@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:-NM1F1MRK-M
  skos:prefLabel "modular arithmetic"@en, "arithmétique modulaire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-C4BL2LXL-9 .

psr:-C4BL2LXL-9
  skos:related psr:-LRPB5V08-Q ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem>, <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_des_trois_carr%C3%A9s> ;
  dc:created "2023-08-28"^^xsd:date ;
  skos:definition """En mathématiques et plus précisément en arithmétique modulaire, le <b>théorème des trois carrés</b> s'énonce de la manière suivante :  <blockquote style="width:90%; border-left: solid #D0D0D0 1px; padding-left:1em;"> Un entier naturel est somme de trois carrés d'entiers <b>si</b> (et seulement si) il n'est <b>pas</b> 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 4^{i}(8k+7)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mn>4</mn>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>           </mrow>         </msup>         <mo stretchy="false">(</mo>         <mn>8</mn>         <mi>k</mi>         <mo>+</mo>         <mn>7</mn>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle 4^{i}(8k+7)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cefd18fd5ddf0a18720febe8bcc0cfd844e649a3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.148ex; height:3.176ex;" alt="{\\\\displaystyle 4^{i}(8k+7)}"></span> 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 i}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>i</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle i}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\\\\displaystyle i}"></span> 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 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="{\\\\displaystyle k}"></span> entiers naturels.</blockquote> Les premiers entiers naturels qui ne sont pas somme de trois carrées sont donc :  <dl><dd>7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... suite A004215 de l'OEIS.</dd></dl> Dit autrement, les racines carrées de ces nombres sont les longueurs interdites des diagonales d'un parallélépipède rectangle à côtés entiers. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_des_trois_carr%C3%A9s">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_des_trois_carr%C3%A9s</a>)"""@fr, """In mathematics, <b>Legendre's three-square theorem</b> states that a natural number  can be represented as the sum of three squares of integers  <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 n=x^{2}+y^{2}+z^{2}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>=</mo>         <msup>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mi>y</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mi>z</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n=x^{2}+y^{2}+z^{2}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d21f8b668e4450e588abd892ed19cae7e2b61835" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.917ex; height:3.009ex;" alt="{\\\\displaystyle n=x^{2}+y^{2}+z^{2}}"></span></dd></dl> if and only if <span class="texhtml mvar" style="font-style:italic;">n</span> is not of the form <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^{a}(8b+7)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>         <mo>=</mo>         <msup>           <mn>4</mn>           <mrow class="MJX-TeXAtom-ORD">             <mi>a</mi>           </mrow>         </msup>         <mo stretchy="false">(</mo>         <mn>8</mn>         <mi>b</mi>         <mo>+</mo>         <mn>7</mn>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n=4^{a}(8b+7)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6eaa543c95adff5bb6bcd86fa1957385c7e14e8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.73ex; height:2.843ex;" alt="{\\\\displaystyle n=4^{a}(8b+7)}"></span> for nonnegative integers <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem">https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem</a>)"""@en ;
  skos:prefLabel "théorème des trois carrés de Legendre"@fr, "Legendre's three-square theorem"@en ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:broader psr:-C7ZLH8LZ-5, psr:-NM1F1MRK-M, psr:-PJSZQ3B9-1 ;
  a skos:Concept ;
  skos:inScheme psr: .

psr:-C7ZLH8LZ-5
  skos:prefLabel "théorie additive des nombres"@fr, "additive number theory"@en ;
  a skos:Concept ;
  skos:narrower psr:-C4BL2LXL-9 .

psr: a skos:ConceptScheme .
psr:-LRPB5V08-Q
  skos:prefLabel "square number"@en, "nombre carré"@fr ;
  a skos:Concept ;
  skos:related psr:-C4BL2LXL-9 .

psr:-PJSZQ3B9-1
  skos:prefLabel "Diophantine equation"@en, "équation diophantienne"@fr ;
  a skos:Concept ;
  skos:narrower psr:-C4BL2LXL-9 .