{"id":1381,"date":"2017-02-17T22:02:30","date_gmt":"2017-02-17T13:02:30","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1381"},"modified":"2021-03-09T15:48:23","modified_gmt":"2021-03-09T06:48:23","slug":"tkr201605","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201605\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2016\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(k\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3057, \\(10\\) \u9032\u6cd5\u3067\u8868\u3055\u308c\u305f\u5c0f\u6570\u70b9\u4ee5\u4e0b \\(k\\) \u6841\u306e\u5b9f\u6570\r\n\\[\r\n0 . a _ 1 a _ 2 \\cdots a _ k = \\dfrac{a _ 1}{10} +\\dfrac{a _ 2}{10^2} +\\cdots +\\dfrac{a _ k}{10^k}\r\n\\]\r\n\u3092 \\(1\\) \u3064\u3068\u308b. \u3053\u3053\u3067, \\(a _ 1 , a _ 2 , \\cdots , a _ k\\) \u306f \\(0\\) \u304b\u3089 \\(9\\) \u307e\u3067\u306e\u6574\u6570\u3067, \\(a _ k \\neq 0\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u307f\u305f\u3059\u6b63\u306e\u6574\u6570 \\(n\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088. \r\n\\[\r\n0 . a _ 1 a _ 2 \\cdots a _ k \\leqq \\sqrt{n} -10^k \\lt 0. a _ 1 a _ 2 \\cdots a _ k +10^{-k}\r\n\\]<\/li>\r\n

  2. (2)<\/strong>\u3000\\(p\\) \u304c \\(5 \\cdot 10^{k-1}\\) \u4ee5\u4e0a\u306e\u6574\u6570\u306a\u3089\u3070, \u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u307f\u305f\u3059\u6b63\u306e\u6574\u6570 \\(m\\) \u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b. \r\n\\[\r\n0 . a _ 1 a _ 2 \\cdots a _ k \\leqq \\sqrt{m} -p \\lt 0. a _ 1 a _ 2 \\cdots a _ k +10^{-k}\r\n\\]<\/li>\r\n

  3. (3)<\/strong>\u3000\u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057, \\(r \\leqq x \\lt r+1\\) \u3092\u307f\u305f\u3059\u6574\u6570 \\(r\\) \u3092 \\([x]\\) \u3067\u8868\u3059. \\(\\sqrt{s} -\\left[ \\sqrt{s} \\right] = 0 . a _ 1 a _ 2 \\cdots a _ k\\) \u3092\u307f\u305f\u3059\u6b63\u306e\u6574\u6570 \\(s\\) \u306f\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\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

    \\(A _ k = \\dfrac{a _ 1}{10} +\\dfrac{a _ 2}{10^2} +\\cdots +\\dfrac{a _ k}{10^k}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n10^{-k} \\leqq A _ k \\leqq 1 -10^{-k} \\quad ... [1]\r\n\\]\r\n\u307e\u305f, \\(10^k A _ k\\) \u306f\u81ea\u7136\u6570\u3067, [1] \u3088\u308a\r\n\\[\r\n1\\leq 10^k A _ k \\leqq 10^k -1 \\quad ... [2]\r\n\\]\r\n\u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n10^k +A _ k & \\leqq \\sqrt{n} \\lt 10^k +A _ k +10^{-k} \\\\\r\n\\text{\u2234} \\quad 10^{2k} +2 \\cdot 10^k A _ k +{A _ k}^2 & \\leqq n \\lt 10^{2k} +2 \\cdot 10^k A _ k +2 +( {A _ k} +10^{-k} )^2\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, [1] \u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n0 \\lt {A _ k}^2 \\lt 1 , \\quad 0 \\lt ( {A _ k} +10^{-k} )^2 \\leqq 1\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n10^{2k} +2 \\cdot 10^k A _ k \\lt n \\lt 10^{2k} +2 \\cdot 10^k A _ k +3\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(n\\) \u306f\r\n\\[\r\nn = \\underline{10^{2k} +2 \\cdot 10^k A _ k +1 , \\ 10^{2k} +2 \\cdot 10^k A _ k +2}\r\n\\]\r\n

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

    \u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\np +A _ k & \\leqq \\sqrt{m} \\lt p +A _ k +10^{-k} \\\\\r\n\\text{\u2234} \\quad \\underline{p^2 +2 p A _ k +{A _ k}^2} _ {[3]} & \\leqq m \\lt \\underline{p^2 +2 p (A _ k +10^{-k} ) +( {A _ k} +10^{-k} )^2} _ {[4]}\r\n\\end{align}\r\n\\]\r\n\u3053\u3053\u3067, [3] \u3068 [4] \u306e\u5dee\u3092\u8003\u3048\u308b\u3068\r\n\\[\\begin{align}\r\n[4] - [3] & = 2p \\cdot 10^{-k} +2 A _ k \\cdot 10^{-k} +10^{-2k} \\\\\r\n& \\gt 2p \\cdot 10^{-k} \\\\\r\n& \\geqq 1 \\quad ( \\ \\text{\u2235} \\ p \\geqq 5 \\cdot 10^{k-1} \\ )\r\n\\end{align}\\]\r\n\u5dee\u304c \\(1\\) \u3088\u308a\u5927\u304d\u306a \\(2\\) \u6570\u306e\u9593\u306b\u306f, \u5c11\u306a\u304f\u3068\u3082 \\(1\\) \u3064\u306e\u6574\u6570\u304c\u542b\u307e\u308c\u308b\u306e\u3067,
    \r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

    \u6761\u4ef6\u3092\u307f\u305f\u3059\u6b63\u306e\u6574\u6570 \\(s\\) \u304c\u5b58\u5728\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b.
    \r\n\\(\\sqrt{s} = \\left[ \\sqrt{s} \\right] +A _ k\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\ns & = \\left[ \\sqrt{s} \\right]^2 +\\underline{2A _ k \\left[ \\sqrt{s} \\right]} _ {[5]} +\\underline{{A _ k}^2} _ {[6]}\r\n\\end{align}\\]\r\n[5] \u306f, \u5c0f\u6570\u70b9\u4ee5\u4e0b\u9ad8\u3005 \\(k\\) \u6841\u306e\u5b9f\u6570\u3060\u304c, [6] \u306f \\(a _ k \\neq 0\\) \u306a\u306e\u3067, \u5c0f\u6570\u70b9\u4ee5\u4e0b \\(2k\\) \u6841\u306e\u5b9f\u6570\u3067\u3042\u308b.
    \r\n\u3059\u306a\u308f\u3061, \\(s\\) \u306f\u5c0f\u6570\u70b9\u4ee5\u4e0b \\(2k\\) \u6841\u306e\u5b9f\u6570\u3068\u306a\u308a, \u77db\u76fe\u3059\u308b.
    \r\n\u3088\u3063\u3066, \u6761\u4ef6\u3092\u307f\u305f\u3059\u6b63\u306e\u6574\u6570 \\(s\\) \u306f\u5b58\u5728\u3057\u306a\u3044.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(k\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3057, \\(10\\) \u9032\u6cd5\u3067\u8868\u3055\u308c\u305f\u5c0f\u6570\u70b9\u4ee5\u4e0b \\(k\\) \u6841\u306e\u5b9f\u6570 \\[ 0 . a _ 1 a _ 2 \\cdots a _ k = \\dfrac{a _ 1}{10} +\\dfrac{a _ … \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":[150],"tags":[139,162],"class_list":["post-1381","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2016","tag-tokyo_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1381","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=1381"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1381\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1381"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1381"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1381"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}