@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: a skos:ConceptScheme .
psr:-LRPB5V08-Q
  skos:prefLabel "square number"@en, "nombre carré"@fr ;
  a skos:Concept ;
  skos:related psr:-R4K99MK9-L .

psr:-R4K99MK9-L
  skos:exactMatch <https://en.wikipedia.org/wiki/Euler%27s_criterion>, <https://fr.wikipedia.org/wiki/Crit%C3%A8re_d%27Euler> ;
  dc:created "2023-08-28"^^xsd:date ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:related psr:-LRPB5V08-Q ;
  skos:prefLabel "Euler's criterion"@en, "critère d'Euler"@fr ;
  skos:definition """En mathématiques et plus précisément en arithmétique modulaire, le critère d'Euler est un théorème utilisé en théorie des nombres pour déterminer si un entier donné est un résidu quadratique (autrement dit, un carré) modulo un nombre premier. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Crit%C3%A8re_d%27Euler">https://fr.wikipedia.org/wiki/Crit%C3%A8re_d%27Euler</a>)"""@fr, """In number theory, <b>Euler's criterion</b> is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let <i>p</i> be an odd prime and <i>a</i> be an integer coprime to <i>p</i>. Then  <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 a^{\\	frac {p-1}{2}}\\\\equiv {\\egin{cases}\\\\;\\\\;\\\\,1{\\\\pmod {p}}&amp;{\\	ext{ if there is an integer }}x{\\	ext{ such that }}x^{2}\\\\equiv a{\\\\pmod {p}},\\\\\\\\-1{\\\\pmod {p}}&amp;{\\	ext{ if there is no such integer.}}\\\\end{cases}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>a</mi>           <mrow class="MJX-TeXAtom-ORD">             <mstyle displaystyle="false" scriptlevel="0">               <mfrac>                 <mrow>                   <mi>p</mi>                   <mo>−<!-- − --></mo>                   <mn>1</mn>                 </mrow>                 <mn>2</mn>               </mfrac>             </mstyle>           </mrow>         </msup>         <mo>≡<!-- ≡ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mrow>             <mo>{</mo>             <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">               <mtr>                 <mtd>                   <mspace width="thickmathspace"></mspace>                   <mspace width="thickmathspace"></mspace>                   <mspace width="thinmathspace"></mspace>                   <mn>1</mn>                   <mrow class="MJX-TeXAtom-ORD">                     <mspace width="0.444em"></mspace>                     <mo stretchy="false">(</mo>                     <mi>mod</mi>                     <mspace width="0.333em"></mspace>                     <mi>p</mi>                     <mo stretchy="false">)</mo>                   </mrow>                 </mtd>                 <mtd>                   <mrow class="MJX-TeXAtom-ORD">                     <mtext> if there is an integer </mtext>                   </mrow>                   <mi>x</mi>                   <mrow class="MJX-TeXAtom-ORD">                     <mtext> such that </mtext>                   </mrow>                   <msup>                     <mi>x</mi>                     <mrow class="MJX-TeXAtom-ORD">                       <mn>2</mn>                     </mrow>                   </msup>                   <mo>≡<!-- ≡ --></mo>                   <mi>a</mi>                   <mrow class="MJX-TeXAtom-ORD">                     <mspace width="0.444em"></mspace>                     <mo stretchy="false">(</mo>                     <mi>mod</mi>                     <mspace width="0.333em"></mspace>                     <mi>p</mi>                     <mo stretchy="false">)</mo>                   </mrow>                   <mo>,</mo>                 </mtd>               </mtr>               <mtr>                 <mtd>                   <mo>−<!-- − --></mo>                   <mn>1</mn>                   <mrow class="MJX-TeXAtom-ORD">                     <mspace width="0.444em"></mspace>                     <mo stretchy="false">(</mo>                     <mi>mod</mi>                     <mspace width="0.333em"></mspace>                     <mi>p</mi>                     <mo stretchy="false">)</mo>                   </mrow>                 </mtd>                 <mtd>                   <mrow class="MJX-TeXAtom-ORD">                     <mtext> if there is no such integer.</mtext>                   </mrow>                 </mtd>               </mtr>             </mtable>             <mo fence="true" stretchy="true" symmetric="true"></mo>           </mrow>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a^{\\	frac {p-1}{2}}\\\\equiv {\\egin{cases}\\\\;\\\\;\\\\,1{\\\\pmod {p}}&amp;{\\	ext{ if there is an integer }}x{\\	ext{ such that }}x^{2}\\\\equiv a{\\\\pmod {p}},\\\\\\\\-1{\\\\pmod {p}}&amp;{\\	ext{ if there is no such integer.}}\\\\end{cases}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1e6e32e4504e7f4f65a3bceea2cee9b48af36f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:75.484ex; height:6.343ex;" alt="{\\\\displaystyle a^{\\	frac {p-1}{2}}\\\\equiv {\\egin{cases}\\\\;\\\\;\\\\,1{\\\\pmod {p}}&amp;{\\	ext{ if there is an integer }}x{\\	ext{ such that }}x^{2}\\\\equiv a{\\\\pmod {p}},\\\\\\\\-1{\\\\pmod {p}}&amp;{\\	ext{ if there is no such integer.}}\\\\end{cases}}}"></span></dd></dl> Euler's criterion can be concisely reformulated using the Legendre symbol:  <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({\\rac {a}{p}}\\
ight)\\\\equiv a^{\\	frac {p-1}{2}}{\\\\pmod {p}}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow>           <mo>(</mo>           <mrow class="MJX-TeXAtom-ORD">             <mfrac>               <mi>a</mi>               <mi>p</mi>             </mfrac>           </mrow>           <mo>)</mo>         </mrow>         <mo>≡<!-- ≡ --></mo>         <msup>           <mi>a</mi>           <mrow class="MJX-TeXAtom-ORD">             <mstyle displaystyle="false" scriptlevel="0">               <mfrac>                 <mrow>                   <mi>p</mi>                   <mo>−<!-- − --></mo>                   <mn>1</mn>                 </mrow>                 <mn>2</mn>               </mfrac>             </mstyle>           </mrow>         </msup>         <mrow class="MJX-TeXAtom-ORD">           <mspace width="1em"></mspace>           <mo stretchy="false">(</mo>           <mi>mod</mi>           <mspace width="0.333em"></mspace>           <mi>p</mi>           <mo stretchy="false">)</mo>         </mrow>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\left({\\rac {a}{p}}\\
ight)\\\\equiv a^{\\	frac {p-1}{2}}{\\\\pmod {p}}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7647139c37a9906c25cf485b38fa070bb08c83e8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.312ex; height:6.343ex;" alt="{\\\\displaystyle \\\\left({\\rac {a}{p}}\\
ight)\\\\equiv a^{\\	frac {p-1}{2}}{\\\\pmod {p}}.}"></span></dd></dl> The criterion dates from a 1748 paper by Leonhard Euler. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_criterion">https://en.wikipedia.org/wiki/Euler%27s_criterion</a>)"""@en ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:broader psr:-NM1F1MRK-M .

psr:-NM1F1MRK-M
  skos:prefLabel "modular arithmetic"@en, "arithmétique modulaire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-R4K99MK9-L .