\u81ea\u7136\u6570 $a , b$ \u306f\u3069\u3061\u3089\u3082 $3$ \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u304c, $a^3 +b^3$ \u306f $81$ \u3067\u5272\u308a\u5207\u308c\u308b.\r\n\u3053\u306e\u3088\u3046\u306a $a , b$ \u306e\u7d44 $( a , b )$ \u306e\u3046\u3061, $a^2 +b^2$ \u306e\u5024\u3092\u6700\u5c0f\u306b\u3059\u308b\u3082\u306e\u3068,\r\n\u305d\u306e\u3068\u304d\u306e $a^2 +b^2$ \u306e\u5024\u3092\u6c42\u3081\u3088. <\/p>\r\n
$$\r\n\\underline{a^3+b^3} _ {\\Large\\maru{1}} = \\underline{(a+b)^3} _ {\\Large\\maru{2}} -\\underline{3ab (a+b)} _ {\\Large\\maru{3}}\r\n$$\r\n$\\maru{1}$ $\\maru{3}$ \u304c $3$ \u306e\u500d\u6570\u306a\u306e\u3067, $\\maru{2}$ \u3082 $3$ \u306e\u500d\u6570\u3067\u3042\u308b.
\r\n\u3064\u307e\u308a, $a+b$ \u304c $3$ \u306e\u500d\u6570\u3068\u306a\u308b\u306e\u3067, $\\maru{2}$ \u306f $3^3 = 27$ \u306e\u500d\u6570\u3067\u3042\u308b.
\r\n\u6761\u4ef6\u3088\u308a, $\\maru{1}$ \u3082 $27$ \u306e\u500d\u6570\u306a\u306e\u3067, $\\maru{3}$ \u3082 $27$ \u306e\u500d\u6570\u3067\u3042\u308b.
\r\n\u6761\u4ef6\u3088\u308a, \u300c $a , b$ \u306f\u3068\u3082\u306b $3$ \u306e\u500d\u6570\u3067\u306f\u306a\u3044 ... $\\maru{4}$ \u300d\u306e\u3067, $a+b$ \u306f $\\dfrac{27}{3} = 9$ \u306e\u500d\u6570\u3067\u3042\u308b.
\r\n\u3057\u305f\u304c\u3063\u3066, $\\maru{2}$ \u306f $9^2 = 81$ \u306e\u500d\u6570\u3067\u3042\u308a,\r\n\u6761\u4ef6\u3088\u308a, $\\maru{1}$ \u3082 $81$ \u306e\u500d\u6570\u3067\u3042\u308b\u304b\u3089, $\\maru{3}$ \u3082 $81$ \u306e\u500d\u6570\u3067\u3042\u308b.
\r\n\u3086\u3048\u306b, \u518d\u5ea6 $\\maru{4}$ \u3088\u308a, $a+b$ \u306f $\\dfrac{81}{3} = 27$ \u306e\u500d\u6570\u3067\u3042\u308b.
\r\n$a+b = 27k$ \uff08 $k$ \u306f\u81ea\u7136\u6570 \uff09\u3068\u304a\u304f.
\r\n\u5bfe\u79f0\u6027\u304b\u3089 $a \\leqq b$ \u3068\u3057\u3066\u3082\u3088\u3044.\r\n$$\\begin{align}\r\na^2+b^2 & = a^2 +(27k-a)^2 \\\\\r\n& = 2a^2 -54ka +27^2k^2 \\\\\r\n& = 2 \\left( a -\\dfrac{27k}{2} \\right)^2 +\\dfrac{27^2k^2}{2}\r\n\\end{align}$$\r\n\u3057\u305f\u304c\u3063\u3066, $a^2+b^2$ \u304c\u6700\u5c0f\u3068\u306a\u308b\u306e\u306f, $k=1$ \u306e\u3068\u304d.
\r\n\u3053\u306e\u3068\u304d, $a$ \u306f\u81ea\u7136\u6570\u306a\u306e\u3067\r\n$$\r\na = 13 \\ \\text{\u306e\u3068\u304d, \u6700\u5c0f\u5024} \\ 13^2+14^2 = 365\r\n$$\r\n\u3092\u3068\u308b.
\r\n\u3088\u3063\u3066, $a^2+b^2$ \u306f\r\n$$\r\n(a,b) = \\underline{( 13 , 14 ) , ( 14 , 13 )}\r\n$$\r\n\u306e\u3068\u304d, \u6700\u5c0f\u5024 $\\underline{365}$ \u3092\u3068\u308b. <\/p>","protected":false},"excerpt":{"rendered":"\u81ea\u7136\u6570 $a , b$ \u306f\u3069\u3061\u3089\u3082 $3$ \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u304c, $a^3 +b^3$ \u306f $81$ \u3067\u5272\u308a\u5207\u308c\u308b. \u3053\u306e\u3088\u3046\u306a $a , b$ \u306e\u7d44 $( a , b )$ \u306e\u3046\u3061, $a^2 +b^2$ \u306e\u5024\u3092\u6700\u5c0f\u306b\u3059 … \u7d9a\u304d\u3092\u8aad\u3080