{"id":182,"date":"2011-12-02T22:46:13","date_gmt":"2011-12-02T13:46:13","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=182"},"modified":"2021-03-24T08:20:58","modified_gmt":"2021-03-23T23:20:58","slug":"kyr200906","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr200906\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb\u4e592009\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\\(a\\) \u3068 \\(b\\) \u3092\u4e92\u3044\u306b\u7d20, \u3059\u306a\u308f\u3061 \\(1\\) \u4ee5\u5916\u306e\u516c\u7d04\u6570\u3092\u6301\u305f\u306a\u3044\u6b63\u306e\u6574\u6570\u3068\u3057, \u3055\u3089\u306b \\(a\\) \u306f\u5947\u6570\u3068\u3059\u308b.\r\n\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\u6574\u6570 \\(a _ n , b _ n\\) \u3092 \\(\\left( a+b \\sqrt{2} \\right)^n = a _ n +b _ n \\sqrt{2}\\) \u3092\u307f\u305f\u3059\u3088\u3046\u306b\u5b9a\u3081\u308b\u3068\u304d, \u6b21\u306e (1)<\/strong> , (2)<\/strong> \u3092\u793a\u305b.\r\n\u305f\u3060\u3057 \\(\\sqrt{2}\\) \u304c\u7121\u7406\u6570\u3067\u3042\u308b\u3053\u3068\u306f\u8a3c\u660e\u306a\u3057\u306b\u7528\u3044\u3066\u3088\u3044.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a _ 2\\) \u306f\u5947\u6570\u3067\u3042\u308a, \\(a _ 2\\) \u3068 \\(b _ 2\\) \u306f\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u3059\u3079\u3066\u306e \\(n\\) \u306b\u5bfe\u3057\u3066, \\(a _ n\\) \u306f\u5947\u6570\u3067\u3042\u308a, \\(a _ n\\) \u3068 \\(b _ n\\) \u306f\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\[\\begin{align}\r\n\\left( a+b \\sqrt{2} \\right)^2 & = a^2+2b^2 +2ab \\sqrt{2} \\\\\r\n\\text{\u2234} \\quad a _ 2 & = a^2 +2b^2 , \\ b _ 2 = 2ab\r\n\\end{align}\\]\r\n\\(a^2\\) \u306f\u5947\u6570, \\(2b^2\\) \u306f\u5076\u6570\u306a\u306e\u3067, \\(a _ 2\\) \u306f\u5947\u6570.
    \r\n\u307e\u305f\r\n\\[\r\naa _ 2 -bb _ 2 = a \\left( a^2 +2b^2 \\right) -b \\cdot 2ab = a^3 \\quad ... [1]\r\n\\]\r\n\u3053\u3053\u3067, \\(a _ 2 , b _ 2\\) \u304c\u516c\u7d04\u6570 \\(m\\) \u3092\u3082\u3064\u3068\u4eee\u5b9a\u3059\u308b\u3068, [1] \u3088\u308a, \\(a\\) \u3082 \\(m\\) \u3092\u7d04\u6570\u306b\u3082\u3061, \\(a\\) \u306f\u5947\u6570\u306a\u306e\u3067, \\(m \\geqq 3\\) .
    \r\n\u3057\u304b\u3057, \\(2b^2 = a _ 2-a^2\\) \u306a\u306e\u3067, \\(b\\) \u3082 \\(m\\) \u3092\u7d04\u6570\u306b\u3082\u3064\u3053\u3068\u306b\u306a\u308a, \\(a\\) \u3068 \\(b\\) \u306f\u4e92\u3044\u306b\u7d20\u3068\u3044\u3046\u6761\u4ef6\u306b\u77db\u76fe\u3059\u308b.
    \r\n\u3088\u3063\u3066, \\(a _ 2\\) \u3068 \\(b _ 2\\) \u306f\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b.<\/p>\r\n

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

    \\(( a+b \\sqrt{2} )^n\\) \u306e\u5c55\u958b\u5f0f\u306e\u3046\u3061, \\(b \\sqrt{2}\\) \u306e\u5947\u6570\u4e57\u306e\u9805, \u5076\u6570\u4e57\u306e\u9805\u3092\u305d\u308c\u305e\u308c\u8db3\u3057\u5408\u308f\u305b\u305f\u3082\u306e\u304c \\(a _ n\\) , \\(b _ n \\sqrt{2}\\) \u3067\u3042\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066,\r\n\\[\r\n( a-b \\sqrt{2} )^n = a _ n -b _ n \\sqrt{2} \\quad ... [2]\r\n\\]\r\n\u3053\u308c\u3068\u6761\u4ef6\u306e\u5f0f\u3092\u8fba\u3005\u639b\u3051\u5408\u308f\u305b\u308b\u3068\r\n\\[\r\n\\left( a^2 -2b^2 \\right)^n = {a _ n}^2 -2{b _ n}^2 \\quad ... [3]\r\n\\]\r\n\u3053\u3053\u3067, \u5de6\u8fba\u306e \\(a^2-2b^2\\) \u306f\u5947\u6570, \u53f3\u8fba\u306e \\(2{b _ n}^2\\) \u306f\u5076\u6570\u306a\u306e\u3067, \\({a _ n}^2\\) \u306f\u5947\u6570.
    \r\n\u3088\u3063\u3066, \\(a _ n\\) \u306f\u5947\u6570\u3067\u3042\u308b.\r\n\\[\\begin{align}\r\n\\left( a+b \\sqrt{2} \\right)^{n+1} & = \\left( a+b \\sqrt{2} \\right) \\left( a _ n+b _ n \\sqrt{2} \\right) \\\\\r\n& = aa _ n +2bb _ n +\\left( ba _ n +ab _ n \\right) \\sqrt{2} \\\\\r\n\\text{\u2234} \\quad & \\left\\{ \\begin{array}{l} a _ {n+1} =aa _ n +2bb _ n \\\\ b _ {n+1} =ba _ n +ab _ n \\end{array} \\right. \\quad ... [4]\r\n\\end{align}\\]\r\n\u3053\u3053\u304b\u3089\r\n\\[\r\n2bb _ n =a _ {n+1} -aa _ n , \\ ba _ n = b _ {n+1} -ab _ n\r\n\\]\r\n\u306a\u306e\u3067, [4] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\r\n\\left\\{ \\begin{array}{ll} a _ {n+2} =2aa _ {n+1} -(a^2-2b^2)a _ n \\\\ b _ {n+2} =2ab _ {n+1} -(a^2-2b^2)b _ n \\end{array} \\right. \\quad ... [5]\r\n\\]\r\n\u3053\u3053\u3067, \\(a _ n\\) \u3068 \\(b _ n\\) , \\(a _ {n+1}\\) \u3068 \\(b _ {n+1}\\) \u5404\u7d44\u304c\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u3068\u304d, \\(a _ {n+2}\\) \u3068 \\(b _ {n+2}\\) \u304c\u516c\u7d04\u6570 \\(m\\) \u3092\u3082\u3064\u3068\u4eee\u5b9a\u3059\u308b.
    \r\n[3] \u3088\u308a, \\(m\\) \u306f \\(a^2 -2b^2\\) \u306e\u500d\u6570\u3067\u3042\u308b.
    \r\n\u3059\u308b\u3068, [5] \u306e \\(2\\) \u5f0f\u306b\u3064\u3044\u3066, \u5de6\u8fba\u306f \\(a^2-2b^2\\) \u3092\u7d04\u6570\u306b\u3082\u3064\u305f\u3081, \\(2aa _ {n+1}\\) \u3068 \\(2ab _ {n+1}\\) \u306f\u3068\u3082\u306b \\(a^2-2b^2\\) \u3092\u7d04\u6570\u306b\u3082\u3064.
    \r\n\u3053\u3053\u3067, \\(a\\) \u3068 \\(b\\) \u304c\u4e92\u3044\u306b\u7d20\u3067\u3042\u308a, \\(a\\) \u304c\u5947\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \\(2a\\) \u3068 \\(a^2-2b^2\\) \u306f\u4e92\u3044\u306b\u7d20\u3067\u3042\u308a, \\(a _ {n+1}\\) \u3068 \\(b _ {n+1}\\) \u304c\u3068\u3082\u306b \\(a^2-2b^2\\) \u3092\u7d04\u6570\u306b\u3082\u3064\u3053\u3068\u306b\u306a\u308a, \u77db\u76fe\u3059\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066, \\(a _ n\\) \u3068 \\(b _ n\\) , \\(a _ {n+1}\\) \u3068 \\(b _ {n+1}\\) \u5404\u7d44\u304c\u4e92\u3044\u306b\u7d20\u3067\u3042\u308c\u3070, \\(a _ {n+2}\\) \u3068 \\(b _ {n+2}\\) \u3082\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b.
    \r\n\u3053\u308c\u3068, \\(a _ 1\\) \u3068 \\(b _ 1\\) , \\(a _ 2\\) \u3068 \\(b _ 2\\) \u306e\u5404\u7d44\u304c\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \u5e30\u7d0d\u7684\u306b\u3059\u3079\u3066\u306e \\(n\\) \u306b\u3064\u3044\u3066, \\(a _ n\\) \u3068 \\(b _ n\\) \u306f\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3068 \\(b\\) \u3092\u4e92\u3044\u306b\u7d20, \u3059\u306a\u308f\u3061 \\(1\\) \u4ee5\u5916\u306e\u516c\u7d04\u6570\u3092\u6301\u305f\u306a\u3044\u6b63\u306e\u6574\u6570\u3068\u3057, \u3055\u3089\u306b \\(a\\) \u306f\u5947\u6570\u3068\u3059\u308b. \u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\u6574\u6570 \\(a _ n , b _ n\\) \u3092 \\(\\ … \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":[21],"tags":[140,15],"class_list":["post-182","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2009","tag-kyoto_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/182","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=182"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/182\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=182"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=182"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}