{"id":1012,"date":"2015-04-04T23:37:53","date_gmt":"2015-04-04T14:37:53","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1012"},"modified":"2021-03-09T16:35:16","modified_gmt":"2021-03-09T07:35:16","slug":"tkr201405","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201405\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2014\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(r\\) \u3092 \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3057, \u6570\u5217 \\(\\{ a _ n \\}\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \r\n\\[\\begin{align}\r\n& a _ 1 = r , \\quad a _ 2 = r+1 , \\\\\r\n& a _ {n+2} = a _ {n+1} ( a _ n +1 ) \\quad ( n = 1 , 2 , 3 , \\cdots )\r\n\\end{align}\\]\r\n\u307e\u305f, \u7d20\u6570 \\(p\\) \u3092 \\(1\\) \u3064\u3068\u308a, \\(a _ n\\) \u3092 \\(p\\) \u3067\u5272\u3063\u305f\u4f59\u308a\u3092 \\(b _ n\\) \u3068\u3059\u308b. \u305f\u3060\u3057, \\(0\\) \u3092 \\(p\\) \u3067\u5272\u3063\u305f\u4f59\u308a\u306f \\(0\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057, \\(b _ {n+2}\\) \u306f \\(b _ {n+1} ( b _ n +1 )\\) \u3092 \\(p\\) \u3067\u5272\u3063\u305f\u4f59\u308a\u3068\u4e00\u81f4\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(r=2\\) , \\(p=17\\) \u306e\u5834\u5408\u306b, \\(10\\) \u4ee5\u4e0b\u306e\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(b _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u3042\u308b \\(2\\) \u3064\u306e\u76f8\u7570\u306a\u308b\u81ea\u7136\u6570 \\(n , m\\) \u306b\u5bfe\u3057\u3066, \r\n\\[\r\nb _ {n+1} = b _ {m+1} \\gt 0 , \\quad b _ {n+2} = b _ {m+2}\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3068\u3059\u308b. \u3053\u306e\u3068\u304d \\(b _ n = b _ m\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  4. (4)<\/strong>\u3000\\(a _ 2 , a _ 3 , a _ 4 , \\cdots\\) \u306b \\(p\\) \u3067\u5272\u308a\u5207\u308c\u308b\u6570\u304c\u73fe\u308c\u306a\u3044\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(a _ 1\\) \u3082 \\(p\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044\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

    \\(a _ n\\) \u3092 \\(p\\) \u3067\u5272\u3063\u305f\u5546\u3092 \\(q _ n\\) \u3068\u304a\u3051\u3070, \\(a _ n = p q _ n +b _ n\\) \u3068\u8868\u305b\u308b.
    \r\n\\(b _ {n+1} ( b _ n +1 )\\) \u3092 \\(p\\) \u3067\u5272\u3063\u305f\u5546\u3092 \\(q'\\) , \u4f59\u308a\u3092 \\(r'\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\na _ {n+2} & = ( p q _ {n+1} +b _ {n+1} ) ( p q _ n +b _ n +1 ) \\\\\r\n& = p \\left\\{ p q _ {n+1} q _ n +q _ {n+1} ( b _ n +1 ) +q _ n b _ {n+1} +q' \\right\\} +r'\r\n\\end{align}\\]\r\n\u3064\u307e\u308a, \\(a _ {n+2}\\) \u3092 \\(p\\) \u3067\u5272\u3063\u305f\u4f59\u308a\u306f \\(r'\\) \u3067\u3042\u308b\u304b\u3089\r\n\\[\r\nb _ {n+2} = r'\r\n\\]\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\r\nb _ 1 = \\underline{2} , \\quad b _ 2 = \\underline{3}\r\n\\]\r\n

    (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nb _ 2 ( b _ 1 +1 ) & = 3 ( 2+1 ) =9 & \\text{\u2234} \\quad b _ 3 = \\underline{9} \\\\\r\nb _ 3 ( b _ 2 +1 ) & = 9 ( 3+1 ) = 36 = 17 \\cdot 2 +2 & \\text{\u2234} \\quad b _ 4 = \\underline{2} \\\\\r\nb _ 4 ( b _ 3 +1 ) & = 2 ( 9+1 ) = 20 = 17 +3 & \\text{\u2234} \\quad b _ 5 = \\underline{3}\r\n\\end{align}\\]\r\n\u3042\u3068\u306f, \u7e70\u8fd4\u3057\u3068\u306a\u308b\u306e\u3067\r\n\\[\r\nb _ 6 = \\underline{9} , \\ b _ 7 = \\underline{2} , \\ b _ 8 = \\underline{3} , \\ b _ 9 = \\underline{9} , \\ b _ {10} = \\underline{2}\r\n\\]\r\n

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

    \u6761\u4ef6\u3088\u308a, \\(b _ {n+1} = b _ {m+1} = k \\gt 0\\) , \\(b _ {n+2} = b _ {m+2} = \\ell\\) ... [1] \u3068\u304a\u304f.
    \r\n\u3042\u308b\u6574\u6570 \\(c _ n , c _ m\\) \u3092\u3068\u308c\u3070\r\n\\[\\begin{align}\r\nb _ {n+2} & = b _ {n+1} ( b _ n +1 ) -p c _ n \\\\\r\nb _ {m+2} & = b _ {m+1} ( b _ m +1 ) -p c _ m\r\n\\end{align}\\]\r\n\u3068\u8868\u305b\u308b.
    \r\n[1] \u3092\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\n\\ell & = k ( b _ n +1 ) -p c _ n \\\\\r\n\\ell & = k ( b _ m +1 ) -p c _ m\r\n\\end{align}\\]\r\n\u8fba\u3005\u3092\u5f15\u3044\u3066, \u6574\u7406\u3059\u308b\u3068\r\n\\[\r\nk \\left( b _ n -b _ m \\right) = p \\left( c _ n -c _ m \\right)\r\n\\]\r\n\u53f3\u8fba\u306f \\(p\\) \u3067\u5272\u308a\u5207\u308c\u308b\u306e\u3067, \u5de6\u8fba\u3082 \\(p\\) \u3067\u5272\u308a\u5207\u308c\u308b.
    \r\n\\(0 \\lt k \\lt p\\) \u306a\u306e\u3067, \\(b _ n -b _ m\\) \u304c \\(p\\) \u3067\u5272\u308a\u5207\u308c\u308b, \u3064\u307e\u308a \\(b _ n\\) \u3068 \\(b _ m\\) \u306f \\(p\\) \u3067\u5272\u3063\u305f\u4f59\u308a\u304c\u7b49\u3057\u3044.
    \r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

    \\(n \\geqq 2\\) \u306b\u304a\u3044\u3066, \\(a _ n\\) \u304c \\(p\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u306a\u3089\u3070\r\n\\[\r\n1 \\leqq b _ n \\leqq p-1\r\n\\]\r\n\u306a\u306e\u3067, \\(b _ n\\) \u3068 \\(b _ {n+1}\\) \u306e\u7d44\u5408\u305b\u306f \\({} _ {p-1} \\text{P} _ 2\\) \u901a\u308a\u3057\u304b\u306a\u3044.
    \r\n\u3057\u305f\u304c\u3063\u3066, \\(n \\gt m\\) \u3068\u3057\u3066, \\(n\\) \u3092\u5341\u5206\u306b\u5927\u304d\u304f\u3068\u308c\u3070, \r\n\\[\r\nb _ {n+1} = b _ {m+1} \\gt 0 , \\ b _ {n+2} = b _ {m+2}\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b \\(m\\) \u3068 \\(n\\) \u306e\u7d44\u304c\u5b58\u5728\u3059\u308b.
    \r\n\u3053\u306e\u3068\u304d, (3)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nb _ n = b _ m\r\n\\]\r\n\u3053\u308c\u3092\u7e70\u8fd4\u3057\u7528\u3044\u308c\u3070\r\n\\[\r\nb _ 1 = b _ {n-m+1} \\gt 0\r\n\\]\r\n\u3088\u3063\u3066, \\(a _ 1\\) \u3082 \\(p\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(r\\) \u3092 \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3057, \u6570\u5217 \\(\\{ a _ n \\}\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \\[\\begin{align} & a _ 1 = r , \\quad a _ 2 = r+1 , \\\\ & 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":[113],"tags":[139,112],"class_list":["post-1012","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2014","tag-tokyo_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1012","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=1012"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1012\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1012"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1012"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1012"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}