{"id":3190,"date":"2025-09-21T01:24:58","date_gmt":"2025-09-20T16:24:58","guid":{"rendered":"https:\/\/math-friend.com\/?p=3190"},"modified":"2025-09-21T07:04:00","modified_gmt":"2025-09-20T22:04:00","slug":"%e3%80%90%e6%9d%b1%e4%ba%ac%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e6%bc%b8%e5%8c%96%e5%bc%8f%e3%81%a8%e6%95%b4%e6%95%b0%e3%81%ae%e8%9e%8d%e5%90%88%e5%95%8f%e9%a1%8c2008","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=3190","title":{"rendered":"\u3010\u6771\u4eac\u5927\u5b66\u5165\u8a66\u3011\u6f38\u5316\u5f0f\u3068\u6574\u6570\u306e\u878d\u5408\u554f\u984c(2008)"},"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

\u81ea\u7136\u6570\\(p\\)\u306b\u5bfe\u3057\u3066, \u6b21\u306e\u6f38\u5316\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u308b\u6570\u5217\\(\\{a_n\\}\\), \\(\\{b_n\\}\\)\u3092\u8003\u3048\u308b.$$
\\left\\{\\begin{aligned} &\\,\\,a_1=p,\\,\\,\\,b_1=p+1\\\\ &\\,\\,a_{n+1}=a_n+pb_n\\,\\,\\,\\,\\,\\qquad\\qquad (n=1,2,3,\\cdots)\\\\ &\\,\\,b_{n+1}=pa_n+(p+1)b_n\\qquad(n=1,2,3,\\cdots) \\end{aligned}\\right.$$(1) \\(n=1,2,3,\\cdots\\)\u306b\u5bfe\u3057, \u6b21\u306e\\(2\\)\u6570\u304c\u3068\u3082\u306b\\(p^3\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u3092\u793a\u305b.$$
a_n-\\frac{n(n-1)}{2}p^2-np,\\,\\,\\,\\,b_n-n(n-1)p^2-np-1$$(2) \\(p\\)\u3092\\(3\\)\u4ee5\u4e0a\u306e\u5947\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(a_p\\)\u306f\\(p^2\\)\u3067\u5272\u308a\u5207\u308c\u308b\u304c, \\(p^3\\)\u3067\u306f\u5272\u308a\u5207\u308c\u306a\u3044\u3053\u3068\u3092\u793a\u305b.
(2008 \u6771\u4eac\u5927\u5b66 \u6587\u7cfb [4])<\/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) $$
\\begin{align}
x_n&=a_n-\\frac{n(n-1)}{2}p^2-np\\\\[1.5ex]
y_n&=b_n-n(n-1)p^2-np-1
\\end{align}$$\u3068\u304a\u304d, \\(n=1,2,3,\\cdots\\)\u306b\u5bfe\u3057\\(x_n\\), \\(y_n\\)\u304c\u5171\u306b\\(p^3\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u8a3c\u660e\u3059\u308b.

\u2460 \\(n=1\\)\u306e\u3068\u304d,$$
\\begin{align}
x_1&=a_1-p=p-p=0\\\\[1.5ex]
y_1&=b_1-p-1=p+1-p-1=0
\\end{align}
$$\u3068\u306a\u308a, \u5171\u306b\\(p^3\\)\u3067\u5272\u308a\u5207\u308c\u308b\u306e\u3067\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308b.

\u2461 \\(n=k\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3066, \\(n=k+1\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059.
\\(n=k\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3066\u3044\u308b\u306e\u3067, \u6574\u6570\\(l\\), \\(m\\)\u3092\u7528\u3044\u3066,$$
\\begin{align}
x_k&=a_k-\\frac{k(k-1)}{2}p^2-kp=lp^3\\\\[1.5ex]
y_k&=b_k-k(k-1)p^2-kp-1=mp^3
\\end{align}
$$\u3068\u8868\u305b\u308b. \u3053\u308c\u304b\u3089,$$
\\begin{align}
a_k&=lp^3+\\frac{k(k-1)}{2}p^2+kp\\\\[1.5ex]
b_k&=mp^3+k(k-1)p^2+kp+1
\\end{align}
$$\u3068\u306a\u308b.

\\(n=k+1\\)\u306e\u3068\u304d, \\(x_{k+1}\\)\u306f,$$
\\begin{align}
x_{k+1}&=a_{k+1}-\\frac{k(k+1)}{2}p^2-(k+1)p\\\\[1.5ex]
&=a_k+pb_k-\\frac{k(k+1)}{2}p^2-(k+1)p\\\\[1.5ex]
&=lp^3+\\frac{k(k-1)}{2}p^2+kp+p\\left\\{mp^3+k(k-1)p^2+kp+1\\right\\}\\\\[1.5ex]
&\\qquad-\\frac{k(k+1)}{2}p^2-(k+1)p\\\\[1.5ex]
&=lp^3+\\frac{k^2p^2}{2}-\\frac{kp^2}{2}+kp+mp^4+k^2p^3-kp^3+kp^2+p\\\\[1.5ex]
&\\qquad-\\frac{k^2p^2}{2}-\\frac{kp^2}{2}-kp-p\\\\[1.5ex]
&=lp^3+mp^4+k^2p^3-kp^3\\\\[1.5ex]
&=p^3(l+mp+k^2-k)
\\end{align}
$$\u3068\u306a\u308a, \\(p^3\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u6b21\u306b\\(y_{k+1}\\)\u306f,$$
\\begin{align}
y_{k+1}&=b_{k+1}-k(k+1)p^2-(k+1)p-1\\\\[1.5ex]
&=pa_k+(p+1)b_k-k(k+1)p^2-(k+1)p-1\\\\[1.5ex]
&=p\\left\\{lp^3+\\frac{k(k-1)}{2}p^2+kp\\right\\}+(p+1)\\left\\{mp^3+k(k-1)p^2+kp+1\\right\\}\\\\[1.5ex]
&\\qquad -k(k+1)p^2-(k+1)p-1\\\\[1.5ex]
&=lp^4+\\frac{k(k-1)}{2}p^3+kp^2+m(p+1)p^3+k^2p^3-kp^3+k^2p^2-kp^2\\\\[1.5ex]
&\\qquad +kp^2+kp+p+1-k^2p^2-kp^2-kp-p-1\\\\[1.5ex]
&=p^3\\left\\{lp+\\frac{k(k-1)}{2}+m(p+1)+k^2-k\\right\\}
\\end{align}
$$\u3068\u306a\u308a, \\(k\\)\u3068\\(k-1\\)\u306f\u3069\u3061\u3089\u304b\u306f\u5076\u6570\u306a\u306e\u3067, \u62ec\u5f27\u5185\u306e\\(\\displaystyle \\frac{k(k-1)}{2}\\)\u306f\u6574\u6570\u3068\u306a\u308a, \\(y_{k+1}\\)\u306f\\(p^3\\)\u3068\u6574\u6570\u3092\u304b\u3051\u305f\u5f62\u3067\u8868\u3055\u308c\u308b\u306e\u3067, \\(p^3\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b.

\u4ee5\u4e0a\u304b\u3089, \\(n=k+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u3055\u308c\u305f\u306e\u3067, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u308a\u5168\u3066\u306e\u81ea\u7136\u6570\\(n\\)\u306b\u5bfe\u3057\u3066, \\(x_n\\), \\(y_n\\)\u304c\\(p^3\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\n\n\n\n

(2) (1)\u304b\u3089\\(x_n\\)\u306b\\(n=p\\)\u3092\u4ee3\u5165\u3057\u305f\u6570\\(x_p\\)\u306f\\(p^3\\)\u3067\u5272\u308a\u5207\u308c\u308b\u306e\u3067, \u6574\u6570\\(l\\)\u3092\u7528\u3044\u3066\\(x_p=lp^3\\)\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b. \u3088\u3063\u3066\u3053\u308c\u304b\u3089, $$
x_p=a_p-\\frac{p(p-1)}{2}p^2-p^2=lp^3
$$\u3068\u306a\u308a, \\(a_p\\)\u306f$$
a_p=lp^3+\\frac{p(p-1)}{2}p^2+p^2
$$\u3068\u8868\u305b\u308b. \\(a_p\\)\u3092\u5909\u5f62\u3059\u308b\u3068,$$
a_p=p^2\\left(lp+\\frac{p(p-1)}{2}+1\\right)
$$\u3068\u306a\u308a, \\(p\\)\u306f\\(3\\)\u4ee5\u4e0a\u306e\u5947\u6570\u306a\u306e\u3067, \\(p-1\\)\u306f\u5076\u6570\u3068\u306a\u308a, \u62ec\u5f27\u5185\u306e\\(\\displaystyle\\frac{p(p-1)}{2}\\)\u306f\u6574\u6570\u3068\u306a\u308b. \u3088\u3063\u3066, \\(a_p\\)\u306f\\(p^2\\)\u3068\u6574\u6570\u3092\u304b\u3051\u305f\u5f62\u3067\u8868\u3055\u308c\u308b\u306e\u3067, \\(p^2\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b.

\u307e\u305f, \\(a_p\\)\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3059\u308b$$
a_p=p^3\\left(l+\\frac{p-1}{2}\\right)+p^2
$$\u307e\u305a, \\(p-1\\)\u304c\u5076\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u62ec\u5f27\u5185\u306e\\(\\displaystyle\\frac{p-1}{2}\\)\u306f\u6574\u6570\u3067\u3042\u308a, \u62ec\u5f27\u5185\u306e\u6570\u306f\u6574\u6570\u3068\u306a\u308b. \u307e\u305f, \\(p\\)\u306f\\(3\\)\u4ee5\u4e0a\u306e\u5947\u6570\u3060\u304b\u3089, \\(0<p^2<p^3\\)\u3067\u3042\u308b. \u3088\u3063\u3066, \\(a_p\\)\u306e\u5f62\u304b\u3089, \\(a_p\\)\u3092\\(p^3\\)\u3067\u5272\u3063\u305f\u3068\u304d\u306e\u4f59\u308a\u306f\\(p^2\\)\u3068\u306a\u308a, \\(a_p\\)\u306f\\(p^3\\)\u3067\u5272\u308c\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

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