{"id":2032,"date":"2021-12-15T00:09:41","date_gmt":"2021-12-14T15:09:41","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=2032"},"modified":"2021-12-15T00:09:41","modified_gmt":"2021-12-14T15:09:41","slug":"wsr202102","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr202102\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52021\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u6574\u5f0f \\(f(x) = x^4 -x^2 +1\\) \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(x^6\\) \u3092 \\(f(x)\\) \u3067\u5272\u3063\u305f\u3068\u304d\u306e\u4f59\u308a\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(x^{2021}\\) \u3092 \\(f(x)\\) \u3067\u5272\u3063\u305f\u3068\u304d\u306e\u4f59\u308a\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u81ea\u7136\u6570 \\(n\\) \u304c \\(3\\) \u306e\u500d\u6570\u3067\u3042\u308b\u3068\u304d, \\(( x^2 -1 )^n -1\\) \u304c \\(f(x)\\) \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

    \\[\r\nx^6 +1 = ( x^2 +1 ) ( x^4 -x^2 +1 )\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nx^6 = ( x^2 +1 ) f(x) -1\r\n\\]\r\n\u3088\u3063\u3066, \u4f59\u308a\u306f\r\n\\[\r\n\\underline{-1}\r\n\\]\r\n

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

    \\(2021 = 6 \\cdot 366 +5\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nx^{2021} & = x^5 \\left\\{ ( x^2 +1 ) f(x) -1 \\right\\}^{366} \\\\\r\n& = x^5 \\left\\{ P(x) f(x) +(-1)^{336} \\right\\} \\\\\r\n& = \\left\\{ x^5 P(x) +1 \\right\\} f(x) +x^3 -x\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u4f59\u308a\u306f\r\n\\[\r\n\\underline{x^3 -x}\r\n\\]\r\n\u305f\u3060\u3057\r\n\\[\r\nP(x) = \\textstyle\\sum\\limits _ {k=0}^{335} {} _ {336} \\text{C}{} _ {k} (-1)^k ( x^2 +1 )^{336-k} \\left\\{ f(x) \\right\\}^{335-k}\r\n\\]\r\n\u3068\u304a\u3044\u305f.<\/p>\r\n

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

    \u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(n=3\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n( x^2 -1 )^3 -1 & = x^6 -3x^4 +3x^2 -2 \\\\\r\n& = ( x^2 -2 ) f(x)\r\n\\end{align}\\]\r\n\u3067, \\(f(x)\\) \u3067\u5272\u308a\u5207\u308c\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = 3m\\) \uff08\\(m\\) \u306f\u81ea\u7136\u6570\uff09\u306e\u3068\u304d\r\n\\[\r\n( x^2 -1 )^{3m} -1 = Q(x) f(x)\r\n\\]\r\n\u3068, \\(f(x)\\) \u3067\u5272\u308a\u5207\u308c\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n( x^2 -1 )^{3m+3} -1 & = ( x^2 -1 )^3 ( x^2 -1 )^{3m} -1 \\\\\r\n& = \\left\\{ ( x^2 -2 ) f(x) +1 \\right\\} \\left\\{ Q(x) f(x) +1 \\right\\} -1 \\\\\r\n& = R(x) f(x) +1 -1 \\\\\r\n& = R(x) f(x)\r\n\\end{align}\\]\r\n\u3067, \\(n = 3m+3\\) \u306e\u3068\u304d\u3082, \\(f(x)\\) \u3067\u5272\u308a\u5207\u308c\u308b.
      \r\n\u305f\u3060\u3057\r\n\\[\r\nR(x) = \\left\\{ ( x^2 -2 ) f(x) +1 \\right\\} Q(x) +( x^2 -2 )\r\n\\]\r\n\u3068\u304a\u3044\u305f.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong> \u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u6574\u5f0f \\(f(x) = x^4 -x^2 +1\\) \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\\(x^6\\) \u3092 \\(f(x)\\) \u3067\u5272\u3063\u305f\u3068\u304d\u306e\u4f59\u308a\u3092\u6c42\u3081\u3088. (2)\u3000\\(x^{2021}\\) \u3092 \\(f(x)\\) \u3067\u5272 … \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":[177],"tags":[147,165],"class_list":["post-2032","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2021","tag-waseda_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/2032","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=2032"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/2032\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=2032"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=2032"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=2032"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}