{"id":852,"date":"2013-08-03T13:43:18","date_gmt":"2013-08-03T04:43:18","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=852"},"modified":"2021-09-15T09:35:35","modified_gmt":"2021-09-15T00:35:35","slug":"ngr201303","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201303\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2013\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(k , m , n\\) \u306f\u6574\u6570\u3068\u3057, \\(n \\geqq 1\\) \u3068\u3059\u308b. \\({} _ {n}\\text{C} {} _ {k}\\) \u3092\u4e8c\u9805\u4fc2\u6570\u3068\u3057\u3066, \\(S_k(n) , T_m(n)\\) \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b.\r\n\\[\\begin{align}\r\nS_k(n) & = 1^k +2^k + \\cdots +n^k , \\ S_k(1) = 1 \\quad ( k \\geqq 0 \uff09 \\\\\r\nT_m(n) & = {} _ {m}\\text{C} {} _ {1} S_1(n) +{} _ {m}\\text{C} {} _ {2} S_2(n) + \\cdots +{} _ {m}\\text{C} {} _ {m-1} S_{m-1}(n) \\\\\r\n& = \\textstyle\\sum\\limits_{k=1}^{m-1} {} _ {m}\\text{C} {} _ {k} S_k(n) \\quad ( m \\geqq 2 )\r\n\\end{align}\\]\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(T_m(1)\\) \u3068 \\(T_m(2)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u4e00\u822c\u306e \\(n\\) \u306b\u5bfe\u3057\u3066 \\(T_m(n)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(p\\) \u304c \\(3\\) \u4ee5\u4e0a\u306e\u7d20\u6570\u306e\u3068\u304d, \\(S_k(p-1) \\ ( k = 1, 2, 3, \\cdots , p-2 )\\) \u306f \\(p\\) \u306e\u500d\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b .<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\r\nS_k(1) = 1 , \\ S_k(2) = 1+2^k\r\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits_{i=0}^m {} _ {m}\\text{C} {} _ {i} & = (1+1)^m = 2^m \\ , \\\\\r\n\\textstyle\\sum\\limits_{i=0}^m 2^i {} _ {m}\\text{C} {} _ {i} & = (2+1)^m = 3^m \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u3053\u308c\u3089\u3092\u7528\u3044\u308c\u3070\r\n\\[ \\begin{align}\r\nT_m(1) & = {} _ {m}\\text{C} {} _ {1} S_1(1) + \\cdots +{} _ {m}\\text{C} {} _ {m-1} S_{m-1}(1) \\\\\r\n& = \\textstyle\\sum\\limits_{i=1}^{m-1} {} _ {m}\\text{C} {} _ {i} \\\\\r\n& = \\textstyle\\sum\\limits_{i=0}^{m} {} _ {m}\\text{C} {} _ {i} -\\left( 1^m +1 \\right) \\\\\r\n& = \\underline{2^m-2} \\ , \\\\\r\nT_m(2) & = {} _ {m}\\text{C} {} _ {1} S_1(2) + \\cdots +{} _ {m}\\text{C} {} _ {m-1} S_{m-1}(2) \\\\\r\n& = \\textstyle\\sum\\limits_{i=1}^{m-1} {} _ {m}\\text{C} {} _ {i} +\\textstyle\\sum\\limits_{i=1}^{m-1} 2^{i} {} _ {m}\\text{C} {} _ {i} \\\\\r\n& = T_m(1) +\\textstyle\\sum\\limits_{i=0}^{m} 2^{i} {} _ {m}\\text{C} {} _ {i} -\\left( 2^m +1 \\right) \\\\\r\n& = \\underline{3^m-3} \\ .\r\n\\end{align} \\]\r\n

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

    (1)<\/strong> \u3068\u540c\u69d8\u306b\u8a08\u7b97\u3059\u308c\u3070\r\n\\[ \\begin{align}\r\nT_m(n) & = {} _ {m}\\text{C} {} _ {1} S_1(n) + \\cdots +{} _ {m}\\text{C} {} _ {m-1} S_{m-1}(n) \\\\\r\n& = \\textstyle\\sum\\limits_{j=1}^{n} \\textstyle\\sum\\limits_{i=1}^{m-1} j^{i} {} _ {m}\\text{C} {} _ {i}\\\\\r\n& = \\textstyle\\sum\\limits_{j=1}^{n} \\textstyle\\sum\\limits_{i=0}^{m} \\left\\{ j^{i} {} _ {m}\\text{C} {} _ {i} -\\left( j^m +1 \\right) \\right\\} \\\\\r\n& = \\textstyle\\sum\\limits_{j=1}^{n} \\left\\{ (j+1)^m - j^m -1 \\right\\} \\\\\r\n& = \\left\\{ (n+1)^m - n^m \\right\\} + \\cdots + ( 2^m -1 ) -n\\\\\r\n& = \\underline{(n+1)^m -(n+1)} \\ .\r\n\\end{align} \\]\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u306b\u3064\u3044\u3066, \\(n = p-1\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nT_{m}(p-1) = p^{p-1}-p \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u300c \\(T_m(p-1)\\) \u306f \\(p\\) \u3067\u5272\u5207\u308c\u308b. \u300d ... [1]\r\n\u307e\u305f, \\(1 \\leqq i \\leqq p-1 , \\ 1 \\leqq j \\leqq i\\) \u3092\u307f\u305f\u3059\u81ea\u7136\u6570 \\(i , j\\) \u306b\u5bfe\u3057\u3066, \\(i !\\) \u304c \\(p\\) \u3067\u5272\u5207\u308c\u306a\u3044\u306e\u3067, \u300c \\({} _ {i}\\text{C} {} _ {j} = \\dfrac{i !}{i! ( i-j )!}\\) \u306f, \\(p\\) \u3067\u5272\u5207\u308c\u306a\u3044. \u300d ... [2]\r\n\u4ee5\u4e0b\u3067\u306f, \u300c \\(S_{k}(p-1)\\) \u304c \\(p\\) \u3067\u5272\u5207\u308c\u308b. \u300d ... [A] \u304c \\(k = 1 , 2 , \\cdots , p-2\\) \u3067\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\\(k = 1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nS_1(p-1) & = 1 +2 + \\cdots +(p-1) \\\\\r\n& = \\dfrac{p(p-1)}{2} \\ .\r\n\\end{align}\\]\r\n\\(p\\) \u306f \\(3\\) \u4ee5\u4e0a\u306e\u7d20\u6570\u306a\u306e\u3067, \\(\\dfrac{p-1}{2}\\) \u306f\u6574\u6570\u3068\u306a\u308a, \\(S_1(p-1)\\) \u306f \\(p\\) \u3067\u5272\u5207\u308c\u308b.
      \r\n\u3064\u307e\u308a, \\(k = 1\\) \u306e\u3068\u304d, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(k = 1 , \\cdots , \\ell \\ (1 \\leqq \\ell \\leqq p-3 )\\) \u306e\u3068\u304d
      \r\n[A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nT_{\\ell +2} & (p-1) \\\\\r\n& = \\textstyle\\sum\\limits_{i=1}^{\\ell +1} {} _ {\\ell+2}\\text{C} {} _ {i} S _ {i}(p-1) \\\\\r\n& = \\textstyle\\sum\\limits_{i=1}^{\\ell} {} _ {\\ell+2}\\text{C} {} _ {i} S _ {i}(p-1) +{} _ {\\ell+2}\\text{C} {} _ {\\ell +1} S _ {\\ell +1}(p-1) \\ .\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\n{} _ {\\ell+2}\\text{C} {} _ {\\ell +1} & S _ {\\ell +1}(p-1) = \\\\\r\n& T _ {\\ell +2} (p-1) -\\textstyle\\sum\\limits_{i=1}^{\\ell} {} _ {\\ell+2}\\text{C} {} _ {i} S _ {i}(p-1) \\ .\r\n\\end{align} \\]\r\n[1] \u3068\u4eee\u5b9a\u3088\u308a\u53f3\u8fba\u306f \\(p\\) \u3067\u5272\u5207\u308c\u308b.
      \r\n\u3057\u305f\u304c\u3063\u3066, [2] \u3088\u308a \\(S_{\\ell +1}(p-1)\\) \u304c \\(p\\) \u3067\u5272\u5207\u308c\u308b.
      \r\n\u3064\u307e\u308a, \\(k = \\ell +1\\) \u306e\u3068\u304d, [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":"\\(k , m , n\\) \u306f\u6574\u6570\u3068\u3057, \\(n \\geqq 1\\) \u3068\u3059\u308b. \\({} _ {n}\\text{C} {} _ {k}\\) \u3092\u4e8c\u9805\u4fc2\u6570\u3068\u3057\u3066, \\(S_k(n) , T_m(n)\\) \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b … \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":[98],"tags":[143,111],"class_list":["post-852","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2013","tag-nagoya_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/852","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=852"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/852\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=852"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=852"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}