@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:-JLMR8X0R-R
  skos:prefLabel "Fibonacci sequence"@en, "suite de Fibonacci"@fr ;
  a skos:Concept ;
  skos:related psr:-LKGD3M5L-0 .

psr: a skos:ConceptScheme .
psr:-LKGD3M5L-0
  dc:modified "2024-10-18"^^xsd:date ;
  skos:prefLabel "identité de Candido"@fr, "Candido's identity"@en ;
  skos:broader psr:-VTR5XXB2-M ;
  dc:created "2023-07-13"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Candido%27s_identity> ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:related psr:-JLMR8X0R-R ;
  skos:definition """<b>Candido's identity</b>, named after the Italian mathematician Giacomo Candido, is an identity for real numbers. It states that for two arbitrary real numbers <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 x}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>x</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle x}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> and <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 y}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>y</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle y}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="y"></span> the following equality holds:  <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 \\\\left[x^{2}+y^{2}+(x+y)^{2}\\
ight]^{2}=2[x^{4}+y^{4}+(x+y)^{4}]}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mrow>             <mo>[</mo>             <mrow>               <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>               <mo stretchy="false">(</mo>               <mi>x</mi>               <mo>+</mo>               <mi>y</mi>               <msup>                 <mo stretchy="false">)</mo>                 <mrow class="MJX-TeXAtom-ORD">                   <mn>2</mn>                 </mrow>               </msup>             </mrow>             <mo>]</mo>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>=</mo>         <mn>2</mn>         <mo stretchy="false">[</mo>         <msup>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mi>y</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msup>         <mo>+</mo>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>+</mo>         <mi>y</mi>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msup>         <mo stretchy="false">]</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\left[x^{2}+y^{2}+(x+y)^{2}\\
ight]^{2}=2[x^{4}+y^{4}+(x+y)^{4}]}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f1815914567ff929ffb72819cf12eac6ec0f09" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.485ex; height:3.843ex;" alt="{\\\\displaystyle \\\\left[x^{2}+y^{2}+(x+y)^{2}\\
ight]^{2}=2[x^{4}+y^{4}+(x+y)^{4}]}"></span></dd></dl> The identity however is not restricted to real numbers but holds in every commutative ring. Candido originally devised the identity to prove the following identity for Fibonacci numbers:  <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 (f_{n}^{2}+f_{n+1}^{2}+f_{n+2}^{2})^{2}=2(f_{n}^{4}+f_{n+1}^{4}+f_{n+2}^{4})}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">(</mo>         <msubsup>           <mi>f</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msubsup>         <mo>+</mo>         <msubsup>           <mi>f</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>         <msubsup>           <mi>f</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>+</mo>             <mn>2</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msubsup>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>=</mo>         <mn>2</mn>         <mo stretchy="false">(</mo>         <msubsup>           <mi>f</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msubsup>         <mo>+</mo>         <msubsup>           <mi>f</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>+</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msubsup>         <mo>+</mo>         <msubsup>           <mi>f</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>+</mo>             <mn>2</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msubsup>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle (f_{n}^{2}+f_{n+1}^{2}+f_{n+2}^{2})^{2}=2(f_{n}^{4}+f_{n+1}^{4}+f_{n+2}^{4})}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee72b7bb71da5443e9cadac4f46e049b56ced11" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:42.878ex; height:3.509ex;" alt="{\\\\displaystyle (f_{n}^{2}+f_{n+1}^{2}+f_{n+2}^{2})^{2}=2(f_{n}^{4}+f_{n+1}^{4}+f_{n+2}^{4})}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Candido%27s_identity">https://en.wikipedia.org/wiki/Candido%27s_identity</a>)"""@en .

psr:-VTR5XXB2-M
  skos:prefLabel "identité"@fr, "identity"@en ;
  a skos:Concept ;
  skos:narrower psr:-LKGD3M5L-0 .