@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:-GP35WQJL-8
  skos:inScheme psr: ;
  skos:broader psr:-C7ZLH8LZ-5, psr:-PJSZQ3B9-1 ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Jacobi>, <https://en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem> ;
  dc:created "2023-08-28"^^xsd:date ;
  skos:prefLabel "théorème des quatre carrés de Jacobi"@fr, "Jacobi's four-square theorem"@en ;
  skos:definition """In number theory, Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares (of integers). 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem">https://en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem</a>)"""@en, """En théorie des nombres, le <b>théorème de Jacobi</b>, dû à Charles Gustave Jacob Jacobi, précise, pour tout entier <span class="texhtml mvar" style="font-style:italic;">n</span> &gt; 0, le nombre <i>r</i><sub>4</sub>(<span class="texhtml mvar" style="font-style:italic;">n</span>) de façons de décomposer <span class="texhtml mvar" style="font-style:italic;">n</span> sous forme d'une somme de quatre carrés (plus précisément : le nombre de quadruplets (<span class="texhtml mvar" style="font-style:italic;">a, b, c, d</span>) d'entiers relatifs tels que <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=a^{2}+b^{2}+c^{2}+d^{2}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>=</mo>         <msup>           <mi>a</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mi>b</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mi>c</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mi>d</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n=a^{2}+b^{2}+c^{2}+d^{2}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f364acfd68371113fea8c967e2ced9cb739d1990" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.684ex; height:2.843ex;" alt="{\\\\displaystyle n=a^{2}+b^{2}+c^{2}+d^{2}}"></span>) :  <center><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_{4}(n)=8\\\\sum _{d\\\\mid n,~4\\
mid d}d.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>r</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mn>8</mn>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>d</mi>             <mo>∣<!-- ∣ --></mo>             <mi>n</mi>             <mo>,</mo>             <mtext> </mtext>             <mn>4</mn>             <mo>∤<!-- ∤ --></mo>             <mi>d</mi>           </mrow>         </munder>         <mi>d</mi>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle r_{4}(n)=8\\\\sum _{d\\\\mid n,~4\\
mid d}d.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612f6e580942a91a02d4afe1f2556e6cac550ff6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:17.685ex; height:6.009ex;" alt="{\\\\displaystyle r_{4}(n)=8\\\\sum _{d\\\\mid n,~4\\
mid d}d.}"></span></center> Le théorème des quatre carrés de Lagrange s'en déduit.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Jacobi">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Jacobi</a>)"""@fr ;
  a skos:Concept ;
  skos:related psr:-LRPB5V08-Q ;
  dc:modified "2024-10-18"^^xsd:date .

psr:-C7ZLH8LZ-5
  skos:prefLabel "théorie additive des nombres"@fr, "additive number theory"@en ;
  a skos:Concept ;
  skos:narrower psr:-GP35WQJL-8 .

psr: a skos:ConceptScheme .
psr:-LRPB5V08-Q
  skos:prefLabel "square number"@en, "nombre carré"@fr ;
  a skos:Concept ;
  skos:related psr:-GP35WQJL-8 .

psr:-PJSZQ3B9-1
  skos:prefLabel "Diophantine equation"@en, "équation diophantienne"@fr ;
  a skos:Concept ;
  skos:narrower psr:-GP35WQJL-8 .