{"id":320,"date":"2012-02-25T12:52:56","date_gmt":"2012-02-25T03:52:56","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=320"},"modified":"2021-11-04T14:47:42","modified_gmt":"2021-11-04T05:47:42","slug":"htb200802","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb200802\/","title":{"rendered":"\u4e00\u6a4b\u59272008\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(3\\) \u6b21\u65b9\u7a0b\u5f0f \\(x^3+ax^2+bx+c = 0\\) \u306f\u7570\u306a\u308b \\(3\\) \u3064\u306e\u89e3 \\(p , q , r\\) \u3092\u3082\u3064.\r\n\u3055\u3089\u306b \\(2p^2-1 , 2q-1 , 2r-1\\) \u3082\u540c\u3058\u65b9\u7a0b\u5f0f\u306e\u7570\u306a\u308b \\(3\\) \u3064\u306e\u89e3\u3067\u3042\u308b. \\(a , b , c , p , q , r\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p>\r\n

\r\n\r\n

\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

\u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\na = -(p+q+r) , \\ b = pq+qr+rp , \\ c = -pqr \\quad ... [1]\r\n\\]\r\n\\(\\left\\{ p , q , r \\right\\}\\) \u3068 \\(\\left\\{ 2p^2-1 , 2q-1 , 2r-1 \\right\\}\\) \u306e\u5bfe\u5fdc\u306b\u3088\u3063\u3066, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.
\r\n\u305f\u3060\u3057\r\n\\[\\begin{align}\r\n\\left\\{ \\begin{array}{l} q=2q-1 \\\\ r=2r-1 \\end{array} \\right. \\ \\text{\u3059\u306a\u308f\u3061} \\ q = r = 1 , \\\\\r\n\\left\\{ \\begin{array}{l} q=2r-1 \\\\ r=2q-1 \\end{array} \\right. \\ \\text{\u3059\u306a\u308f\u3061} \\ q = r = 1\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u3053\u308c\u3089\u306e\u5bfe\u5fdc\u3092\u542b\u3080\u7d44\u5408\u305b\u306f\u6761\u4ef6\u3092\u307f\u305f\u3055\u306a\u3044.
\r\n\u3055\u3089\u306b, \\(q\\) \u3068 \\(r\\) \u306e\u5bfe\u79f0\u6027\u3092\u8003\u616e\u3059\u308c\u3070, \u4ee5\u4e0b\u306e \\(2\\) \u901a\u308a\u306e\u7d44\u5408\u305b\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.<\/p>\r\n

    \r\n

  1. 1*<\/strong>\u3000\\(\\left\\{ \\begin{array}{ll} p = 2q-1 & ... [2] \\\\ q = 2p^2-1 & ... [3] \\\\ r = 2r-1 & ... [4] \\end{array} \\right.\\)<\/p><\/li>\r\n

  2. 2*<\/strong>\u3000\\(\\left\\{ \\begin{array}{ll} p = 2q-1 & ... [5] \\\\ q = 2r-1 & ... [6] \\\\ r = 2p^2-1 & ... [7] \\end{array} \\right.\\)<\/p><\/li>\r\n

  3. 1*<\/strong>\u306e\u3068\u304d
    \r\n[4] \u3092\u3068\u304f\u3068, \\(r = 1\\) .
    \r\n[3] \u3092 [2] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\np = 4p^2 & -3 \\\\\r\n(4p+3)(p-1) & = 0 \\\\\r\n\\text{\u2234} \\quad p = -\\dfrac{3}{4} & \\quad ( \\ \\text{\u2235} \\ p \\neq r \\ )\r\n\\end{align}\\]\r\n[3] \u3088\u308a\r\n\\[\r\nq = 2 \\left( -\\dfrac{3}{4} \\right)^2 -1 = \\dfrac{1}{8}\r\n\\]\r\n\u3053\u308c\u3089\u3092 [1] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\na & = -\\left( -\\dfrac{3}{4} +\\dfrac{1}{8} +1 \\right) = -\\dfrac{3}{8} , \\\\\r\nb & = -\\dfrac{3}{4} \\cdot \\dfrac{1}{8} +\\dfrac{1}{8} \\cdot 1 +1 \\cdot \\left( -\\dfrac{3}{4} \\right) = -\\dfrac{23}{32} , \\\\\r\nc & = -\\left( -\\dfrac{3}{4} \\right) \\cdot \\dfrac{1}{8} \\cdot 1 = \\dfrac{3}{32}\r\n\\end{align}\\]<\/li>\r\n

  4. 2*<\/strong>\u306e\u3068\u304d
    \r\n[7] \u3092 [6] \u306b\u4ee3\u5165\u3059\u308b\u3068, \\(q = 4p^2-3\\) .
    \r\n\u3053\u308c\u3092 [5] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\np = 8p^2 & -7 \\\\\r\n(8p+7)(p-1) & = 0 \\\\\r\n\\text{\u2234} \\quad p = -\\dfrac{7}{8} &\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nr & = 2 \\left( -\\dfrac{7}{8} \\right)^2 -1 = \\dfrac{17}{32} \\\\\r\nq & = 2 \\cdot \\dfrac{17}{32} -1 = \\dfrac{1}{16}\r\n\\end{align}\\]\r\n\u3053\u308c\u3089\u3092 [1] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\na & = -\\left( -\\dfrac{7}{8} +\\dfrac{1}{16} +\\dfrac{17}{32} \\right) = -\\dfrac{11}{32} , \\\\\r\nb & = -\\dfrac{7}{8} \\cdot \\dfrac{1}{16} +\\dfrac{1}{16} \\cdot \\dfrac{17}{32} +\\dfrac{17}{32} \\left( -\\dfrac{7}{8} \\right) = -\\dfrac{249}{512} , \\\\\r\nc & = -\\left( -\\dfrac{7}{8} \\right) \\cdot \\dfrac{1}{16} \\cdot \\dfrac{17}{32} = \\dfrac{119}{4096}\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

    \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u7d44\u5408\u305b\u306f \\(q\\) \u3068 \\(r\\) \u3092\u5165\u66ff\u3048\u305f\u3082\u306e\u3082\u542b\u3081\u3066 \\(4\\) \u7d44\u3042\u308a\r\n\\[\\begin{align}\r\n(p,q,r,a,b,c) & = \\underline{\\left( -\\dfrac{3}{4} , \\dfrac{1}{8} , 1 , -\\dfrac{3}{8} , -\\dfrac{23}{32} , \\dfrac{3}{32} \\right) , } \\\\\r\n& \\qquad \\underline{\\left( -\\dfrac{3}{4} , 1 ,\\dfrac{1}{8} , -\\dfrac{3}{8} , -\\dfrac{23}{32} , \\dfrac{3}{32} \\right) , } \\\\\r\n& \\qquad \\quad \\underline{\\left( -\\dfrac{7}{8} , \\dfrac{17}{32} , \\dfrac{1}{16} , -\\dfrac{1}{32} , -\\dfrac{249}{512} , \\dfrac{119}{4096} \\right) , } \\\\\r\n& \\qquad \\qquad \\underline{\\left( -\\dfrac{7}{8} , \\dfrac{1}{16} , \\dfrac{17}{32} , -\\dfrac{1}{32} , -\\dfrac{249}{512} , \\dfrac{119}{4096} \\right)}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(3\\) \u6b21\u65b9\u7a0b\u5f0f \\(x^3+ax^2+bx+c = 0\\) \u306f\u7570\u306a\u308b \\(3\\) \u3064\u306e\u89e3 \\(p , q , r\\) \u3092\u3082\u3064. \u3055\u3089\u306b \\(2p^2-1 , 2q-1 , 2r-1\\) \u3082\u540c\u3058\u65b9\u7a0b\u5f0f\u306e\u7570\u306a\u308b \\(3 … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[57],"tags":[146,16],"class_list":["post-320","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2008","tag-hitotsubashi","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/320","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=320"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/320\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=320"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=320"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}