@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:-J1P7B30L-P
  skos:broader psr:-VTR5XXB2-M ;
  skos:prefLabel "identité de Brahmagupta"@fr, "Brahmagupta-Fibonacci identity"@en ;
  skos:definition """En mathématiques, l'identité de Brahmagupta est une formule utilisée pour la résolution d'équations diophantiennes. Elle est ancienne ; Diophante d'Alexandrie, un mathématicien grec vivant probablement au IIIe siècle avant J.C., en établit un cas particulier pour l'étude d'un ancêtre du théorème des deux carrés de Fermat. Brahmagupta (598-668) l'établit dans toute sa généralité pour résoudre une question associée à l'équation de Pell-Fermat. L'école indienne élabora par la suite un algorithme appelé « méthode chakravala », dont un ingrédient de base est l'identité de Brahmagupta. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Identit%C3%A9_de_Brahmagupta">https://fr.wikipedia.org/wiki/Identit%C3%A9_de_Brahmagupta</a>)"""@fr, """In algebra, the <b>Brahmagupta–Fibonacci identity</b> expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says  <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 {\\egin{aligned}\\\\left(a^{2}+b^{2}\\
ight)\\\\left(c^{2}+d^{2}\\
ight)&amp;{}=\\\\left(ac-bd\\
ight)^{2}+\\\\left(ad+bc\\
ight)^{2}&amp;&amp;(1)\\\\\\\\&amp;{}=\\\\left(ac+bd\\
ight)^{2}+\\\\left(ad-bc\\
ight)^{2}.&amp;&amp;(2)\\\\end{aligned}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">             <mtr>               <mtd>                 <mrow>                   <mo>(</mo>                   <mrow>                     <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>                   </mrow>                   <mo>)</mo>                 </mrow>                 <mrow>                   <mo>(</mo>                   <mrow>                     <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>                   </mrow>                   <mo>)</mo>                 </mrow>               </mtd>               <mtd>                 <mi></mi>                 <mrow class="MJX-TeXAtom-ORD">                  </mrow>                 <mo>=</mo>                 <msup>                   <mrow>                     <mo>(</mo>                     <mrow>                       <mi>a</mi>                       <mi>c</mi>                       <mo>−<!-- − --></mo>                       <mi>b</mi>                       <mi>d</mi>                     </mrow>                     <mo>)</mo>                   </mrow>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>2</mn>                   </mrow>                 </msup>                 <mo>+</mo>                 <msup>                   <mrow>                     <mo>(</mo>                     <mrow>                       <mi>a</mi>                       <mi>d</mi>                       <mo>+</mo>                       <mi>b</mi>                       <mi>c</mi>                     </mrow>                     <mo>)</mo>                   </mrow>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>2</mn>                   </mrow>                 </msup>               </mtd>               <mtd></mtd>               <mtd>                 <mi></mi>                 <mo stretchy="false">(</mo>                 <mn>1</mn>                 <mo stretchy="false">)</mo>               </mtd>             </mtr>             <mtr>               <mtd></mtd>               <mtd>                 <mi></mi>                 <mrow class="MJX-TeXAtom-ORD">                  </mrow>                 <mo>=</mo>                 <msup>                   <mrow>                     <mo>(</mo>                     <mrow>                       <mi>a</mi>                       <mi>c</mi>                       <mo>+</mo>                       <mi>b</mi>                       <mi>d</mi>                     </mrow>                     <mo>)</mo>                   </mrow>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>2</mn>                   </mrow>                 </msup>                 <mo>+</mo>                 <msup>                   <mrow>                     <mo>(</mo>                     <mrow>                       <mi>a</mi>                       <mi>d</mi>                       <mo>−<!-- − --></mo>                       <mi>b</mi>                       <mi>c</mi>                     </mrow>                     <mo>)</mo>                   </mrow>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>2</mn>                   </mrow>                 </msup>                 <mo>.</mo>               </mtd>               <mtd></mtd>               <mtd>                 <mi></mi>                 <mo stretchy="false">(</mo>                 <mn>2</mn>                 <mo stretchy="false">)</mo>               </mtd>             </mtr>           </mtable>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\egin{aligned}\\\\left(a^{2}+b^{2}\\
ight)\\\\left(c^{2}+d^{2}\\
ight)&amp;{}=\\\\left(ac-bd\\
ight)^{2}+\\\\left(ad+bc\\
ight)^{2}&amp;&amp;(1)\\\\\\\\&amp;{}=\\\\left(ac+bd\\
ight)^{2}+\\\\left(ad-bc\\
ight)^{2}.&amp;&amp;(2)\\\\end{aligned}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab35734d306c0d9f5c78d5604dd76a2c52eb694" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.259ex; height:7.009ex;" alt="{\\\\displaystyle {\\egin{aligned}\\\\left(a^{2}+b^{2}\\
ight)\\\\left(c^{2}+d^{2}\\
ight)&amp;{}=\\\\left(ac-bd\\
ight)^{2}+\\\\left(ad+bc\\
ight)^{2}&amp;&amp;(1)\\\\\\\\&amp;{}=\\\\left(ac+bd\\
ight)^{2}+\\\\left(ad-bc\\
ight)^{2}.&amp;&amp;(2)\\\\end{aligned}}}"></span></dd></dl> For example,  <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 (1^{2}+4^{2})(2^{2}+7^{2})=26^{2}+15^{2}=30^{2}+1^{2}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">(</mo>         <msup>           <mn>1</mn>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mn>4</mn>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo stretchy="false">)</mo>         <mo stretchy="false">(</mo>         <msup>           <mn>2</mn>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mn>7</mn>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo stretchy="false">)</mo>         <mo>=</mo>         <msup>           <mn>26</mn>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mn>15</mn>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>=</mo>         <msup>           <mn>30</mn>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mn>1</mn>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle (1^{2}+4^{2})(2^{2}+7^{2})=26^{2}+15^{2}=30^{2}+1^{2}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d470c3a2531335f129c219b580432aefb1b0803" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.045ex; height:3.176ex;" alt="{\\\\displaystyle (1^{2}+4^{2})(2^{2}+7^{2})=26^{2}+15^{2}=30^{2}+1^{2}.}"></span></dd></dl> The identity is also known as the <b>Diophantus identity</b>,  as it was first proved by Diophantus of Alexandria.  It is a special case of Euler's four-square identity, and also of Lagrange's identity.  Brahmagupta proved and used a more general Brahmagupta identity, stating  <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 {\\egin{aligned}\\\\left(a^{2}+nb^{2}\\
ight)\\\\left(c^{2}+nd^{2}\\
ight)&amp;{}=\\\\left(ac-nbd\\
ight)^{2}+n\\\\left(ad+bc\\
ight)^{2}&amp;&amp;(3)\\\\\\\\&amp;{}=\\\\left(ac+nbd\\
ight)^{2}+n\\\\left(ad-bc\\
ight)^{2}.&amp;&amp;(4)\\\\end{aligned}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">             <mtr>               <mtd>                 <mrow>                   <mo>(</mo>                   <mrow>                     <msup>                       <mi>a</mi>                       <mrow class="MJX-TeXAtom-ORD">                         <mn>2</mn>                       </mrow>                     </msup>                     <mo>+</mo>                     <mi>n</mi>                     <msup>                       <mi>b</mi>                       <mrow class="MJX-TeXAtom-ORD">                         <mn>2</mn>                       </mrow>                     </msup>                   </mrow>                   <mo>)</mo>                 </mrow>                 <mrow>                   <mo>(</mo>                   <mrow>                     <msup>                       <mi>c</mi>                       <mrow class="MJX-TeXAtom-ORD">                         <mn>2</mn>                       </mrow>                     </msup>                     <mo>+</mo>                     <mi>n</mi>                     <msup>                       <mi>d</mi>                       <mrow class="MJX-TeXAtom-ORD">                         <mn>2</mn>                       </mrow>                     </msup>                   </mrow>                   <mo>)</mo>                 </mrow>               </mtd>               <mtd>                 <mi></mi>                 <mrow class="MJX-TeXAtom-ORD">                  </mrow>                 <mo>=</mo>                 <msup>                   <mrow>                     <mo>(</mo>                     <mrow>                       <mi>a</mi>                       <mi>c</mi>                       <mo>−<!-- − --></mo>                       <mi>n</mi>                       <mi>b</mi>                       <mi>d</mi>                     </mrow>                     <mo>)</mo>                   </mrow>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>2</mn>                   </mrow>                 </msup>                 <mo>+</mo>                 <mi>n</mi>                 <msup>                   <mrow>                     <mo>(</mo>                     <mrow>                       <mi>a</mi>                       <mi>d</mi>                       <mo>+</mo>                       <mi>b</mi>                       <mi>c</mi>                     </mrow>                     <mo>)</mo>                   </mrow>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>2</mn>                   </mrow>                 </msup>               </mtd>               <mtd></mtd>               <mtd>                 <mi></mi>                 <mo stretchy="false">(</mo>                 <mn>3</mn>                 <mo stretchy="false">)</mo>               </mtd>             </mtr>             <mtr>               <mtd></mtd>               <mtd>                 <mi></mi>                 <mrow class="MJX-TeXAtom-ORD">                  </mrow>                 <mo>=</mo>                 <msup>                   <mrow>                     <mo>(</mo>                     <mrow>                       <mi>a</mi>                       <mi>c</mi>                       <mo>+</mo>                       <mi>n</mi>                       <mi>b</mi>                       <mi>d</mi>                     </mrow>                     <mo>)</mo>                   </mrow>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>2</mn>                   </mrow>                 </msup>                 <mo>+</mo>                 <mi>n</mi>                 <msup>                   <mrow>                     <mo>(</mo>                     <mrow>                       <mi>a</mi>                       <mi>d</mi>                       <mo>−<!-- − --></mo>                       <mi>b</mi>                       <mi>c</mi>                     </mrow>                     <mo>)</mo>                   </mrow>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>2</mn>                   </mrow>                 </msup>                 <mo>.</mo>               </mtd>               <mtd></mtd>               <mtd>                 <mi></mi>                 <mo stretchy="false">(</mo>                 <mn>4</mn>                 <mo stretchy="false">)</mo>               </mtd>             </mtr>           </mtable>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\egin{aligned}\\\\left(a^{2}+nb^{2}\\
ight)\\\\left(c^{2}+nd^{2}\\
ight)&amp;{}=\\\\left(ac-nbd\\
ight)^{2}+n\\\\left(ad+bc\\
ight)^{2}&amp;&amp;(3)\\\\\\\\&amp;{}=\\\\left(ac+nbd\\
ight)^{2}+n\\\\left(ad-bc\\
ight)^{2}.&amp;&amp;(4)\\\\end{aligned}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee00cd06f0c8be0bd47debfcb9db0d93417dcd2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.838ex; height:7.009ex;" alt="{\\\\displaystyle {\\egin{aligned}\\\\left(a^{2}+nb^{2}\\
ight)\\\\left(c^{2}+nd^{2}\\
ight)&amp;{}=\\\\left(ac-nbd\\
ight)^{2}+n\\\\left(ad+bc\\
ight)^{2}&amp;&amp;(3)\\\\\\\\&amp;{}=\\\\left(ac+nbd\\
ight)^{2}+n\\\\left(ad-bc\\
ight)^{2}.&amp;&amp;(4)\\\\end{aligned}}}"></span></dd></dl> This shows that, for any fixed <i>A</i>, the set of all numbers of the form <i>x</i><sup>2</sup> + <i>Ay</i><sup>2</sup> is closed under multiplication. These identities hold for all integers, as well as all rational numbers; more generally, they are true in any commutative ring.  All four forms of the identity can be verified by expanding each side of the equation.  Also, (2) can be obtained from (1), or (1) from (2), by changing <i>b</i> to −<i>b</i>, and likewise with (3) and (4). 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity">https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity</a>)"""@en ;
  skos:related psr:-PJSZQ3B9-1, psr:-LRPB5V08-Q ;
  a skos:Concept ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity>, <https://fr.wikipedia.org/wiki/Identit%C3%A9_de_Brahmagupta> ;
  skos:altLabel "Diophantus identity"@en, "identité de Diophante"@fr ;
  skos:inScheme psr: ;
  dc:created "2023-07-13"^^xsd:date ;
  dc:modified "2024-10-18"^^xsd:date .

psr: a skos:ConceptScheme .
psr:-VTR5XXB2-M
  skos:prefLabel "identité"@fr, "identity"@en ;
  a skos:Concept ;
  skos:narrower psr:-J1P7B30L-P .

psr:-LRPB5V08-Q
  skos:prefLabel "square number"@en, "nombre carré"@fr ;
  a skos:Concept ;
  skos:related psr:-J1P7B30L-P .

psr:-PJSZQ3B9-1
  skos:prefLabel "Diophantine equation"@en, "équation diophantienne"@fr ;
  a skos:Concept ;
  skos:related psr:-J1P7B30L-P .