@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-CVDPQB0Q-M
  skos:prefLabel "natural numbers"@en, "entier naturel"@fr ;
  a skos:Concept ;
  skos:narrower psr:-PVX57VVP-X .

psr:-PVX57VVP-X
  dc:modified "2024-10-18"^^xsd:date ;
  skos:prefLabel "Sierpiński number"@en, "nombre de Sierpiński"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Nombre_de_Sierpi%C5%84ski>, <https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number> ;
  a skos:Concept ;
  skos:definition """In number theory, a <b>Sierpiński number</b> is an odd natural number <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\\	imes 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\\	imes 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" aria-hidden="true" style="vertical-align: -0.505ex; width:10.435ex; height:2.509ex;" alt="{\\\\displaystyle k\\	imes 2^{n}+1}"></span> is composite for all natural numbers <i>n</i>. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers <i>k</i> which have this property. In other words, when <i>k</i> is a Sierpiński number, all members of the following set are composite:  <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\\\\cdot 2^{n}+1:n\\\\in \\\\mathbb {N} \\\\,\\
ight\\\\}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow>           <mo>{</mo>           <mrow>             <mspace width="thinmathspace"></mspace>             <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"></mspace>           </mrow>           <mo>}</mo>         </mrow>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\left\\\\{\\\\,k\\\\cdot 2^{n}+1:n\\\\in \\\\mathbb {N} \\\\,\\
ight\\\\}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb2622114cc5aa284290724a1fc45b65b35f8f8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.258ex; height:2.843ex;" alt="{\\\\displaystyle \\\\left\\\\{\\\\,k\\\\cdot 2^{n}+1:n\\\\in \\\\mathbb {N} \\\\,\\
ight\\\\}.}"></span></dd></dl> 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\\	imes 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\\	imes 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" aria-hidden="true" style="vertical-align: -0.505ex; width:10.435ex; height:2.509ex;" alt="{\\\\displaystyle k\\	imes 2^{n}-1}"></span>, then <i>k</i> is a Riesel number. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number">https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number</a>)"""@en, """En mathématiques, un <b>nombre de Sierpiński</b> est un entier naturel impair <i><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}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>k</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle k}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span></i> pour lequel tous les nombres <i><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 N}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>N</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle N}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="N"></span></i> de la forme <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 N=k2^{n}+1}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>N</mi>         <mo>=</mo>         <mi>k</mi>         <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 N=k2^{n}+1}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99caf4003a06e8eb0603ba214cef67338cf8f475" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.757ex; height:2.509ex;" alt="{\\\\displaystyle N=k2^{n}+1}"></span> sont composés (c'est-à-dire non premiers), quel que soit l'entier naturel <i><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 n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span></i>. En 1960, Wacław Sierpiński montra qu'il existe une infinité de ces nombres. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_de_Sierpi%C5%84ski">https://fr.wikipedia.org/wiki/Nombre_de_Sierpi%C5%84ski</a>)"""@fr ;
  skos:broader psr:-FM1M1PDT-5, psr:-CVDPQB0Q-M, psr:-VHDD6KJX-8 ;
  skos:inScheme psr: ;
  dc:created "2023-07-26"^^xsd:date .

psr: a skos:ConceptScheme .
psr:-VHDD6KJX-8
  skos:prefLabel "analytic number theory"@en, "théorie analytique des nombres"@fr ;
  a skos:Concept ;
  skos:narrower psr:-PVX57VVP-X .

psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-PVX57VVP-X .