{"id":399,"date":"2012-05-18T23:29:06","date_gmt":"2012-05-18T14:29:06","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=399"},"modified":"2021-09-15T07:56:05","modified_gmt":"2021-09-14T22:56:05","slug":"ngr201204","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201204\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2012\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(m , p\\) \u3092 \\(3\\) \u4ee5\u4e0a\u306e\u5947\u6570\u3068\u3057, \\(m\\) \u306f \\(p\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\((x-1)^{101}\\) \u306e\u5c55\u958b\u5f0f\u306b\u304a\u3051\u308b \\(x^2\\) \u306e\u9805\u306e\u4fc2\u6570\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\((p-1)^m +1\\) \u306f \\(p\\) \u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\((p-1)^m +1\\) \u306f \\(p^2\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  4. (4)<\/strong>\u3000\\(r\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3057, \\(s = 3^{r-1}m\\) \u3068\u3059\u308b. \\(2^s +1\\) \u306f \\(3^r\\) \u3067\u5272\u308a\u5207\u308c\u308b\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

    \\(x^2\\) \u306e\u9805\u306f\r\n\\[\r\n{} _ {101} \\text{C} {} _ 2 x^2 (-1)^{99} = -5050x^2\n\\]\r\n\u3088\u3063\u3066, \u4fc2\u6570\u306f\r\n\\[\r\n\\underline{-5050}\n\\]\r\n

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

    \\[\\begin{align}\r\n(p-1)^m+1 & = \\textstyle\\sum\\limits _ {k=0}^m {} _ {m} \\text{C} {} _ {k} (-1)^{m-k} p^k +1 \\\\\r\n& = (-1)^m +\\textstyle\\sum\\limits _ {k=1}^{m} {} _ {m} \\text{C} {} _ {k} (-1)^{m-k} p^k +1 \\\\\r\n& = -1 +\\textstyle\\sum\\limits _ {k=1}^{m} {} _ {m} \\text{C} {} _ {k} (-1)^{m-k} p^k +1 \\quad ( \\ \\text{\u2235} \\ m \\text{\u306f\u5947\u6570} ) \\\\\r\n& = p \\textstyle\\sum\\limits _ {k=1}^{m} {} _ {m} \\text{C} {} _ {k} (-1)^{m-k} p^{k-1}\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u3053\u308c\u306f \\(p\\) \u3067\u5272\u308a\u5207\u308c\u308b.<\/p>\r\n

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

    \\[\\begin{align}\r\n(p-1)^m+1 & = mp +\\textstyle\\sum\\limits _ {k=2}^{m} {} _ {m} \\text{C} {} _ {k} (-1)^{m-k} p^k \\\\\r\n& = mp +p^2 \\textstyle\\sum\\limits _ {k=2}^m {} _ {m} \\text{C} {} _ {k} (-1)^{m-k} p^{k-2}\n\\end{align}\\]\r\n\\(m\\) \u306f \\(p\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u306e\u3067, \u3053\u308c\u306f \\(p^2\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044.<\/p>\r\n

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

    \\(s(r) = 3^{r-1}m\\) \u3068\u304a\u304f.
    \r\n\u300c \\(2^{s(r)}+1\\) \u306f \\(3^r\\) \u3067\u5272\u308a\u5207\u308c\u308b. \u300d... [A] \u304c, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(r\\) \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\\(r = 1\\) \u306e\u3068\u304d
      \r\n\\(s(1) = m\\) \u306a\u306e\u3067, (2)<\/strong> \u306e\u7d50\u679c\u306b\u304a\u3044\u3066 \\(p=3\\) \u3068\u304a\u3051\u3070, \\(2^m+1\\) \u306f \\(3\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u306e\u3067, [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(r = k \\ ( k \\geqq 1 )\\) \u306e\u3068\u304d
      \r\n[A]\u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061, \\(2^{s(k)}+1 = 3^k M\\) \uff08 \\(M\\) \u306f\u6574\u6570\uff09\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n2^{s(k+1)}+1 & = 2^{3 s(k)} +1 \\\\\r\n& = \\left( 2^{s(k)}+1 \\right) \\left( 2^{2 s(k)} -2^{s(k)} +1 \\right) \\\\\r\n& = \\left( 2^{s(k)}+1 \\right) \\left\\{ \\left( 2^{s(k)} +1 \\right)^2 -3 \\cdot 2^{s(k)} \\right\\} \\\\\r\n& = 3^k M \\left( 3^{2k} M^2 -3 \\cdot 2^{s(k)} \\right) \\\\\r\n& = 3^{k+1} M \\left( 3^{2k-1} M^2 -2^{s(k)} \\right)\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(r = k+1\\) \u306e\u3068\u304d\u3082 [A] \u306f\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":"\\(m , p\\) \u3092 \\(3\\) \u4ee5\u4e0a\u306e\u5947\u6570\u3068\u3057, \\(m\\) \u306f \\(p\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u3068\u3059\u308b. (1)\u3000\\((x-1)^{101}\\) \u306e\u5c55\u958b\u5f0f\u306b\u304a\u3051\u308b \\(x^2\\) \u306e\u9805\u306e\u4fc2\u6570\u3092\u6c42\u3081\u3088. (2)\u3000\\((p- … \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":[63],"tags":[143,68],"class_list":["post-399","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2012","tag-nagoya_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/399","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=399"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/399\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=399"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=399"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=399"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}