{"id":1018,"date":"2015-04-07T19:55:47","date_gmt":"2015-04-07T10:55:47","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1018"},"modified":"2021-11-04T08:55:12","modified_gmt":"2021-11-03T23:55:12","slug":"htb201301","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb201301\/","title":{"rendered":"\u4e00\u6a4b\u59272013\uff1a\u7b2c1\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(3p^3-p^2q-pq^2+3q^3 = 2013\\) \u3092\u6e80\u305f\u3059\u6b63\u306e\u6574\u6570 \\(p , q\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p>\r\n<hr \/>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p>\u4e0e\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n3 ( p^3 +q^3 ) -pq(p+q) & = 2013 \\\\\r\n\\text{\u2234} \\quad (p+q) ( 3p^2 -4pq +3q^2 ) & = 2013 \\quad ... [1]\r\n\\end{align}\\]\r\n\\(p+q = k\\) \uff08 \\(k\\) \u306f \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\uff09\u3068\u304a\u3044\u3066, [1]\u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n3p^2 -4p(k-p) +3(k-p)^2 & = \\dfrac{2013}{k} \\\\\r\n\\text{\u2234} \\quad 10p^2 -10kp +3k^2 -\\dfrac{2013}{k} & = 0 \\quad ... [2]\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u307f\u305f\u3059\u6574\u6570 \\(p\\) \u304c\u5b58\u5728\u3059\u308b\u306e\u3067, \u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D}{4} & = (5k)^2 -10 \\left( 3k^2 -\\dfrac{2013}{k} \\right) \\geqq 0 \\\\\r\n& -k^2 +\\dfrac{4026}{k} \\geqq 0 \\\\\r\n& \\text{\u2234} \\quad k^3 \\leqq 4046\r\n\\end{align}\\]\r\n\\(15^3 = 3375\\) , \\(16^3 = 4096\\) \u306a\u306e\u3067\r\n\\[\r\n1 \\leqq k \\leqq 15\r\n\\]\r\n\u3053\u308c\u3068, \\(2013 = 3 \\cdot 11 \\cdot 61\\) \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070, [1] \u3092\u307f\u305f\u3059 \\(k\\) \u306e\u5019\u88dc\u306f\r\n\\[\r\nk = 3 , 11\r\n\\]\r\n\u305d\u308c\u305e\u308c\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u8abf\u3079\u308b.<\/p>\r\n<ul>\r\n<li><p>\\(k = 3\\) \u306e\u3068\u304d, [2] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n10p^2 -30p +27 -671 & = 0 \\\\\r\n5p^2 -15p -322 & = 0 \\\\\r\n\\end{align}\\]\r\n\u3053\u308c\u306f, \u6574\u6570\u89e3\u3092\u3082\u305f\u305a\u4e0d\u9069.<\/p><\/li>\r\n<li><p>\\(k = 11\\) \u306e\u3068\u304d, [2] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n10p^2 -110p +363 -183 & = 0 \\\\\r\np^2 -11p -18 & = 0 \\\\\r\n(p-2)(p-9) & = 0 \\\\\r\n\\text{\u2234} \\quad p = 2 , 9 &\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, \\(q = 11-p\\) \u3082\u6574\u6570\u3067\u3042\u308b.<\/p><\/li>\r\n<\/ul>\r\n<p>\u3088\u3063\u3066, \u6c42\u3081\u308b\u89e3\u306f\r\n\\[\r\n( p , q ) = \\underline{( 2 , 9 ) , \\ ( 9 , 2 )}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(3p^3-p^2q-pq^2+3q^3 = 2013\\) \u3092\u6e80\u305f\u3059\u6b63\u306e\u6574\u6570 \\(p , q\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.","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":[108],"tags":[146,111],"class_list":["post-1018","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2013","tag-hitotsubashi","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1018","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=1018"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1018\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1018"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1018"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1018"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}