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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Riesel_problem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Riesel_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Riesel problem</span> </div> </a> <ul id="toc-Riesel_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Known_Riesel_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Known_Riesel_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Known Riesel numbers</span> </div> </a> <ul id="toc-Known_Riesel_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Covering_set" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Covering_set"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Covering set</span> </div> </a> <ul id="toc-Covering_set-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_smallest_n_for_which_k_·_2n_−_1_is_prime" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_smallest_n_for_which_k_·_2n_−_1_is_prime"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>The smallest <i>n</i> for which <i>k</i> · 2<i><sup>n</sup></i> − 1 is prime</span> </div> </a> <ul id="toc-The_smallest_n_for_which_k_·_2n_−_1_is_prime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simultaneously_Riesel_and_Sierpiński" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Simultaneously_Riesel_and_Sierpiński"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Simultaneously Riesel and Sierpiński</span> </div> </a> <ul id="toc-Simultaneously_Riesel_and_Sierpiński-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_dual_Riesel_problem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_dual_Riesel_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>The dual Riesel problem</span> </div> </a> <ul id="toc-The_dual_Riesel_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riesel_number_base_b" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Riesel_number_base_b"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Riesel number base <i>b</i></span> </div> </a> <ul id="toc-Riesel_number_base_b-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </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" > <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 " 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class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_de_Riesel" title="Nombre de Riesel – French" lang="fr" hreflang="fr" data-title="Nombre de Riesel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_di_Riesel" title="Numero di Riesel – Italian" lang="it" hreflang="it" data-title="Numero di Riesel" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rieselgetal" title="Rieselgetal – Dutch" lang="nl" hreflang="nl" data-title="Rieselgetal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Rieseltall" title="Rieseltall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Rieseltall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Problem_Riesela" title="Problem Riesela – Polish" lang="pl" hreflang="pl" data-title="Problem Riesela" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_Riesel" title="Număr Riesel – Romanian" lang="ro" hreflang="ro" data-title="Număr Riesel" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Odd number with specific properties</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>Riesel number</b> is an <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a> <a href="/wiki/Natural_number" title="Natural number">natural number</a> <i>k</i> for which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\times 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\times 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae941aacbc931e26a3f0a1c99967336a39c3ec30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.435ex; height:2.509ex;" alt="{\displaystyle k\times 2^{n}-1}"></span> is <a href="/wiki/Composite_number" title="Composite number">composite</a> for all natural numbers <i>n</i> (sequence <span class="nowrap external"><a href="//oeis.org/A101036" class="extiw" title="oeis:A101036">A101036</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). In other words, when <i>k</i> is a Riesel number, all members of the following <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> are composite: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\,k\times 2^{n}-1:n\in \mathbb {N} \,\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mspace width="thinmathspace" /> <mi>k</mi> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\,k\times 2^{n}-1:n\in \mathbb {N} \,\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a9080629895cd4b69ec09a766d8618e0930902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.419ex; height:2.843ex;" alt="{\displaystyle \left\{\,k\times 2^{n}-1:n\in \mathbb {N} \,\right\}.}"></span></dd></dl> <p>If the form is instead <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\times 2^{n}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\times 2^{n}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786f0137f00467b1690b5736c5d142842b1a9807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.435ex; height:2.509ex;" alt="{\displaystyle k\times 2^{n}+1}"></span>, then <i>k</i> is a <a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński number</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Riesel_problem">Riesel problem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=1" title="Edit section: Riesel problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 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><span class="unsolved-label" id="unsolved-label-mathematics">Unsolved problem in mathematics</span>:</div> <div class="unsolved-body">Is 509,203 the smallest Riesel number?</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> <p>In 1956, <a href="/wiki/Hans_Riesel" title="Hans Riesel">Hans Riesel</a> showed that there are an <a href="/wiki/Infinite_set" title="Infinite set">infinite</a> number of integers <i>k</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\times 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\times 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae941aacbc931e26a3f0a1c99967336a39c3ec30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.435ex; height:2.509ex;" alt="{\displaystyle k\times 2^{n}-1}"></span> is not <a href="/wiki/Prime_number" title="Prime number">prime</a> for any integer <i>n</i>. He showed that the number 509203 has this property, as does 509203 plus any positive <a href="/wiki/Integer" title="Integer">integer</a> multiple of 11184810.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The <b>Riesel problem</b> consists in determining the smallest Riesel number. Because no <a href="/wiki/Covering_set" title="Covering set">covering set</a> has been found for any <i>k</i> less than 509203, it is <a href="/wiki/Conjecture" title="Conjecture">conjectured</a> to be the smallest Riesel number. </p><p>To check if there are <i>k</i> < 509203, the <a href="/wiki/Riesel_Sieve" class="mw-redirect" title="Riesel Sieve">Riesel Sieve project</a> (analogous to <a href="/wiki/Seventeen_or_Bust" class="mw-redirect" title="Seventeen or Bust">Seventeen or Bust</a> for <a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński numbers</a>) started with 101 candidates <i>k</i>. As of December 2022, 57 of these <i>k</i> had been eliminated by Riesel Sieve, <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a>, or outside persons.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The remaining 42 values of <i>k</i> that have yielded only composite numbers for all values of <i>n</i> so far tested are </p> <dl><dd>23669, 31859, 38473, 46663, 67117, 74699, 81041, 107347, 121889, 129007, 143047, 161669, 206231, 215443, 226153, 234343, 245561, 250027, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 409753, 444637, 470173, 474491, 477583, 485557, 494743.</dd></dl> <p>The most recent elimination was in April 2023, when 97139 × 2<sup>18397548</sup> − 1 was found to be prime by Ryan Propper. This number is 5,538,219 digits long. </p><p>As of January 2023, PrimeGrid has searched the remaining candidates up to <i>n</i> = 14,900,000.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Known_Riesel_numbers">Known Riesel numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=2" title="Edit section: Known Riesel numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sequence of currently <i>known</i> Riesel numbers begins with: </p> <dl><dd>509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, ... (sequence <span class="nowrap external"><a href="//oeis.org/A101036" class="extiw" title="oeis:A101036">A101036</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Covering_set">Covering set</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=3" title="Edit section: Covering set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A number can be shown to be a Riesel number by exhibiting a <i><a href="/wiki/Covering_set" title="Covering set">covering set</a></i>: a set of prime numbers that will divide any member of the sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have covering sets as follows: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 509203\times 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>509203</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 509203\times 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d00fb13e902314069e90b21b65ac1067b7f0d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.199ex; height:2.509ex;" alt="{\displaystyle 509203\times 2^{n}-1}"></span> has covering set {3, 5, 7, 13, 17, 241}</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 762701\times 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>762701</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 762701\times 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/096fe251c790368d220e670023c2be6f826fe0ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.199ex; height:2.509ex;" alt="{\displaystyle 762701\times 2^{n}-1}"></span> has covering set {3, 5, 7, 13, 17, 241}</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 777149\times 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>777149</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 777149\times 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8f0d1973605a24bd32df6f78d3e70efddd4c9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.199ex; height:2.509ex;" alt="{\displaystyle 777149\times 2^{n}-1}"></span> has covering set {3, 5, 7, 13, 19, 37, 73}</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 790841\times 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>790841</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 790841\times 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e1ac1cd06a539ab23dda6f438e251af377cbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.199ex; height:2.509ex;" alt="{\displaystyle 790841\times 2^{n}-1}"></span> has covering set {3, 5, 7, 13, 19, 37, 73}</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 992077\times 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>992077</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 992077\times 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f174e87a1fbda2a6c3f08133f82fa3bf48bce8d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.199ex; height:2.509ex;" alt="{\displaystyle 992077\times 2^{n}-1}"></span> has covering set {3, 5, 7, 13, 17, 241}.</li></ul> <div class="mw-heading mw-heading2"><h2 id="The_smallest_n_for_which_k_·_2n_−_1_is_prime"><span id="The_smallest_n_for_which_k_.C2.B7_2n_.E2.88.92_1_is_prime"></span>The smallest <i>n</i> for which <i>k</i> · 2<i><sup>n</sup></i> − 1 is prime</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=4" title="Edit section: The smallest n for which k · 2n − 1 is prime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Here is a sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c7111dd18cc74a15cc32e63405ff8514cbd759d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.25ex; height:2.843ex;" alt="{\displaystyle a(k)}"></span> for <i>k</i> = 1, 2, .... It is defined as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c7111dd18cc74a15cc32e63405ff8514cbd759d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.25ex; height:2.843ex;" alt="{\displaystyle a(k)}"></span> is the smallest <i>n</i> ≥ 0 such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\cdot 2^{n}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\cdot 2^{n}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8846578230e948d4112047ff65b2db7f3c4da3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.274ex; height:2.509ex;" alt="{\displaystyle k\cdot 2^{n}-1}"></span> is prime, or −1 if no such prime exists. </p> <dl><dd>2, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 1, 0, 1, 1, 4, 0, 3, 2, 1, 3, 4, 0, 1, 0, 2, 1, 2, 1, 1, 0, 3, 1, 2, 0, 7, 0, 1, 3, 4, 0, 1, 2, 1, 1, 2, 0, 1, 2, 1, 3, 12, 0, 3, 0, 2, 1, 4, 1, 5, 0, 1, 1, 2, 0, 7, 0, 1, ... (sequence <span class="nowrap external"><a href="//oeis.org/A040081" class="extiw" title="oeis:A040081">A040081</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). The first unknown <i>n</i> is for that <i>k</i> = 23669.</dd></dl> <p>Related sequences are <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A050412" class="extiw" title="oeis:A050412">A050412</a></span> (not allowing <i>n</i> = 0), for odd <i>k</i>s, see <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A046069" class="extiw" title="oeis:A046069">A046069</a></span> or <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A108129" class="extiw" title="oeis:A108129">A108129</a></span> (not allowing <i>n</i> = 0). </p> <div class="mw-heading mw-heading2"><h2 id="Simultaneously_Riesel_and_Sierpiński"><span id="Simultaneously_Riesel_and_Sierpi.C5.84ski"></span>Simultaneously Riesel and Sierpiński</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=5" title="Edit section: Simultaneously Riesel and Sierpiński"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A number both Riesel and <a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a> is a <a href="/wiki/%C3%89ric_Brier" title="Éric Brier">Brier</a> number. The five smallest known examples (and note that some might be smaller, i.e. that the sequence might not be comprehensive) are: 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, ... (<a href="//oeis.org/A076335" class="extiw" title="oeis:A076335">A076335</a>).<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="The_dual_Riesel_problem">The dual Riesel problem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=6" title="Edit section: The dual Riesel problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>dual Riesel numbers</b> are defined as the odd natural numbers <i>k</i> such that |2<sup><i>n</i></sup> - <i>k</i>| is composite for all natural numbers <i>n</i>. There is a conjecture that the set of this numbers is the same as the set of Riesel numbers. For example, |2<sup><i>n</i></sup> - 509203| is composite for all natural numbers <i>n</i>, and 509203 is conjectured to be the smallest dual Riesel number. </p><p>The smallest <i>n</i> which 2<sup><i>n</i></sup> - <i>k</i> is prime are (for odd <i>k</i>s, and this sequence requires that 2<sup><i>n</i></sup> > <i>k</i>) </p> <dl><dd>2, 3, 3, 39, 4, 4, 4, 5, 6, 5, 5, 6, 5, 5, 5, 7, 6, 6, 11, 7, 6, 29, 6, 6, 7, 6, 6, 7, 6, 6, 6, 8, 8, 7, 7, 10, 9, 7, 8, 9, 7, 8, 7, 7, 8, 7, 8, 10, 7, 7, 26, 9, 7, 8, 7, 7, 10, 7, 7, 8, 7, 7, 7, 47, 8, 14, 9, 11, 10, 9, 10, 8, 9, 8, 8, ... (sequence <span class="nowrap external"><a href="//oeis.org/A096502" class="extiw" title="oeis:A096502">A096502</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>The odd <i>k</i>s which <i>k</i> - 2<sup><i>n</i></sup> are all composite for all 2<sup><i>n</i></sup> < <i>k</i> (the <b>de Polignac numbers</b>) are </p> <dl><dd>1, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 905, 907, 959, 977, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1271, 1477, ... (sequence <span class="nowrap external"><a href="//oeis.org/A006285" class="extiw" title="oeis:A006285">A006285</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <p>The unknown values<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (August 2020)">clarification needed</span></a></i>]</sup> of <i>k</i>s are (for which 2<sup><i>n</i></sup> > <i>k</i>) </p> <dl><dd>1871, 2293, 25229, 31511, 36971, 47107, 48959, 50171, 56351, 63431, 69427, 75989, 81253, 83381, 84491, ...</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Riesel_number_base_b">Riesel number base <i>b</i></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=7" title="Edit section: Riesel number base b"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can generalize the Riesel problem to an integer base <i>b</i> ≥ 2. A <b>Riesel number base <i>b</i></b> is a positive integer <i>k</i> such that <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">gcd</a>(<i>k</i> − 1, <i>b</i> − 1) = 1. (if gcd(<i>k</i> − 1, <i>b</i> − 1) > 1, then gcd(<i>k</i> − 1, <i>b</i> − 1) is a trivial factor of <i>k</i>×<i>b</i><sup><i>n</i></sup> − 1 (Definition of trivial factors for the conjectures: Each and every <i>n</i>-value has the same factor))<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> For every integer <i>b</i> ≥ 2, there are infinitely many Riesel numbers base <i>b</i>. </p><p>Example 1: All numbers congruent to 84687 mod 10124569 and not congruent to 1 mod 5 are Riesel numbers base 6, because of the covering set {7, 13, 31, 37, 97}. Besides, these <i>k</i> are not trivial since gcd(<i>k</i> + 1, 6 − 1) = 1 for these <i>k</i>. (The Riesel base 6 conjecture is not proven, it has 3 remaining <i>k</i>, namely 1597, 9582 and 57492) </p><p>Example 2: 6 is a Riesel number to all bases <i>b</i> congruent to 34 mod 35, because if <i>b</i> is congruent to 34 mod 35, then 6×<i>b</i><sup><i>n</i></sup> − 1 is divisible by 5 for all even <i>n</i> and divisible by 7 for all odd <i>n</i>. Besides, 6 is not a trivial <i>k</i> in these bases <i>b</i> since gcd(6 − 1, <i>b</i> − 1) = 1 for these bases <i>b</i>. </p><p>Example 3: All squares <i>k</i> congruent to 12 mod 13 and not congruent to 1 mod 11 are Riesel numbers base 12, since for all such <i>k</i>, <i>k</i>×12<sup><i>n</i></sup> − 1 has algebraic factors for all even <i>n</i> and divisible by 13 for all odd <i>n</i>. Besides, these <i>k</i> are not trivial since gcd(<i>k</i> + 1, 12 − 1) = 1 for these <i>k</i>. (The Riesel base 12 conjecture is proven) </p><p>Example 4: If <i>k</i> is between a multiple of 5 and a multiple of 11, then <i>k</i>×109<sup><i>n</i></sup> − 1 is divisible by either 5 or 11 for all positive integers <i>n</i>. The first few such <i>k</i> are 21, 34, 76, 89, 131, 144, ... However, all these <i>k</i> < 144 are also trivial <i>k</i> (i. e. gcd(<i>k</i> − 1, 109 − 1) is not 1). Thus, the smallest Riesel number base 109 is 144. (The Riesel base 109 conjecture is not proven, it has one remaining <i>k</i>, namely 84) </p><p>Example 5: If <i>k</i> is square, then <i>k</i>×49<sup><i>n</i></sup> − 1 has algebraic factors for all positive integers <i>n</i>. The first few positive squares are 1, 4, 9, 16, 25, 36, ... However, all these <i>k</i> < 36 are also trivial <i>k</i> (i. e. gcd(<i>k</i> − 1, 49 − 1) is not 1). Thus, the smallest Riesel number base 49 is 36. (The Riesel base 49 conjecture is proven) </p><p>We want to find and proof the smallest Riesel number base <i>b</i> for every integer <i>b</i> ≥ 2. It is a conjecture that if <i>k</i> is a Riesel number base <i>b</i>, then at least one of the three conditions holds: </p> <ol><li>All numbers of the form <i>k</i>×<i>b</i><sup><i>n</i></sup> − 1 have a factor in some covering set. (For example, <i>b</i> = 22, <i>k</i> = 4461, then all numbers of the form <i>k</i>×<i>b</i><sup><i>n</i></sup> − 1 have a factor in the covering set: {5, 23, 97})</li> <li><i>k</i>×<i>b</i><sup><i>n</i></sup> − 1 has algebraic factors. (For example, <i>b</i> = 9, <i>k</i> = 4, then <i>k</i>×<i>b</i><sup><i>n</i></sup> − 1 can be factored to (2×3<sup><i>n</i></sup> − 1) × (2×3<sup><i>n</i></sup> + 1))</li> <li>For some <i>n</i>, numbers of the form <i>k</i>×<i>b</i><sup><i>n</i></sup> − 1 have a factor in some covering set; and for all other <i>n</i>, <i>k</i>×<i>b</i><sup><i>n</i></sup> − 1 has algebraic factors. (For example, <i>b</i> = 19, <i>k</i> = 144, then if <i>n</i> is odd, then <i>k</i>×<i>b</i><sup><i>n</i></sup> − 1 is divisible by 5, if <i>n</i> is even, then <i>k</i>×<i>b</i><sup><i>n</i></sup> − 1 can be factored to (12×19<sup><i>n</i>/2</sup> − 1) × (12×19<sup><i>n</i>/2</sup> + 1))</li></ol> <p>In the following list, we only consider those positive integers <i>k</i> such that gcd(<i>k</i> − 1, <i>b</i> − 1) = 1, and all integer <i>n</i> must be ≥ 1. </p><p>Note: <i>k</i>-values that are a multiple of <i>b</i> and where <i>k</i>−1 is not prime are included in the conjectures (and included in the remaining <i>k</i> with <style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style><span class="tmp-color" style="color:red">red</span> color if no primes are known for these <i>k</i>-values) but excluded from testing (Thus, never be the <i>k</i> of "largest 5 primes found"), since such <i>k</i>-values will have the same prime as <i>k</i> / <i>b</i>. </p> <table class="wikitable"> <tbody><tr> <td><i>b</i> </td> <td>conjectured smallest Riesel <i>k</i> </td> <td>covering set / algebraic factors </td> <td>remaining <i>k</i> with no known primes (red indicates the <i>k</i>-values that are a multiple of <i>b</i> and <i>k</i>−1 is not prime) </td> <td>number of remaining <i>k</i> with no known primes<br />(excluding the red <i>k</i>s) </td> <td>testing limit of <i>n</i><br />(excluding the red <i>k</i>s) </td> <td>largest 5 primes found<br />(excluding red <i>k</i>s) </td></tr> <tr> <td>2 </td> <td>509203 </td> <td>{3, 5, 7, 13, 17, 241} </td> <td>23669, 31859, 38473, 46663, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">47338</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">63718</span>, 67117, 74699, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">76946</span>, 81041, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">93326</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">94676</span>, 107347, 121889, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">127436</span>, 129007, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">134234</span>, 143047, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">149398</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">153892</span>, 161669, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">162082</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">186652</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">189352</span>, 206231, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">214694</span>, 215443, 226153, 234343, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">243778</span>, 245561, 250027, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">254872</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">258014</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">268468</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">286094</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">298796</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">307784</span>, 315929, 319511, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">323338</span>, 324011, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">324164</span>, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">373304</span>, 384539, 386801, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">388556</span>, 397027, 409753, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">412462</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">429388</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">430886</span>, 444637, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">452306</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">468686</span>, 470173, 474491, 477583, 485557, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">487556</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">491122</span>, 494743, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">500054</span> </td> <td>42 </td> <td><a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a> is currently searching every remaining <i>k</i> at <i>n</i> > 14.5M </td> <td>97139×2<sup>18397548</sup>−1<br />93839×2<sup>15337656</sup>−1<br />192971×2<sup>14773498</sup>−1<br />206039×2<sup>13104952</sup>−1<br />2293×2<sup>12918431</sup>−1 </td></tr> <tr> <td>3 </td> <td>63064644938 </td> <td>{5, 7, 13, 17, 19, 37, 41, 193, 757} </td> <td>3677878, 6878756, 10463066, 10789522, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">11033634</span>, 16874152, 18137648, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">20636268</span>, 21368582, 29140796, 31064666, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">31389198</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">32368566</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">33100902</span>, 38394682, 40175404, 40396658, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">50622456</span>, 51672206, 52072432, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">54412944</span>, 56244334, 59254534, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">61908864</span>, 62126002, 62402206, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">64105746</span>, 65337866, 71248336, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">87422388</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">93193998</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">94167594</span>, 94210372, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">97105698</span>, 97621124, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">99302706</span>, 103101766, 103528408, 107735486, 111036578, 115125596, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">115184046</span>, ... </td> <td>100714 </td> <td><i>k</i> = 3677878 at <i>n</i> = 5M, 4M < <i>k</i> ≤ 2.147G at <i>n</i> = 1.07M, 2.147G < <i>k</i> ≤ 6G at <i>n</i> = 500K, 6G < <i>k</i> ≤ 10G at <i>n</i> = 250K, 10G < <i>k</i> ≤ 63G at <i>n</i> = 100K, , <i>k</i> > 63G at <i>n</i> = 655K </td> <td> <p>676373272×3<sup>1072675</sup>−1<br />1068687512×3<sup>1067484</sup>−1<br />1483575692×3<sup>1067339</sup>−1<br />780548926×3<sup>1064065</sup>−1<br />1776322388×3<sup>1053069</sup>−1 </p> </td></tr> <tr> <td>4 </td> <td>9 </td> <td>9×4<sup><i>n</i></sup> − 1 = (3×2<sup><i>n</i></sup> − 1) × (3×2<sup><i>n</i></sup> + 1) </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>8×4<sup>1</sup>−1<br />6×4<sup>1</sup>−1<br />5×4<sup>1</sup>−1<br />3×4<sup>1</sup>−1<br />2×4<sup>1</sup>−1 </td></tr> <tr> <td>5 </td> <td>346802 </td> <td>{3, 7, 13, 31, 601} </td> <td>4906, 23906, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">24530</span>, 26222, 35248, 68132, 71146, 76354, 81134, 92936, 102952, 109238, 109862, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">119530</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">122650</span>, 127174, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">131110</span>, 131848, 134266, 143632, 145462, 145484, 146756, 147844, 151042, 152428, 154844, 159388, 164852, 170386, 170908, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">176240</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">179080</span>, 182398, 187916, 189766, 190334, 195872, 201778, 204394, 206894, 231674, 239062, 239342, 246238, 248546, 259072, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">264610</span>, 265702, 267298, 271162, 285598, 285728, 298442, 304004, 313126, 318278, 325922, 335414, 338866, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">340660</span> </td> <td>54 </td> <td><a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a> is currently searching every remaining <i>k</i> at <i>n</i> > 4.8M </td> <td>3622×5<sup>7558139</sup>-1<br /> <p>136804×5<sup>4777253</sup>-1<br /> 52922×5<sup>4399812</sup>-1<br /> 177742×5<sup>4386703</sup>-1<br /> 213988×5<sup>4138363</sup>-1<br /> </p> </td></tr> <tr> <td>6 </td> <td>84687 </td> <td>{7, 13, 31, 37, 97} </td> <td>1597, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">9582</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">57492</span> </td> <td>1 </td> <td>5M </td> <td>36772×6<sup>1723287</sup>−1<br />43994×6<sup>569498</sup>−1<br />77743×6<sup>560745</sup>−1<br />51017×6<sup>528803</sup>−1<br />57023×6<sup>483561</sup>−1 </td></tr> <tr> <td>7 </td> <td>408034255082 </td> <td>{5, 13, 19, 43, 73, 181, 193, 1201} </td> <td>315768, 1356018, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">2210376</span>, 2494112, 2631672, 3423408, 4322834, 4326672, 4363418, 4382984, 4870566, 4990788, 5529368, 6279074, 6463028, 6544614, 7446728, 7553594, 8057622, 8354966, 8389476, 8640204, 8733908, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">9492126</span>, 9829784, 10096364, 10098716, 10243424, 10289166, 10394778, 10494794, 10965842, 11250728, 11335962, 11372214, 11522846, 11684954, 11943810, 11952888, 11983634, 12017634, 12065672, 12186164, 12269808, 12291728, 12801926, 13190732, 13264728, 13321148, 13635266, 13976426, ... </td> <td>16399 <i>k</i>s ≤ 1G </td> <td><i>k</i> ≤ 2M at <i>n</i> = 1M, 2M < <i>k</i> ≤ 10M at <i>n</i> = 500K, 10M < <i>k</i> ≤ 110M at <i>n</i> = 150K, 110M < <i>k</i> ≤ 300M at <i>n</i> = 100K, 300M < <i>k</i> ≤ 1G at <i>n</i> = 25K </td> <td>1620198×7<sup>684923</sup>−1<br />7030248×7<sup>483691</sup>−1<br />7320606×7<sup>464761</sup>−1<br />5646066×7<sup>460533</sup>−1<br />9012942×7<sup>425310</sup>−1 </td></tr> <tr> <td>8 </td> <td>14 </td> <td>{3, 5, 13} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>11×8<sup>18</sup>−1<br />5×8<sup>4</sup>−1<br />12×8<sup>3</sup>−1<br />7×8<sup>3</sup>−1<br />2×8<sup>2</sup>−1 </td></tr> <tr> <td>9 </td> <td>4 </td> <td>4×9<sup><i>n</i></sup> − 1 = (2×3<sup><i>n</i></sup> − 1) × (2×3<sup><i>n</i></sup> + 1) </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>2×9<sup>1</sup>−1 </td></tr> <tr> <td>10 </td> <td>10176 </td> <td>{7, 11, 13, 37} </td> <td>4421 </td> <td>1 </td> <td>1.72M </td> <td>7019×10<sup>881309</sup>−1<br />8579×10<sup>373260</sup>−1<br />6665×10<sup>60248</sup>−1<br />1935×10<sup>51836</sup>−1<br />1803×10<sup>45882</sup>−1 </td></tr> <tr> <td>11 </td> <td>862 </td> <td>{3, 7, 19, 37} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>62×11<sup>26202</sup>−1<br />308×11<sup>444</sup>−1<br />172×11<sup>187</sup>−1<br />284×11<sup>186</sup>−1<br />518×11<sup>78</sup>−1 </td></tr> <tr> <td>12 </td> <td>25 </td> <td>{13} for odd <i>n</i>, 25×12<sup><i>n</i></sup> − 1 = (5×12<sup><i>n</i>/2</sup> − 1) × (5×12<sup><i>n</i>/2</sup> + 1) for even <i>n</i> </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>24×12<sup>4</sup>−1<br />18×12<sup>2</sup>−1<br />17×12<sup>2</sup>−1<br />13×12<sup>2</sup>−1<br />10×12<sup>2</sup>−1 </td></tr> <tr> <td>13 </td> <td>302 </td> <td>{5, 7, 17} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>288×13<sup>109217</sup>−1<br />146×13<sup>30</sup>−1<br />92×13<sup>23</sup>−1<br />102×13<sup>20</sup>−1<br />300×13<sup>10</sup>−1 </td></tr> <tr> <td>14 </td> <td>4 </td> <td>{3, 5} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>2×14<sup>4</sup>−1<br />3×14<sup>1</sup>−1 </td></tr> <tr> <td>15 </td> <td>36370321851498 </td> <td>{13, 17, 113, 211, 241, 1489, 3877} </td> <td>381714, 4502952, 5237186, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">5725710</span>, 7256276, 8524154, 11118550, 11176190, 12232180, 15691976, 16338798, 16695396, 18267324, 18709072, 19615792, ... </td> <td>14 <i>k</i>s ≤ 20M </td> <td><i>k</i> ≤ 10M at <i>n</i> = 1M, 10M < <i>k</i> ≤ 20M at <i>n</i> = 250K </td> <td>4242104×15<sup>728840</sup>−1<br />9756404×15<sup>527590</sup>−1<br />9105446×15<sup>496499</sup>−1<br />5854146×15<sup>428616</sup>−1<br />9535278×15<sup>375675</sup>−1 </td></tr> <tr> <td>16 </td> <td>9 </td> <td>9×16<sup><i>n</i></sup> − 1 = (3×4<sup><i>n</i></sup> − 1) × (3×4<sup><i>n</i></sup> + 1) </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>8×16<sup>1</sup>−1<br />5×16<sup>1</sup>−1<br />3×16<sup>1</sup>−1<br />2×16<sup>1</sup>−1 </td></tr> <tr> <td>17 </td> <td>86 </td> <td>{3, 5, 29} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>44×17<sup>6488</sup>−1<br />36×17<sup>243</sup>−1<br />10×17<sup>117</sup>−1<br />26×17<sup>110</sup>−1<br />58×17<sup>35</sup>−1 </td></tr> <tr> <td>18 </td> <td>246 </td> <td>{5, 13, 19} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>151×18<sup>418</sup>−1<br />78×18<sup>172</sup>−1<br />50×18<sup>110</sup>−1<br />79×18<sup>63</sup>−1<br />237×18<sup>44</sup>−1 </td></tr> <tr> <td>19 </td> <td>144 </td> <td>{5} for odd <i>n</i>, 144×19<sup><i>n</i></sup> − 1 = (12×19<sup><i>n</i>/2</sup> − 1) × (12×19<sup><i>n</i>/2</sup> + 1) for even <i>n</i> </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>134×19<sup>202</sup>−1<br />104×19<sup>18</sup>−1<br />38×19<sup>11</sup>−1<br />128×19<sup>10</sup>−1<br />108×19<sup>6</sup>−1 </td></tr> <tr> <td>20 </td> <td>8 </td> <td>{3, 7} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>2×20<sup>10</sup>−1<br />6×20<sup>2</sup>−1<br />5×20<sup>2</sup>−1<br />7×20<sup>1</sup>−1<br />3×20<sup>1</sup>−1 </td></tr> <tr> <td>21 </td> <td>560 </td> <td>{11, 13, 17} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>64×21<sup>2867</sup>−1<br />494×21<sup>978</sup>−1<br />154×21<sup>103</sup>−1<br />84×21<sup>88</sup>−1<br />142×21<sup>48</sup>−1 </td></tr> <tr> <td>22 </td> <td>4461 </td> <td>{5, 23, 97} </td> <td>3656 </td> <td>1 </td> <td>2M </td> <td>3104×22<sup>161188</sup>−1<br />4001×22<sup>36614</sup>−1<br />2853×22<sup>27975</sup>−1<br />1013×22<sup>26067</sup>−1<br />4118×22<sup>12347</sup>−1 </td></tr> <tr> <td>23 </td> <td>476 </td> <td>{3, 5, 53} </td> <td>404 </td> <td>1 </td> <td>1.35M </td> <td>194×23<sup>211140</sup>−1<br />134×23<sup>27932</sup>−1<br />394×23<sup>20169</sup>−1<br />314×23<sup>17268</sup>−1<br />464×23<sup>7548</sup>−1 </td></tr> <tr> <td>24 </td> <td>4 </td> <td>{5} for odd <i>n</i>, 4×24<sup><i>n</i></sup> − 1 = (2×24<sup><i>n</i>/2</sup> − 1) × (2×24<sup><i>n</i>/2</sup> + 1) for even <i>n</i> </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>3×24<sup>1</sup>−1<br />2×24<sup>1</sup>−1 </td></tr> <tr> <td>25 </td> <td>36 </td> <td>36×25<sup><i>n</i></sup> − 1 = (6×5<sup><i>n</i></sup> − 1) × (6×5<sup><i>n</i></sup> + 1) </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>32×25<sup>4</sup>−1<br />30×25<sup>2</sup>−1<br />26×25<sup>2</sup>−1<br />12×25<sup>2</sup>−1<br />2×25<sup>2</sup>−1 </td></tr> <tr> <td>26 </td> <td>149 </td> <td>{3, 7, 31, 37} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>115×26<sup>520277</sup>−1<br />32×26<sup>9812</sup>−1<br />73×26<sup>537</sup>−1<br />80×26<sup>382</sup>−1<br />128×26<sup>300</sup>−1 </td></tr> <tr> <td>27 </td> <td>8 </td> <td>8×27<sup><i>n</i></sup> − 1 = (2×3<sup><i>n</i></sup> − 1) × (4×9<sup><i>n</i></sup> + 2×3<sup><i>n</i></sup> + 1) </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>6×27<sup>2</sup>−1<br />4×27<sup>1</sup>−1<br />2×27<sup>1</sup>−1 </td></tr> <tr> <td>28 </td> <td>144 </td> <td>{29} for odd <i>n</i>, 144×28<sup><i>n</i></sup> − 1 = (12×28<sup><i>n</i>/2</sup> − 1) × (12×28<sup><i>n</i>/2</sup> + 1) for even <i>n</i> </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>107×28<sup>74</sup>−1<br />122×28<sup>71</sup>−1<br />101×28<sup>53</sup>−1<br />14×28<sup>47</sup>−1<br />90×28<sup>36</sup>−1 </td></tr> <tr> <td>29 </td> <td>4 </td> <td>{3, 5} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>2×29<sup>136</sup>−1 </td></tr> <tr> <td>30 </td> <td>1369 </td> <td>{7, 13, 19} for odd <i>n</i>, 1369×30<sup><i>n</i></sup> − 1 = (37×30<sup><i>n</i>/2</sup> − 1) × (37×30<sup><i>n</i>/2</sup> + 1) for even <i>n</i> </td> <td>659, 1024 </td> <td>2 </td> <td>500K </td> <td>239×30<sup>337990</sup>−1<br />249×30<sup>199355</sup>−1<br />225×30<sup>158755</sup>−1<br />774×30<sup>148344</sup>−1<br />25×30<sup>34205</sup>−1 </td></tr> <tr> <td>31 </td> <td>134718 </td> <td>{7, 13, 19, 37, 331} </td> <td>55758 </td> <td>1 </td> <td>3M </td> <td>6962×31<sup>2863120</sup>−1<br />126072×31<sup>374323</sup>−1<br />43902×31<sup>251859</sup>−1<br />55940×31<sup>197599</sup>−1<br />101022×31<sup>133208</sup>−1 </td></tr> <tr> <td>32 </td> <td>10 </td> <td>{3, 11} </td> <td>none (proven) </td> <td>0 </td> <td>− </td> <td>3×32<sup>11</sup>−1<br />2×32<sup>6</sup>−1<br />9×32<sup>3</sup>−1<br />8×32<sup>2</sup>−1<br />5×32<sup>2</sup>−1 </td></tr></tbody></table> <p>Conjectured smallest Riesel number base <i>n</i> are (start with <i>n</i> = 2) </p> <dl><dd>509203, 63064644938, 9, 346802, 84687, 408034255082, 14, 4, 10176, 862, 25, 302, 4, 36370321851498, 9, 86, 246, 144, 8, 560, 4461, 476, 4, 36, 149, 8, 144, 4, 1369, 134718, 10, 16, 6, 287860, 4, 7772, 13, 4, 81, 8, 15137, 672, 4, 22564, 8177, 14, 3226, 36, 16, 64, 900, 5392, 4, 6852, 20, 144, 105788, 4, 121, 13484, 8, 187258666, 9, ... (sequence <span class="nowrap external"><a href="//oeis.org/A273987" class="extiw" title="oeis:A273987">A273987</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.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 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id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFRiesel1956" class="citation journal cs1"><a href="/wiki/Hans_Riesel" title="Hans Riesel">Riesel, Hans</a> (1956). "Några stora primtal". <i>Elementa</i>. <b>39</b>: 258–260.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Elementa&rft.atitle=N%C3%A5gra+stora+primtal&rft.volume=39&rft.pages=258-260&rft.date=1956&rft.aulast=Riesel&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiesel+number" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.primegrid.com/stats_trp_llr.php">"The Riesel Problem statistics"</a>. PrimeGrid.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Riesel+Problem+statistics&rft.pub=PrimeGrid&rft_id=http%3A%2F%2Fwww.primegrid.com%2Fstats_trp_llr.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiesel+number" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.primegrid.com/stats_trp_llr.php">"The Riesel Problem statistics"</a>. <i>PrimeGrid</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240115171321/https://www.primegrid.com/stats_trp_llr.php">Archived</a> from the original on 15 January 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">15 January</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PrimeGrid&rft.atitle=The+Riesel+Problem+statistics&rft_id=https%3A%2F%2Fwww.primegrid.com%2Fstats_trp_llr.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiesel+number" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.primepuzzles.net/problems/prob_029.htm">"Problem 29.- Brier Numbers"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Problem+29.-+Brier+Numbers&rft_id=http%3A%2F%2Fwww.primepuzzles.net%2Fproblems%2Fprob_029.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiesel+number" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm">"Riesel conjectures and proofs"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Riesel+conjectures+and+proofs&rft_id=http%3A%2F%2Fwww.noprimeleftbehind.net%2Fcrus%2FRiesel-conjectures.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiesel+number" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm">"Riesel conjectures & proofs powers of 2"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Riesel+conjectures+%26+proofs+powers+of+2&rft_id=http%3A%2F%2Fwww.noprimeleftbehind.net%2Fcrus%2FRiesel-conjectures-powers2.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiesel+number" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=10" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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). <i>Unsolved Problems in Number Theory</i>. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. 120. <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>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+Problems+in+Number+Theory&rft.place=Berlin&rft.pages=120&rft.pub=Springer-Verlag&rft.date=2004&rft.isbn=0-387-20860-7&rft.aulast=Guy&rft.aufirst=Richard+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiesel+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1996" class="citation book cs1"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (1996). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/newbookprimenumb00ribe_623"><i>The New Book of Prime Number Records</i></a></span>. New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/newbookprimenumb00ribe_623/page/n381">357</a>–358. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94457-5" title="Special:BookSources/0-387-94457-5"><bdi>0-387-94457-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+New+Book+of+Prime+Number+Records&rft.place=New+York&rft.pages=357-358&rft.pub=Springer-Verlag&rft.date=1996&rft.isbn=0-387-94457-5&rft.aulast=Ribenboim&rft.aufirst=Paulo&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnewbookprimenumb00ribe_623&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARiesel+number" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Riesel_number&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.primegrid.com">PrimeGrid</a></li> <li><a rel="nofollow" class="external text" href="http://www.prothsearch.com/rieselprob.html">The Riesel Problem: Definition and Status</a></li> <li><a rel="nofollow" class="external text" href="http://primes.utm.edu/glossary/xpage/RieselNumber.html">The Prime Glossary: Riesel number</a></li> <li><a rel="nofollow" class="external text" href="http://www.prothsearch.com/riesel2.html">List of primes of the form: k*2^n-1, k<300</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20210715115849/http://www.15k.org/riesellist.html">List of primes of the form: k*2^n-1, k<300, Project Riesel Prime Search</a></li> <li><a rel="nofollow" class="external text" href="https://www.rieselprime.de/">Riesel and Proth Prime Database</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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4em">Classes of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of 2</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of 3</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Fourth power</a></li> <li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">Sixth power</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">Seventh power</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">Eighth power</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a class="mw-selflink selflink">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics" style="font-size:114%;margin:0 4em"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li> <li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclid_number" title="Euclid number">Euclid</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers" style="font-size:114%;margin:0 4em">Other <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="/wiki/Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blum_integer" title="Blum integer">Blum</a></li> <li><a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Giuga_number" title="Giuga number">Giuga</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/St%C3%B8rmer_number" title="Størmer number">Størmer</a></li> <li><a href="/wiki/Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br />and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Persistence_of_a_number" title="Persistence of a number">Persistence</a> <ul><li><a href="/wiki/Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li> <li><a href="/wiki/Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></li> <li><a href="/wiki/Digital_root" title="Digital root">Digital root</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Meertens_number" title="Meertens number">Meertens</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dudeney_number" title="Dudeney number">Dudeney</a></li> <li><a href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit</a></li> <li><a href="/wiki/Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li> <li><a href="/wiki/Undulating_number" title="Undulating number">Undulating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="/wiki/Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyclic_number" title="Cyclic number">Cyclic</a></li> <li><a href="/wiki/Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Transposable_integer" title="Transposable integer">Transposable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li> <li><a href="/wiki/Vampire_number" title="Vampire number">Vampire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Friedman_number" title="Friedman number">Friedman</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td 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