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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Mills' constant</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://de.wikipedia.org/wiki/Mills%E2%80%99_Konstante" title="Mills’ Konstante – German" lang="de" hreflang="de" data-title="Mills’ Konstante" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Constante_de_Mills" title="Constante de Mills – French" lang="fr" hreflang="fr" data-title="Constante de Mills" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%80%EC%8A%A4_%EC%83%81%EC%88%98" title="밀스 상수 – Korean" lang="ko" hreflang="ko" data-title="밀스 상수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Costante_di_Mills" title="Costante di Mills – Italian" lang="it" hreflang="it" data-title="Costante di Mills" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%91%D7%95%D7%A2_%D7%9E%D7%99%D7%9C%D7%A1" title="קבוע מילס – Hebrew" lang="he" hreflang="he" data-title="קבוע מילס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Constante_van_Mills" title="Constante van Mills – Dutch" lang="nl" hreflang="nl" data-title="Constante van Mills" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9F%E3%83%AB%E3%82%BA%E3%81%AE%E5%AE%9A%E6%95%B0" title="ミルズの定数 – Japanese" lang="ja" hreflang="ja" data-title="ミルズの定数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Constante_de_Mills" title="Constante de Mills – Portuguese" lang="pt" hreflang="pt" data-title="Constante de Mills" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%81%D1%82%D0%B0%D0%BD%D1%82%D0%B0_%D0%9C%D0%B8%D0%BB%D0%BB%D1%81%D0%B0" title="Константа Миллса – Russian" lang="ru" hreflang="ru" data-title="Константа Миллса" data-language-autonym="Русский" 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theory">number theory</a>, <b>Mills' constant</b> is defined as the smallest positive <a href="/wiki/Real_number" title="Real number">real number</a> <i>A</i> such that the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a> of the <a href="/wiki/Double_exponential_function" title="Double exponential function">double exponential function</a> </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\lfloor A^{3^{n}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌊</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lfloor A^{3^{n}}\right\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e20de2708e2f6bb1c5ae4bffd0925d01268a10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.217ex; height:4.843ex;" alt="{\displaystyle \left\lfloor A^{3^{n}}\right\rfloor }"></span></dd></dl> <p>is a <a href="/wiki/Prime_number" title="Prime number">prime number</a> for all positive <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <i>n</i>. This constant is named after <a href="/wiki/William_Harold_Mills" class="mw-redirect" title="William Harold Mills">William Harold Mills</a> who proved in 1947 the existence of <i>A</i> based on results of <a href="/wiki/Guido_Hoheisel" title="Guido Hoheisel">Guido Hoheisel</a> and <a href="/wiki/Albert_Ingham" title="Albert Ingham">Albert Ingham</a> on the <a href="/wiki/Prime_gap" title="Prime gap">prime gaps</a>.<sup id="cite_ref-Mills1947_1-0" class="reference"><a href="#cite_note-Mills1947-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Its value is unproven, but if the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> is true, it is approximately 1.3063778838630806904686144926... (sequence <span class="nowrap external"><a href="//oeis.org/A051021" class="extiw" title="oeis:A051021">A051021</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Mills_primes">Mills primes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mills%27_constant&action=edit&section=1" title="Edit section: Mills primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins </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 2,11,1361,2521008887,16022236204009818131831320183,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>1361</mn> <mo>,</mo> <mn>2521008887</mn> <mo>,</mo> <mn>16022236204009818131831320183</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,11,1361,2521008887,16022236204009818131831320183,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b885fc8f134bae529f8c63f4db64ebc72da3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:58.256ex; height:2.509ex;" alt="{\displaystyle 2,11,1361,2521008887,16022236204009818131831320183,}"></span></dd> <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 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4113101149215104800030529537915953170486139623539759933135949994882770404074832568499</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d806e19ff921196edb5f61a78c351523d70ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:102.566ex; height:2.509ex;" alt="{\displaystyle 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499,\ldots }"></span> (sequence <span class="nowrap external"><a href="//oeis.org/A051254" class="extiw" title="oeis:A051254">A051254</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>If <i>a<sub>i</sub></i> denotes the <i>i</i><sup> th</sup> prime in this sequence, then <i>a<sub>i</sub></i> can be calculated as the smallest prime number larger than <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_{i-1}^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i-1}^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ede03563a4ef126f6f58f11168a6bb71ca14bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.13ex; height:3.343ex;" alt="{\displaystyle a_{i-1}^{3}}"></span>. In order to ensure that rounding <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^{3^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{3^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43bcf3adc17cbed55d3bc6a20917719a7c1c5201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.762ex; height:2.676ex;" alt="{\displaystyle A^{3^{n}}}"></span>, for <i>n</i> = 1, 2, 3, ..., produces this sequence of primes, it must be the case 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 a_{i}<(a_{i-1}+1)^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}<(a_{i-1}+1)^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b65e2b568405617662b5beaecc35fc8b249e6e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.124ex; height:3.176ex;" alt="{\displaystyle a_{i}<(a_{i-1}+1)^{3}}"></span>. The Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently large <a href="/wiki/Cube_number" class="mw-redirect" title="Cube number">cube numbers</a>, which is sufficient to prove this inequality if we start from a sufficiently large first prime <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"></span>. The Riemann hypothesis implies that there exists a prime between any two consecutive cubes, allowing the <i>sufficiently large</i> condition to be removed, and allowing the sequence of Mills primes to begin at <i>a</i><sub>1</sub> = 2. </p><p>For all a > <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 e^{e^{34}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{e^{34}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff74f76f5bd36af6dc3f3bbe7afca5906af8cbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.581ex; height:3.009ex;" alt="{\displaystyle e^{e^{34}}}"></span>, there is at least one prime between <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^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd83c98f7301a720f69dd6d4043461e4cc83daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.676ex;" alt="{\displaystyle a^{3}}"></span> and <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+1)^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+1)^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b07d2c69587b08d28b43db7251f9eb34f4d9cb00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.096ex; height:3.176ex;" alt="{\displaystyle (a+1)^{3}}"></span>.<sup id="cite_ref-Dudek2016_2-0" class="reference"><a href="#cite_note-Dudek2016-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This upper bound is much too large to be practical, as it is infeasible to check every number below that figure. However, the value of Mills' constant can be verified by calculating the first prime in the sequence that is greater than that figure. </p><p>As of April 2017, the 11th number in the sequence is the largest one that has been <a href="/wiki/Provable_prime" title="Provable prime"><i>proved</i> prime</a>. It is </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 \displaystyle (((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>30</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>80</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>450</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>894</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3636</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>70756</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>97220</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle (((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9ce1e0c775638f27e1ff726520a2594216ec90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:88.941ex; height:3.176ex;" alt="{\displaystyle \displaystyle (((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220}"></span></dd></dl> <p>and has 20562 digits.<sup id="cite_ref-Caldwell2006_3-0" class="reference"><a href="#cite_note-Caldwell2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>As of 2024<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Mills%27_constant&action=edit">[update]</a></sup>, the largest known Mills <a href="/wiki/Probable_prime" title="Probable prime"><i>probable</i> prime</a> (under the Riemann hypothesis) is </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 \displaystyle (((((((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220)^{3}+66768)^{3}+300840)^{3}+1623568)^{3}+8436308}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>30</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>80</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>450</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>894</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3636</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>70756</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>97220</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>66768</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>300840</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1623568</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>8436308</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle (((((((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220)^{3}+66768)^{3}+300840)^{3}+1623568)^{3}+8436308}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38255785f638af15f2bc1a8b5f49b53c6599cfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:140.818ex; height:3.176ex;" alt="{\displaystyle \displaystyle (((((((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220)^{3}+66768)^{3}+300840)^{3}+1623568)^{3}+8436308}"></span></dd></dl> <p>(sequence <span class="nowrap external"><a href="//oeis.org/A108739" class="extiw" title="oeis:A108739">A108739</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), which is 1,665,461 digits long. </p> <div class="mw-heading mw-heading2"><h2 id="Numerical_calculation">Numerical calculation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mills%27_constant&action=edit&section=2" title="Edit section: Numerical calculation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By calculating the sequence of Mills primes, one can approximate Mills' constant as </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 A\approx a(n)^{1/3^{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≈</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\approx a(n)^{1/3^{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe7fea97e8ac25cd28a7ef81e14c60ff750f1f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.585ex; height:3.343ex;" alt="{\displaystyle A\approx a(n)^{1/3^{n}}.}"></span></dd></dl> <p>Caldwell and Cheng used this method to compute 6850 base 10 digits of Mills' constant under the assumption that the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> is true.<sup id="cite_ref-CaldwellCheng2005_4-0" class="reference"><a href="#cite_note-CaldwellCheng2005-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> There is no closed-form formula known for Mills' constant, and it is not even known whether this number is <a href="/wiki/Rational_number" title="Rational number">rational</a>.<sup id="cite_ref-Finch2003_5-0" class="reference"><a href="#cite_note-Finch2003-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalisations">Generalisations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mills%27_constant&action=edit&section=3" title="Edit section: Generalisations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is nothing special about the middle exponent value of 3. It is possible to produce similar prime-generating <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> for different middle exponent values. In fact, for any real number above 2.106..., it is possible to find a different constant <i>A</i> that will work with this middle exponent to always produce primes. Moreover, if <a href="/wiki/Legendre%27s_conjecture" title="Legendre's conjecture">Legendre's conjecture</a> is true, the middle exponent can be replaced<sup id="cite_ref-WarrenJr2013_6-0" class="reference"><a href="#cite_note-WarrenJr2013-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> with value 2 (sequence <span class="nowrap external"><a href="//oeis.org/A059784" class="extiw" title="oeis:A059784">A059784</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p><p>Matomäki showed unconditionally (without assuming Legendre's conjecture) the existence of a (possibly large) constant <i>A</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 \lfloor A^{2^{n}}\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor A^{2^{n}}\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df10397bff01ec81c3e649a19dfa5ce83b760ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.827ex; height:3.176ex;" alt="{\displaystyle \lfloor A^{2^{n}}\rfloor }"></span> is prime for all <i>n</i>.<sup id="cite_ref-Matomäki2010_7-0" class="reference"><a href="#cite_note-Matomäki2010-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Additionally, Tóth proved that the floor function in the formula could be replaced with the <a href="/wiki/Ceiling_function" class="mw-redirect" title="Ceiling function">ceiling function</a>, so that there exists a constant <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> such that </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 \lceil B^{r^{n}}\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌈</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo fence="false" stretchy="false">⌉</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lceil B^{r^{n}}\rceil }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ced528c6b31da254e18541a1bf3c50225fa0387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.767ex; height:3.176ex;" alt="{\displaystyle \lceil B^{r^{n}}\rceil }"></span></dd></dl> <p>is also prime-representing for <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 r>2.106\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>2.106</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>2.106\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a9941b7866377ef24150212758c4eca2b763c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.554ex; height:2.176ex;" alt="{\displaystyle r>2.106\ldots }"></span>.<sup id="cite_ref-Tóth2017_8-0" class="reference"><a href="#cite_note-Tóth2017-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In the case <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 r=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2452d01ccc3bc930e3e2bff9fb18d2a425272ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=3}"></span>, the value of the constant <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> begins with 1.24055470525201424067... The first few primes generated are: </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 2,7,337,38272739,56062005704198360319209,176199995814327287356671209104585864397055039072110696028654438846269,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>337</mn> <mo>,</mo> <mn>38272739</mn> <mo>,</mo> <mn>56062005704198360319209</mn> <mo>,</mo> <mn>176199995814327287356671209104585864397055039072110696028654438846269</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,7,337,38272739,56062005704198360319209,176199995814327287356671209104585864397055039072110696028654438846269,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c57615fd5ab9d9216f789b933bf7f008c81e2dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:130.985ex; height:2.509ex;" alt="{\displaystyle 2,7,337,38272739,56062005704198360319209,176199995814327287356671209104585864397055039072110696028654438846269,\ldots }"></span></dd></dl> <p><i>Without</i> assuming the Riemann hypothesis, Elsholtz proved 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 \lfloor A^{10^{10n}}\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor A^{10^{10n}}\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79d390ac44a57ccbfb5259c4a917ba858eacc02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.984ex; height:3.509ex;" alt="{\displaystyle \lfloor A^{10^{10n}}\rfloor }"></span> is prime for all positive integers <span class="texhtml mvar" style="font-style:italic;">n</span>, where <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\approx 1.00536773279814724017}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≈</mo> <mn>1.00536773279814724017</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\approx 1.00536773279814724017}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a08bce129292169e463291c43e9e3391b80a8ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:29.9ex; height:2.176ex;" alt="{\displaystyle A\approx 1.00536773279814724017}"></span>, and 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 \lfloor B^{3^{13n}}\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> <mi>n</mi> </mrow> </msup> </mrow> </msup> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor B^{3^{13n}}\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e94296e3c568d16dba94af865afc0b9dc9a2dbc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.183ex; height:3.509ex;" alt="{\displaystyle \lfloor B^{3^{13n}}\rfloor }"></span> is prime for all positive integers <span class="texhtml mvar" style="font-style:italic;">n</span>, where <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 B\approx 3.8249998073439146171615551375}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>≈</mo> <mn>3.8249998073439146171615551375</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\approx 3.8249998073439146171615551375}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee427d38c4b223405ee894192286cb4c7ac5c0e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:39.221ex; height:2.176ex;" alt="{\displaystyle B\approx 3.8249998073439146171615551375}"></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <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=Mills%27_constant&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Formula_for_primes" title="Formula for primes">Formula for primes</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mills%27_constant&action=edit&section=5" 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-Mills1947-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mills1947_1-0">^</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="CITEREFMills1947" class="citation journal cs1">Mills, W. H. (1947). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1947-53-06/S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf">"A prime-representing function"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>. <b>53</b> (6): 604. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1947-08849-2">10.1090/S0002-9904-1947-08849-2</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=A+prime-representing+function&rft.volume=53&rft.issue=6&rft.pages=604&rft.date=1947&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1947-08849-2&rft.aulast=Mills&rft.aufirst=W.+H.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1947-53-06%2FS0002-9904-1947-08849-2%2FS0002-9904-1947-08849-2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMills%27+constant" class="Z3988"></span></span> </li> <li id="cite_note-Dudek2016-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dudek2016_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudek2016" class="citation journal cs1">Dudek, Adrian W. 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(2003). "Mills' Constant". <i>Mathematical Constants</i>. Cambridge University Press. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalcons0000finc/page/130">pp. 130–133</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-81805-2" title="Special:BookSources/0-521-81805-2"><bdi>0-521-81805-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mills%27+Constant&rft.btitle=Mathematical+Constants&rft.pages=pp.+130-133&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=0-521-81805-2&rft.aulast=Finch&rft.aufirst=Steven+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMills%27+constant" class="Z3988"></span></span> </li> <li id="cite_note-WarrenJr2013-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-WarrenJr2013_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarren_Jr.2013" class="citation book cs1">Warren Jr., Henry S. (2013). <i>Hacker's Delight</i> (2nd ed.). Addison-Wesley Professional. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780321842688" title="Special:BookSources/9780321842688"><bdi>9780321842688</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hacker%27s+Delight&rft.edition=2nd&rft.pub=Addison-Wesley+Professional&rft.date=2013&rft.isbn=9780321842688&rft.aulast=Warren+Jr.&rft.aufirst=Henry+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMills%27+constant" class="Z3988"></span></span> </li> <li id="cite_note-Matomäki2010-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Matomäki2010_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatomäki2010" class="citation journal cs1"><a href="/wiki/Kaisa_Matom%C3%A4ki" title="Kaisa Matomäki">Matomäki, K.</a> (2010). <a rel="nofollow" class="external text" href="http://users.utu.fi/ksmato/papers/Primerepfunc.pdf">"Prime-representing functions"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Acta_Mathematica_Hungarica" title="Acta Mathematica Hungarica">Acta Mathematica Hungarica</a></i>. <b>128</b> (4): 307–314. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10474-010-9191-x">10.1007/s10474-010-9191-x</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18960874">18960874</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica+Hungarica&rft.atitle=Prime-representing+functions&rft.volume=128&rft.issue=4&rft.pages=307-314&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2Fs10474-010-9191-x&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18960874%23id-name%3DS2CID&rft.aulast=Matom%C3%A4ki&rft.aufirst=K.&rft_id=http%3A%2F%2Fusers.utu.fi%2Fksmato%2Fpapers%2FPrimerepfunc.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMills%27+constant" class="Z3988"></span></span> </li> <li id="cite_note-Tóth2017-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tóth2017_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTóth2017" class="citation journal cs1">Tóth, László (2017). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth2/toth32.pdf">"A Variation on Mills-Like Prime-Representing Functions"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Integer Sequences</i>. <b>20</b>. p. 17.9.8. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1801.08014">1801.08014</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Integer+Sequences&rft.atitle=A+Variation+on+Mills-Like+Prime-Representing+Functions&rft.volume=20&rft.pages=p.+17.9.8&rft.date=2017&rft_id=info%3Aarxiv%2F1801.08014&rft.aulast=T%C3%B3th&rft.aufirst=L%C3%A1szl%C3%B3&rft_id=https%3A%2F%2Fcs.uwaterloo.ca%2Fjournals%2FJIS%2FVOL20%2FToth2%2Ftoth32.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMills%27+constant" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElsholtz2020" class="citation journal cs1">Elsholtz, Christian (2020). "Unconditional Prime-Representing Functions, Following Mills". <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>. <b>127</b> (7): 639–642. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2004.01285">2004.01285</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2020.1751560">10.1080/00029890.2020.1751560</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:214795216">214795216</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Unconditional+Prime-Representing+Functions%2C+Following+Mills&rft.volume=127&rft.issue=7&rft.pages=639-642&rft.date=2020&rft_id=info%3Aarxiv%2F2004.01285&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A214795216%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F00029890.2020.1751560&rft.aulast=Elsholtz&rft.aufirst=Christian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMills%27+constant" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mills%27_constant&action=edit&section=6" title="Edit section: Further reading"><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="CITEREFCheng2010" class="citation journal cs1">Cheng, Yuanyou Furui 2010 (2010). "Explicit estimate on primes between consecutive cubes". <i>The Rocky Mountain Journal of Mathematics</i>. <b>40</b> (1): 117–153. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0810.2113">0810.2113</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1216%2FRMJ-2010-40-1-117">10.1216/RMJ-2010-40-1-117</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2607111">2607111</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15502941">15502941</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Rocky+Mountain+Journal+of+Mathematics&rft.atitle=Explicit+estimate+on+primes+between+consecutive+cubes&rft.volume=40&rft.issue=1&rft.pages=117-153&rft.date=2010&rft_id=info%3Aarxiv%2F0810.2113&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2607111%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15502941%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1216%2FRMJ-2010-40-1-117&rft.aulast=Cheng&rft.aufirst=Yuanyou+Furui+2010&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMills%27+constant" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>)</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=Mills%27_constant&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Mills'_Constant"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/MillsConstant.html">"Mills' Constant"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Mills%27+Constant&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMillsConstant.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMills%27+constant" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://blogs.ethz.ch/kowalski/2009/04/02/who-remembers-the-mills-number/">Who remembers the Mills number?</a>, E. Kowalski.</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=6ltrPVPEwfo">Awesome Prime Number Constant</a>, Numberphile.</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:": 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.navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Prime_number_classes" title="Template:Prime number classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Prime_number_classes" title="Template talk:Prime number classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Prime_number_classes" title="Special:EditPage/Template:Prime number classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Prime_number_classes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Prime number</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By formula</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fermat_number" title="Fermat number">Fermat (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup>2<sup><i>n</i></sup></sup> + 1</span>)</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>p</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup>2<sup><i>p</i></sup>−1</sup> − 1</span>)</a></li> <li><a href="/wiki/Wagstaff_prime" title="Wagstaff prime">Wagstaff <span class="texhtml texhtml-big" style="font-size:110%;">(2<sup><i>p</i></sup> + 1)/3</span></a></li> <li><a href="/wiki/Proth_prime" title="Proth prime">Proth (<span class="texhtml texhtml-big" style="font-size:110%;"><i>k</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Factorial_prime" title="Factorial prime">Factorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>! ± 1</span>)</a></li> <li><a href="/wiki/Primorial_prime" title="Primorial prime">Primorial (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># ± 1</span>)</a></li> <li><a href="/wiki/Euclid_number" title="Euclid number">Euclid (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p<sub>n</sub></i># + 1</span>)</a></li> <li><a href="/wiki/Pythagorean_prime" title="Pythagorean prime">Pythagorean (<span class="texhtml texhtml-big" style="font-size:110%;">4<i>n</i> + 1</span>)</a></li> <li><a href="/wiki/Pierpont_prime" title="Pierpont prime">Pierpont (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup>·3<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Quartan_prime" title="Quartan prime">Quartan (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>4</sup> + <i>y</i><sup>4</sup></span>)</a></li> <li><a href="/wiki/Solinas_prime" title="Solinas prime">Solinas (<span class="texhtml texhtml-big" style="font-size:110%;">2<sup><i>m</i></sup> ± 2<sup><i>n</i></sup> ± 1</span>)</a></li> <li><a href="/wiki/Cullen_number" title="Cullen number">Cullen (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> + 1</span>)</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i>·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Cuban_prime" title="Cuban prime">Cuban (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x</i><sup>3</sup> − <i>y</i><sup>3</sup>)/(<i>x</i> − <i>y</i></span>)</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland (<span class="texhtml texhtml-big" style="font-size:110%;"><i>x<sup>y</sup></i> + <i>y<sup>x</sup></i></span>)</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit (<span class="texhtml texhtml-big" style="font-size:110%;">3·2<sup><i>n</i></sup> − 1</span>)</a></li> <li><a href="/wiki/Williams_number" title="Williams number">Williams (<span class="texhtml texhtml-big" style="font-size:110%;">(<i>b</i>−1)·<i>b</i><sup><i>n</i></sup> − 1</span>)</a></li> <li><a class="mw-selflink selflink">Mills (<span class="texhtml texhtml-big" style="font-size:110%;"><span style="font-size:1em">⌊</span><i>A</i><sup>3<sup><i>n</i></sup></sup><span style="font-size:1em">⌋</span></span>)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By integer sequence</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_prime" title="Fibonacci prime">Fibonacci</a></li> <li><a href="/wiki/Lucas_prime" class="mw-redirect" title="Lucas prime">Lucas</a></li> <li><a href="/wiki/Pell_prime" class="mw-redirect" title="Pell prime">Pell</a></li> <li><a href="/wiki/Newman%E2%80%93Shanks%E2%80%93Williams_prime" title="Newman–Shanks–Williams prime">Newman–Shanks–Williams</a></li> <li><a href="/wiki/Perrin_prime" class="mw-redirect" title="Perrin prime">Perrin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By property</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime" title="Wieferich prime">Wieferich</a> (<a href="/wiki/Wieferich_pair" title="Wieferich pair">pair</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</a></li> <li><a href="/wiki/Wilson_prime" title="Wilson prime">Wilson</a></li> <li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li> <li><a href="/wiki/Ramanujan_prime" title="Ramanujan prime">Ramanujan</a></li> <li><a href="/wiki/Pillai_prime" title="Pillai prime">Pillai</a></li> <li><a href="/wiki/Regular_prime" title="Regular prime">Regular</a></li> <li><a href="/wiki/Strong_prime" title="Strong prime">Strong</a></li> <li><a href="/wiki/Stern_prime" title="Stern prime">Stern</a></li> <li><a href="/wiki/Supersingular_prime_(algebraic_number_theory)" title="Supersingular prime (algebraic number theory)">Supersingular (elliptic curve)</a></li> <li><a href="/wiki/Supersingular_prime_(moonshine_theory)" title="Supersingular prime (moonshine theory)">Supersingular (moonshine theory)</a></li> <li><a href="/wiki/Good_prime" title="Good prime">Good</a></li> <li><a href="/wiki/Super-prime" title="Super-prime">Super</a></li> <li><a href="/wiki/Higgs_prime" title="Higgs prime">Higgs</a></li> <li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Reciprocals_of_primes#Unique_primes" title="Reciprocals of primes">Unique</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_prime" title="Palindromic prime">Palindromic</a></li> <li><a href="/wiki/Emirp" title="Emirp">Emirp</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit <span class="texhtml texhtml-big" style="font-size:110%;">(10<sup><i>n</i></sup> − 1)/9</span></a></li> <li><a href="/wiki/Permutable_prime" title="Permutable prime">Permutable</a></li> <li><a href="/wiki/Circular_prime" title="Circular prime">Circular</a></li> <li><a href="/wiki/Truncatable_prime" title="Truncatable prime">Truncatable</a></li> <li><a href="/wiki/Minimal_prime_(recreational_mathematics)" title="Minimal prime (recreational mathematics)">Minimal</a></li> <li><a href="/wiki/Delicate_prime" title="Delicate prime">Delicate</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Full_reptend_prime" title="Full reptend prime">Full reptend</a></li> <li><a href="/wiki/Unique_prime_number" class="mw-redirect" title="Unique prime number">Unique</a></li> <li><a href="/wiki/Happy_number#Happy_primes" title="Happy number">Happy</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_prime" class="mw-redirect" title="Smarandache–Wellin prime">Smarandache–Wellin</a></li> <li><a href="/wiki/Strobogrammatic_prime" class="mw-redirect" title="Strobogrammatic prime">Strobogrammatic</a></li> <li><a href="/wiki/Dihedral_prime" title="Dihedral prime">Dihedral</a></li> <li><a href="/wiki/Tetradic_number" title="Tetradic number">Tetradic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Patterns</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="k-tuples" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_k-tuple" title="Prime k-tuple"><i>k</i>-tuples</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/Twin_prime" title="Twin prime">Twin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2</span>)</a></li> <li><a href="/wiki/Prime_triplet" title="Prime triplet">Triplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2 or <i>p</i> + 4, <i>p</i> + 6</span>)</a></li> <li><a href="/wiki/Prime_quadruplet" title="Prime quadruplet">Quadruplet (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 2, <i>p</i> + 6, <i>p</i> + 8</span>)</a></li> <li><a href="/wiki/Cousin_prime" title="Cousin prime">Cousin (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 4</span>)</a></li> <li><a href="/wiki/Sexy_prime" title="Sexy prime">Sexy (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, <i>p</i> + 6</span>)</a></li> <li><a href="/wiki/Primes_in_arithmetic_progression" title="Primes in arithmetic progression">Arithmetic progression (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i> + <i>a·n</i>, <i>n</i> = 0, 1, 2, 3, ...</span>)</a></li> <li><a href="/wiki/Balanced_prime" title="Balanced prime">Balanced (<span class="texhtml texhtml-big" style="font-size:110%;">consecutive <i>p</i> − <i>n</i>, <i>p</i>, <i>p</i> + <i>n</i></span>)</a></li></ul> </div></td></tr></tbody></table><div> <ul><li><a href="/wiki/Bi-twin_chain" title="Bi-twin chain">Bi-twin chain (<span class="texhtml texhtml-big" style="font-size:110%;"><i>n</i> ± 1, 2<i>n</i> ± 1, 4<i>n</i> ± 1, …</span>)</a></li> <li><a href="/wiki/Chen_prime" title="Chen prime">Chen</a></li> <li><a href="/wiki/Safe_and_Sophie_Germain_primes" title="Safe and Sophie Germain primes">Sophie Germain/Safe (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, 2<i>p</i> + 1</span>)</a></li> <li><a href="/wiki/Cunningham_chain" title="Cunningham chain">Cunningham (<span class="texhtml texhtml-big" style="font-size:110%;"><i>p</i>, 2<i>p</i> ± 1, 4<i>p</i> ± 3, 8<i>p</i> ± 7, ...</span>)</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By size</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Megaprime" title="Megaprime">Mega (1,000,000+ digits)</a></li> <li><a href="/wiki/Largest_known_prime_number" title="Largest known prime number">Largest known</a> <ul><li><a href="/wiki/List_of_largest_known_primes_and_probable_primes" title="List of largest known primes and probable primes">list</a></li></ul></li> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Eisenstein_prime" class="mw-redirect" title="Eisenstein prime">Eisenstein prime</a></li> <li><a href="/wiki/Gaussian_integer#Gaussian_primes" title="Gaussian integer">Gaussian prime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprime</a> <ul><li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas</a></li> <li><a href="/wiki/Perrin_pseudoprime" class="mw-redirect" title="Perrin pseudoprime">Perrin</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong</a></li></ul></li> <li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic number</a></li> <li><a href="/wiki/Interprime" title="Interprime">Interprime</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Probable_prime" title="Probable prime">Probable prime</a></li> <li><a href="/wiki/Industrial-grade_prime" title="Industrial-grade prime">Industrial-grade prime</a></li> <li><a href="/wiki/Illegal_prime" class="mw-redirect" title="Illegal prime">Illegal prime</a></li> <li><a href="/wiki/Formula_for_primes" title="Formula for primes">Formula for primes</a></li> <li><a href="/wiki/Prime_gap" title="Prime gap">Prime gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">First 60 primes</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2" title="2">2</a></li> <li><a href="/wiki/3" title="3">3</a></li> <li><a href="/wiki/5" title="5">5</a></li> <li><a href="/wiki/7" title="7">7</a></li> <li><a href="/wiki/11_(number)" title="11 (number)">11</a></li> <li><a href="/wiki/13_(number)" title="13 (number)">13</a></li> <li><a href="/wiki/17_(number)" title="17 (number)">17</a></li> <li><a href="/wiki/19_(number)" title="19 (number)">19</a></li> <li><a href="/wiki/23_(number)" title="23 (number)">23</a></li> <li><a href="/wiki/29_(number)" title="29 (number)">29</a></li> <li><a href="/wiki/31_(number)" title="31 (number)">31</a></li> <li><a href="/wiki/37_(number)" title="37 (number)">37</a></li> <li><a href="/wiki/41_(number)" title="41 (number)">41</a></li> <li><a href="/wiki/43_(number)" title="43 (number)">43</a></li> <li><a href="/wiki/47_(number)" title="47 (number)">47</a></li> <li><a href="/wiki/53_(number)" title="53 (number)">53</a></li> <li><a href="/wiki/59_(number)" title="59 (number)">59</a></li> <li><a href="/wiki/61_(number)" title="61 (number)">61</a></li> <li><a href="/wiki/67_(number)" title="67 (number)">67</a></li> <li><a href="/wiki/71_(number)" title="71 (number)">71</a></li> <li><a href="/wiki/73_(number)" title="73 (number)">73</a></li> <li><a href="/wiki/79_(number)" title="79 (number)">79</a></li> <li><a href="/wiki/83_(number)" title="83 (number)">83</a></li> <li><a href="/wiki/89_(number)" title="89 (number)">89</a></li> <li><a href="/wiki/97_(number)" title="97 (number)">97</a></li> <li><a href="/wiki/101_(number)" title="101 (number)">101</a></li> <li><a href="/wiki/103_(number)" title="103 (number)">103</a></li> <li><a href="/wiki/107_(number)" title="107 (number)">107</a></li> <li><a href="/wiki/109_(number)" title="109 (number)">109</a></li> <li><a href="/wiki/113_(number)" title="113 (number)">113</a></li> <li><a href="/wiki/127_(number)" title="127 (number)">127</a></li> <li><a href="/wiki/131_(number)" title="131 (number)">131</a></li> <li><a href="/wiki/137_(number)" title="137 (number)">137</a></li> <li><a href="/wiki/139_(number)" title="139 (number)">139</a></li> <li><a href="/wiki/149_(number)" title="149 (number)">149</a></li> <li><a href="/wiki/151_(number)" title="151 (number)">151</a></li> <li><a href="/wiki/157_(number)" title="157 (number)">157</a></li> <li><a href="/wiki/163_(number)" title="163 (number)">163</a></li> <li><a href="/wiki/167_(number)" title="167 (number)">167</a></li> <li><a href="/wiki/173_(number)" title="173 (number)">173</a></li> <li><a href="/wiki/179_(number)" title="179 (number)">179</a></li> <li><a href="/wiki/181_(number)" title="181 (number)">181</a></li> <li><a href="/wiki/191_(number)" title="191 (number)">191</a></li> <li><a href="/wiki/193_(number)" title="193 (number)">193</a></li> <li><a href="/wiki/197_(number)" title="197 (number)">197</a></li> <li><a href="/wiki/199_(number)" title="199 (number)">199</a></li> <li><a href="/wiki/211_(number)" title="211 (number)">211</a></li> <li><a href="/wiki/223_(number)" title="223 (number)">223</a></li> <li><a href="/wiki/227_(number)" title="227 (number)">227</a></li> <li><a href="/wiki/229_(number)" title="229 (number)">229</a></li> <li><a href="/wiki/233_(number)" title="233 (number)">233</a></li> <li><a href="/wiki/239_(number)" title="239 (number)">239</a></li> <li><a href="/wiki/241_(number)" title="241 (number)">241</a></li> <li><a href="/wiki/251_(number)" title="251 (number)">251</a></li> <li><a href="/wiki/257_(number)" title="257 (number)">257</a></li> <li><a href="/wiki/263_(number)" title="263 (number)">263</a></li> <li><a href="/wiki/269_(number)" title="269 (number)">269</a></li> <li><a href="/wiki/271_(number)" title="271 (number)">271</a></li> <li><a href="/wiki/277_(number)" title="277 (number)">277</a></li> <li><a href="/wiki/281_(number)" title="281 (number)">281</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div><a href="/wiki/List_of_prime_numbers" title="List of prime numbers">List of prime numbers</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" 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