{"id":1326,"date":"2015-11-09T00:58:51","date_gmt":"2015-11-08T15:58:51","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1326"},"modified":"2021-10-21T08:25:01","modified_gmt":"2021-10-20T23:25:01","slug":"wsr201502","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201502\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52015\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u6574\u6570 \\(x , y\\) \u304c \\(x^2 -2y^2 = 1\\) \u3092\u307f\u305f\u3059\u3068\u304d, \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u6574\u6570 \\(a , b , u , v\\) \u304c \\(( a +b \\sqrt{2} ) ( x +y \\sqrt{2} ) = u +v \\sqrt{2}\\) \u3092\u307f\u305f\u3059\u3068\u304d, \\(u , v\\) \u3092 \\(a , b , x , y\\) \u3067\u8868\u305b. \u3055\u3089\u306b \\(a^2 -2b^2 = 1\\) \u306e\u3068\u304d\u306e \\(u^2 -2v^2\\) \u306e\u5024\u3092\u6c42\u3081\u3088. \u3068\u3082\u306b\u7b54\u306e\u307f\u3067\u3088\u3044.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(1 \\lt x +y \\sqrt{2} \\leqq 3 +2 \\sqrt{2}\\) \u306e\u3068\u304d, \\(x = 3 , \\ y = 2\\) \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(( 3 +2 \\sqrt{2} )^{n-1} \\lt x +y \\sqrt{2} \\leqq ( 3 +2 \\sqrt{2} )^n\\) \u306e\u3068\u304d, \\(x +y \\sqrt{2} = ( 3 +2 \\sqrt{2} )^n\\) \u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\[\r\n( a +b \\sqrt{2} ) ( x +y \\sqrt{2} ) = ax +2by +( ay +bx ) \\sqrt{2}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nu = \\underline{ax +2by} , \\ v = \\underline{ay +bx}\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nu^2 -2v^2 & = ( ax +2by )^2 -2 ( ay +bx )^2 \\\\\r\n& = a^2 x^2 +4 b^2 y^2 -2 a^2 y^2 -2 b^2 x^2 \\\\\r\n& = ( a^2 -2b^2 ) ( x^2 -2y^2 ) \\\\\r\n& = 1 \\cdot 1 = \\underline{1}\r\n\\end{align}\\]\r\n

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

    \u4e0e\u3048\u3089\u308c\u305f\u6761\u4ef6\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\n-\\dfrac{x}{\\sqrt{2}} +\\dfrac{1}{\\sqrt{2}} \\lt y \\leqq -\\dfrac{x}{\\sqrt{2}} +\\dfrac{3}{\\sqrt{2}} +2\r\n\\]\r\n\u3053\u306e\u9818\u57df \\(R\\) \u306b\u542b\u307e\u308c\u308b\u683c\u5b50\u70b9\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.
    \r\n\\(R\\) \u306f, \u50be\u304d \\(-\\dfrac{1}{\\sqrt{2}}\\) \u3067, \u305d\u308c\u305e\u308c\u70b9 \\(( 1 , 0 )\\) , \\(( 3 , 2 )\\) \u3092\u901a\u308b\u76f4\u7dda \\(\\ell _ 1 , \\ell _ 2\\) \u306b\u631f\u307e\u308c\u305f\u9818\u57df\u3067\u3042\u308b\uff08\u305f\u3060\u3057, \\(\\ell _ 1\\) \u4e0a\u306e\u70b9\u306f\u542b\u307e\u306a\u3044\uff09.
    \r\n\u53cc\u66f2\u7dda \\(C : \\ x^2 -2y^2 = 1\\) \u306e\u6f38\u8fd1\u7dda\u304c \\(y = \\pm \\dfrac{x}{\\sqrt{2}}\\) \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070, \\(C\\) \u3068 \\(R\\) \u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b\u306e\u3067<\/p>\r\n\"wsr20150201\"\r\n

    \\(R\\) \u306b\u542b\u307e\u308c\u308b\u683c\u5b50\u70b9\u306f, \u70b9 \\(( 3 , 2 )\\) \u306e\u307f\u3067\u3042\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

      \r\n
    1. [A] ... \u300c \\(( 3 +2 \\sqrt{2} )^{n-1} \\lt x +y \\sqrt{2} \\leqq ( 3 +2 \\sqrt{2} )^n\\) \u306e\u3068\u304d, \\(x +y \\sqrt{2} = ( 3 +2 \\sqrt{2} )^n\\) \u300d<\/li>\r\n<\/ol>\r\n

      \u304c, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d
        \r\n(2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

      2. 2*<\/strong>\u3000\\(n = k\\) \u306e\u3068\u304d, [A] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061,
        \r\n\u300c \\(( 3 +2 \\sqrt{2} )^{k-1} \\lt x +y \\sqrt{2} \\leqq ( 3 +2 \\sqrt{2} )^k\\) ... [1] \u306e\u3068\u304d, \\(x +y \\sqrt{2} = ( 3 +2 \\sqrt{2} )^k\\) \u300d\u3068\u4eee\u5b9a\u3059\u308b.
        \r\n[1] \u306e\u8fba\u3005\u306b \\(3 +2 \\sqrt{2}\\) \u3092\u639b\u3051\u308c\u3070, \\(n = k+1\\) \u306e\u3068\u304d, \\(x +y \\sqrt{2} = ( 3 +2 \\sqrt{2} )^{k+1}\\) ... [2] \u306f\r\n\\[\r\n( 3 +2 \\sqrt{2} )^k \\lt x +y \\sqrt{2} \\leqq ( 3 +2 \\sqrt{2} )^{k+1} \\quad ... [3]\r\n\\]\r\n\u3092\u307f\u305f\u3059\u89e3\u306e\u3072\u3068\u3064\u3067\u3042\u308b.
        \r\n\u6b21\u306b, [2] \u4ee5\u5916\u306b [3] \u3092\u307f\u305f\u3059\u89e3 \\(x' +y' \\sqrt{2}\\) \uff08 \\(x' , y'\\) \u306f\u6574\u6570\uff09\u304c\u5b58\u5728\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b.
        \r\n\\(( 3 +2 \\sqrt{2} ) ( 3 -2 \\sqrt{2} ) = 1\\) \u306b\u6ce8\u610f\u3059\u308c\u3070, [3] \u306e\u8fba\u3005\u306b \\(3 -2 \\sqrt{2}\\) \u3092\u304b\u3051\u308b\u3068\r\n\\[\r\n( 3 +2 \\sqrt{2} )^{k-1} \\lt \\underline{( 3 -2 \\sqrt{2} ) ( x' +y' \\sqrt{2} ) } _ {[4]} \\leqq ( 3 +2 \\sqrt{2} )^k\r\n\\]\r\n\u3053\u3053\u3067, \\([4] \\neq ( 3 +2 \\sqrt{2} )^k\\) \u3067\u3042\u308b\u304c, \u3053\u308c\u306f, \\(n = k\\) \u306e\u3068\u304d\u306b [A] \u304c\u6210\u7acb\u3059\u308b\u3068\u3057\u305f\u4eee\u5b9a\u306b\u77db\u76fe\u3059\u308b.
        \r\n\u3057\u305f\u304c\u3063\u3066, [2] \u306f [3] \u306e\u552f\u4e00\u306e\u89e3\u3067\u3042\u308a, \\(n = k+1\\) \u306e\u3068\u304d\u3082 [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u6574\u6570 \\(x , y\\) \u304c \\(x^2 -2y^2 = 1\\) \u3092\u307f\u305f\u3059\u3068\u304d, \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\u6574\u6570 \\(a , b , u , v\\) \u304c \\(( a +b \\sqrt{2} ) ( x +y \\sqrt{ … \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":[134],"tags":[147,137],"class_list":["post-1326","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2015","tag-waseda_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1326","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=1326"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1326\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1326"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1326"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1326"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}