{"id":893,"date":"2018-05-08T15:33:58","date_gmt":"2018-05-08T06:33:58","guid":{"rendered":"https:\/\/it1.jp\/?p=893"},"modified":"2018-12-24T13:53:42","modified_gmt":"2018-12-24T04:53:42","slug":"post-893","status":"publish","type":"post","link":"https:\/\/it1.jp\/?p=893","title":{"rendered":"\u30ac\u30a6\u30b9\u306e\u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247\u306f\u3069\u3053\u304c\u7f8e\u3057\u3044\u306e\u304b"},"content":{"rendered":"\n

\u30ac\u30a6\u30b9\u306e\u9ec4\u91d1\u5b9a\u7406\u3068\u3057\u3066\u3082\u77e5\u3089\u308c\u308b\u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247\u3002\u4f55\u5341\u500b\u3082\u306e\u8a3c\u660e\u304c\u306a\u3055\u308c\u3066\u3044\u308b\u305d\u3046\u3067\u3059\u304c\u3001\u3053\u306e\u5b9a\u7406\u304c\u3069\u306e\u3088\u3046\u306b\u7f8e\u3057\u3044\u306e\u304b\u3068\u3044\u3046\u8aac\u660e\u306f\u96e3\u3057\u3044\u3067\u3059\u3002<\/p>\n

\u5b9f\u969b\u306b\u50d5\u304c\u9ad8\u6821\u751f\u3050\u3089\u3044\u306e\u6642\u306b\u521d\u3081\u3066\u898b\u305f\u6642\u3082\u3001\u306a\u304b\u306a\u304b\u7dba\u9e97\u306a\u6027\u8cea\u304c\u3042\u308b\u306a\u3068\u306f\u611f\u3058\u305f\u3082\u306e\u306e\u3001\u4f8b\u3048\u3070\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3092\u898b\u305f\u3068\u304d\u306e\u3088\u3046\u306a\u885d\u6483\u306f\u306a\u304b\u3063\u305f\u3088\u3046\u306b\u601d\u3044\u307e\u3059\u3002<\/p>\n

\u3057\u304b\u3057\u30ac\u30a6\u30b9\u304c\u9ec4\u91d1\u5b9a\u7406\u3068\u547c\u3076\u306b\u306f\u305d\u308c\u306a\u308a\u306e\u8a33\u304c\u3042\u3063\u305f\u306e\u3067\u3057\u3087\u3046\u3002\u6539\u3081\u3066\u3053\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066\u306e\u8a3c\u660e\u3068\u8aac\u660e\u3092\u8aad\u3093\u3067\u3044\u305f\u6642\u306b\u3001\u3053\u306e\u5b9a\u7406\u306e\u7f8e\u3057\u3044\u3068\u3053\u308d\u304c\u3088\u3046\u3084\u304f\u81ea\u5206\u306a\u308a\u306b\u7406\u89e3\u3067\u304d\u305f\u305f\u3081\u3001\u89e3\u8aac\u8a18\u4e8b\u3092\u66f8\u304f\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n

\u3053\u306e\u8a18\u4e8b\u306b\u304a\u3044\u3066\u306f\u3001\u8a73\u3057\u3044\u8a3c\u660e\u306f\u3057\u307e\u305b\u3093\u3002\u5404\u81ea\u6c17\u306b\u306a\u308b\u3068\u3053\u308d\u306f\u81ea\u5206\u3067\u8abf\u3079\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n

\u57fa\u672c\u7684\u306a\u3053\u3068<\/h2>\n

\u8a3c\u660e\u306f\u3057\u307e\u305b\u3093\u304c\u3001\u8a18\u53f7\u306e\u610f\u5473\u304c\u308f\u304b\u3089\u306a\u3051\u308c\u3070\u4f55\u3082\u308f\u304b\u3089\u306a\u3044\u304b\u3068\u601d\u3044\u307e\u3059\u306e\u3067\u3001\u7c21\u5358\u306b\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n

\u5408\u540c\u5f0f<\/h3>\n

\u300c\\(\\equiv\\)\u300d\u306f\u5408\u540c\u3092\u8868\u3057\u307e\u3059\u3002
\n\\(7 \\equiv 2\\;(5)\\)
\n\u3053\u308c\u306f\u30017\u30822\u30825\u3067\u5272\u3063\u305f\u4f59\u308a\u306f\u540c\u3058\u3067\u3042\u308b\u3068\u3044\u3046\u610f\u5473\u3067\u3059\u3002\u8a00\u3044\u63db\u3048\u308b\u3068\u30017\u30682\u306e\u5dee\u306f5\u3067\u5272\u308a\u5207\u308c\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u6307\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n

\u4e00\u822c\u7684\u306b\u306f\\(a \\equiv b\\;(p)\\)\u3068\u3044\u3046\u4f7f\u3044\u65b9\u3092\u3057\u3066\u3001\\(a-b\\)\u304c\\(p\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002
\n\u3053\u306e\u3088\u3046\u306b\u3001\u6574\u6570\u8ad6\u3067\u306f\u4f59\u308a\u306e\u4e16\u754c\u3067\u8b70\u8ad6\u3059\u308b\u3053\u3068\u304c\u3057\u3070\u3057\u3070\u3042\u308a\u307e\u3059\u3002<\/p>\n

\u305d\u3057\u3066\u6574\u6570\u3084\u5b9f\u6570\u3084\u8907\u7d20\u6570\u306e\u4e16\u754c\u3067\u300c\u3053\u306e\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306f\u3067\u304d\u308b\u306e\u304b\uff1f\u300d\u3068\u3044\u3046\u3053\u3068\u304c\u91cd\u8981\u306a\u7591\u554f\u3067\u3042\u3063\u305f\u3088\u3046\u306b\u5270\u4f59\uff08\u4f59\u308a\uff09\u306e\u4e16\u754c\u306b\u304a\u3044\u3066\u3082\u65b9\u7a0b\u5f0f\u304c\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u304b\u3069\u3046\u304b\u3068\u3044\u3046\u554f\u984c\u306f\u5927\u304d\u306a\u554f\u984c\u3067\u3059\u3002<\/p>\n

\u4f8b\u3048\u3070<\/p>\n

\\(ax \\equiv b\\;(m)\\)<\/p>\n

\u3068\u3044\u3046\u5408\u540c\u5f0f\u306f\\(a\\)\u3068\\(x\\)\u306e\u6700\u5927\u516c\u7d04\u6570\u304c\\(b\\)\u3092\u5272\u308a\u5207\u3063\u305f\u6642\u306e\u307f\u89e3\u3092\u6301\u3061\u3001\u89e3\u306e\u500b\u6570\u306f\u305d\u306e\u6700\u5927\u516c\u7d04\u6570\u306e\u6570\u3068\u7b49\u3057\u3044\u3053\u3068\u304c\u78ba\u304b\u3081\u3089\u308c\u307e\u3059\u3002<\/p>\n

\u30eb\u30b8\u30e3\u30f3\u30c9\u30eb\u8a18\u53f7<\/h3>\n

\u81ea\u7136\u306a\u6d41\u308c\u3067<\/p>\n

\\(x^2 \\equiv a\\;(p)\\)<\/p>\n

\u3068\u3044\u3046\u5408\u540c\u5f0f\u3092\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u3053\u306e\u5f0f\u306f\u3001\\(a\\)\u3068\\(p\\)\u306e\u7d44\u307f\u5408\u308f\u305b\u306b\u3088\u3063\u3066\u89e3\u3092\u6301\u3063\u305f\u308a\u6301\u305f\u306a\u304b\u3063\u305f\u308a\u3057\u307e\u3059\u3002\u3053\u306e\u5f0f\u304c\u89e3\u3092\u6301\u3064\u306e\u304b\u3069\u3046\u304b\u306b\u95a2\u3059\u308b\u5b9a\u7406\u304c\u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247\u3067\u3059\u3002<\/p>\n

\u300c\\( \\left( \\frac{ a }{ p } \\right ) \\)\u300d\u306f\u30eb\u30b8\u30e3\u30f3\u30c9\u30eb\u8a18\u53f7\u3068\u8a00\u3044\u307e\u3059\u3002\u3053\u3053\u3067\\(p\\)\u306f\u7d20\u6570\u3067\u3059\u3002<\/p>\n

\\(x^2 \\equiv a\\;(p)\\) \u304c\u89e3\u3092\u6301\u3064\u6642\u306b\u00a0\\( \\left( \\frac{ a }{ p } \\right ) = 1\\) \u3068\u5b9a\u7fa9\u3055\u308c\u3001
\n\\(x^2 \\equiv a\\;(p)\\) \u304c\u89e3\u3092\u6301\u305f\u306a\u3044\u6642\u306b\u00a0\\( \\left( \\frac{ a }{ p } \\right ) = -1\\) \u3068\u5b9a\u7fa9\u3055\u308c\u3001
\n\\(p\\)\u304c\\(a\\)\u3092\u5272\u308a\u5207\u308b\u6642\u306b\u00a0\\( \\left( \\frac{ a }{ p } \\right ) = 0\\) \u3068\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<\/p>\n

    \n
  1. \\( \\left( \\frac{ a }{ p } \\right ) \\equiv a^{(p-1)\/2} \\;(p) \\)<\/span><\/li>\n
  2. \\( \\left( \\frac{ ab }{ p } \\right ) =\u00a0\\left( \\frac{ a }{ p } \\right )\u00a0\\left( \\frac{ b }{ p } \\right ) \\)<\/span><\/li>\n
  3. \\(a \\equiv b\\;(p)\\) \u306a\u3089\u3070 \\( \\left( \\frac{ a }{ p } \\right ) = \\left( \\frac{ b }{ p } \\right ) \\)<\/span><\/li>\n<\/ol>\n

    \u3068\u3044\u3063\u305f\u6027\u8cea\u3092\u6301\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n

    \u5468\u8fba\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066<\/h2>\n

    \u5148\u307b\u3069\u7d39\u4ecb\u3057\u305f\u5f0f\u3092\u898b\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n

    \\( \\left( \\frac{ a }{ p } \\right ) \\equiv a^{(p-1)\/2} \\;(p) \\)<\/span><\/p>\n

    \u3053\u308c\u3067\u3059\u3002\u3053\u306e\u5f0f\u306f\u7d50\u69cb\u5f37\u529b\u3067\u3059\u3002\u4eca\u306f\u00a0\\(x^2 \\equiv a\\;(p)\\) \u3068\u3044\u3046\u5408\u540c\u5f0f\u3092\u8003\u3048\u3066\u3044\u307e\u3057\u305f\u3002\u3053\u306e\u5f0f\u306b\u4f8b\u3048\u3070\u5177\u4f53\u7684\u306b\u6570\u3092\u5f53\u3066\u306f\u3081\u3066\u3044\u3063\u305f\u3068\u3057\u307e\u3057\u3087\u3046\u3002\u4f8b\u3048\u3070\u00a0\\(x^2 \\equiv 3\\;(7)\\) \u3068\u3044\u3046\u5f0f\u304c\u89e3\u3092\u6301\u3064\u304b\u8abf\u3079\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n

    \\( \\left( \\frac{ 3 }{ 7 } \\right ) \\equiv 3^{(7-1)\/2} \\equiv -1\\;(p) \\) \u3068\u306a\u308b\u305f\u3081\u3001\u3053\u308c\u306f\u89e3\u3092\u6301\u3061\u307e\u305b\u3093\u3002<\/span><\/p>\n

    \u3053\u306e\u3088\u3046\u306b\u3001\u5177\u4f53\u7684\u306a\u5f0f\u304c\u4e0e\u3048\u3089\u308c\u305f\u5834\u5408\u306b\u306f\u3001\u89e3\u3092\u6301\u3064\u304b\u6301\u305f\u306a\u3044\u304b\u306e\u5224\u5b9a\u304c\u3059\u3050\u306b\u3067\u304d\u3066\u3057\u307e\u3046\u306e\u3067\u3059\u3002<\/p>\n

    \u3082\u3057\u3082\u3042\u306a\u305f\u304c7\u67087\u65e5\u751f\u307e\u308c\u3067\u30017\u3068\u3044\u3046\u6570\u5b57\u3092\u6df1\u304f\u611b\u3057\u3066\u3044\u308b\u3068\u3057\u305f\u3089\u3001\u3042\u306a\u305f\u306f\u6b21\u306b\u3053\u3046\u601d\u3046\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002<\/p>\n

    \u300c\\(x^2 \\equiv a\\;(7)\\) \u3068\u3044\u3046\u5f0f\u304c\u89e3\u3092\u6301\u3064\u3088\u3046\u306a\\(a\\)\u3092\u5168\u3066\u898b\u3064\u3051\u51fa\u3057\u305f\u3044\u3002\u300d
    \n\u3064\u307e\u308a\u3001\\( \\left( \\frac{ a }{ p } \\right )=1 \\) \u3068\u306a\u308b\u3088\u3046\u306a\\(a\\)\u3092\u5168\u3066\u898b\u3064\u3051\u51fa\u3057\u305f\u3044\u306e\u3067\u3059\u3002
    \n\\(1\\)\u304b\u3089\\(6\\)\u307e\u3067\u3092\u540c\u69d8\u306b\u8a08\u7b97\u3059\u308b\u3068\u3001\\(a=1,2,4\\)\u306e\u6642\u306b\u89e3\u3092\u6301\u3064\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\\(8\\)\u4ee5\u4e0a\u306e\\(a\\)\u306b\u95a2\u3057\u3066\u306f\u3001\\(1\\)\u304b\u3089\\(6\\)\u306e\u3069\u308c\u304b\u3068\u5408\u540c\u3067\u3042\u308b\u306e\u3067\u3001\\(1\\)\u304b\u3089\\(6\\)\u307e\u3067\u3092\u8a08\u7b97\u3057\u3055\u3048\u3059\u308c\u3070\u5341\u5206\u306a\u306e\u3067\u3059\u3002<\/span><\/p>\n

    \u8003\u3048\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\\(a\\)\u304c\u6709\u9650\u306a\u306e\u3067\u3001
    \n<\/span>\u300c\u3042\u308b\u00a0\\(a\\) \u306b\u5bfe\u3057\u3066 \\(x^2 \\equiv a\\;(p)\\) \u3068\u3044\u3046\u5f0f\u304c\u89e3\u3092\u6301\u3064\u3088\u3046\u306a\\(a\\)\u3092\u5168\u3066\u898b\u3064\u3051\u51fa\u3057\u305f\u3044\u3002\u300d
    \n\u3068\u3044\u3046\u6b32\u6c42\u306f\u6e80\u305f\u3055\u308c\u308b\u306e\u3067\u3059\u3002<\/span><\/p>\n

    \u6b21\u306b\\(7\\)\u3068\u3044\u3046\u6570\u5b57\u304c\u597d\u304d\u306a\u3042\u306a\u305f\u306f\u3053\u3046\u601d\u3046\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002
    \n\u300c\\(x^2 \\equiv 7\\;(p)\\) \u3068\u3044\u3046\u5f0f\u304c\u89e3\u3092\u6301\u3064\u3088\u3046\u306a\u7d20\u6570\\(p\\)\u3092\u5168\u3066\u898b\u3064\u3051\u51fa\u3057\u305f\u3044\u3002\u300d
    \n\u4f55\u3092\u96a0\u305d\u3046\u3001\u3053\u308c\u304c\u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247\u306e\u52d5\u6a5f\u306a\u306e\u3067\u3059\u3002<\/p>\n

    \u3053\u306e\u8b70\u8ad6\u306e\u96e3\u3057\u3044\u3068\u3053\u308d\u306f\u8003\u3048\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u7d20\u6570\\(p\\)\u304c\u6709\u9650\u3067\u306f\u306a\u304f\u3001\u7121\u9650\u306b\u5e83\u304c\u3063\u3066\u3044\u308b\u3053\u3068\u3067\u3059\u3002\u5148\u307b\u3069\u306e
    \n\\( \\left( \\frac{ a }{ p } \\right ) \\equiv a^{(p-1)\/2} \\;(p) \\)
    \n\u3068\u3044\u3046\u6027\u8cea\u3067\u306f\u3001\u4e00\u3064\u4e00\u3064\u306e\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u3059\u304c\u3001<\/span>\\(p\\)\u304c\u5e73\u65b9\u5270\u4f59\u3067\u3042\u308b\uff08\u89e3\u3092\u6301\u3064\uff09\u305f\u3081\u306e\u96a0\u3055\u308c\u305f\u6761\u4ef6\u3092\u898b\u3064\u3051\u51fa\u3059\u3053\u3068\u306f\u3067\u304d\u306a\u3044\u306e\u3067\u3059\u3002<\/p>\n

    \u305d\u3057\u3066\u3001\u305d\u308c\u306f\u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247\u306b\u3088\u3063\u3066\u5929\u5730\u304c\u3072\u3063\u304f\u308a\u8fd4\u308b\u3088\u3046\u306a\u65b9\u6cd5\u3067\u9054\u6210\u3055\u308c\u307e\u3059\u3002<\/p>\n

    \u305d\u306e\u524d\u306b\u3001\u300c\u3042\u308b \\(a\\) \u306b\u5bfe\u3057\u3066 \\(x^2 \\equiv a\\;(p)\\) \u3068\u3044\u3046\u5f0f\u304c\u89e3\u3092\u6301\u3064\u3088\u3046\u306a\u7d20\u6570\\(p\\)\u3092\u5168\u3066\u898b\u3064\u3051\u51fa\u3057\u305f\u3044\u3002\u300d\u3068\u3044\u3046\u554f\u984c\u306e\u7279\u5225\u306a\u5834\u5408\u306b\u3064\u3044\u3066\u7d39\u4ecb\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n

    \\(a = -1\\) \u306e\u6642\u3068\u00a0\\(a = 2\\) \u306e\u6642\u306b\u95a2\u3059\u308b\u6cd5\u5247\u3067\u7b2c1\u88dc\u5145\u6cd5\u5247\u3068\u7b2c2\u88dc\u5145\u6cd5\u5247\u3068\u547c\u3070\u308c\u3066\u3044\u308b\u3082\u306e\u3067\u3059\u3002<\/p>\n

      \n
    1. \\( \\left( \\frac{ -1 }{ p } \\right ) = (-1)^{(p-1)\/2} \\)<\/span><\/li>\n
    2. \\( \\left( \\frac{ 2 }{ p } \\right ) = (-1)^{(p^2-1)\/8} \\)<\/span><\/li>\n<\/ol>\n

      \u7b2c1\u88dc\u5145\u6cd5\u5247\u306f \\( \\left( \\frac{ a }{ p } \\right ) \\equiv a^{(p-1)\/2} \\;(p) \\) \u3068\u3044\u3046\u5148\u307b\u3069\u306e\u5f0f\u306b\\(a = -1\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\u51fa\u3066\u304d\u307e\u3059\u3002
      \n\u3053\u3053\u3067\u30dd\u30a4\u30f3\u30c8\u306f \\(p=4k+1\\) \u3068\u8868\u3055\u308c\u308b\u6642\u306f\u5e73\u65b9\u5270\u4f59\u3067\u3042\u308a\u3001\\(p=4k+3\\) \u3068\u8868\u3055\u308c\u308b\u6642\u306f\u5e73\u65b9\u975e\u5270\u4f59\u3067\u3042\u308b\u3001\u3068\u8a00\u3044\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/span><\/p>\n

      \u307e\u305f\u7b2c2\u88dc\u5145\u6cd5\u5247\u306f\u5c11\u3057\u5c0e\u51fa\u304c\u8907\u96d1\u3067\u3059\u304c\u3001\u3053\u3061\u3089\u3082\u540c\u69d8\u306b\u8a00\u3044\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 \\(p=8k+1\\) \u53c8\u306f\u00a0\\(p=8k+7\\) \u3068\u8868\u3055\u308c\u308b\u6642\u306f\u5e73\u65b9\u5270\u4f59\u3067\u3042\u308a\u3001\\(p=8k+3\\) \u53c8\u306f\u00a0\\(p=8k+5\\) \u3068\u8868\u3055\u308c\u308b\u6642\u306f\u5e73\u65b9\u975e\u5270\u4f59\u3067\u3042\u308b\u306e\u3067\u3059\u3002<\/span><\/p>\n

      \u4e00\u898b\u7121\u9650\u306b\u3042\u308b\u3068\u601d\u308f\u308c\u305f\u7d20\u6570\\(p\\)\u3082\u3001\\(4\\)\u3084\\(8\\)\u306b\u3088\u308b\u5270\u4f59\u3092\u8003\u3048\u3066\u6709\u9650\u901a\u308a\u306e\u5834\u5408\u5206\u3051\u3092\u3057\u3066\u7279\u5fb4\u4ed8\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b\u306e\u3067\u3059\u3002<\/p>\n

      \u305d\u3057\u3066\u3001\u3068\u3046\u3068\u3046\\(-1\\)\u3084\\(2\\)\u3060\u3051\u3067\u306f\u306a\u304f\u3001\u4e00\u822c\u306e\u5024\\(a\\)\u306b\u3064\u3044\u3066\\(a\\)\u304c\u5e73\u65b9\u5270\u4f59\u3068\u306a\u308b\u3088\u3046\u306a\u7d20\u6570\u3092\u63a2\u3057\u3001\u305d\u306e\u7d20\u6570\u3092\u5270\u4f59\u306b\u3088\u3063\u3066\u7279\u5fb4\u4ed8\u3051\u307e\u3057\u3087\u3046\u3002<\/p>\n

      \u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247<\/h2>\n

      \u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247\u3068\u306f\u4ee5\u4e0b\u306e\u7b49\u5f0f\u306e\u3053\u3068\u3067\u3001\\(p\\)\u3068\\(q\\)\u304c\u5947\u6570\u306e\u7d20\u6570\u3067\u3042\u308b\u3068\u304d\u306b\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n

      \n

      \\( \\left( \\frac{ q }{ p } \\right ) \\left( \\frac{ p }{ q } \\right )\u00a0= (-1)^{ \\frac{p-1}{2} \\frac{q-1}{2}} \\)<\/span><\/p>\n<\/div>\n

      \u53f3\u8fba\u306f\u3001\\(p\\)\u304b\\(q\\)\u306e\u3069\u3061\u3089\u304b\u304c \\(p=4k+1\\) \u3068\u8868\u3055\u308c\u308b\u6642\u306f1\u3068\u306a\u308a\u3001\u3069\u3061\u3089\u3082\\(p=4k+3\\)\u3068\u8868\u3055\u308c\u308b\u6642\u306f\\(-1\\)\u3068\u306a\u308a\u307e\u3059\u3002<\/span><\/p>\n

      \u3053\u308c\u3060\u3051\u3092\u898b\u3066\u3082\u4f55\u304c\u51c4\u3044\u306e\u304b\u3042\u307e\u308a\u308f\u304b\u3089\u306a\u3044\u3053\u3068\u3067\u3057\u3087\u3046\u3002
      \n\u3053\u306e\u5b9a\u7406\u306e\u7d20\u6674\u3089\u3057\u3055\u3092\u4f1d\u3048\u308b\u305f\u3081\u306b\u3001\u3057\u3070\u3057\u3070\u00a0\\( \\left( \\frac{ 79 }{ 101 } \\right ) \\) \u306a\u3069\u306e\u5177\u4f53\u7684\u306a\u5024\u3092\u5b9f\u969b\u306b\u6c42\u3081\u3066\u307f\u3088\u3046\uff01\u3068\u3044\u3063\u305f\u4f8b\u984c\u304c\u51fa\u3055\u308c\u307e\u3059\u3002\u5b9f\u969b\u306b\u6c42\u3081\u3066\u307f\u308b\u3068\u3001\u5206\u6bcd\u3068\u5206\u5b50\u3092\u9006\u8ee2\u3055\u305b\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u624b\u8a08\u7b97\u306b\u3088\u3063\u3066\u3042\u308b\u7a0b\u5ea6\u7c21\u5358\u306b\u8a08\u7b97\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002
      \n<\/span>\u3053\u306e\u624b\u6cd5\u306f\u3001\u5b9f\u969b\u306b\u4f7f\u3063\u3066\u307f\u308b\u3068\u305d\u3053\u305d\u3053\u611f\u52d5\u3059\u308b\u306e\u3067\u3059\u304c\u3001\u5b9f\u969b\u306b\u3053\u308c\u3050\u3089\u3044\u306e\u8a08\u7b97\u306a\u3089\u3070\u4f55\u5ea6\u3082\u51fa\u3066\u304d\u305f
      \n\\( \\left( \\frac{ a }{ p } \\right ) \\equiv a^{(p-1)\/2} \\;(p) \\)
      \n\u3068\u3044\u3046\u5f0f\u3067\u3082\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u30fc\u3092\u4f7f\u3048\u3070\u4e00\u77ac\u3067\u7b54\u3048\u304c\u51fa\u307e\u3059\u3057\u3001\u3053\u306e\u5b9a\u7406\u306e\u672c\u5f53\u306e\u611f\u52d5\u30dd\u30a4\u30f3\u30c8\u306f\u3053\u3053\u3067\u306f\u306a\u3044\u3068\u601d\u3046\u306e\u3067\u3059\u3002<\/span><\/p>\n

      \u5148\u307b\u3069\u306e\u554f\u984c\u3092\u601d\u3044\u51fa\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u79c1\u305f\u3061\u306f\u300c\u3042\u308b \\(a\\) \u306b\u5bfe\u3057\u3066 \\(x^2 \\equiv a\\;(p)\\) \u3068\u3044\u3046\u5f0f\u304c\u89e3\u3092\u6301\u3064\u3088\u3046\u306a\u7d20\u6570\\(p\\)\u3092\u5168\u3066\u898b\u3064\u3051\u51fa\u3057\u305f\u3044\u3002\u300d\u3068\u3044\u3046\u554f\u984c\u3092\u8003\u3048\u3066\u3044\u307e\u3057\u305f\u3002
      \n\u3053\u306e\u3053\u3068\u306b\u5bfe\u3057\u3066\u3001\u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247\u304b\u3089\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u8a00\u3048\u307e\u3059\u3002<\/p>\n

      \n

      \\(q\\)\u304c\\(4k+1\\)\u3068\u8868\u3055\u308c\u308b\u7d20\u6570\u306e\u6642\u3001
      \n\\( \\left( \\frac{ q }{ p } \\right ) = 1 \\)\u00a0<\/span>\u3067\u3042\u308b\u3053\u3068\u3068\u3001\u3042\u308b\u00a0\\( \\left( \\frac{ r }{ q } \\right ) = 1 \\)<\/span>\u00a0\u3068\u306a\u308b\u3088\u3046\u306a\\(r\\) \u306b\u5bfe\u3057\u3066 \\(p \\equiv r \\;(q) \\) \u3068\u306a\u308b\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b\u3002<\/p>\n

      \\(q\\)\u304c\\(4k+3\\)\u3068\u8868\u3055\u308c\u308b\u7d20\u6570\u306e\u6642\u3001
      \n\\( \\left( \\frac{ q }{ p } \\right ) = 1 \\)\u00a0<\/span>\u3067\u3042\u308b\u3053\u3068\u3068\u3001\u3042\u308b\\(q\\)\u3068\u4e92\u3044\u306b\u7d20\u306a\u5947\u6570\\(b\\) \u306b\u5bfe\u3057\u3066 \\(p \\equiv \\pm b^2 \\;(4q)\\) \u3068\u306a\u308b\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b\u3002<\/p>\n<\/div>\n

      \u3044\u304d\u306a\u308a\u8aad\u3093\u3067\u3082\u5c11\u3057\u7406\u89e3\u3057\u3065\u3089\u3044\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002
      \n\\(q=3\\)\u306e\u6642\u306b\u3064\u3044\u3066\u307e\u305a\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u3053\u308c\u306f\\(4k+3\\)\u3068\u8868\u3055\u308c\u308b\u7d20\u6570\u306e\u5834\u5408\u306b\u5f53\u3066\u306f\u307e\u308a\u307e\u3059\u3002
      \n\u3053\u306e\u6642\u3001\\(b\\) \u3068\u3057\u3066\u53d6\u308a\u3046\u308b12(\\(3 \\times 4\\))\u4ee5\u4e0b\u306e\u5024\u306f\u3001\\(1,5,7,11\\)\u3067\u3042\u308a\u3001\u3053\u308c\u3089\u306f2\u4e57\u3059\u308b\u3068\u5168\u306612\u306b\u3088\u308b\u5270\u4f59\u306e\u4e16\u754c\u3067\u306f1\u306b\u7b49\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u30013\u304c\u5e73\u65b9\u5270\u4f59\u3068\u306a\u308b\u3088\u3046\u306a\u7d20\u6570\u306f12\u3067\u5272\u3063\u305f\u4f59\u308a\u304c\\( \\pm 1\\) \u3067\u3042\u308b\u3088\u3046\u306a\u7d20\u6570\u3067\u3042\u308b\u3068\u7279\u5fb4\u4ed8\u3051\u3089\u308c\u307e\u3059\u3002
      \n\u6b21\u306b\\(q=5\\)\u306e\u6642\u306b\u3064\u3044\u3066\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u3053\u308c\u306f\\(4k+1\\)\u3068\u8868\u3055\u308c\u308b\u7d20\u6570\u306e\u5834\u5408\u306b\u5f53\u3066\u306f\u307e\u308a\u307e\u3059\u3002\\( \\left( \\frac{ r }{ 5 } \\right ) = 1 \\)<\/span>\u00a0\u3068\u306a\u308b\u3088\u3046\u306a\\(r\\)\u306f1\u30684\u306a\u306e\u3067\u30015\u304c\u5e73\u65b9\u5270\u4f59\u3068\u306a\u308b\u3088\u3046\u306a\u7d20\u6570\u306f5\u3067\u5272\u3063\u305f\u4f59\u308a\u304c\\(1\\) \u304b\u00a0\\(4\\) \u3067\u3042\u308b\u3088\u3046\u306a\u7d20\u6570\u3067\u3042\u308b\u3068\u7279\u5fb4\u4ed8\u3051\u3089\u308c\u307e\u3059\u3002<\/p>\n

      \u4ee5\u4e0a\u306e\u4f8b\u3067\u898b\u305f\u3088\u3046\u306b\u3001\\(q\\)\u304c\u5e73\u65b9\u5270\u4f59\u3068\u306a\u308b\u3088\u3046\u306a\\(p\\)\u306e\u6761\u4ef6\u3092\u63a2\u3059\u3068\u3044\u3046\u884c\u70ba\u3092\u3001\\(q\\)\u3084\\(4q\\)\u306e\u5270\u4f59\u306e\u4e16\u754c\u306b\u304a\u3044\u3066\\(p\\)\u306e\u6761\u4ef6\u3092\u63a2\u3059\u3068\u3044\u3046\u98a8\u306b\u3072\u3063\u304f\u308a\u8fd4\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u3044\u3046\u306e\u304c\u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247\u306e\u5f37\u307f\u306a\u306e\u3067\u3059\u3002<\/p>\n

      \u3064\u307e\u308a\u3001\u7121\u9650\u306b\u7d9a\u304f\u7d20\u6570\u306e\u4e2d\u3092\u6761\u4ef6\u306b\u5408\u3046\u7d20\u6570\u304c\u898b\u3064\u304b\u308b\u3088\u3046\u306b\u679c\u3066\u3057\u306a\u304f\u63a2\u7d22\u3059\u308b\u306e\u3067\u306f\u306a\u304f\u3001\u6709\u9650\u306a\u5270\u4f59\u306e\u4e16\u754c\u306b\u6301\u3063\u3066\u3044\u3063\u3066\u7279\u5fb4\u3065\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u3044\u3046\u5b9a\u7406\u306a\u306e\u3067\u3059\u3002<\/p>\n

      \u3044\u3064\u306e\u9593\u306b\u304b\u3072\u3063\u304f\u308a\u8fd4\u3055\u308c\u3066\u3044\u305f\u5270\u4f59\u306b\u50d5\u306f\u3053\u306e\u5b9a\u7406\u306e\u7f8e\u3057\u3055\u3092\u898b\u305f\u306e\u3067\u3057\u305f\u3002<\/p>\n

      \u6700\u5f8c\u306b\u300c\\(x^2 \\equiv 7\\;(p)\\) \u3068\u3044\u3046\u5f0f\u304c\u89e3\u3092\u6301\u3064\u3088\u3046\u306a\u7d20\u6570\\(p\\)\u3092\u5168\u3066\u898b\u3064\u3051\u51fa\u3057\u305f\u3044\u3002\u300d\u3068\u3044\u3046\u554f\u984c\u306b\u7b54\u3048\u3092\u4e0e\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002\\(q=3\\)\u306e\u6642\u3068\u540c\u3058\u65b9\u6cd5\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n

      \u3053\u308c\u306f\\(4k+3\\)\u3068\u8868\u3055\u308c\u308b\u7d20\u6570\u306e\u5834\u5408\u306b\u5f53\u3066\u306f\u307e\u308a\u307e\u3059\u3002
      \n\u3053\u306e\u6642\u3001\\(b\\) \u3068\u3057\u3066\u53d6\u308a\u3046\u308b28(\\(7 \\times 4\\))\u4ee5\u4e0b\u306e\u5024\u306f\u3001\\(1,3,5,9,11,13\\)\u3067\u3059\u3002\\(b^2 = (-b)^2\\)\u306a\u306e\u3067\u300115\u4ee5\u4e0a\u306b\u3064\u3044\u3066\u306f\u8003\u3048\u306a\u304f\u3066\u3082\u826f\u3044\u306e\u3067\u3059\u3002
      \n\u3053\u308c\u3089\u306f2\u4e57\u3059\u308b\u3068\\(1,9,-3,3,-1,1\\)\u3068\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u30017\u304c\u5e73\u65b9\u5270\u4f59\u3068\u306a\u308b\u3088\u3046\u306a\u7d20\u6570\u306f28\u3067\u5272\u3063\u305f\u4f59\u308a\u304c\\( \\pm 1, \\pm 3, \\pm 9\\) \u3067\u3042\u308b\u3088\u3046\u306a\u7d20\u6570\u3067\u3042\u308b\u3068\u7279\u5fb4\u4ed8\u3051\u3089\u308c\u307e\u3059\u3002<\/p>\n

      \u5177\u4f53\u7684\u306b\u3042\u3052\u308b\u3068\u3001\\(3,19,29,31,37,47,53,…\\)\u306e\u3088\u3046\u306a\u6574\u6570\u306e\u5217\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

      \u53c2\u8003\u56f3\u66f8<\/h2>\n

      \u4ee5\u4e0a\u306e\u5185\u5bb9\u306f\u4ee5\u4e0b\u306e\u672c\u3092\u53c2\u8003\u306b\u66f8\u304d\u307e\u3057\u305f\u3002\u6c17\u306b\u306a\u308b\u4eba\u306f\u305c\u3072\u8cb7\u3063\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n

      <\/span>A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics)<\/a><\/span><\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

      \u30ac\u30a6\u30b9\u306e\u9ec4\u91d1\u5b9a\u7406\u3068\u3057\u3066\u3082\u77e5\u3089\u308c\u308b\u5e73\u65b9\u5270\u4f59\u306e\u76f8\u4e92\u6cd5\u5247\u3002\u4f55\u5341\u500b\u3082\u306e\u8a3c\u660e\u304c\u306a\u3055\u308c\u3066\u3044\u308b\u305d\u3046\u3067\u3059\u304c\u3001\u3053\u306e\u5b9a\u7406\u304c\u3069\u306e\u3088\u3046\u306b\u7f8e\u3057\u3044\u306e\u304b\u3068\u3044\u3046\u8aac\u660e\u306f\u96e3\u3057\u3044\u3067\u3059\u3002 \u5b9f\u969b\u306b\u50d5\u304c\u9ad8\u6821\u751f\u3050\u3089\u3044\u306e\u6642\u306b\u521d\u3081\u3066\u898b\u305f\u6642\u3082\u3001\u306a\u304b\u306a\u304b\u7dba\u9e97\u306a\u6027\u8cea\u304c\u3042\u308b\u306a\u3068\u306f\u611f\u3058\u305f\u3082\u306e\u306e\u3001\u4f8b\u3048\u3070\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3092\u898b\u305f\u3068\u304d\u306e\u3088\u3046\u306a\u885d\u6483\u306f\u306a\u304b\u3063\u305f\u3088\u3046\u306b\u601d\u3044\u307e\u3059\u3002 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