{"id":2070,"date":"2025-07-29T20:38:45","date_gmt":"2025-07-29T11:38:45","guid":{"rendered":"https:\/\/math-friend.com\/?p=2070"},"modified":"2025-08-01T10:11:21","modified_gmt":"2025-08-01T01:11:21","slug":"%e3%80%90%e5%90%8d%e5%8f%a4%e5%b1%8b%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e5%b9%b3%e6%96%b9%e5%b7%ae%e3%81%ae%e5%bd%a2%e3%81%a7%e6%95%b4%e6%95%b0%e3%81%ae%e7%b5%84%e3%82%92%e5%85%a8%e3%81%a6","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=2070","title":{"rendered":"\u3010\u540d\u53e4\u5c4b\u5927\u5b66\u5165\u8a66\u3011\u5e73\u65b9\u5dee\u306e\u5f62\u3067\u6574\u6570\u306e\u7d44\u3092\u5168\u3066\u6c42\u3081\u308b\u554f\u984c(2025)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\u6574\u6570\\(a\\), \\(b\\), \\(c\\)\u306b\u5bfe\u3057, \u6b21\u306e\u6761\u4ef6\u3092\u8003\u3048\u308b.
$$
(\\ast)\\,\\,\\,\\, a\\geq b\\geq 0\\,\\,\\,\u304b\u3064\\,\\,\\,a^2-b^2=c
$$\u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.
(1) \\(c=24, 25, 26\\)\u306e\u5404\u5834\u5408\u306b\u304a\u3044\u3066, \u6761\u4ef6\\((\\ast)\\)\u3092\u6e80\u305f\u3059\u6574\u6570\u306e\u7d44\\((a, b)\\)\u3092\u5168\u3066\u6c42\u3081\u3088.
(2) \\(c=4p^{2n}\\)\u3068\u3059\u308b\u3068\u304d, \u6761\u4ef6\\((\\ast)\\)\u3092\u6e80\u305f\u3059\u6574\u6570\u306e\u7d44\\((a, b)\\)\u3092\u5168\u3066\u6c42\u3081\u3088.
(2025 \u540d\u53e4\u5c4b\u5927\u5b66 \u6587\u7cfb [2])<\/span><\/p>\n\n\n\n

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

(1) \\(a^2-b^2=c\\)\u306e\u5de6\u8fba\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u56e0\u6570\u5206\u89e3\u3059\u308b.
$$
(a+b)(a-b)=c
$$\u3053\u3053\u3067, \\(a\\geq b \\geq 0\\)\u3088\u308a, \\(a+b\\geq a-b\\geq 0\\)\u3067\u3042\u308b. \u307e\u305f, \\(a+b=k\\), \\(a-b=l\\)\u3068\u304a\u304f\u3068,
$$
\\begin{align}
a&=\\frac{k+l}{2}\\\\[1.5ex]
b&=\\frac{k-l}{2}
\\end{align}
$$\u3068\u8868\u305b\u308b. \u3053\u3053\u3067, \\(a\\), \\(b\\)\u306f\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \\(k\\), \\(l\\)\u306f\u3044\u305a\u308c\u3082\u5947\u6570, \u307e\u305f\u306f, \u3044\u305a\u308c\u3082\u5076\u6570\u3068\u306a\u308b\u5fc5\u8981\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u306a\u305c\u306a\u3089, \\(k\\), \\(l\\)\u306e\u3044\u305a\u308c\u304b\u4e00\u65b9\u304c\u5076\u6570, \u4ed6\u65b9\u304c\u5947\u6570\u306e\u3068\u304d, \\(k+l\\), \\(k-l\\)\u306f\u5947\u6570\u3068\u306a\u308a, \u4e0a\u5f0f\u3092\u6e80\u305f\u3059\u6574\u6570\\(a\\), \\(b\\)\u306f\u5b58\u5728\u3057\u306a\u3044\u304b\u3089\u3067\u3042\u308b.

\u4ee5\u4e0a\u304b\u3089, \u4ee5\u4e0b\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059, \u6574\u6570\u306e\u7d44\\((k,l)\\)\u3092\u6d17\u3044\u51fa\u3057, \\(\\displaystyle a=\\frac{k+l}{2}, b=\\frac{k-l}{2} \\)\u3068\u3057\u3066, \\((a,b)\\) \u306e\u7d44\u3092\u6c42\u3081\u308b\u3053\u3068\u3068\u3059\u308b.
$$
\\begin{align}
\u30fb&c=kl\\\\[1.5ex]
\u30fb&k\\geq l\\geq 0\\\\[1.5ex]
\u30fb& k, l \u306f\u5171\u306b\u5076\u6570, \u307e\u305f\u306f, \u5171\u306b\u5947\u6570
\\end{align}
$$

\u2460 \\(c=24\\)\u306e\u3068\u304d
\u6761\u4ef6\u3092\u6e80\u305f\u3059\\((k,l)\\)\u306f,
$$
(k,l)=\\left\\{(12,2), (6,4)\\right\\}
$$\u306e2\u901a\u308a\u3067\u3042\u308b. \u3088\u3063\u3066\u3053\u308c\u304b\u3089, \\((a, b)\\)\u3092\u6c42\u3081\u308b\u3068,
$$
(a,b)=\\left\\{(7,5), (5,1)\\right\\}
$$\u3068\u3057\u3066, 2\u901a\u308a\u304c\u6c42\u307e\u308b.

\u2461 \\(c=25\\)\u306e\u3068\u304d
\u6761\u4ef6\u3092\u6e80\u305f\u3059\\((k,l)\\)\u306f,
$$
(k,l)=\\left\\{(25,1), (5,5)\\right\\}
$$\u306e2\u901a\u308a\u3067\u3042\u308b. \u3088\u3063\u3066\u3053\u308c\u304b\u3089, \\((a, b)\\)\u3092\u6c42\u3081\u308b\u3068,
$$
(a,b)=\\left\\{(13,12), (5,0)\\right\\}
$$\u3068\u3057\u3066, 2\u901a\u308a\u304c\u6c42\u307e\u308b.

\u2462 \\(c=26\\)\u306e\u3068\u304d
\u6761\u4ef6\u3092\u6e80\u305f\u3059\\((k,l)\\)\u306f, \u5b58\u5728\u3057\u306a\u3044. \u3088\u3063\u3066, \\((a,b)\\)\u306b\u3064\u3044\u3066\u3082\u5b58\u5728\u3057\u306a\u3044.<\/p>\n\n\n\n

(2) (1)\u3068\u540c\u69d8\u306b, \\(k,l\\)\u3092\u6c7a\u3081\u3066\u3044\u304f. \\(c=4p^{2n}\\)\u3088\u308a, \\(c\\)\u306f\\(4\\)\u306e\u500d\u6570, \u3064\u307e\u308a, \u5076\u6570\u3067\u3042\u308b. \\(c=kl\\)\u304b\u3089, \\(k\\), \\(l\\)\u306e\u3044\u305a\u308c\u304b\u4e00\u65b9\u306f\u5fc5\u305a\u5076\u6570\u306b\u306a\u308b\u304c, (1)\u3067\u4e0e\u3048\u305f\\(k\\), \\(l\\)\u306e\u6761\u4ef6\u3088\u308a, \\(k\\), \\(l\\)\u3044\u305a\u308c\u3082\u5076\u6570\u3068\u306a\u308b\u5fc5\u8981\u304c\u3042\u308b. \u3088\u3063\u3066, \u6574\u6570\\(k^\\prime\\), \\(l^\\prime\\)\u3092\u7528\u3044\u3066, \\(k=2k^\\prime\\), \\(l=2l^\\prime\\)\u3068\u304b\u3051\u308b. \\(p\\)\u304c\u5076\u6570\u3067\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068,
$$
\\begin{align}
&c=kl\\\\[1.5ex]
&\\iff 4p^{2n}=4k^\\prime l^\\prime\\\\[1.5ex]
&\\iff p^{2n}=k^\\prime l^\\prime
\\end{align}
$$\u3068\u306a\u308a, \\(k\\geq l\\)\u3067\u3042\u308b\u304b\u3089, \\(0\\leq m \\leq l\\)\u3068\u306a\u308b\u6574\u6570\\(m\\)\u3092\u7528\u3044\u3066, \\(k^\\prime=p^{2n-m}\\), \\(l^\\prime=p^m\\)\u3068\u8868\u305b\u308b. \u3088\u3063\u3066, \\(k=2p^{2n-m}\\), \\(l=2p^m\\) (\\(m=0,1,2,\\cdots n\\))\u3068\u8868\u305b\u308b\u304b\u3089,
$$
\\begin{align}
a&=\\frac{2p^{2n-m}+2p^m}{2}=p^{2n-m}+p^m\\\\[1.5ex]
b&=\\frac{2p^{2n-m}-2p^m}{2}=p^{2n-m}-p^m,\\,\\,\\,\\,(m=0,1,2,\\cdots n)
\\end{align}
$$\u3068\u306a\u308a, \u6761\u4ef6\u3092\u6e80\u305f\u3059\\((a,b)\\)\u306e\u7d44\u306f,
$$
(a,b)=\\left\\{(p^{2n-m}+p^m, p^{2n-m}-p^m)|m=0,1,2,\\cdots n\\right\\}
$$\u306e\\(n+1\\)\u901a\u308a\u304c\u6c42\u307e\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n