@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:-C5QSJ3KK-V .

psr:-C5QSJ3KK-V
  skos:related psr:-LRPB5V08-Q ;
  skos:definition """En arithmétique, on appelle parfois <b>nombre premier de Pythagore</b> (ou <b>nombre premier pythagoricien</b>)</span> un nombre premier <i>p</i> qui est l'hypoténuse d'un triangle rectangle à côtés entiers, c'est-à-dire (théorème de Pythagore) :  <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 p^{2}=a^{2}+b^{2}\\\\quad (a,b\\\\in \\\\mathbb {N} ^{*}).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <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>         <mspace width="1em"></mspace>         <mo stretchy="false">(</mo>         <mi>a</mi>         <mo>,</mo>         <mi>b</mi>         <mo>∈<!-- ∈ --></mo>         <msup>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="double-struck">N</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mo>∗<!-- ∗ --></mo>           </mrow>         </msup>         <mo stretchy="false">)</mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p^{2}=a^{2}+b^{2}\\\\quad (a,b\\\\in \\\\mathbb {N} ^{*}).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12568ca8be5193769a663de3cb81eb7db151b5b1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:26.201ex; height:3.176ex;" alt="{\\\\displaystyle p^{2}=a^{2}+b^{2}\\\\quad (a,b\\\\in \\\\mathbb {N} ^{*}).}"></span></center> D'après la caractérisation des  triplets pythagoriciens primitifs, les nombres premiers de Pythagore sont donc les nombres premiers impairs sommes de deux carrés :  <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 p=x^{2}+y^{2}>2\\\\quad (x,y\\\\in \\\\mathbb {N} ),}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>p</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>&gt;</mo>         <mn>2</mn>         <mspace width="1em"></mspace>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>,</mo>         <mi>y</mi>         <mo>∈<!-- ∈ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">N</mi>         </mrow>         <mo stretchy="false">)</mo>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p=x^{2}+y^{2}&gt;2\\\\quad (x,y\\\\in \\\\mathbb {N} ),}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b2f9a12d3a78da0dc340eed135e19c5197df53" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:28.874ex; height:3.176ex;" alt="{\\\\displaystyle p=x^{2}+y^{2}>2\\\\quad (x,y\\\\in \\\\mathbb {N} ),}"></span></center> c'est-à-dire, par le théorème des deux carrés de Fermat dans le cas des nombres premiers, les nombres premiers congrus à 1 modulo 4</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 p=4k+1\\\\quad (k\\\\in \\\\mathbb {N} ).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>p</mi>         <mo>=</mo>         <mn>4</mn>         <mi>k</mi>         <mo>+</mo>         <mn>1</mn>         <mspace width="1em"></mspace>         <mo stretchy="false">(</mo>         <mi>k</mi>         <mo>∈<!-- ∈ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">N</mi>         </mrow>         <mo stretchy="false">)</mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p=4k+1\\\\quad (k\\\\in \\\\mathbb {N} ).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc5807207a9598af739709d48fb6376d102c4f7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:21.243ex; height:2.843ex;" alt="{\\\\displaystyle p=4k+1\\\\quad (k\\\\in \\\\mathbb {N} ).}"></span></center> Par exemple, le nombre premier 5 est de Pythagore : 5<sup>2</sup> = 3<sup>2</sup> + 4<sup>2</sup>, 5 = 1<sup>2</sup> + 2<sup>2</sup>, 5 = 4×1 + 1. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_premier_pythagoricien">https://fr.wikipedia.org/wiki/Nombre_premier_pythagoricien</a>)"""@fr, """A <b>Pythagorean prime</b> is a prime number of the <span class="nowrap">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 4n+1}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mn>4</mn>         <mi>n</mi>         <mo>+</mo>         <mn>1</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle 4n+1}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/337ba7558e90ed245894e9b56d49cbfcaa3acf6b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.56ex; height:2.343ex;" alt="4n+1"></span>.</span> Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they are the odd prime 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 p}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>p</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span> for which <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 {\\\\sqrt {p}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <msqrt>             <mi>p</mi>           </msqrt>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\sqrt {p}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0527785cd1ad7fa60789e172c720affdcdb28b7f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.105ex; height:3.009ex;" alt="{\\\\displaystyle {\\\\sqrt {p}}}"></span> is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime 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 p}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>p</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span>  for which <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 p}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>p</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span>  itself is the hypotenuse of a primitive Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; <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 {\\\\sqrt {5}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <msqrt>             <mn>5</mn>           </msqrt>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\sqrt {5}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b78ccdb7e18e02d4fc567c66aac99bf524acb5f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\\\\displaystyle {\\\\sqrt {5}}}"></span>  is the hypotenuse of a right triangle with legs 1 and 2, and 5 itself is the hypotenuse of a right triangle with legs 3 and 4. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Pythagorean_prime">https://en.wikipedia.org/wiki/Pythagorean_prime</a>)"""@en ;
  skos:prefLabel "Pythagorean prime"@en, "nombre premier pythagoricien"@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Pythagorean_prime>, <https://fr.wikipedia.org/wiki/Nombre_premier_pythagoricien> ;
  skos:broader psr:-CVDPQB0Q-M ;
  dc:modified "2024-10-18"^^xsd:date ;
  a skos:Concept ;
  dc:created "2023-08-28"^^xsd:date ;
  skos:inScheme psr: ;
  skos:altLabel "nombre premier de Pythagore"@fr .

psr:-CVDPQB0Q-M
  skos:prefLabel "natural numbers"@en, "entier naturel"@fr ;
  a skos:Concept ;
  skos:narrower psr:-C5QSJ3KK-V .