\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(\\sqrt{2}\\) \u3068 \\(\\sqrt[3]{3}\\) \u304c\u7121\u7406\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(p , q , \\sqrt{2} +\\sqrt[3]{3} q\\) \u304c\u3059\u3079\u3066\u6709\u7406\u6570\u3067\u3042\u308b\u3068\u3059\u308b. \u305d\u306e\u3068\u304d, \\(p = q = 0\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(p\\) \u3092\u7d20\u6570, \\(k\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3057\u3066\r\n\\[\r\n\\sqrt[k]{p} \\ \\text{\u306f\u7121\u7406\u6570} \\quad ... [ \\text{A} ] \\ .\r\n\\]\r\n\u3067\u3042\u308b\u3053\u3068\u3092\u80cc\u7406\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059. (2)<\/strong><\/p>\r\n \\(r = \\sqrt{2} p +\\sqrt[3]{3} q\\) \u3068\u304a\u304f. 1*<\/strong>\u3000\\(3r^2 +2p^2 = 0\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(p = 0\\) \u306e\u3068\u304d \u3088\u3063\u3066, \u3044\u305a\u308c\u306e\u5834\u5408\u306b\u3082\r\n\\[\r\np = q = r = 0 \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(\\sqrt{2}\\) \u3068 \\(\\sqrt[3]{3}\\) \u304c\u7121\u7406\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b. (2)\u3000\\(p , q , \\sqrt{2} +\\sqrt[3]{3} q\\) \u304c\u3059\u3079\u3066\u6709\u7406\u6570\u3067\u3042\u308b … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(\\sqrt[k]{p} = \\dfrac{m}{n}\\) \uff08 \\(m , n\\) \u306f\u4e92\u3044\u306b\u7d20 ... [1] \uff09\u3068\u304a\u3044\u3066, \u7121\u7406\u6570\u3067\u3042\u308b\u3068\u4eee\u5b9a\u3059\u308b.
\r\n\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\nn^k p = m^k \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(m = p m'\\) \uff08 \\(m'\\) \u306f\u81ea\u7136\u6570\uff09\u3068\u8868\u305b\u308b.
\r\n\u3053\u308c\u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nn^k p & = p^k {m'}^k \\\\\r\n\\text{\u2234} \\quad n^k & = p^{k-1} {m'}^k \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n = p n'\\) \uff08 \\(n'\\) \u306f\u81ea\u7136\u6570\uff09\u3068\u8868\u305b\u308b\u304c, \u3053\u308c\u306f [1] \u306b\u77db\u76fe\u3059\u308b.
\r\n\u3088\u3063\u3066, [A] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c, \\(\\sqrt{2} , \\sqrt[3]{3}\\) \u306f\u3068\u3082\u306b\u7121\u7406\u6570\u3067\u3042\u308b\u3068\u3044\u3048\u308b.<\/p>\r\n
\r\n\u3053\u308c\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\left( r -\\sqrt{2} p \\right)^3 = 3 q^3 & \\\\\r\nr^3 -3 \\sqrt{2} r^2 p +6r p^2 -2 \\sqrt{2} p^3 = 3 q^3 & \\\\\r\n\\text{\u2234} \\quad r^3 +6rp^2 -3q^3 -\\sqrt{2} p ( 3r^2 +2p^2 ) & = 0 \\ .\r\n\\end{align}\\]\r\n\\(p , q , r\\) \u306f\u3059\u3079\u3066\u6709\u7406\u6570\u3067, \\(\\sqrt{2}\\) \u306f\u7121\u7406\u6570\u306a\u306e\u3067\r\n\\[\r\n\\left\\{ \\begin{array}{ll} r^3 +6rp^2 -3q^3= 0 & ... [2] \\\\ p ( 3r^2 +2p^2 ) = 0 & ... [3] \\end{array} \\right. \\ .\r\n\\]\r\n[3] \u304b\u3089, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n\r\n
\r\n\\[\r\np = r = 0 \\ .\r\n\\]\r\n[3] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n-3q^3 & = 0 \\\\\r\n\\text{\u2234} \\quad q & = 0 \\ .\r\n\\end{align}\\]<\/li>\r\n
\r\n[3] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\r\nr^3 = 3 q^3 \\ .\r\n\\]\r\n\\(q \\neq 0\\) \u3068\u4eee\u5b9a\u3059\u308c\u3070, \u3053\u308c\u3092\u5909\u5f62\u3057\u3066\r\n\\[\\begin{align}\r\n\\dfrac{r^3}{q^3} & = 3 \\\\\r\n\\text{\u2234} \\quad \\dfrac{r}{q} & = \\sqrt[3]{3} \\ .\r\n\\end{align}\\]\r\n\\(\\dfrac{r}{q}\\) \u306f\u6709\u7406\u6570, \\(\\sqrt[3]{3}\\) \u306f\u7121\u7406\u6570\u306a\u306e\u3067, \u77db\u76fe\u3059\u308b.
\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nq = r = 0 \\ .\r\n\\]<\/li>\r\n<\/ol>\r\n