{"id":690,"date":"2013-03-20T09:33:21","date_gmt":"2013-03-20T00:33:21","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=690"},"modified":"2021-10-30T15:40:07","modified_gmt":"2021-10-30T06:40:07","slug":"wsr200701","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr200701\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52007\uff1a\u7b2c1\u554f"},"content":{"rendered":"
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\u8907\u7d20\u6570 \\(\\alpha , \\beta \\ ( \\alpha , \\beta \\neq 0 )\\) \u306b\u5bfe\u3057\u3066, \\(p _ 1 = 3\\) \u3092\u521d\u9805\u3068\u3059\u308b\u6570\u5217 \\(\\{ p _ n \\}\\) \u3092\r\n\\[\r\np _ n = 1 +\\alpha^{n-1} +\\beta^{n-1} \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u3067\u5b9a\u3081\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(p _ 2 \\neq 0\\) , \\(p _ 4 \\neq 0\\) \u306e\u3069\u3061\u3089\u304b\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u6570\u5217 \\(\\{ p _ n \\}\\) \u304c\u3055\u3089\u306b\u6b21\u306e\u6761\u4ef6\u3092\u307f\u305f\u3059\u3068\u3059\u308b.\r\n

      \r\n

    1. (\uff0a)\u3000\u96a3\u63a5\u3059\u308b \\(2\\) \u9805\u306e\u7a4d\u306f\u3059\u3079\u3066 \\(0\\) \u3068\u306a\u308b. \u3059\u306a\u308f\u3061\r\n\\[\r\np _ n p _ {n+1} = 0 \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]<\/li>\r\n<\/ol>\r\n\u3053\u306e\u3068\u304d \\(\\alpha , \\beta\\) \u304a\u3088\u3073 \\(p _ n\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

      \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

      (1)<\/strong><\/p>\r\n

      \u5bfe\u5076\u3092\u3068\u3063\u3066<\/p>\r\n

        \r\n
      1. [A] ... \u300c \\(p _ 2 = 0\\) \u304b\u3064 \\(p _ 4 = 0\\) \u306a\u3089\u3070, \\(\\alpha = 0\\) \u307e\u305f\u306f \\(\\beta = 0\\) \u300d<\/li>\r\n<\/ol>\r\n

        \u3092\u793a\u305b\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\np _ 2 & = 1 +\\alpha +\\beta = 0 \\quad ... [1] , \\\\\r\np _ 4 & = 1 +{\\alpha}^3 +{\\beta}^3 = 0 \\quad ... [2]\r\n\\end{align}\\]\r\n[1] \u3088\u308a\r\n\\[\r\n\\alpha +\\beta = -1\r\n\\]\r\n[2] \u3088\u308a\r\n\\[\\begin{align}\r\n( \\alpha +\\beta )^3 -3 \\alpha \\beta ( \\alpha +\\beta ) & = -1 \\\\\r\n-1 +3 \\alpha \\beta & = -1 \\\\\r\n\\text{\u2234} \\quad \\alpha \\beta & = 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, [A] \u306f\u793a\u3055\u308c, \u984c\u610f\u3082\u793a\u3055\u308c\u305f.<\/p>\r\n

        (2)<\/strong><\/p>\r\n

        \\(p _ 1 = 3\\) \u306a\u306e\u3067, \u6761\u4ef6 (\uff0a) \u3088\u308a\r\n\\[\r\np _ 2 = 1 +\\alpha +\\beta = 0 \\quad ... [3]\r\n\\]\r\n\u306a\u306e\u3067, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\np _ 4 \\neq 0\r\n\\]\r\n\u3055\u3089\u306b, \u6761\u4ef6 (\uff0a) \u304b\u3089\r\n\\[\r\np _ 3 = 1 +{\\alpha}^2 +{\\beta}^2 = 0 \\quad ... [4]\r\n\\]\r\n[3] \u3088\u308a\r\n\\[\r\n\\beta = -( 1 +\\alpha )\r\n\\]\r\n[4] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n1 +{\\alpha}^2 +( 1 +\\alpha )^2 & = 0 \\\\\r\n{\\alpha}^2 +\\alpha +1 & = 0 \\quad ... [5] \\\\\r\n\\text{\u2234} \\quad \\alpha & = \\dfrac{-1 \\pm \\sqrt{3} i}{2}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\beta & = -\\left( 1 +\\dfrac{-1 \\pm \\sqrt{3} i}{2} \\right) \\\\\r\n& = \\dfrac{-1 \\mp \\sqrt{3} i}{2}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n( \\alpha , \\beta ) = \\underline{\\left( \\dfrac{-1 \\pm \\sqrt{3} i}{2} , \\dfrac{-1 \\mp \\sqrt{3} i}{2} \\right) \\quad ( \\text{\u8907\u53f7\u540c\u9806} )}\r\n\\]\r\n\u3053\u3053\u3067, \\(1\\) \u306e \\(3\\) \u4e57\u6839 \\(\\omega = \\dfrac{-1 +\\sqrt{3} i}{2}\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n( \\alpha , \\beta ) = ( \\omega , {\\omega}^2 ) , ( {\\omega}^2 , \\omega )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\np _ n = 1 +{\\omega}^{n-1} +{\\omega}^{2(n-1)}\r\n\\]\r\n\u3053\u3053\u3067 \\(k\\) \u3092 \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3057\u3066<\/p>\r\n