-like conjecture. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output 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class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-4"><a href="#cite_ref-4">^</a> It is also known as the 3n + 1 problem (or conjecture), the 3x + 1 problem (or conjecture), the Ulam conjecture (after <a href="/wiki/Stanis%C5%82aw_Ulam" title="Stanisław Ulam">Stanisław Ulam</a>), Kakutani's problem (after <a href="/wiki/Shizuo_Kakutani" title="Shizuo Kakutani">Shizuo Kakutani</a>), the Thwaites conjecture (after <a href="/wiki/Bryan_Thwaites" title="Bryan Thwaites">Bryan Thwaites</a>), Hasse's algorithm (after <a href="/wiki/Helmut_Hasse" title="Helmut Hasse">Helmut Hasse</a>), or the Syracuse problem (after <a href="/wiki/Syracuse_University" title="Syracuse University">Syracuse University</a>).<a href="#cite_note-1">[1]</a><a href="#cite_note-3">[3]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMadduxJohnson1997" class="citation book cs1">Maddux, Cleborne D.; Johnson, D. Lamont (1997). Logo: A Retrospective. New York: Haworth Press. p. 160. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7890-0374-0" title="Special:BookSources/0-7890-0374-0"><bdi>0-7890-0374-0</bdi></a>. <q>The problem is also known by several other names, including: Ulam's conjecture, the Hailstone problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and the Collatz problem.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logo%3A+A+Retrospective&rft.place=New+York&rft.pages=160&rft.pub=Haworth+Press&rft.date=1997&rft.isbn=0-7890-0374-0&rft.aulast=Maddux&rft.aufirst=Cleborne+D.&rft.au=Johnson%2C+D.+Lamont&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Lagarias_(1985)-2">^ <a href="#cite_ref-Lagarias_(1985)_2-0">a</a> <a href="#cite_ref-Lagarias_(1985)_2-1">b</a> <a href="#cite_ref-Lagarias_(1985)_2-2">c</a> <a href="#cite_ref-Lagarias_(1985)_2-3">d</a> <a href="#cite_ref-Lagarias_(1985)_2-4">e</a> <a href="#cite_ref-Lagarias_(1985)_2-5">f</a> <a href="#cite_ref-Lagarias_(1985)_2-6">g</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLagarias1985" class="citation journal cs1">Lagarias, Jeffrey C. (1985). "The 3x + 1 problem and its generalizations". <a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 92 (1): 3–23. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.1985.11971528">10.1080/00029890.1985.11971528</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2322189">2322189</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=The+3x+%2B+1+problem+and+its+generalizations&rft.volume=92&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3-%3C%2Fspan%3E23&rft.date=1985&rft_id=info%3Adoi%2F10.1080%2F00029890.1985.11971528&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2322189%23id-name%3DJSTOR&rft.aulast=Lagarias&rft.aufirst=Jeffrey+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> According to Lagarias (1985),<a href="#cite_note-Lagarias_(1985)-2">[2]</a> p. 4, the name "Syracuse problem" was proposed by Hasse in the 1950s, during a visit to <a href="/wiki/Syracuse_University" title="Syracuse University">Syracuse University</a>. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" class="mw-redirect" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Collatz.html">"Lothar Collatz"</a>, <a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lothar+Collatz&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FCollatz.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPickover2001" class="citation book cs1">Pickover, Clifford A. (2001). <a rel="nofollow" class="external text" href="https://archive.org/details/wondersnumbersad00pick">Wonders of Numbers</a>. Oxford: Oxford University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/wondersnumbersad00pick/page/n136">116</a>–118. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-513342-0" title="Special:BookSources/0-19-513342-0"><bdi>0-19-513342-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wonders+of+Numbers&rft.place=Oxford&rft.pages=116-118&rft.pub=Oxford+University+Press&rft.date=2001&rft.isbn=0-19-513342-0&rft.aulast=Pickover&rft.aufirst=Clifford+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fwondersnumbersad00pick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHofstadter1979" class="citation book cs1"><a href="/wiki/Douglas_Hofstadter" title="Douglas Hofstadter">Hofstadter, Douglas R.</a> (1979). <a href="/wiki/G%C3%B6del,_Escher,_Bach" title="Gödel, Escher, Bach">Gödel, Escher, Bach</a>. New York: Basic Books. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/godelescherbach00doug/page/400">400–2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-465-02685-0" title="Special:BookSources/0-465-02685-0"><bdi>0-465-02685-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=G%C3%B6del%2C+Escher%2C+Bach&rft.place=New+York&rft.pages=400-2&rft.pub=Basic+Books&rft.date=1979&rft.isbn=0-465-02685-0&rft.aulast=Hofstadter&rft.aufirst=Douglas+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Guy_(2004)-8"><a href="#cite_ref-Guy_(2004)_8-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuy2004" class="citation book cs1"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1AP2CEGxTkgC&pg=PA330">""E16: The 3x+1 problem""</a>. Unsolved Problems in Number Theory (3rd ed.). <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. 330–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-20860-7" title="Special:BookSources/0-387-20860-7"><bdi>0-387-20860-7</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1058.11001">1058.11001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%22E16%3A+The+3x%2B1+problem%22&rft.btitle=Unsolved+Problems+in+Number+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E330-%3C%2Fspan%3E6&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1058.11001%23id-name%3DZbl&rft.isbn=0-387-20860-7&rft.aulast=Guy&rft.aufirst=Richard+K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1AP2CEGxTkgC%26pg%3DPA330&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Lagarias_(2010)-9">^ <a href="#cite_ref-Lagarias_(2010)_9-0">a</a> <a href="#cite_ref-Lagarias_(2010)_9-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLagarias2010" class="citation book cs1"><a href="/wiki/Jeffrey_Lagarias" title="Jeffrey Lagarias">Lagarias, Jeffrey C.</a>, ed. (2010). The Ultimate Challenge: The 3x + 1 Problem. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4940-8" title="Special:BookSources/978-0-8218-4940-8"><bdi>978-0-8218-4940-8</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1253.11003">1253.11003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Ultimate+Challenge%3A+The+3x+%2B+1+Problem&rft.pub=American+Mathematical+Society&rft.date=2010&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1253.11003%23id-name%3DZbl&rft.isbn=978-0-8218-4940-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Tao-10">^ <a href="#cite_ref-Tao_10-0">a</a> <a href="#cite_ref-Tao_10-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTao2022" class="citation journal cs1">Tao, Terence (2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Ffmp.2022.8">"Almost all orbits of the Collatz map attain almost bounded values"</a>. Forum of Mathematics, Pi. 10: e12. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1909.03562">1909.03562</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Ffmp.2022.8">10.1017/fmp.2022.8</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2050-5086">2050-5086</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Forum+of+Mathematics%2C+Pi&rft.atitle=Almost+all+orbits+of+the+Collatz+map+attain+almost+bounded+values&rft.volume=10&rft.pages=e12&rft.date=2022&rft_id=info%3Aarxiv%2F1909.03562&rft.issn=2050-5086&rft_id=info%3Adoi%2F10.1017%2Ffmp.2022.8&rft.aulast=Tao&rft.aufirst=Terence&rft_id=https%3A%2F%2Fdoi.org%2F10.1017%252Ffmp.2022.8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeavensVermeulen1992" class="citation journal cs1">Leavens, Gary T.; Vermeulen, Mike (December 1992). "3x + 1 search programs". Computers & Mathematics with Applications. 24 (11): 79–99. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0898-1221%2892%2990034-F">10.1016/0898-1221(92)90034-F</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computers+%26+Mathematics+with+Applications&rft.atitle=3x+%2B+1+search+programs&rft.volume=24&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E79-%3C%2Fspan%3E99&rft.date=1992-12&rft_id=info%3Adoi%2F10.1016%2F0898-1221%2892%2990034-F&rft.aulast=Leavens&rft.aufirst=Gary+T.&rft.au=Vermeulen%2C+Mike&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Roosendaal-12"><a href="#cite_ref-Roosendaal_12-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoosendaal" class="citation web cs1">Roosendaal, Eric. <a rel="nofollow" class="external text" href="http://www.ericr.nl/wondrous/delrecs.html">"3x+1 delay records"</a>. Retrieved 14 March 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=3x%2B1+delay+records&rft.aulast=Roosendaal&rft.aufirst=Eric&rft_id=http%3A%2F%2Fwww.ericr.nl%2Fwondrous%2Fdelrecs.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> (Note: "Delay records" are total stopping time records.) </li> <li id="cite_note-Barina-13"><a href="#cite_ref-Barina_13-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarina2020" class="citation journal cs1">Barina, David (2020). "Convergence verification of the Collatz problem". The Journal of Supercomputing. 77 (3): 2681–2688. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11227-020-03368-x">10.1007/s11227-020-03368-x</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220294340">220294340</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Supercomputing&rft.atitle=Convergence+verification+of+the+Collatz+problem&rft.volume=77&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2681-%3C%2Fspan%3E2688&rft.date=2020&rft_id=info%3Adoi%2F10.1007%2Fs11227-020-03368-x&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A220294340%23id-name%3DS2CID&rft.aulast=Barina&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Garner_(1981)-14">^ <a href="#cite_ref-Garner_(1981)_14-0">a</a> <a href="#cite_ref-Garner_(1981)_14-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarner1981" class="citation journal cs1">Garner, Lynn E. (1981). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1981-0603593-2">"On the Collatz 3n + 1 algorithm"</a>. <a href="/wiki/Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a>. 82 (1): 19–22. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1981-0603593-2">10.1090/S0002-9939-1981-0603593-2</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2044308">2044308</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=On+the+Collatz+3n+%2B+1+algorithm&rft.volume=82&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E19-%3C%2Fspan%3E22&rft.date=1981&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-1981-0603593-2&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2044308%23id-name%3DJSTOR&rft.aulast=Garner&rft.aufirst=Lynn+E.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9939-1981-0603593-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Eliahou_(1993)-15">^ <a href="#cite_ref-Eliahou_(1993)_15-0">a</a> <a href="#cite_ref-Eliahou_(1993)_15-1">b</a> <a href="#cite_ref-Eliahou_(1993)_15-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEliahou1993" class="citation journal cs1">Eliahou, Shalom (1993). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2893%2990052-U">"The 3x + 1 problem: new lower bounds on nontrivial cycle lengths"</a>. Discrete Mathematics. 118 (1): 45–56. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2893%2990052-U">10.1016/0012-365X(93)90052-U</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=The+3x+%2B+1+problem%3A+new+lower+bounds+on+nontrivial+cycle+lengths&rft.volume=118&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E45-%3C%2Fspan%3E56&rft.date=1993&rft_id=info%3Adoi%2F10.1016%2F0012-365X%2893%2990052-U&rft.aulast=Eliahou&rft.aufirst=Shalom&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0012-365X%252893%252990052-U&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Simons_&_de_Weger_(2005)-16">^ <a href="#cite_ref-Simons_&_de_Weger_(2005)_16-0">a</a> <a href="#cite_ref-Simons_&_de_Weger_(2005)_16-1">b</a> <a href="#cite_ref-Simons_&_de_Weger_(2005)_16-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimonsde_Weger2005" class="citation journal cs1">Simons, J.; de Weger, B. (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220318094356/http://deweger.xs4all.nl/papers/%5B35%5DSidW-3n+1-ActaArith%5B2005%5D.pdf">"Theoretical and computational bounds for m-cycles of the 3n + 1 problem"</a> (PDF). Acta Arithmetica. 117 (1): 51–70. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005AcAri.117...51S">2005AcAri.117...51S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa117-1-3">10.4064/aa117-1-3</a>. Archived from the original on 2022-03-18. Retrieved 2023-03-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Arithmetica&rft.atitle=Theoretical+and+computational+bounds+for+m-cycles+of+the+3n+%2B+1+problem&rft.volume=117&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E51-%3C%2Fspan%3E70&rft.date=2005&rft_id=info%3Adoi%2F10.4064%2Faa117-1-3&rft_id=info%3Abibcode%2F2005AcAri.117...51S&rft.aulast=Simons&rft.aufirst=J.&rft.au=de+Weger%2C+B.&rft_id=http%3A%2F%2Fdeweger.xs4all.nl%2Fpapers%2F%5B35%5DSidW-3n%2B1-ActaArith%5B2005%5D.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>) </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Lagarias (1985),<a href="#cite_note-Lagarias_(1985)-2">[2]</a> section "<a rel="nofollow" class="external text" href="http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/node3.html">A heuristic argument"</a>. </li> <li id="cite_note-Terras_(1976)-18">^ <a href="#cite_ref-Terras_(1976)_18-0">a</a> <a href="#cite_ref-Terras_(1976)_18-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTerras1976" class="citation journal cs1">Terras, Riho (1976). <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/aa/aa30/aa3034.pdf">"A stopping time problem on the positive integers"</a> (PDF). Acta Arithmetica. 30 (3): 241–252. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa-30-3-241-252">10.4064/aa-30-3-241-252</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0568274">0568274</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Arithmetica&rft.atitle=A+stopping+time+problem+on+the+positive+integers&rft.volume=30&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E241-%3C%2Fspan%3E252&rft.date=1976&rft_id=info%3Adoi%2F10.4064%2Faa-30-3-241-252&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0568274%23id-name%3DMR&rft.aulast=Terras&rft.aufirst=Riho&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Faa%2Faa30%2Faa3034.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartnett2019" class="citation web cs1">Hartnett, Kevin (December 11, 2019). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/mathematician-proves-huge-result-on-dangerous-problem-20191211/">"Mathematician Proves Huge Result on 'Dangerous' Problem"</a>. Quanta Magazine.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quanta+Magazine&rft.atitle=Mathematician+Proves+Huge+Result+on+%27Dangerous%27+Problem&rft.date=2019-12-11&rft.aulast=Hartnett&rft.aufirst=Kevin&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fmathematician-proves-huge-result-on-dangerous-problem-20191211%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrasikovLagarias2003" class="citation journal cs1">Krasikov, Ilia; <a href="/wiki/Jeffrey_Lagarias" title="Jeffrey Lagarias">Lagarias, Jeffrey C.</a> (2003). <a rel="nofollow" class="external text" href="https://www.impan.pl/download/pdf/aa109-3-4">"Bounds for the 3x + 1 problem using difference inequalities"</a>. Acta Arithmetica. 109 (3): 237–258. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0205002">math/0205002</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003AcAri.109..237K">2003AcAri.109..237K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa109-3-4">10.4064/aa109-3-4</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1980260">1980260</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18467460">18467460</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Arithmetica&rft.atitle=Bounds+for+the+3x+%2B+1+problem+using+difference+inequalities&rft.volume=109&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E237-%3C%2Fspan%3E258&rft.date=2003&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18467460%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2003AcAri.109..237K&rft_id=info%3Aarxiv%2Fmath%2F0205002&rft_id=info%3Adoi%2F10.4064%2Faa109-3-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1980260%23id-name%3DMR&rft.aulast=Krasikov&rft.aufirst=Ilia&rft.au=Lagarias%2C+Jeffrey+C.&rft_id=https%3A%2F%2Fwww.impan.pl%2Fdownload%2Fpdf%2Faa109-3-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Hercher_(2023)-21">^ <a href="#cite_ref-Hercher_(2023)_21-0">a</a> <a href="#cite_ref-Hercher_(2023)_21-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHercher2023" class="citation journal cs1">Hercher, C. (2023). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Hercher/hercher5.pdf">"There are no Collatz m-cycles with m <= 91"</a> (PDF). Journal of Integer Sequences. 26 (3): Article 23.3.5.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Integer+Sequences&rft.atitle=There+are+no+Collatz+m-cycles+with+m+%3C%3D+91&rft.volume=26&rft.issue=3&rft.pages=Article+23.3.5&rft.date=2023&rft.aulast=Hercher&rft.aufirst=C.&rft_id=https%3A%2F%2Fcs.uwaterloo.ca%2Fjournals%2FJIS%2FVOL26%2FHercher%2Fhercher5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Steiner_(1977)-22"><a href="#cite_ref-Steiner_(1977)_22-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteiner1977" class="citation book cs1">Steiner, R. P. (1977). "A theorem on the syracuse problem". Proceedings of the 7th Manitoba Conference on Numerical Mathematics. pp. 553–9. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0535032">0535032</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+theorem+on+the+syracuse+problem&rft.btitle=Proceedings+of+the+7th+Manitoba+Conference+on+Numerical+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E553-%3C%2Fspan%3E9&rft.date=1977&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D535032%23id-name%3DMR&rft.aulast=Steiner&rft.aufirst=R.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimons2005" class="citation journal cs1">Simons, John L. (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0025-5718-04-01728-4">"On the nonexistence of 2-cycles for the 3x + 1 problem"</a>. Math. Comp. 74: 1565–72. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005MaCom..74.1565S">2005MaCom..74.1565S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0025-5718-04-01728-4">10.1090/s0025-5718-04-01728-4</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2137019">2137019</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Comp.&rft.atitle=On+the+nonexistence+of+2-cycles+for+the+3x+%2B+1+problem&rft.volume=74&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1565-%3C%2Fspan%3E72&rft.date=2005&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2137019%23id-name%3DMR&rft_id=info%3Adoi%2F10.1090%2Fs0025-5718-04-01728-4&rft_id=info%3Abibcode%2F2005MaCom..74.1565S&rft.aulast=Simons&rft.aufirst=John+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252Fs0025-5718-04-01728-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Colussi2011-24"><a href="#cite_ref-Colussi2011_24-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFColussi2011" class="citation journal cs1">Colussi, Livio (9 September 2011). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2011.05.056">"The convergence classes of Collatz function"</a>. Theoretical Computer Science. 412 (39): 5409–5419. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2011.05.056">10.1016/j.tcs.2011.05.056</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=The+convergence+classes+of+Collatz+function&rft.volume=412&rft.issue=39&rft.pages=%3Cspan+class%3D%22nowrap%22%3E5409-%3C%2Fspan%3E5419&rft.date=2011-09-09&rft_id=info%3Adoi%2F10.1016%2Fj.tcs.2011.05.056&rft.aulast=Colussi&rft.aufirst=Livio&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.tcs.2011.05.056&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Hew2016-25"><a href="#cite_ref-Hew2016_25-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHew2016" class="citation journal cs1">Hew, Patrick Chisan (7 March 2016). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2015.12.033">"Working in binary protects the repetends of 1/3h: Comment on Colussi's 'The convergence classes of Collatz function'"</a>. Theoretical Computer Science. 618: 135–141. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2015.12.033">10.1016/j.tcs.2015.12.033</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Working+in+binary+protects+the+repetends+of+1%2F3%3Csup%3Eh%3C%2Fsup%3E%3A+Comment+on+Colussi%27s+%27The+convergence+classes+of+Collatz+function%27&rft.volume=618&rft.pages=%3Cspan+class%3D%22nowrap%22%3E135-%3C%2Fspan%3E141&rft.date=2016-03-07&rft_id=info%3Adoi%2F10.1016%2Fj.tcs.2015.12.033&rft.aulast=Hew&rft.aufirst=Patrick+Chisan&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.tcs.2015.12.033&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLagarias1990" class="citation journal cs1">Lagarias, Jeffrey (1990). <a rel="nofollow" class="external text" href="https://eudml.org/doc/206298">"The set of rational cycles for the 3x+1 problem"</a>. Acta Arithmetica. 56 (1): 33–53. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa-56-1-33-53">10.4064/aa-56-1-33-53</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0065-1036">0065-1036</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Arithmetica&rft.atitle=The+set+of+rational+cycles+for+the+3x%2B1+problem&rft.volume=56&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E33-%3C%2Fspan%3E53&rft.date=1990&rft_id=info%3Adoi%2F10.4064%2Faa-56-1-33-53&rft.issn=0065-1036&rft.aulast=Lagarias&rft.aufirst=Jeffrey&rft_id=https%3A%2F%2Feudml.org%2Fdoc%2F206298&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Belaga_(1998a)-27"><a href="#cite_ref-Belaga_(1998a)_27-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBelagaMignotte1998" class="citation journal cs1">Belaga, Edward G.; Mignotte, Maurice (1998). <a rel="nofollow" class="external text" href="http://www.emis.de/journals/EM/expmath/volumes/7/7.html">"Embedding the 3x+1 Conjecture in a 3x+d Context"</a>. Experimental Mathematics. 7 (2): 145–151. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10586458.1998.10504364">10.1080/10586458.1998.10504364</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17925995">17925995</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Experimental+Mathematics&rft.atitle=Embedding+the+3x%2B1+Conjecture+in+a+3x%2Bd+Context&rft.volume=7&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E145-%3C%2Fspan%3E151&rft.date=1998&rft_id=info%3Adoi%2F10.1080%2F10586458.1998.10504364&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17925995%23id-name%3DS2CID&rft.aulast=Belaga&rft.aufirst=Edward+G.&rft.au=Mignotte%2C+Maurice&rft_id=http%3A%2F%2Fwww.emis.de%2Fjournals%2FEM%2Fexpmath%2Fvolumes%2F7%2F7.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernsteinLagarias1996" class="citation journal cs1">Bernstein, Daniel J.; Lagarias, Jeffrey C. (1996). <a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-1996-060-x">"The 3x + 1 conjugacy map"</a>. <a href="/wiki/Canadian_Journal_of_Mathematics" title="Canadian Journal of Mathematics">Canadian Journal of Mathematics</a>. 48 (6): 1154–1169. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-1996-060-x">10.4153/CJM-1996-060-x</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0008-414X">0008-414X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Mathematics&rft.atitle=The+3x+%2B+1+conjugacy+map&rft.volume=48&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1154-%3C%2Fspan%3E1169&rft.date=1996&rft_id=info%3Adoi%2F10.4153%2FCJM-1996-060-x&rft.issn=0008-414X&rft.aulast=Bernstein&rft.aufirst=Daniel+J.&rft.au=Lagarias%2C+Jeffrey+C.&rft_id=https%3A%2F%2Fdoi.org%2F10.4153%252FCJM-1996-060-x&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Chamberland_(1996)-29"><a href="#cite_ref-Chamberland_(1996)_29-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChamberland1996" class="citation journal cs1">Chamberland, Marc (1996). "A continuous extension of the 3x + 1 problem to the real line". Dynam. Contin. Discrete Impuls Systems. 2 (4): 495–509.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Dynam.+Contin.+Discrete+Impuls+Systems&rft.atitle=A+continuous+extension+of+the+3x+%2B+1+problem+to+the+real+line&rft.volume=2&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E495-%3C%2Fspan%3E509&rft.date=1996&rft.aulast=Chamberland&rft.aufirst=Marc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-Letherman,_Schleicher,_and_Wood_(1999)-30"><a href="#cite_ref-Letherman,_Schleicher,_and_Wood_(1999)_30-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLethermanSchleicherWood1999" class="citation journal cs1">Letherman, Simon; Schleicher, Dierk; Wood, Reg (1999). "The (3n + 1)-problem and holomorphic dynamics". Experimental Mathematics. 8 (3): 241–252. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10586458.1999.10504402">10.1080/10586458.1999.10504402</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Experimental+Mathematics&rft.atitle=The+%283n+%2B+1%29-problem+and+holomorphic+dynamics&rft.volume=8&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E241-%3C%2Fspan%3E252&rft.date=1999&rft_id=info%3Adoi%2F10.1080%2F10586458.1999.10504402&rft.aulast=Letherman&rft.aufirst=Simon&rft.au=Schleicher%2C+Dierk&rft.au=Wood%2C+Reg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScollo2007" class="citation web cs1">Scollo, Giuseppe (2007). <a rel="nofollow" class="external text" href="http://www.dmi.unict.it/~scollo/seminars/gridpa2007/CR3x+1paper.pdf">"Looking for class records in the 3x + 1 problem by means of the COMETA grid infrastructure"</a> (PDF). Grid Open Days at the University of Palermo.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Grid+Open+Days+at+the+University+of+Palermo&rft.atitle=Looking+for+class+records+in+the+3x+%2B+1+problem+by+means+of+the+COMETA+grid+infrastructure&rft.date=2007&rft.aulast=Scollo&rft.aufirst=Giuseppe&rft_id=http%3A%2F%2Fwww.dmi.unict.it%2F~scollo%2Fseminars%2Fgridpa2007%2FCR3x%2B1paper.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> Lagarias (1985),<a href="#cite_note-Lagarias_(1985)-2">[2]</a> Theorem D. </li> <li id="cite_note-Clay-33"><a href="#cite_ref-Clay_33-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClay" class="citation web cs1">Clay, Oliver Keatinge. <a rel="nofollow" class="external text" href="https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=2052&context=jhm">"The Long Search for Collatz Counterexamples"</a>. p. 208. Retrieved 26 July 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Long+Search+for+Collatz+Counterexamples&rft.pages=208&rft.aulast=Clay&rft.aufirst=Oliver+Keatinge&rft_id=https%3A%2F%2Fscholarship.claremont.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D2052%26context%3Djhm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConway1972" class="citation conference cs1">Conway, John H. (1972). "Unpredictable iterations". Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder. pp. 49–52.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Unpredictable+iterations&rft.btitle=Proc.+1972+Number+Theory+Conf.%2C+Univ.+Colorado%2C+Boulder&rft.pages=%3Cspan+class%3D%22nowrap%22%3E49-%3C%2Fspan%3E52&rft.date=1972&rft.aulast=Conway&rft.aufirst=John+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-KurtzSimon-35"><a href="#cite_ref-KurtzSimon_35-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurtzSimon2007" class="citation book cs1">Kurtz, Stuart A.; Simon, Janos (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mhrOkx-xyJIC&pg=PA542">"The undecidability of the generalized Collatz problem"</a>. In Cai, J.-Y.; Cooper, S. B.; Zhu, H. (eds.). Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. pp. 542–553. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-72504-6_49">10.1007/978-3-540-72504-6_49</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-72503-9" title="Special:BookSources/978-3-540-72503-9"><bdi>978-3-540-72503-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+undecidability+of+the+generalized+Collatz+problem&rft.btitle=Proceedings+of+the+4th+International+Conference+on+Theory+and+Applications+of+Models+of+Computation%2C+TAMC+2007%2C+held+in+Shanghai%2C+China+in+May+2007&rft.pages=%3Cspan+class%3D%22nowrap%22%3E542-%3C%2Fspan%3E553&rft.date=2007&rft_id=info%3Adoi%2F10.1007%2F978-3-540-72504-6_49&rft.isbn=978-3-540-72503-9&rft.aulast=Kurtz&rft.aufirst=Stuart+A.&rft.au=Simon%2C+Janos&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmhrOkx-xyJIC%26pg%3DPA542&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> As <a rel="nofollow" class="external text" href="http://www.cs.uchicago.edu/~simon/RES/collatz.pdf">PDF</a> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBen-Amram2015" class="citation journal cs1">Ben-Amram, Amir M. (2015). "Mortality of iterated piecewise affine functions over the integers: Decidability and complexity". Computability. 1 (1): 19–56. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3233%2FCOM-150032">10.3233/COM-150032</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computability&rft.atitle=Mortality+of+iterated+piecewise+affine+functions+over+the+integers%3A+Decidability+and+complexity&rft.volume=1&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E19-%3C%2Fspan%3E56&rft.date=2015&rft_id=info%3Adoi%2F10.3233%2FCOM-150032&rft.aulast=Ben-Amram&rft.aufirst=Amir+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichel,_Pascal1993" class="citation journal cs1">Michel, Pascal (1993). "Busy beaver competition and Collatz-like problems". Archive for Mathematical Logic. 32 (5): 351–367. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01409968">10.1007/BF01409968</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+Mathematical+Logic&rft.atitle=Busy+beaver+competition+and+Collatz-like+problems&rft.volume=32&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E351-%3C%2Fspan%3E367&rft.date=1993&rft_id=info%3Adoi%2F10.1007%2FBF01409968&rft.au=Michel%2C+Pascal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://arxiv.org/html/2107.12475v2">"Hardness of busy beaver value BB(15)"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Hardness+of+busy+beaver+value+BB%2815%29&rft_id=https%3A%2F%2Farxiv.org%2Fhtml%2F2107.12475v2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatthews" class="citation web cs1">Matthews, Keith. <a rel="nofollow" class="external text" href="http://www.numbertheory.org/3x+1/">"3 x + 1 page"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=%3Cspan+class%3D%22nowrap%22%3E3+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3Ex%3C%2Fspan%3E+%2B+1%3C%2Fspan%3E+page&rft.aulast=Matthews&rft.aufirst=Keith&rft_id=http%3A%2F%2Fwww.numbertheory.org%2F3x%2B1%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"></li> <li>An ongoing <a href="/wiki/Volunteer_computing" title="Volunteer computing">volunteer computing</a> <a rel="nofollow" class="external text" href="https://collatz-problem.org/">project</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210830110430/https://collatz-problem.org/">Archived</a> 2021-08-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> by David Bařina verifies Convergence of the Collatz conjecture for large values. (furthest progress so far)</li> <li>(<a href="/wiki/Berkeley_Open_Infrastructure_for_Network_Computing" title="Berkeley Open Infrastructure for Network Computing">BOINC</a>) volunteer computing <a rel="nofollow" class="external text" href="http://boinc.thesonntags.com/collatz/">project</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171204131813/http://boinc.thesonntags.com/collatz/">Archived</a> 2017-12-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> that verifies the Collatz conjecture for larger values.</li> <li>An ongoing volunteer computing <a rel="nofollow" class="external text" href="http://www.ericr.nl/wondrous/index.html">project</a> by Eric Roosendaal verifies the Collatz conjecture for larger and larger values.</li> <li>Another ongoing volunteer computing <a rel="nofollow" class="external text" href="http://sweet.ua.pt/tos/3x+1.html">project</a> by Tomás Oliveira e Silva continues to verify the Collatz conjecture (with fewer statistics than Eric Roosendaal's page but with further progress made).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CollatzProblem.html">"Collatz Problem"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Collatz+Problem&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCollatzProblem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/CollatzProblem">Collatz Problem</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>..</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNochella" class="citation web cs1">Nochella, Jesse. <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/CollatzPaths/">"Collatz Paths"</a>. <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+Demonstrations+Project&rft.atitle=Collatz+Paths&rft.aulast=Nochella&rft.aufirst=Jesse&rft_id=http%3A%2F%2Fdemonstrations.wolfram.com%2FCollatzPaths%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation audio-visual cs1"><a href="/wiki/David_Eisenbud" title="David Eisenbud">Eisenbud, D.</a> (8 August 2016). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=5mFpVDpKX70">Uncrackable? The Collatz conjecture</a> (short video). Numberphile. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211211/5mFpVDpKX70">Archived</a> from the original on 2021-12-11 – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Uncrackable%3F+The+Collatz+conjecture&rft.series=Numberphile&rft.date=2016-08-08&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3D5mFpVDpKX70&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation audio-visual cs1"><a href="/wiki/David_Eisenbud" title="David Eisenbud">Eisenbud, D.</a> (August 9, 2016). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=O2_h3z1YgEU">Uncrackable? Collatz conjecture</a> (extra footage). Numberphile. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211211/O2_h3z1YgEU">Archived</a> from the original on 2021-12-11 – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Uncrackable%3F+Collatz+conjecture&rft.series=Numberphile&rft.date=2016-08-09&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DO2_h3z1YgEU&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation audio-visual cs1"><a href="/wiki/Alex_Kontorovich" title="Alex Kontorovich">Alex Kontorovich</a> (featuring) (30 July 2021). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=094y1Z2wpJg">The simplest math problem no one can solve</a> (short video). Veritasium – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+simplest+math+problem+no+one+can+solve&rft.series=Veritasium&rft.date=2021-07-30&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3D094y1Z2wpJg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACollatz+conjecture" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://www.technologyreview.com/2021/07/02/1027475/computers-ready-solve-this-notorious-math-problem/">Are computers ready to solve this notoriously unwieldy math problem?</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Collatz_conjecture&oldid=1267439306">https://en.wikipedia.org/w/index.php?title=Collatz_conjecture&oldid=1267439306</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Conjectures" title="Category:Conjectures">Conjectures</a></li><li><a href="/wiki/Category:Arithmetic_dynamics" title="Category:Arithmetic dynamics">Arithmetic dynamics</a></li><li><a href="/wiki/Category:Integer_sequences" title="Category:Integer sequences">Integer sequences</a></li><li><a href="/wiki/Category:Unsolved_problems_in_number_theory" title="Category:Unsolved problems in number theory">Unsolved problems in number theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">CS1 maint: bot: original URL status unknown</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_indefinitely_semi-protected_pages" title="Category:Wikipedia indefinitely semi-protected pages">Wikipedia indefinitely semi-protected pages</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_September_2024" title="Category:Wikipedia articles needing clarification from September 2024">Wikipedia articles needing clarification from September 2024</a></li><li><a href="/wiki/Category:Commons_category_link_from_Wikidata" title="Category:Commons category link from Wikidata">Commons category link from Wikidata</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 5 January 2025, at 03:11 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Collatz conjecture - Wikipedia