@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:-W42D202L-K
  skos:prefLabel "inégalité"@fr, "inequality"@en ;
  a skos:Concept ;
  skos:narrower psr:-C39K2VPB-T .

psr:-T0WTK17L-B
  skos:prefLabel "nombre premier"@fr, "prime number"@en ;
  a skos:Concept ;
  skos:related psr:-C39K2VPB-T .

psr:-C39K2VPB-T
  skos:related psr:-T0WTK17L-B ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9_de_Bonse>, <https://en.wikipedia.org/wiki/Bonse%27s_inequality> ;
  a skos:Concept ;
  skos:definition """En théorie des nombres, l'<b>inégalité de Bonse</b>, du nom de H. Bonse</span>, permet une comparaison entre un nombre primoriel et le plus petit nombre premier qui ne figure pas dans sa décomposition.  Elle déclare que si <i>p</i><sub>1</sub>, ..., <i>p</i><sub><i>n</i></sub>, <i>p</i><sub><i>n</i>+1</sub> sont les <i>n</i> + 1 plus petits nombres premiers et <i>n</i> ≥ 4, alors  <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 p_{n}\\\\#=p_{1}\\\\cdots p_{n}>p_{n+1}^{2}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mi mathvariant="normal">#<!-- # --></mi>         <mo>=</mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>1</mn>           </mrow>         </msub>         <mo>⋯<!-- ⋯ --></mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>&gt;</mo>         <msubsup>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>+</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msubsup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p_{n}\\\\#=p_{1}\\\\cdots p_{n}&gt;p_{n+1}^{2}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1371ddb5ca32f227588f132216eb4befc9d2b6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:23.208ex; height:3.343ex;" alt="{\\\\displaystyle p_{n}\\\\#=p_{1}\\\\cdots p_{n}>p_{n+1}^{2}}"></span> ou <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 p_{n+1}<{\\\\sqrt {p_{n}\\\\#}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>+</mo>             <mn>1</mn>           </mrow>         </msub>         <mo>&lt;</mo>         <mrow class="MJX-TeXAtom-ORD">           <msqrt>             <msub>               <mi>p</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>n</mi>               </mrow>             </msub>             <mi mathvariant="normal">#<!-- # --></mi>           </msqrt>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p_{n+1}&lt;{\\\\sqrt {p_{n}\\\\#}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3062d24bd997e5bd06ca3d62ccb40d2060bb3633" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:14.324ex; height:3.509ex;" alt="{\\\\displaystyle p_{n+1}<{\\\\sqrt {p_{n}\\\\#}}}"></span>.</dd></dl> Elle est une conséquence facile du postulat de Bertrand : <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 p_{n+1}<2p_{n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>+</mo>             <mn>1</mn>           </mrow>         </msub>         <mo>&lt;</mo>         <mn>2</mn>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p_{n+1}&lt;2p_{n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8127e1b918a2ee6ab4ce2e6fd6082759e9baef" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.227ex; height:2.509ex;" alt="{\\\\displaystyle p_{n+1}<2p_{n}}"></span> ; en effet  <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 p_{n+1}^{2}<4p_{n}^{2}<8p_{n-1}p_{n}<2\\	imes 3\\	imes 5\\	imes p_{n-1}p_{n}\\\\leqslant p_{1}p_{2}...p_{n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msubsup>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>+</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msubsup>         <mo>&lt;</mo>         <mn>4</mn>         <msubsup>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msubsup>         <mo>&lt;</mo>         <mn>8</mn>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </msub>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>&lt;</mo>         <mn>2</mn>         <mo>×<!-- × --></mo>         <mn>3</mn>         <mo>×<!-- × --></mo>         <mn>5</mn>         <mo>×<!-- × --></mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </msub>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>⩽<!-- ⩽ --></mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>1</mn>           </mrow>         </msub>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mo>.</mo>         <mo>.</mo>         <mo>.</mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p_{n+1}^{2}&lt;4p_{n}^{2}&lt;8p_{n-1}p_{n}&lt;2\\	imes 3\\	imes 5\\	imes p_{n-1}p_{n}\\\\leqslant p_{1}p_{2}...p_{n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb45bb16954208f04ad1b0b682bd20cacbc07403" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:57.383ex; height:3.343ex;" alt="{\\\\displaystyle p_{n+1}^{2}<4p_{n}^{2}<8p_{n-1}p_{n}<2\\	imes 3\\	imes 5\\	imes p_{n-1}p_{n}\\\\leqslant p_{1}p_{2}...p_{n}}"></span> pour <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\\\\geqslant 5}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>⩾<!-- ⩾ --></mo>         <mn>5</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n\\\\geqslant 5}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b29e0bd313fa15456f5bdd732a8200b7d2456fb4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\\\\displaystyle n\\\\geqslant 5}"></span>, le cas <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=4}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>=</mo>         <mn>4</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n=4}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d928ec15aeef83aade867992ee473933adb6139d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\\\\displaystyle n=4}"></span> se montrant à la main. Mais elle possède une démonstration élémentaire directe plus courte que celle du postulat de Bertrand. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9_de_Bonse">https://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9_de_Bonse</a>)"""@fr, """In number theory, <b>Bonse's inequality</b>, named after H. Bonse, relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that if <i>p</i><sub>1</sub>, ..., <i>p</i><sub><i>n</i></sub>, <i>p</i><sub><i>n</i>+1</sub> are the smallest <i>n</i> + 1 prime numbers and <i>n</i> ≥ 4, then  <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 p_{n}\\\\#=p_{1}\\\\cdots p_{n}>p_{n+1}^{2}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mi mathvariant="normal">#<!-- # --></mi>         <mo>=</mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>1</mn>           </mrow>         </msub>         <mo>⋯<!-- ⋯ --></mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>&gt;</mo>         <msubsup>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>+</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msubsup>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p_{n}\\\\#=p_{1}\\\\cdots p_{n}&gt;p_{n+1}^{2}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d13977b8d1701ccec2a9c078e0d570d3c8c33e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:23.854ex; height:3.343ex;" alt="{\\\\displaystyle p_{n}\\\\#=p_{1}\\\\cdots p_{n}>p_{n+1}^{2}.}"></span></dd></dl> (the middle product is short-hand for the primorial <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 p_{n}\\\\#}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mi mathvariant="normal">#<!-- # --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p_{n}\\\\#}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b8e5884d7e5c55d72c2d5c78023ec5db9d3ed4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.413ex; height:2.509ex;" alt="{\\\\displaystyle p_{n}\\\\#}"></span> of <i>p</i><sub><i>n</i></sub>) Mathematician Denis Hanson showed an upper bound 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 n\\\\#\\\\leq 3^{n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mi mathvariant="normal">#<!-- # --></mi>         <mo>≤<!-- ≤ --></mo>         <msup>           <mn>3</mn>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n\\\\#\\\\leq 3^{n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2c547e8a1b3502148c53b352815ef3656ab49d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.81ex; height:2.676ex;" alt="{\\\\displaystyle n\\\\#\\\\leq 3^{n}}"></span>.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Bonse%27s_inequality">https://en.wikipedia.org/wiki/Bonse%27s_inequality</a>)"""@en ;
  skos:inScheme psr: ;
  skos:broader psr:-W42D202L-K, psr:-VHDD6KJX-8 ;
  skos:prefLabel "Bonse's inequality"@en, "inégalité de Bonse"@fr ;
  dc:modified "2024-10-18"^^xsd:date ;
  dc:created "2023-08-11"^^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:-C39K2VPB-T .