@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:-F0DSFWFM-H
  dc:modified "2024-10-18"^^xsd:date ;
  skos:prefLabel "Thue's lemma"@en, "lemme de Thue"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Lemme_de_Thue>, <https://en.wikipedia.org/wiki/Thue%27s_lemma> ;
  skos:definition """In modular arithmetic, <b>Thue's lemma</b> roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of the modulus. More precisely, for every pair of integers <span class="texhtml">(<i>a</i>, <i>m</i>)</span> with <span class="texhtml"><i>m</i> &gt; 1</span>, given two positive integers <span class="texhtml"><i>X</i></span> and <span class="texhtml"><i>Y</i></span> such that <span class="texhtml"><i>X</i> ≤ <i>m</i> &lt; <i>XY</i></span>, there are two integers <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> such that   <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 ay\\\\equiv x{\\\\pmod {m}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>         <mi>y</mi>         <mo>≡<!-- ≡ --></mo>         <mi>x</mi>         <mrow class="MJX-TeXAtom-ORD">           <mspace width="1em"></mspace>           <mo stretchy="false">(</mo>           <mi>mod</mi>           <mspace width="0.333em"></mspace>           <mi>m</mi>           <mo stretchy="false">)</mo>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle ay\\\\equiv x{\\\\pmod {m}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dddb533536f74f196d386c9afb4c04406d214024" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.538ex; height:2.843ex;" alt="{\\\\displaystyle ay\\\\equiv x{\\\\pmod {m}}}"></span></dd></dl> and   <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 |x|<X,\\\\quad 0<y<Y.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mi>x</mi>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mo>&lt;</mo>         <mi>X</mi>         <mo>,</mo>         <mspace width="1em"></mspace>         <mn>0</mn>         <mo>&lt;</mo>         <mi>y</mi>         <mo>&lt;</mo>         <mi>Y</mi>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle |x|&lt;X,\\\\quad 0&lt;y&lt;Y.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec58e4538ac46b98bdbf91ff3387fcf3ce26c42" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.993ex; height:2.843ex;" alt="{\\\\displaystyle |x|<X,\\\\quad 0<y<Y.}"></span></dd></dl> Usually, one takes <span class="texhtml"><i>X</i></span> and <span class="texhtml"><i>Y</i></span> equal to the smallest integer greater than the square root of <span class="texhtml"><i>m</i></span>, but the general form is sometimes useful, and makes the uniqueness theorem (below) easier to state. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Thue%27s_lemma">https://en.wikipedia.org/wiki/Thue%27s_lemma</a>)"""@en, """En arithmétique modulaire, le <b>lemme de Thue</b> établit que tout entier modulo <span class="texhtml"><i>m</i></span> peut être représenté par une « fraction modulaire » dont le numérateur et le dénominateur sont, en valeur absolue, majorés par la racine carrée de <span class="texhtml"><i>m</i></span>. La première démonstration, attribuée à Axel Thue</span>, utilise le principe des tiroirs</span>. Appliqué à un entier <span class="texhtml"><i>m</i></span> modulo lequel –1 est un carré (en particulier à un nombre premier <span class="texhtml"><i>m</i></span> congru à 1 modulo 4) et à un entier <span class="texhtml"><i>a</i></span> tel que <span class="texhtml"><i>a</i><sup>2</sup> + 1 ≡ 0 mod <i>m</i></span>, ce lemme fournit une expression de <span class="texhtml"><i>m</i></span> comme somme de deux carrés premiers entre eux </span>.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Lemme_de_Thue">https://fr.wikipedia.org/wiki/Lemme_de_Thue</a>)"""@fr ;
  skos:broader psr:-Z1B19BG4-0, psr:-NM1F1MRK-M ;
  dc:created "2023-08-30"^^xsd:date ;
  skos:inScheme psr: ;
  a skos:Concept .

psr:-Z1B19BG4-0
  skos:prefLabel "approximation diophantienne"@fr, "Diophantine approximation"@en ;
  a skos:Concept ;
  skos:narrower psr:-F0DSFWFM-H .

psr:-NM1F1MRK-M
  skos:prefLabel "modular arithmetic"@en, "arithmétique modulaire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-F0DSFWFM-H .