@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: a skos:ConceptScheme .
psr:-CVDPQB0Q-M
  skos:prefLabel "natural numbers"@en, "entier naturel"@fr ;
  a skos:Concept ;
  skos:narrower psr:-M584XKQZ-7 .

psr:-M584XKQZ-7
  skos:prefLabel "nombre de Carmichael"@fr, "Carmichael number"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Carmichael_number>, <https://fr.wikipedia.org/wiki/Nombre_de_Carmichael> ;
  skos:altLabel "nombre absolument pseudo-premier"@fr ;
  skos:broader psr:-CVDPQB0Q-M, psr:-FM1M1PDT-5 ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:definition """In number theory, a <b>Carmichael number</b> is a composite number <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>, which in modular arithmetic satisfies the congruence relation:  <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 b^{n}\\\\equiv b{\\\\pmod {n}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>b</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>         <mo>≡<!-- ≡ --></mo>         <mi>b</mi>         <mrow class="MJX-TeXAtom-ORD">           <mspace width="1em"></mspace>           <mo stretchy="false">(</mo>           <mi>mod</mi>           <mspace width="0.333em"></mspace>           <mi>n</mi>           <mo stretchy="false">)</mo>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle b^{n}\\\\equiv b{\\\\pmod {n}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee96e9be7fb8ce02619e5cb4be7a7368a3c592d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.391ex; height:2.843ex;" alt="{\\\\displaystyle b^{n}\\\\equiv b{\\\\pmod {n}}}"></span></dd></dl> for all integers <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/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="b"></span>. The relation may also be expressed in the form:  <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 b^{n-1}\\\\equiv 1{\\\\pmod {n}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>b</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </msup>         <mo>≡<!-- ≡ --></mo>         <mn>1</mn>         <mrow class="MJX-TeXAtom-ORD">           <mspace width="1em"></mspace>           <mo stretchy="false">(</mo>           <mi>mod</mi>           <mspace width="0.333em"></mspace>           <mi>n</mi>           <mo stretchy="false">)</mo>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle b^{n-1}\\\\equiv 1{\\\\pmod {n}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a865f5eed65bd5f1a68ddb9ad0f35cfe9aa1208" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.656ex; height:3.176ex;" alt="b^{{n-1}}\\\\equiv 1{\\\\pmod  {n}}"></span>.</dd></dl> for all integers <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/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="b"></span> which are relatively prime to <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>. Carmichael numbers are named after American mathematician Robert Carmichael, the term having been introduced by Nicolaas Beeger in 1950 (Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "<i>F</i> numbers" for short). They are infinite in number. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Carmichael_number">https://en.wikipedia.org/wiki/Carmichael_number</a>)"""@en, """En théorie des nombres, un <b>nombre de Carmichael</b> (portant le nom du mathématicien américain Robert Daniel Carmichael), ou <b>nombre absolument pseudo-premier</b>, est un nombre composé <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> qui vérifie la propriété suivante, satisfaite par tous les nombres premiers d'après le petit théorème de Fermat :  <dl><dd>pour tout entier <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 a}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="a"></span></i> premier avec <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>, <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> est un diviseur de <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^{n-1}-1}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>a</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </msup>         <mo>−<!-- − --></mo>         <mn>1</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a^{n-1}-1}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c043605383213fefcd0b5a906b8637ec44ea506" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.552ex; height:2.843ex;" alt="{\\\\displaystyle a^{n-1}-1}"></span>.</dd></dl> C'est donc un nombre pseudo-premier de Fermat en toute base première avec lui (on peut d'ailleurs se restreindre aux entiers <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 a}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="a"></span></i> de 2 à <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-1}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>1</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n-1}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="n-1"></span> dans cette définition). 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_de_Carmichael">https://fr.wikipedia.org/wiki/Nombre_de_Carmichael</a>)"""@fr ;
  skos:inScheme psr: ;
  a skos:Concept .

psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-M584XKQZ-7 .