@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:-VHDD6KJX-8
  skos:prefLabel "analytic number theory"@en, "théorie analytique des nombres"@fr ;
  a skos:Concept ;
  skos:narrower psr:-SLCRLH7Z-J .

psr: a skos:ConceptScheme .
psr:-T0WTK17L-B
  skos:prefLabel "nombre premier"@fr, "prime number"@en ;
  a skos:Concept ;
  skos:related psr:-SLCRLH7Z-J .

psr:-SLCRLH7Z-J
  skos:exactMatch <https://fr.wikipedia.org/wiki/Conjecture_de_Cram%C3%A9r>, <https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture> ;
  skos:related psr:-T0WTK17L-B ;
  skos:prefLabel "Cramér's conjecture"@en, "conjecture de Cramér"@fr ;
  skos:definition """In number theory, <b>Cramér's conjecture</b>, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be.  It states that  <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+1}-p_{n}=O((\\\\log p_{n})^{2}),\\\\ }">   <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>−<!-- − --></mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <mi>O</mi>         <mo stretchy="false">(</mo>         <mo stretchy="false">(</mo>         <mi>log</mi>         <mo>⁡<!-- ⁡ --></mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo stretchy="false">)</mo>         <mo>,</mo>         <mtext> </mtext>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p_{n+1}-p_{n}=O((\\\\log p_{n})^{2}),\\\\ }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2291ad3e886fd19e6b5c1d6ee6762bfef6e028fa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:26.712ex; height:3.176ex;" alt="p_{n+1}-p_n=O((\\\\log p_n)^2),\\\\ "></span></dd></dl> where <i>p</i><sub><i>n</i></sub> denotes the <i>n</i>th prime number, <i>O</i> is big O notation, and "log" is the natural logarithm. While this is the statement explicitly conjectured by Cramér, his heuristic actually supports the stronger statement  <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 \\\\limsup _{n\\
ightarrow \\\\infty }{\\rac {p_{n+1}-p_{n}}{(\\\\log p_{n})^{2}}}=1,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <munder>           <mo movablelimits="true" form="prefix">lim sup</mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo stretchy="false">→<!-- → --></mo>             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munder>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <msub>                 <mi>p</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                   <mo>+</mo>                   <mn>1</mn>                 </mrow>               </msub>               <mo>−<!-- − --></mo>               <msub>                 <mi>p</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msub>             </mrow>             <mrow>               <mo stretchy="false">(</mo>               <mi>log</mi>               <mo>⁡<!-- ⁡ --></mo>               <msub>                 <mi>p</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msub>               <msup>                 <mo stretchy="false">)</mo>                 <mrow class="MJX-TeXAtom-ORD">                   <mn>2</mn>                 </mrow>               </msup>             </mrow>           </mfrac>         </mrow>         <mo>=</mo>         <mn>1</mn>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\limsup _{n\\
ightarrow \\\\infty }{\\rac {p_{n+1}-p_{n}}{(\\\\log p_{n})^{2}}}=1,}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e48a75b9326a4bc5a5f68fd6e54a2956cf95fbd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.966ex; height:6.009ex;" alt="\\\\limsup_{n\\
ightarrow\\\\infty} \\rac{p_{n+1}-p_n}{(\\\\log p_n)^2} = 1,"></span></dd></dl> and sometimes this formulation is called Cramér's conjecture. However, this stronger version is not supported by more accurate heuristic models, which nevertheless support the first version of Cramér's conjecture. Neither form has yet been proven or disproven. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture">https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture</a>)"""@en, """En mathématiques, la <b>conjecture de Cramér</b>, formulée par le mathématicien suédois Harald Cramér en 1936</span>, pronostique l'asymptotique suivante pour l'écart entre nombres premiers :  <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 g_{n}=p_{n+1}-p_{n}=O((\\\\ln p_{n})^{2}),}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>g</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>+</mo>             <mn>1</mn>           </mrow>         </msub>         <mo>−<!-- − --></mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <mi>O</mi>         <mo stretchy="false">(</mo>         <mo stretchy="false">(</mo>         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo stretchy="false">)</mo>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle g_{n}=p_{n+1}-p_{n}=O((\\\\ln p_{n})^{2}),}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb1caeeec10e4b29924be3b798a495c2b1b972f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.048ex; height:3.176ex;" alt="{\\\\displaystyle g_{n}=p_{n+1}-p_{n}=O((\\\\ln p_{n})^{2}),}"></span></dd></dl> où <i>g</i><sub><i>n</i></sub> est le <i>n</i>-ième écart, <i>p</i><sub><i>n</i></sub> est le <i>n</i>-ième nombre premier et <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 O}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>O</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle O}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="O"></span> désigne le symbole de Bachmann-Landau ; cette conjecture n'est pas démontrée à ce jour. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Conjecture_de_Cram%C3%A9r">https://fr.wikipedia.org/wiki/Conjecture_de_Cram%C3%A9r</a>)"""@fr ;
  dc:created "2023-08-17"^^xsd:date ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:inScheme psr: ;
  skos:broader psr:-VHDD6KJX-8 ;
  a skos:Concept .