\\(ad -bc = 1 , \\ a \\gt 0\\) \u3092\u6e80\u305f\u3059\u6574\u6570 \\(a , b , c , d\\) \u3092\u8003\u3048\u308b. \u884c\u5217\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} 6 & 10 \\\\ 10 & 17 \\end{array} \\right) , \\quad B = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 2 \\end{array} \\right) , \\\\\r\nM & = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) , \\quad N = \\left( \\begin{array}{cc} a & c \\\\ b & d \\end{array} \\right) \\ .\r\n\\end{align}\\]\r\n\u304c \\(NA = BM^{-1}\\) \u3092\u6e80\u305f\u3059\u3068\u304d, \u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057, \\(M^{-1}\\) \u306f \\(M\\) \u306e\u9006\u884c\u5217\u3092\u8868\u3059.<\/p>\r\n
(1)<\/strong>\u3000\\(6a^2+20ac+17c^2\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(2a^2+b^2\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(a , b , c , d\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (4)<\/strong>\u3000\\(6x^2+20xy+17y^2 = 59\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570 \\(x , y\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\n\\left\\{ \\begin{array}{l} X = dx-by \\\\ Y = -cx+ay \\end{array} \\right. \\ .\r\n\\]\r\n\u3068\u304a\u304f\u3068\u304d, \\(X^2+2Y^2\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (5)<\/strong>\u3000\\(6x^2+20xy+17y^2 = 59\\) \u3092\u6e80\u305f\u3059\u6574\u6570\u306e\u7d44 \\((x,y)\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6761\u4ef6\u3088\u308a, \\(NAM = B\\) .\r\n\\[\\begin{align}\r\nNAM & = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) \\left( \\begin{array}{cc} 6 & 10 \\\\ 10 & 17 \\end{array} \\right) \\left( \\begin{array}{cc} a & c \\\\ b & d \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 6a+10b & 10a+17b \\\\ 6c+10d & 10c+17d \\end{array} \\right) \\left( \\begin{array}{cc} a & c \\\\ b & d \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 6a^2+20ab+17b^2 & 6ac+10(ad+bc)+17bd \\\\ 6ac+10(ad+bc)+17bd & 6c^2+20cd+17d^2 \\end{array} \\right) \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\((1,1)\\) \u6210\u5206\u3092\u6bd4\u8f03\u3057\u3066\r\n\\[\r\n6a^2+20ab+17b^2 = \\underline{1} \\quad ... [1] \\ .\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\(ad -bc = 1\\) \u3088\u308a\r\n\\[\r\nM^{-1} = \\left( \\begin{array}{cc} d & -b \\\\ -c & a \\end{array} \\right) , \\ N^{-1} = \\left( \\begin{array}{cc} d & -c \\\\ -b & a \\end{array} \\right) \\ .\r\n\\]\r\n\u307e\u305f, \u6761\u4ef6\u3088\u308a, \\(A = N^{-1}BM^{-1}\\) .\r\n\\[\\begin{align}\r\nN^{-1}BM^{-1} & = \\left( \\begin{array}{cc} d & -c \\\\ -b & a \\end{array} \\right) \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 2 \\end{array} \\right) \\left( \\begin{array}{cc} d & -c \\\\ -b & a \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} d & -2c \\\\ -b & 2a \\end{array} \\right) \\left( \\begin{array}{cc} d & -b \\\\ -c & a \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 2c^2+d^2 & -bd-2ac \\\\ -bd-2ac & 2a^2+b^2 \\end{array} \\right) \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\((2,2)\\) \u6210\u5206\u3092\u6bd4\u8f03\u3057\u3066\r\n\\[\r\n2a^2+b^2 = \\underline{17} \\quad ... [2] \\ .\r\n\\]\r\n (3)<\/strong><\/p>\r\n (1)<\/strong> (2)<\/strong> \u306b\u3064\u3044\u3066, \u4ed6\u306e\u6210\u5206\u306b\u3064\u3044\u3066\u3082\u6bd4\u8f03\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n6c^2+20cd+17d^2 & = 2 \\quad ... [3] , \\\\\r\n6ac+10(ad+bc)+17bd & = 0 \\quad ... [4] , \\\\\r\n2c^2+d^2 & = 6 \\quad ... [5] , \\\\\r\n-bd -2ac & = 10 \\quad ... [6] \\ .\r\n\\end{align}\\]\r\n\\(a , b, c , d\\) \u306f\u3059\u3079\u3066\u6574\u6570\u306a\u306e\u3067, [5] \u3088\u308a\r\n\\[\r\n(c,d) = ( \\pm 1 , \\pm 2 ) \\ .\r\n\\]\r\n\u540c\u69d8\u306b\u8003\u3048\u3066, \\(a \\gt 0\\) \u3068 [2] \u3088\u308a\r\n\\[\r\n(a,b) = ( 2 , \\pm 3 ) \\ .\r\n\\]\r\n\u3053\u306e\u3046\u3061, \\(ad -bc = 1\\) \u3092\u307f\u305f\u3059\u7d44\u306f, \\(|ad| = 4\\) , \\(|bc| = 3\\) \u306a\u306e\u3067\r\n\\[\r\n(a,b,c,d) = ( 2 , \\pm 3 , \\pm 1 , 2 ) , ( 2 , \\mp 3 , \\mp 1 , 2 ) \\quad ( \\text{\u8907\u53f7\u540c\u9806} ) \\ .\r\n\\]\r\n\u3055\u3089\u306b, [6] \u3092\u307f\u305f\u3059\u7d44\u306f\r\n\\[\r\n(a,b,c,d) = ( 2 , -3 , -1 , 2 ) \\ .\r\n\\]\r\n\u3053\u308c\u306f [3] [4] \u3082\u307f\u305f\u3057\u3066\u3044\u308b. (4)<\/strong><\/p>\r\n \u6761\u4ef6\u3088\u308a\r\n\\[\r\n\\left( \\begin{array}{c} X \\\\ Y \\end{array} \\right) = M^{-1} \\left( \\begin{array}{c} x \\\\ y \\end{array} \\right) \\ .\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\left( \\begin{array}{c} x \\\\ y \\end{array} \\right) = M \\left( \\begin{array}{c} X \\\\ Y \\end{array} \\right) = \\left( \\begin{array}{c} 2X-3Y \\\\ -X+2Y \\end{array} \\right) \\quad ... [7] \\ .\r\n\\]\r\n\u3053\u308c\u3092 \\(6x^2+20xy+17y^2 = 59\\) \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n6 (2X-3Y)^2 +20(2X-3Y)(-X+2Y) +17(-X+2Y)^2 & = 59 \\\\\r\n24X^2-72XY+54Y^2 -40X^2+140XY \\qquad & \\\\\r\n-120Y^2 +17X^2-68XY+68Y^2 & = 59 \\\\\r\n\\text{\u2234} \\quad X^2+2Y^2 = \\underline{59} \\quad ... [8] & \\ .\r\n\\end{align}\\]\r\n (5)<\/strong><\/p>\r\n \\(x , y\\) \u304c\u6574\u6570\u306a\u3089\u3070, [7] \u3088\u308a \\(X , Y\\) \u3082\u6574\u6570\u3067\u3042\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\u3088\u3063\u3066\r\n\\[\r\n(a,b,c,d) =\\underline{( 2 , -3 , -1 , 2 )} \\ .\r\n\\]\r\n
\r\n[8] \u3088\u308a, \\(0 \\leqq Y^2 \\leqq \\dfrac{59}{2}\\) \u306a\u306e\u3067, [8] \u306e\u89e3\u306f\r\n\\[\r\n(X,Y) = ( \\pm 3 , \\pm 5 ) \\ .\r\n\\]\r\n\u3088\u3063\u3066, [7] \u3088\u308a\u6c42\u3081\u308b\u89e3\u306f\r\n\\[\r\n(x,y) = \\underline{( \\pm 9 , \\mp 7 ) , ( \\pm 21 , \\mp 13 )} \\quad ( \\text{\u8907\u53f7\u540c\u9806} ) \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(ad -bc = 1 , \\ a \\gt 0\\) \u3092\u6e80\u305f\u3059\u6574\u6570 \\(a , b , c , d\\) \u3092\u8003\u3048\u308b. \u884c\u5217 \\[\\begin{align} A & = \\left( \\begin{array}{cc} 6 … \u7d9a\u304d\u3092\u8aad\u3080