@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:-RBFVN7DN-2
  skos:prefLabel "mathematical constant"@en, "constante mathématique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-JG72VR41-K .

psr:-P93ST75Z-8
  skos:prefLabel "théorie des nombres"@fr, "number theory"@en ;
  a skos:Concept ;
  skos:narrower psr:-JG72VR41-K .

psr: a skos:ConceptScheme .
psr:-L3LNPG9M-Q
  skos:prefLabel "nombre transcendant"@fr, "transcendental number"@en ;
  a skos:Concept ;
  skos:narrower psr:-JG72VR41-K .

psr:-JG72VR41-K
  skos:broader psr:-RBFVN7DN-2, psr:-L3LNPG9M-Q, psr:-P93ST75Z-8 ;
  dc:modified "2023-08-24"^^xsd:date ;
  skos:prefLabel "constante de Prouhet-Thue-Morse"@fr, "Prouhet-Thue-Morse constant"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Prouhet%E2%80%93Thue%E2%80%93Morse_constant>, <https://fr.wikipedia.org/wiki/Constante_de_Prouhet-Thue-Morse> ;
  skos:definition """En mathématiques et dans ses applications, la <b>constante de Prouhet-Thue-Morse</b>, portant les noms de Eugène Prouhet, Axel Thue et Marston Morse, est le nombre <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 \\	au \\\\,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>τ<!-- τ --></mi>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	au \\\\,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8661cd2a09f95bb46a58ad86eebc9c7d81a31bea" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.589ex; height:1.676ex;" alt="\\	au\\\\,"></span> dont le développement binaire est la suite de Prouhet-Thue-Morse. En d'autres termes,
<br/>
<br/><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 \\	au =\\\\sum _{i=0}^{\\\\infty }{\\rac {t_{i}}{2^{i+1}}}=0,412~454~033~640\\\\ldots }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>τ<!-- τ --></mi>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <msub>
<br/>              <mi>t</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>i</mi>
<br/>              </mrow>
<br/>            </msub>
<br/>            <msup>
<br/>              <mn>2</mn>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>i</mi>
<br/>                <mo>+</mo>
<br/>                <mn>1</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>        <mn>412</mn>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mn>454</mn>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mn>033</mn>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mn>640</mn>
<br/>        <mo>…<!-- … --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	au =\\\\sum _{i=0}^{\\\\infty }{\\rac {t_{i}}{2^{i+1}}}=0,412~454~033~640\\\\ldots }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353651b6ffbd1905789187c64f6a4dc00c97e616" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:37.038ex; height:6.843ex;" alt="{\\\\displaystyle \\	au =\\\\sum _{i=0}^{\\\\infty }{\\rac {t_{i}}{2^{i+1}}}=0,412~454~033~640\\\\ldots }"></span></dd></dl>
<br/>où <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 t_{i}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle t_{i}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="t_i"></span> est le <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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>i</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle i}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="i"></span><sup>e</sup> terme de la suite de Prouhet-Thue-Morse.
<br/>Elle est répertoriée comme la suite A014571 de l'OEIS.
<br/>La série génératrice pour <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 t_{i}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle t_{i}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="t_i"></span> est donnée par 
<br/>
<br/><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 \\	au (x)=\\\\sum _{i=0}^{\\\\infty }(-1)^{t_{i}}\\\\,x^{i}={\\rac {1}{1-x}}-2\\\\sum _{i=0}^{\\\\infty }t_{i}\\\\,x^{i}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>τ<!-- τ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>1</mn>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <msub>
<br/>              <mi>t</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>i</mi>
<br/>              </mrow>
<br/>            </msub>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mn>1</mn>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>2</mn>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	au (x)=\\\\sum _{i=0}^{\\\\infty }(-1)^{t_{i}}\\\\,x^{i}={\\rac {1}{1-x}}-2\\\\sum _{i=0}^{\\\\infty }t_{i}\\\\,x^{i}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1649cc97159d00715c2c6ad00115f93649e17d1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:41.096ex; height:6.843ex;" alt="{\\\\displaystyle \\	au (x)=\\\\sum _{i=0}^{\\\\infty }(-1)^{t_{i}}\\\\,x^{i}={\\rac {1}{1-x}}-2\\\\sum _{i=0}^{\\\\infty }t_{i}\\\\,x^{i}}"></span></dd></dl>
<br/>et peut être exprimée par
<br/>
<br/><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 \\	au (x)=\\\\prod _{n=0}^{\\\\infty }(1-x^{2^{n}}).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>τ<!-- τ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∏<!-- ∏ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mo>−<!-- − --></mo>
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <msup>
<br/>              <mn>2</mn>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	au (x)=\\\\prod _{n=0}^{\\\\infty }(1-x^{2^{n}}).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edc75456c2f145a74bb30755287bd20b8b05c5bc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:20.334ex; height:6.843ex;" alt="{\\\\displaystyle \\	au (x)=\\\\prod _{n=0}^{\\\\infty }(1-x^{2^{n}}).}"></span></dd></dl>
<br/>Ceci est un produit de polynômes de Frobenius, et ainsi se généralise aux corps commutatifs arbitraires.
<br/>Kurt Mahler a montré que ce nombre est transcendant en 1929. Comme la suite de Prouhet-Thue-Morse est une suite automatique, ce fait résulte maintenant du théorème général que tout nombre défini par une suite automatique est soit rationnel, soit transcendant. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Constante_de_Prouhet-Thue-Morse">https://fr.wikipedia.org/wiki/Constante_de_Prouhet-Thue-Morse</a>)"""@fr, """In mathematics, the <b>Prouhet–Thue–Morse constant</b>, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by <span class="texhtml mvar" style="font-style:italic;">τ</span>—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence.  That is,
<br/>
<br/><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 \\	au =\\\\sum _{n=0}^{\\\\infty }{\\rac {t_{n}}{2^{n+1}}}=0.412454033640\\\\ldots }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>τ<!-- τ --></mi>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <msub>
<br/>              <mi>t</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>            </msub>
<br/>            <msup>
<br/>              <mn>2</mn>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>                <mo>+</mo>
<br/>                <mn>1</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mn>0.412454033640</mn>
<br/>        <mo>…<!-- … --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	au =\\\\sum _{n=0}^{\\\\infty }{\\rac {t_{n}}{2^{n+1}}}=0.412454033640\\\\ldots }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b491a90c381c2543ef4387077648e8c62f7836" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:35.328ex; height:6.843ex;" alt="{\\\\displaystyle \\	au =\\\\sum _{n=0}^{\\\\infty }{\\rac {t_{n}}{2^{n+1}}}=0.412454033640\\\\ldots }"></span></dd></dl>
<br/>where <span class="texhtml"><i>t<sub>n</sub></i></span> is the <span class="texhtml"><i>n</i><sup>th</sup></span> element of the Prouhet–Thue–Morse sequence. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Prouhet%E2%80%93Thue%E2%80%93Morse_constant">https://en.wikipedia.org/wiki/Prouhet%E2%80%93Thue%E2%80%93Morse_constant</a>)"""@en ;
  skos:inScheme psr: ;
  dc:created "2023-08-03"^^xsd:date ;
  a skos:Concept .