{"id":601,"date":"2009-01-28T13:38:34","date_gmt":"2009-01-28T04:38:34","guid":{"rendered":"http:\/\/njet.oops.jp\/wordpress\/2009\/01\/28\/%e3%81%a8%e3%81%82%e3%82%8b%e3%83%ac%e3%83%b3%e3%83%9e\/"},"modified":"2011-06-17T20:43:11","modified_gmt":"2011-06-17T11:43:11","slug":"%e3%81%a8%e3%81%82%e3%82%8b%e3%83%ac%e3%83%b3%e3%83%9e","status":"publish","type":"post","link":"https:\/\/njet.oops.jp\/wordpress\/2009\/01\/28\/%e3%81%a8%e3%81%82%e3%82%8b%e3%83%ac%e3%83%b3%e3%83%9e\/","title":{"rendered":"\u3068\u3042\u308b\u30ec\u30f3\u30de"},"content":{"rendered":"


\n[ \u5099\u5fd8\u9332 ]\u3000(Cox\u306e\u672c\u306e\u8a3c\u660e\u304c\u6c17\u306b\u5165\u3089\u306a\u304b\u3063\u305f(?)\u306e\u3067\u3001\u81ea\u524d\u306e\u8a3c\u660e\u3002\u3082\u3063\u3068\u3082\u3001\u81ea\u5206\u3067\u8003\u3048\u305f\u3042\u3068\u3067\u3082\u3046\u4e00\u5ea6\u8aad\u3093\u3067\u307f\u305f\u3089\u3001\u5b9f\u306f\u540c\u3058\u3060\u3063\u305f(\u7206)\u3002\u3044\u3084\u3001\u3088\u304f\u898b\u308c\u3070\u5185\u5bb9\u7684\u306b\u306f\u540c\u3058\u306a\u306e\u3060\u304c\u3001\u63d0\u793a\u306e\u4ed5\u65b9\u304c\u306d\u3048\u30fb\u30fb\u30fb\u3002
\n\u3068\u3044\u3046\u3053\u3068\u3067\u3001\u81ea\u5206\u306e\u70ba\u306b\u8a18\u9332\u3002)<\/p>\n

\u521d\u7b49\u7684\u8b70\u8ad6\u3067\u6b21\u306e Lemma (\u88dc\u52a9\u5b9a\u7406) \u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

2\u500b\u306e\u5e73\u65b9\u6570\u306e\u548c\u3068\u3057\u3066\u8868\u3055\u308c\u308b\u81ea\u7136\u6570$N=x^2+y^2$\u3068\u3001\u305d\u306e\u7d20\u56e0\u6570$p$\u3092\u8003\u3048\u308b\u3002
\n\u3082\u3057\u3001$p$\u3082$p=a^2+b^2$\u30682\u500b\u306e\u5e73\u65b9\u6570\u306e\u548c\u306b\u306a\u308b\u306a\u3089\u3070\u3001
\n$\\frac{N}{p}$\u30822\u500b\u306e\u5e73\u65b9\u6570\u306e\u548c\u3068\u3057\u3066\u8868\u3055\u308c\u308b\u3002<\/p>\n

\u4f8b\u3048\u3070\u3001$N=65$\u3068\u305d\u306e\u7d20\u56e0\u6570$p=5$\u306f
\n\\[ N=65=16+49=4^2+7^2, \\qquad p=5=1+4=1^2+2^2 \\]
\n\u30682\u500b\u306e\u5e73\u65b9\u6570\u306e\u548c\u3068\u3057\u3066\u8868\u3055\u308c\u308b\u3002\u3088\u3063\u3066\u3001$\\frac{N}{p}=13$\u3082\u305d\u3046\u306a\u306e\u3067\u3042\u308b\u3002\u5b9f\u969b\u3001
\n\\[ \\frac{N}{p}=13=4+9=2^2+3^2 \\]
\n\u3068\u306a\u308b\u3002\u3053\u308c\u304c\u3044\u3064\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u30ec\u30f3\u30de\u306f\u4e3b\u5f35\u3059\u308b\u3082\u306e\u3067\u3042\u308b\u3002<\/p>\n

2\u500b\u306e\u5e73\u65b9\u6570\u306e\u548c\u306e\u5168\u4f53\u304c\u300c\u7a4d\u306b\u95a2\u3057\u3066\u9589\u3058\u3066\u3044\u308b\u300d\u3053\u3068\u306f\u3001\u30d6\u30e9\u30cf\u30de\u30b0\u30d7\u30bf\u306e\u6052\u7b49\u5f0f
\n\\[ (a^2+b^2)(z^2+w^2)=(az+bw)^2+(aw-bz)^2 \\]
\n\u304b\u3089\u76f4\u3061\u306b\u5206\u304b\u308b\u304c\u3001\u5546\u306b\u95a2\u3057\u3066\u3082\u540c\u69d8\u306e\u4e8b\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u306f\u81ea\u660e\u3067\u306f\u306a\u3044\u3002
\n\u3068\u306f\u3044\u3048\u3001\u8a3c\u660e\u306e\u30a2\u30a4\u30c7\u30a3\u30a2\u306f\u3084\u306f\u308a\u3053\u306e\u6052\u7b49\u5f0f\u306b\u3042\u308b\u3002
\n<\/latex><\/p>\n

<\/p>\n


\n$p=a^2+b^2$\u3068$N=x^2+y^2$\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u3057\u3066\u3001$\\frac{N}{p}=m$\u3068\u304a\u304f\u3002
\n\u76ee\u6a19\u306f$m=z^2+w^2$\u3092\u6e80\u305f\u3059\u6574\u6570$z$, $w$\u3092\u898b\u51fa\u3059\u3053\u3068\u3067\u3042\u308b\u3002
\n\u30d6\u30e9\u30cf\u30de\u30b0\u30d7\u30bf\u306e\u6052\u7b49\u5f0f
\n\\[ (a^2+b^2)(z^2+w^2)=(az+bw)^2+(aw-bz)^2 \\]
\n\u3068\u898b\u6bd4\u3079\u3066\u307f\u308c\u3070\u3001
\n\\[ \\left\\lbrace\\begin{eqnarray} az+bw&=x\\\\ -bz+aw&=y \\end{eqnarray}\\right. \\]
\n\u3092\u6e80\u305f\u3059$z$, $w$\u304c\u3042\u308c\u3070\u3001\u305d\u308c\u3067\u76ee\u7684\u304c\u679c\u305f\u305b\u308b\u3053\u3068\u304c\u5206\u304b\u308b\u3002
\n\u5b9f\u969b\u3001\u3053\u308c\u304c\u6210\u308a\u7acb\u3063\u3066\u3044\u308b\u3068\u3057\u3066\u3001\u6052\u7b49\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001
\n\\begin{align*}
\np(z^2+w^2)&=(a^2+b^2)(z^2+w^2)\\\\
\n&=(az+bw)^2+(aw-bz)^2\\\\
\n&=x^2+y^2=N=pm
\n\\end{align*}
\n\u3068\u306a\u308b\u306e\u3067\u3001\u4e21\u8fba\u3092$p$\u3067\u5272\u308c\u3070 $z^2+w^2=m$ \u3092\u5f97\u308b\u3002<\/p>\n

\u3057\u304b\u3057\u3001\u554f\u984c\u306f\u4e0a\u8a18\u306e\u9023\u7acb1\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3$z$, $w$\u304c\u306f\u305f\u3057\u3066\u6574\u6570\u306b\u306a\u308b\u304b\u3001\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002
\n\u5177\u4f53\u7684\u306b\u89e3\u3092\u8868\u793a\u3059\u308c\u3070\u3001
\n\\[ z=\\frac{ax-by}{a^2+b^2}=\\frac{ax-by}{p}, \\qquad y=\\frac{bx+ay}{a^2+b^2}=\\frac{bx+ay}{p} \\]
\n\u3067\u3042\u308b\u304b\u3089\u3001\u5206\u5b50\u306e2\u3064\u306e\u6574\u6570
\n\\[ K=ax-by, \\qquad L=bx+ay \\]
\n\u304c\u679c\u305f\u3057\u3066\u7d20\u6570$p$\u3067\u5272\u308a\u5207\u308c\u308b\u3060\u308d\u3046\u304b\u3001\u3068\u3044\u3046\u554f\u984c\u306b\u306a\u308b\u3002<\/p>\n

\u3053\u308c\u306f\u6b21\u306e\u3088\u3046\u306b\u3057\u3066\u89e3\u6c7a\u3067\u304d\u308b\u3002\u30dd\u30a4\u30f3\u30c8\u306f$K=ax-by$\u306e\u4ed6\u306b\u3001$b$\u3092$(-b)$\u306b\u53d6\u308a\u66ff\u3048\u305f$K’=ax+by$\u3092\u8003\u3048\u308b\u3053\u3068\u3067\u3042\u308b\u3002
\n$p=a^2+b^2$\u306e\u8868\u793a\u306b\u304a\u3044\u3066\u3001$a$\u3068$b$\u306f\u7b26\u53f7\u3092\u5909\u3048\u3066\u3082\u69cb\u308f\u306a\u3044\u306e\u3060\u304b\u3089\u3001\u3082\u3057\u3001$K$\u304c$p$\u3067\u5272\u308a\u5207\u308c\u306a\u304f\u3068\u3082\u3001$b$\u306e\u7b26\u53f7\u3092\u5909\u3048\u305f$K’$\u306e\u65b9\u306a\u3089\u671b\u307f\u304c\u3042\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u3001\u3068\u3044\u3046\u8a33\u3067\u3042\u3063\u305f\u3002
\n\u305d\u306e\u969b\u306b\u306f\u3001$L$\u306e\u65b9\u3082$b$\u306e\u7b26\u53f7\u3092\u5909\u3048\u305f$L’=-bx+ay$\u306b\u3059\u308b\u306e\u3067\u3042\u308b\u3002<\/p>\n

\u3055\u3066\u3001$K$\u3068$K’$\u306e\u3044\u305a\u308c\u304b\u304c$p$\u3067\u5272\u308a\u5207\u308c\u308c\u3070\u826f\u3044\u3068\u3044\u3046\u72b6\u6cc1\u306b\u306a\u3063\u305f\u3002\u305d\u3053\u3067\u3001\u3053\u306e2\u3064\u306e\u7a4d\u3092\u4f5c\u3063\u3066\u307f\u308b\u3002\u7d50\u679c\u306f\u3001
\n\\begin{align*}
\n KK’ &=(ax-by)(ax+by)\\\\
\n&=a^2x^2-b^2y^2 \\\\
\n&=a^2x^2+a^2y^2-a^2y^2-b^2y^2\\\\
\n&=a^2(x^2+y^2)-(a^2+b^2)y^2\\\\
\n&=a^2 N-py^2
\n\\end{align*}
\n\u3068\u306a\u308b\u304b\u3089\u3001$KK’$\u306f$p$\u3067\u5272\u308a\u5207\u308c\u308b\u3002$p$\u306f\u7d20\u6570\u3067\u3042\u308b\u304b\u3089\u3001$K$\u3068$K’$\u306e\u5c11\u306a\u304f\u3068\u3082\u4e00\u65b9\u306f$p$\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

\u3055\u3042\u3001\u30b4\u30fc\u30eb\u76ee\u524d\u3002\u3044\u3084\u3042\u3001\u5206\u304b\u3063\u3066\u3044\u308b\u3053\u3068\u3067\u3082\u5b9f\u969b\u306b\u66f8\u3044\u3066\u307f\u308b\u3068\u9577\u304f\u306a\u308a\u307e\u3059\u3067\u3059\u306d\u3048\u3002\u3042\u3042\u3001\u59cb\u3081\u306a\u304d\u3083\u3088\u304b\u3063\u305f \ud83d\ude25 <\/p>\n

\u3068\u3044\u3046\u3053\u3068\u3067\u3001\u5fc5\u8981\u306a\u3089$b$\u306e\u7b26\u53f7\u3092\u5909\u3048\u308b\u3053\u3068\u3067\u3001$K=ax-by$\u304c$p$\u3067\u5272\u308a\u5207\u308c\u308b\u3068\u4eee\u5b9a\u3057\u3066\u3082\u3088\u3044\u3002
\n\u3053\u306e\u3068\u304d\u3001$L=bx+ay$\u306e\u65b9\u3082$p$\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u3092\u793a\u3057\u305f\u3044\u306e\u3067\u3042\u308b\u304c\u3001\u6b63\u76f4\u8a00\u3046\u3068\u3001\u3053\u306e\u7b87\u6240\u3067\u304b\u306a\u308a\u82e6\u52b4\u3057\u305f\u3002\u5929\u4e0b\u308a\u306b\u5f0f\u3092\u4e0e\u3048\u3089\u308c\u308b\u306e\u306f\u30ef\u30bf\u30b7\u304c\u6700\u3082\u5acc\u3044\u306a\u3068\u3053\u308d\u306a\u306e\u3067(\u82e6\u7b11)\u3001\u5f8c\u77e5\u6075\u3068\u306f\u8a00\u3044\u306a\u304c\u3089\u3082\u3001\u305d\u308c\u306a\u308a\u306b\u3082\u3063\u3068\u3082\u3089\u3057\u3044\u81ea\u7136\u306a\u8b70\u8ad6\u3092\u3057\u305f\u3044\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u7121\u99c4\u53e3\u306f\u3053\u308c\u304f\u3089\u3044\u306b\u3057\u3066\u3001\u305d\u308d\u305d\u308d\u30b4\u30fc\u30eb\u306b\u5165\u308d\u3046\u3002
\n\\[ \\left\\lbrace\\begin{eqnarray} ax-by&=K\\\\ bx+ay&=L\\end{eqnarray}\\right. \\]
\n\u3092$x$, $y$\u306b\u3064\u3044\u3066\u89e3\u3044\u3066\u307f\u308b\u3068\u3001
\n\\[ x=\\frac{aK+bL}{a^2+b^2}=\\frac{aK+bL}{p}, \\qquad y=\\frac{-bK+aL}{p} \\]
\n\u3068\u306a\u308b\u3002\u3064\u307e\u308a\u3001
\n\\[ px=aK+bL, \\qquad\\qquad bL=px-aK \\]
\n\u3067\u3042\u308b\u3002\u3053\u306e\u5f0f\u304b\u3089$K$\u304c$p$\u306e\u500d\u6570\u306a\u3089\u3070\u3001$bL$\u3082$p$\u306e\u500d\u6570\u3067\u3042\u308b\u3053\u3068\u304c\u8a00\u3048\u308b\u3002
\n$p=a^2+b^2$\u306f\u7d20\u6570\u3067\u3042\u3063\u305f\u304b\u3089\u3001\u3082\u3061\u308d\u3093$p$\u3068$b$\u306f\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u3002\u5f93\u3063\u3066\u3001$L$\u306f$p$\u306e\u500d\u6570\u3068\u306a\u308b\u3002<\/p>\n

\u4ee5\u4e0a\u304b\u3089\u3001$z=\\frac{K}{p}$, $w=\\frac{L}{p}$ \u306f\u3001\u3044\u305a\u308c\u3082\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u3001$m=\\frac{N}{p}$\u3082\u307e\u305f2\u500b\u306e\u5e73\u65b9\u6570\u306e\u548c\u3068\u3057\u3066\u8868\u3055\u308c\u308b\u3053\u3068\u304c\u5206\u304b\u3063\u305f\u3002QED
\n<\/latex><\/p>\n","protected":false},"excerpt":{"rendered":"

[ \u5099\u5fd8\u9332 ]\u3000(Cox\u306e\u672c\u306e\u8a3c\u660e\u304c\u6c17\u306b\u5165\u3089\u306a\u304b\u3063\u305f(?)\u306e\u3067\u3001\u81ea\u524d\u306e\u8a3c\u660e\u3002\u3082\u3063\u3068\u3082\u3001\u81ea\u5206\u3067\u8003\u3048\u305f\u3042\u3068\u3067\u3082\u3046\u4e00\u5ea6\u8aad\u3093\u3067\u307f\u305f\u3089\u3001\u5b9f\u306f\u540c\u3058\u3060\u3063\u305f(\u7206)\u3002\u3044\u3084\u3001\u3088\u304f\u898b\u308c\u3070\u5185\u5bb9\u7684\u306b\u306f\u540c\u3058\u306a\u306e\u3060\u304c\u3001\u63d0\u793a\u306e\u4ed5\u65b9\u304c\u306d\u3048\u30fb\u30fb\u30fb\u3002 \u3068\u3044\u3046 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[37,5],"tags":[],"class_list":["post-601","post","type-post","status-publish","format-standard","hentry","category-37","category-5"],"_links":{"self":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/601","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/comments?post=601"}],"version-history":[{"count":6,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/601\/revisions"}],"predecessor-version":[{"id":1799,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/601\/revisions\/1799"}],"wp:attachment":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/media?parent=601"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/categories?post=601"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/tags?post=601"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}