{"id":7461,"date":"2022-04-14T15:25:00","date_gmt":"2022-04-14T13:25:00","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=7461"},"modified":"2023-07-22T10:46:08","modified_gmt":"2023-07-22T08:46:08","slug":"les-carres-de-dirichlet","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/","title":{"rendered":"Les carr\u00e9s de Dirichlet"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Les carr\u00e9s de Dirichlet constituent une famille de carr\u00e9s math\u00e9matiquement int\u00e9ressants.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nulle question ici de g\u00e9om\u00e9trie, mais plut\u00f4t d&rsquo;arithm\u00e9tique&#8230;<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_89 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Carres_de_Dirichlet_definition\" >Carr\u00e9s de Dirichlet: d\u00e9finition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Un_premier_resultat_sur_les_carres_de_Dirichlet\" >Un premier r\u00e9sultat sur les carr\u00e9s de Dirichlet<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Unicite_du_carre_de_Dirichlet\" >Unicit\u00e9 du carr\u00e9 de Dirichlet<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Une_methode_pour_completer_un_carre_de_Dirichlet\" >Une m\u00e9thode pour compl\u00e9ter un carr\u00e9 de Dirichlet<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Etape_initiale_pour_la_resolution_des_carres_de_Dirichlet\" >\u00c9tape initiale pour la r\u00e9solution des carr\u00e9s de Dirichlet<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Etape_1\" >\u00c9tape 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Etape_2\" >\u00c9tape 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Implementation_en_Python\" >Impl\u00e9mentation en Python<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Pourquoi_cette_methode_fonctionne\" >Pourquoi cette m\u00e9thode fonctionne ?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.mathweb.fr\/euclide\/2022\/04\/14\/les-carres-de-dirichlet\/#Epilogue\" >\u00c9pilogue<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Carres_de_Dirichlet_definition\"><\/span>Carr\u00e9s de Dirichlet: d\u00e9finition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">En une phrase, on pourrait d\u00e9finir un carr\u00e9 de Dirichlet comme une grille (carr\u00e9e) de nombres o\u00f9 chacun des nombres est la moyenne des nombres se trouvant au-dessus, en dessous, \u00e0 droite et \u00e0 gauche. Par exemple:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"142\" height=\"120\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2023\/07\/carre-dirichlet.png\" alt=\"\" class=\"wp-image-8634\"\/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Le nombre \u00ab\u00a04\u00a0\u00bb est la moyenne arithm\u00e9tique des quatre nombres 7, 3, 0 et 2:$$3 = \\frac{7+3+0+2}{4}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Bien entendu, vous vous en doutez, il y aura un l\u00e9ger soucis aux bords de la grille; c&rsquo;est la raison pour laquelle figurent des nombres \u00e0 l&rsquo;ext\u00e9rieur du carr\u00e9.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Voici un exemple complet:<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-7387b849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"214\" height=\"187\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-8.png\" alt=\"carr\u00e9 Dirichlet\" class=\"wp-image-7465\"\/><figcaption class=\"wp-element-caption\">Exemple de carr\u00e9 de Dirichlet<\/figcaption><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"latex\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"false\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">\\documentclass{article}\n\\usepackage{array,cellspace}\n\\setlength{\\cellspacetoplimit}{4pt}\n\\setlength{\\cellspacebottomlimit}{4pt}\n\n\\begin{document}\n\n\\begin{tabular}{c|Sc|Sc|Sc|c}\n\\multicolumn{1}{c}{} &amp; \\multicolumn{1}{c}{7} &amp; \\multicolumn{1}{c}{9} &amp; \\multicolumn{1}{c}{9} &amp;  \\\\\\cline{2-4}\n0 &amp; 5 &amp; 7 &amp; 8 &amp; 9 \\\\\\cline{2-4}\n9&amp; 6 &amp; 6 &amp; 7 &amp; 8 \\\\\\cline{2-4}\n0 &amp; 4 &amp; 4 &amp; 6 &amp; 4 \\\\\\cline{2-4}\n\\multicolumn{1}{c}{} &amp; \\multicolumn{1}{c}{6} &amp; \\multicolumn{1}{c}{0} &amp; \\multicolumn{1}{c}{9} &amp; \\\\\n\\end{tabular}\n\n\\end{document}\n<\/pre>\n<\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_premier_resultat_sur_les_carres_de_Dirichlet\"><\/span>Un premier r\u00e9sultat sur les carr\u00e9s de Dirichlet<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-pullquote\"><blockquote><p>Tous les nombres \u00e0 l&rsquo;int\u00e9rieur du carr\u00e9 de Dirichlet sont inf\u00e9rieurs ou \u00e9gaux au maximum de ceux qui se trouvent \u00e0 l&rsquo;ext\u00e9rieur.<\/p><\/blockquote><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Si l&rsquo;on regarde l&rsquo;exemple pr\u00e9c\u00e9dent, le maximum des nombres se trouvant \u00e0 l&rsquo;ext\u00e9rieur est \u00e9gal \u00e0 9. On constate bien que tous les nombre \u00e0 l&rsquo;int\u00e9rieur du carr\u00e9 sont inf\u00e9rieurs ou \u00e9gaux \u00e0 9.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pourquoi est-ce vrai ? On peut raisonner par l&rsquo;absurde. Supposons que le maximum soit \u00e0 l&rsquo;int\u00e9rieur, et notons-le \u00ab\u00a0M\u00a0\u00bb. Pour plus de facilit\u00e9 dans les explications, je vais d\u00e9signer par \u00ab\u00a0les nombres autour\u00a0\u00bb ceux qui se trouvent au-dessus, en dessous, \u00e0 droite et \u00e0 gauche.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Alors, comme moyenne de quatre nombres, M est inf\u00e9rieur ou \u00e9gal aux nombres qui l&rsquo;entourent. Mais comme il est le maximum de la grille, il est n\u00e9cessairement \u00e9gal \u00e0 tous les nombres qui l&rsquo;entourent. On se retrouve donc localement avec un sch\u00e9ma comme celui-ci:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"172\" height=\"121\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-9.png\" alt=\"\" class=\"wp-image-7467\"\/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Par un raisonnement analogue sur les autres cases (qui contiennent M), on arrive \u00e0 voir que n\u00e9cessairement, le carr\u00e9 de Dirichlet ne comporterait que des M.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Unicite_du_carre_de_Dirichlet\"><\/span>Unicit\u00e9 du carr\u00e9 de Dirichlet<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-pullquote\"><blockquote><p>\u00c9tant donn\u00e9s des nombres \u00e0 l&rsquo;ext\u00e9rieur du carr\u00e9 de Dirichlet, il n&rsquo;existe qu&rsquo;une seule fa\u00e7on de le compl\u00e9ter.<\/p><\/blockquote><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Pour s&rsquo;en convaincre, on va consid\u00e9rer deux carr\u00e9s de Dirichlet d&rsquo;ordre 2 ayant les m\u00eames nombres ext\u00e9rieurs, mais pas n\u00e9cessairement les m\u00eames nombres int\u00e9rieurs:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"502\" height=\"148\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-10.png\" alt=\"carr\u00e9s de Dirichlet\" class=\"wp-image-7468\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-10.png 502w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-10-300x88.png 300w\" sizes=\"auto, (max-width: 502px) 100vw, 502px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons maintenant un autre carr\u00e9 de m\u00eames dimensions o\u00f9 tous les nombres sont les diff\u00e9rences de ceux du deuxi\u00e8me et ceux du premier :<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"275\" height=\"148\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-11.png\" alt=\"\" class=\"wp-image-7469\"\/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Souvenons-nous maintenant que nous avions les \u00e9galit\u00e9s suivantes:$$\\begin{cases}x = \\frac{A+B+y+z}{4}\\\\x&rsquo;=\\frac{A+H+y&rsquo;+z&rsquo;}{4}\\end{cases}$$Donc:$$x-x&rsquo;=\\frac{0 + 0 + (y-y&rsquo;) + (z-z&rsquo;)}{4}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Donc \\(x-x&rsquo;\\) est la moyenne des quatre nombres qui l&rsquo;entourent dans le dernier carr\u00e9.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Il en est de m\u00eame pour les trois autres nombres int\u00e9rieurs de ce dernier carr\u00e9. Donc c&rsquo;est un carr\u00e9 de Dirichlet.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D&rsquo;apr\u00e8s le \u00ab\u00a0principe du maximum\u00a0\u00bb vu dans la section pr\u00e9c\u00e9dente, cela signifie que \u00ab\u00a00\u00a0\u00bb est le maximum de tous les nombres de ce carr\u00e9, et donc que \\(x=x&rsquo;\\), \\(y=y&rsquo;\\), \\(z=z&rsquo;\\) et \\(t=t&rsquo;\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Cela montre donc que la solution d&rsquo;un carr\u00e9 de Dirichlet est unique.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_methode_pour_completer_un_carre_de_Dirichlet\"><\/span>Une m\u00e9thode pour compl\u00e9ter un carr\u00e9 de Dirichlet<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_initiale_pour_la_resolution_des_carres_de_Dirichlet\"><\/span>\u00c9tape initiale pour la r\u00e9solution des carr\u00e9s de Dirichlet<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons chercher \u00e0 compl\u00e9ter le carr\u00e9 de Dirichlet suivant :<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"208\" height=\"187\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-12.png\" alt=\"\" class=\"wp-image-7471\"\/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">J&rsquo;ai ins\u00e9r\u00e9 des \u00ab\u00a00\u00a0\u00bb \u00e0 l&rsquo;int\u00e9rieur par d\u00e9faut, mais ils vont tr\u00e8s vite dispara\u00eetre.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;id\u00e9e est de se dire que ceci n&rsquo;est que l&rsquo;\u00e9tape initiale d&rsquo;un (long) processus. Appelons-la l&rsquo;\u00e9tape 0.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_1\"><\/span>\u00c9tape 1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;\u00e9tape suivante consiste \u00e0 transformer le premier \u00ab\u00a00\u00a0\u00bb (en haut \u00e0 gauche) en la moyenne des nombres qui l&rsquo;entourent.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Il devient alors (6 + 5 + 0 + 0)\/4 = 2,75.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On passe alors au \u00ab\u00a00\u00a0\u00bb qui se trouve \u00e0 sa droite, qui se transforme en (0 + 2,75 + 0 + 0)\/4 = 0,5375.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ensuite, on passe au \u00ab\u00a00\u00a0\u00bb \u00e0 sa droite, qui devient : (4 + 0,5375 + 0 + 5)\/4.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On parcourt ainsi toute la grille de haut en bas, de gauche \u00e0 droite.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etape_2\"><\/span>\u00c9tape 2<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Si les nombres int\u00e9rieurs obtenus \u00e0 l&rsquo;\u00e9tape pr\u00e9c\u00e9dente ne sont pas tous \u00e9gaux \u00e0 la moyenne des nombres qui les entourent, on recommence&#8230; jusqu&rsquo;\u00e0 obtenir ce que l&rsquo;on veut.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Implementation_en_Python\"><\/span>Impl\u00e9mentation en Python<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">square = [[0,  6,  0,  4, 0],\n     [5, 0, 0, 0, 5],\n     [5, 0, 0, 0, 5],\n     [3, 0, 0, 0, 4],\n     [0, 2, 2, 3, 0]]\n\ndef dirichlet_square_solver():\n    end = True\n    for line in range(1,len(square)-1):\n        for col in range(1,len(square)-1):\n            temp = square[line][col]\n            square[line][col] = ( square[line-1][col] + square[line][col-1] + square[line][col+1] + square[line+1][col] ) \/ 4\n            if square[line][col] != temp:\n                end = False\n    \n    if not end:\n        dirichlet_square_solver()\n    else:\n        c = ''\n        for line in range(len(square)):\n            for col in range(len(square)):\n                c += '{:^5}'.format(square[line][col])\n            c += '\\n'\n            \n        print(c)\n            \ndirichlet_square_solver()\n<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>  0    6    0    4    0  \n  5   4.5  3.0  4.0   5  \n  5   4.0  3.5  4.0   5  \n  3   3.0  3.0  3.5   4  \n  0    2    2    3    0  <\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Pourquoi_cette_methode_fonctionne\"><\/span>Pourquoi cette m\u00e9thode fonctionne ?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Notons \\(x^{(k)}_n\\) les suites de nombres d\u00e9finies ainsi:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"296\" height=\"228\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/04\/image-13.png\" alt=\"\" class=\"wp-image-7473\"\/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">\\(x^{(k)}_1\\) est la moyenne des nombres qui l&rsquo;entourent, donc sa valeur sera n\u00e9cessairement plus grande que 0. Donc \u00e0 la fin de l&rsquo;\u00e9tape 1, toutes les valeurs int\u00e9rieures seront plus grandes que les valeurs initiales.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On comprend alors qu&rsquo;\u00e0 la fin de l&rsquo;\u00e9tape <em>n<\/em>, toutes les valeurs de \\(x^{(k)}_n\\) seront sup\u00e9rieures \u00e0 celles de l&rsquo;\u00e9tape pr\u00e9c\u00e9dente et ainsi de suite (un raisonnement par r\u00e9currence peut nous en convaincre). <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Les suites \\( (x^{(k)}_n) \\) sont donc croissantes. De plus, elles sont toutes major\u00e9es (par le maximum, qui est \u00e0 l&rsquo;ext\u00e9rieur du carr\u00e9). Donc, elles convergent vers des limites finies, n\u00e9cessairement la moyenne des nombres qui les entourent. Ainsi, le carr\u00e9 des valeurs limites est bien un carr\u00e9 de Dirichlet.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Epilogue\"><\/span>\u00c9pilogue<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Je me suis fortement inspir\u00e9 de la <a href=\"https:\/\/video.math.cnrs.fr\/carres-magiques-de-dirichlet\/\" target=\"_blank\" rel=\"noreferrer noopener\">vid\u00e9o<\/a> d&rsquo;Olivier Druet, directeur de recherches au CNRS (Institut Camille Jordan, Universit\u00e9 Lyon 1).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Vous comprendrez, \u00e0 travers l&rsquo;introduction cette vid\u00e9o, l&rsquo;histoire de ce probl\u00e8me, qui n&rsquo;est autre qu&rsquo;une histoire de temp\u00e9ratures.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En effet, supposons que la grille initiale (o\u00f9 l&rsquo;on ne conna\u00eet que les nombres ext\u00e9rieurs) repr\u00e9sente un groupe de pi\u00e8ces et que les nombres ext\u00e9rieurs repr\u00e9sentent la temp\u00e9rature dans chacune des pi\u00e8ces \u00ab\u00a0ext\u00e9rieures\u00a0\u00bb. Alors, le carr\u00e9 de Dirichlet repr\u00e9sente les temp\u00e9ratures de chacune des pi\u00e8ces une fois que toutes les pi\u00e8ces aient une temp\u00e9rature constante, moyenne des temp\u00e9ratures des pi\u00e8ces qui l&rsquo;entourent avec lesquelles elle a un mur en commun&#8230; Ouais, je sais, je n&rsquo;explique pas super bien&#8230; les transferts thermiques ne sont clairement pas ma sp\u00e9cialit\u00e9 !&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Les carr\u00e9s de Dirichlet constituent une famille de carr\u00e9s math\u00e9matiquement int\u00e9ressants. Nulle question ici de g\u00e9om\u00e9trie, mais plut\u00f4t d&rsquo;arithm\u00e9tique&#8230;<\/p>\n","protected":false},"author":1,"featured_media":7479,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,6,5],"tags":[341,190,101],"class_list":["post-7461","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-latex","category-mathematiques","category-python","tag-dirichlet","tag-moyenne","tag-python"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Les carr\u00e9s de Dirichlet - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Que sont les carr\u00e9s de Dirichlet ? Et comment les compl\u00e9ter quand on connait les nombres ext\u00e9rieurs ? 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