{"id":102321,"date":"2023-12-20T14:23:24","date_gmt":"2023-12-20T14:23:24","guid":{"rendered":"https:\/\/science-hub.click\/%E3%82%AB%E3%83%BC%E3%83%9E%E3%82%A4%E3%82%B1%E3%83%AB%E6%95%B0-%E5%AE%9A%E7%BE%A9\/"},"modified":"2023-12-20T14:23:24","modified_gmt":"2023-12-20T14:23:24","slug":"%E3%82%AB%E3%83%BC%E3%83%9E%E3%82%A4%E3%82%B1%E3%83%AB%E6%95%B0-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=102321","title":{"rendered":"\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570 – \u5b9a\u7fa9"},"content":{"rendered":"

\u5c0e\u5165<\/h2>

\u6570\u8ad6\u3067\u306f\u3001\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f<\/b>\u6b21\u306e\u6027\u8cea\u3092\u6e80\u305f\u3059\u6b63\u306e\u5408\u6210\u6574\u6570n<\/i>\u3067\u3059\u3002

$$ {\\mathcal P} $$<\/div>\u7d9a\u304f\uff1a<\/p>
\u4efb\u610f\u306e\u6574\u6570a<\/i><\/span>\u306b\u3064\u3044\u3066\u3001 n \u306f<\/i><\/span>a<\/i> n<\/i><\/sup> \u2212 a<\/i><\/span>\u306e\u7d04\u6570\u3067\u3059<\/dd><\/dl>
\n\"\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570<\/figure>

\u6982\u8981<\/h2>

\u30d5\u30a7\u30eb\u30de\u30fc\u306e\u5c0f\u5b9a\u7406\u306f\u3001<\/a><\/span>\u7d20\u6570\u306b\u306f\u6b21\u306e\u6027\u8cea\u304c\u3042\u308b\u3068\u8ff0\u3079\u3066\u3044\u307e\u3059\u3002

$$ {\\mathcal P} $$<\/div> \u3002\u305d\u306e\u9006\u306f\u507d\u3067\u3042\u308a\u3001\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u6b63\u306e\u6570\u3067\u3059\u3002
$$ {\\mathcal P} $$<\/div>\u6700\u521d\u3067\u306a\u304f\u3066\u3082\u3001\u5f7c\u3089\u306f\u30d5\u30a7\u30eb\u30de\u30fc\u306e\u5618\u3064\u304d<\/i>\u3067\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u7d76\u5bfe\u7684\u306a\u64ec\u4f3c\u7d20\u6570<\/b>\u306e\u5b58\u5728\u306f\u3001\u30d5\u30a7\u30eb\u30de\u30fc\u306e\u7d20\u6570\u30c6\u30b9\u30c8\u3092\u4f7f\u7528\u3057\u3066\u6570\u5024<\/a><\/span>\u304c\u5408\u6210\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u3053\u3068\u3092\u59a8\u3052\u307e\u3059\u3002<\/p>

\u6570\u304c\u5927\u304d\u304f\u306a\u308a\u3001\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u304c\u5e0c\u5c11\u306b\u306a\u308b\u306b\u3064\u308c\u3066\u3001\u305d\u306e\u5927\u591a\u6570\u306b\u3088\u308a\u3001\u30d5\u30a7\u30eb\u30de\u30fc\u306e\u7d20\u6570\u6027\u30c6\u30b9\u30c8\u306f\u3001\u30bd\u30ed\u30d9\u30a4\u30fb\u30b9\u30c8\u30e9\u30c3\u30bb\u30f3\u306e\u7d20\u6570\u6027\u30c6\u30b9\u30c8\u306e\u3088\u3046\u306a\u4ed6\u306e\u7d20\u6570\u6027\u30c6\u30b9\u30c8\u3068\u6bd4\u8f03\u3057\u3066\u307b\u3068\u3093\u3069\u5f79\u306b\u7acb\u305f\u306a\u304f\u306a\u308a\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001 646 \u756a\u76ee\u306e<\/sup>\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570<\/a><\/span>\u306f 993,905,641 \u3067\u30011 \u304b\u3089 10 15<\/sup>\u307e\u3067\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f 105,212 \u500b\u3042\u308a\u307e\u3059\u3002<\/p>

\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306e\u7279\u5fb4\u4ed8\u3051\u306f\u30011899 \u5e74\u306b\u30b3\u30fc\u30bb\u30eb\u30c8\u306e\u5b9a\u7406\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u307e\u3057\u305f<\/span>\u3002<\/p>

\u5b9a\u7406<\/b>(Korselt 1899): \u5408\u6210\u6b63\u306e\u6574\u6570n<\/i>\u306f\u3001 n \u3092<\/i>\u5272\u308b\u7d20\u6570<\/a><\/span>\u306e\u4e8c\u4e57<\/a><\/span>\u304c\u5b58\u5728\u3057\u306a\u3044\u5834\u5408 ( n \u306f<\/i>quadratfrei (\u4e8c\u4e57\u306a\u3057)<\/i>\u3067\u3042\u308b\u3068\u8a00\u3044\u307e\u3059)\u3001\u304a\u3088\u3073n<\/i>\u306e\u5404\u7d20\u7d04\u6570<\/a><\/span>p<\/i>\u306b\u3064\u3044\u3066\u3001\u6570p<\/i>\u3067\u3042\u308b\u5834\u5408\u306b\u9650\u308a\u3001\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306b\u306a\u308a\u307e\u3059\u3002 \u2212 1 \u306fn<\/i> \u2212 1 \u3092\u9664\u7b97\u3057\u307e\u3059\u3002<\/p>

\u3053\u306e\u5b9a\u7406\u304b\u3089\u3001\u3059\u3079\u3066\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f\u5c11\u306a\u304f\u3068\u3082 3 \u3064\u306e\u7570\u306a\u308b\u5947\u6570\u7d20\u6570\u306e\u7a4d\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>

\u30b3\u30fc\u30bb\u30eb\u30c8\u306f\u3053\u308c\u3089\u306e\u7279\u6027\u3092\u6700\u521d\u306b\u89b3\u5bdf\u3057\u307e\u3057\u305f\u304c\u3001\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306e\u4f8b\u3092\u898b\u3064\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3067\u3057\u305f\u3002 1910 \u5e74\u306b\u30ed\u30d0\u30fc\u30c8 \u30c0\u30cb\u30a8\u30eb \u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u306f\u3053\u308c\u3089\u306e\u6570\u5b57\u306e\u4e2d\u3067\u6700\u3082\u5c0f\u3055\u3044 561 \u3092\u767a\u898b\u3057\u3001\u3053\u308c\u3089\u306f\u5f7c\u306e\u540d\u8a89\u3092\u8b83\u3048\u3066\u547d\u540d\u3055\u308c\u307e\u3057\u305f\u3002<\/p>

\u3053\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570 561 \u306f\u3001\u30b3\u30eb\u30bb\u30eb\u30c8\u306e\u5b9a\u7406\u3067\u691c\u8a3c\u3067\u304d\u307e\u3059\u3002\u5b9f\u969b\u3001561 = 3 \u00b7 11 \u00b7 17 \u306f\u5e73\u65b9\u7d20\u6570\u3067\u5272\u308a\u5207\u308c\u305a\u30013 – 1 = 2\u300111 – 1 = 10\u300117 – 1 = 16 \u306f\u3059\u3079\u3066 560 \u306e\u7d04\u6570\u3067\u3059\u3002<\/p>

\u6700\u521d\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002<\/p>

561 = 3 \u00b7 11 \u00b7 17<\/dd>
1105 = 5 \u00b7 13 \u00b7 17<\/dd>
1729 = 7 \u00b7 13 \u00b7 19<\/dd>
2465 = 5 \u00b7 17 \u00b7 29<\/dd>
2,821 = 7 \u00b7 13 \u00b7 31<\/dd>
6,601 = 7 \u00b7 23 \u00b7 41<\/dd>
8,911 = 7 \u00b7 19 \u00b7 67<\/dd><\/dl>

\u6700\u521d\u306e 33 \u500b\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306e\u6607\u9806\u30b7\u30fc\u30b1\u30f3\u30b9\u306f\u3001\u96fb\u5b50\u6574\u6570\u5217\u767e\u79d1\u4e8b\u5178 (OEIS) \u306e\u30b7\u30fc\u30b1\u30f3\u30b9 n\u00b0A002997 \u3068\u3001\u6700\u521d\u306e 8241 \u500b\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306e\u30b7\u30fc\u30b1\u30f3\u30b9 (\u6607\u9806\u306b\u5206\u985e\u3055\u308c\u3001\u6700\u521d\u306b\u305d\u306e\u56e0\u6570\u306b\u5206\u89e3\u3055\u308c\u307e\u3059) \u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002 \uff09\u3092\u3053\u3053\u306b\u793a\u3057\u307e\u3059\u3002<\/p>

J. Chernick \u306f 1939 \u5e74\u306b\u3001\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306e\u30b5\u30d6\u30bb\u30c3\u30c8<\/a><\/span>\u306e\u69cb\u7bc9\u306b\u4f7f\u7528\u3067\u304d\u308b\u5b9a\u7406\u3092\u5b9f\u8a3c\u3057\u307e\u3057\u305f\u3002\u6570\u5024 (6 k<\/i> + 1)(12 k<\/i> + 1)(18 k<\/i> + 1) \u306f\u3001\u305d\u306e 3 \u3064\u306e\u7d04\u6570\u304c\u3059\u3079\u3066\u7d20\u6570\u3067\u3042\u308c\u3070\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u3067\u3059\u3002 k \u304c\u6574\u6570\u306e\u96c6\u5408<\/a><\/span>\u3092\u8868\u3059\u5834\u5408\u3001\u3053\u306e\u5f0f\u307e\u305f\u306f\u305d\u308c\u306b\u985e\u4f3c\u3057\u305f\u4ed6\u306e\u5f0f\u304c\u7121\u9650\u306b\u591a\u304f\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u3092\u751f\u6210\u3059\u308b\u304b\u3069\u3046\u304b\u306f\u4e0d\u660e\u3067\u3059\u3002<\/p>

\u30dd\u30fc\u30eb\u30fb\u30a8\u30eb\u30c7\u30b7\u30e5\u306f\u767a\u898b\u7684\u306b\u3001<\/a><\/span>\u7121\u9650\u306b\u591a\u304f\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u304c\u5b58\u5728\u3059\u308b\u306f\u305a\u3060\u3068\u4e3b\u5f35\u3057\u307e\u3057\u305f\u3002\u3053\u306e\u4e88\u60f3\u306f<\/a><\/span>1994 \u5e74\u306b William Alford\u3001Andrew Granville\u3001Carl Pomerance \u306b\u3088\u3063\u3066\u5b9f\u8a3c\u3055\u308c\u3001\u3088\u308a\u6b63\u78ba\u306b\u306f\u3001\u5341\u5206\u306b\u5927\u304d\u306an<\/i>\u306b\u5bfe\u3057\u3066\u30011 \u3068n<\/i>\u306e\u9593\u306b\u5c11\u306a\u304f\u3068\u3082n<\/i> 2\/7<\/sup>\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u304c\u5b58\u5728\u3057\u307e\u3059\u3002<\/p>

\u30ea\u30c1\u30e3\u30fc\u30c9\u30fbGE\u30fb\u30d4\u30f3\u30c1\u306f\u3001\u3059\u3079\u3066\u306e<\/a><\/span>n<\/i>\u306b\u5bfe\u3057\u3066\u3001\u305d\u308c\u4ee5\u4e0a\u306e\u3082\u306e\u306f\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u3092\u5b9f\u8a3c\u3057\u307e\u3057\u305f\u3002

$$ {n*{exp({-K{\\frac {\\ln \\left( n \\right) \\ln \\left( \\ln \\left( \\ln \\left( n \\right) \\right) \\right) }{\\ln \\left( \\ln \\left( n \\right) \\right) }}}})} $$<\/div>\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u306f\u3001 K<\/i><\/span>\u3092\u6240\u5b9a\u306e\u5b9a\u6570\u3068\u3057\u3066 1 \u304b\u3089n<\/i>\u307e\u3067\u306e\u756a\u53f7\u3092\u4ed8\u3051\u307e\u3059\u3002<\/p>

\u6b21\u306e\u8868\u306b\u3001<\/a><\/span>\u3053\u306e\u5b9a\u6570\u306e\u304a\u304a\u3088\u305d\u306e\u5024\u3092\u793a\u3057\u307e\u3059\u3002<\/p>
n<\/i><\/td> K<\/td><\/tr>
4<\/td> 2.19547<\/td><\/tr>
6<\/td> 1.97946<\/td><\/tr>
8<\/td> 1.90495<\/td><\/tr>
10<\/td> 1.86870<\/td><\/tr>
12<\/td> 1.86377<\/td><\/tr>
14<\/td> 1.86293<\/td><\/tr>
16<\/td> 1.86406<\/td><\/tr>
18<\/td> 1.86522<\/td><\/tr>
20<\/td> 1.86598<\/td><\/tr><\/table>

2007 \u5e74 12 \u6708\u3068\u540c\u69d8\u306b\u3001 10 20<\/sup><\/span>\u307e\u3067\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f 8220777 \u500b\u3042\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u307e\u3057\u305f\u3002<\/p>

\n\"\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570<\/figure>

\u30d7\u30ed\u30d1\u30c6\u30a3<\/span><\/h3>

\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306b\u306f\u5c11\u306a\u304f\u3068\u3082 3 \u3064\u306e\u7d20\u56e0\u6570\u304c\u3042\u308a\u307e\u3059\u3002<\/p>

\u305d\u308c\u305e\u308c\u5c11\u306a\u304f\u3068\u3082k<\/i> = 3\u30014\u30015\u3001… \u306e\u7d20\u56e0\u6570\u3092\u6301\u3064\u6700\u521d\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059 (OEIS \u306e\u30b9\u30a4\u30fc\u30c8 A006931)\u3002<\/p>
k<\/i><\/th><\/th><\/tr>
3<\/td> 561 = 3 \u00b7 11 \u00b7 17<\/td><\/tr>
4<\/td> 41041 = 7 \u00b7 11 \u00b7 13 \u00b7 41<\/td><\/tr>
5<\/td> 825265 = 5 \u00b7 7 \u00b7 17 \u00b7 19 \u00b7 73<\/td><\/tr>
6<\/td> 321197185 = 5 \u00b7 19 \u00b7 23 \u00b7 29 \u00b7 37 \u00b7 137<\/td><\/tr>
7<\/td> 5394826801 = 7 \u00b7 13 \u00b7 17 \u00b7 23 \u00b7 31 \u00b7 67 \u00b7 73<\/td><\/tr>
8<\/td> 232250619601 = 7 \u00b7 11 \u00b7 13 \u00b7 17 \u00b7 31 \u00b7 37 \u00b7 73 \u00b7 163<\/td><\/tr>
9<\/td> 9746347772161 = 7 \u00b7 11 \u00b7 13 \u00b7 17 \u00b7 19 \u00b7 31 \u00b7 37 \u00b7 41 \u00b7 641<\/td><\/tr><\/table>

4 \u3064\u306e\u7d20\u56e0\u6570\u3092\u542b\u3080\u6700\u521d\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059 (OEIS \u30b9\u30a4\u30fc\u30c8 A074379)\u3002<\/p>
\u79c1<\/i><\/th><\/th><\/tr>
1<\/td> 41041 = 7 \u00b7 11 \u00b7 13 \u00b7 41<\/td><\/tr>
2<\/td> 62745 = 3 \u00b7 5 \u00b7 47 \u00b7 89<\/td><\/tr>
3<\/td> 63973 = 7 \u00b7 13 \u00b7 19 \u00b7 37<\/td><\/tr>
4<\/td> 75361 = 11 \u00b7 13 \u00b7 17 \u00b7 31<\/td><\/tr>
5<\/td> 101101 = 7 \u00b7 11 \u00b7 13 \u00b7 101<\/td><\/tr>
6<\/td> 126217 = 7 \u00b7 13 \u00b7 19 \u00b7 73<\/td><\/tr>
7<\/td> 172081 = 7 \u00b7 13 \u00b7 31 \u00b7 61<\/td><\/tr>
8<\/td> 188461 = 7 \u00b7 13 \u00b7 19 \u00b7 109<\/td><\/tr>
9<\/td> 278545 = 5 \u00b7 17 \u00b7 29 \u00b7 113<\/td><\/tr>
10<\/td> 340561 = 13 \u00b7 17 \u00b7 23 \u00b7 67<\/td><\/tr><\/table>

\u9762\u767d\u3044\u5076\u7136\u306e\u4e00\u81f4\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u30023 \u756a\u76ee\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u3068\u6700\u521d\u306e\u30c1\u30a7\u30eb\u30cb\u30c3\u30af\u6570 1729 \u306f\u3001\u30cf\u30fc\u30c7\u30a3\u30fb\u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u6570\u306b\u307b\u304b\u306a\u308a\u307e\u305b\u3093\u3002\u3064\u307e\u308a\u30012 \u3064\u306e\u7570\u306a\u308b\u65b9\u6cd5\u3067 2 \u3064\u306e\u7acb\u65b9\u4f53\u306e\u5408\u8a08\u3068\u3057\u3066\u66f8\u304d\u8fbc\u3080\u3053\u3068\u304c\u3067\u304d\u308b\u6700\u5c0f\u306e\u6b63\u306e\u6574\u6570\u3067\u3059\u3002 (1729 = 7 * 13* 19 = 10 3<\/sup> + 9 3<\/sup> = 12 3<\/sup> + 1 3<\/sup> )\u3002\u540c\u69d8<\/a><\/span>\u306b\u30012 \u756a\u76ee\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570 1105 \u306f\u3001\u305d\u308c\u3088\u308a\u5c0f\u3055\u3044\u6574\u6570\u3088\u308a\u3082\u591a\u304f\u306e\u65b9\u6cd5\u3067 2 \u3064\u306e\u5e73\u65b9\u306e\u548c\u3068\u3057\u3066\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u30b7\u30fc\u30b1\u30f3\u30b9\u3092\u5b8c\u4e86\u3059\u308b\u306b\u306f\u3001\u6700\u521d\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570 561 \u3092 (\u4ed6\u306e\u81ea\u7136\u6570\u3068\u540c\u69d8\u306b) \u3088\u308a\u5c0f\u3055\u306a\u6b63\u306e\u6574\u6570\u3088\u308a\u3082\u591a\u304f\u306e\u65b9\u6cd5\u3067\u6b63\u306e\u6574\u6570\u306e\u5358\u9805\u7d2f\u4e57\u306e\u5408\u8a08\u3068\u3057\u3066\u8a18\u8ff0\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>

\n\"\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570<\/figure>

\u30b3\u30fc\u30bb\u30eb\u30c8\u306e\u5b9a\u7406\u306e\u8a3c\u660e<\/a><\/span><\/span><\/h3>

p \u3092<\/i>\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570n \u3092<\/i>\u5272\u308b\u7d20\u6570\u3068\u3057\u307e\u3057\u3087\u3046\u3002 ( p<\/i> +1) n<\/i><\/sup> =1+ np<\/i> +C n<\/i><\/sub> 2<\/sup> p<\/i> 2<\/sup> +…\u22611 (mod p<\/i> 2<\/sup> ) \u3068\u306a\u308a\u307e\u3059\u3002\u4e00\u65b9\u3001 ( p<\/i> +1) n<\/i><\/sup> \u2261 p<\/i> n<\/i><\/sup> +1 \u2261 p<\/i> + 1(mod n<\/i> ) (n \u304c\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u3067\u3042\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u304b\u3089 2 \u756a\u76ee\u306e\u5408\u540c\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059)\u3002 p<\/i> 2<\/sup>\u304cn<\/i>\u3092\u5272\u3063\u305f\u5834\u5408\u3001 1\u2261( p<\/i> +1) n<\/i><\/sup> \u2261 p<\/i> +1 (mod p<\/i> 2<\/sup> ) \u306b\u306a\u308a\u307e\u3059\u304c\u3001\u3053\u308c\u306f\u4e0d\u53ef\u80fd\u3067\u3059\u3002\u3053\u308c\u306f\u3001 p<\/i>\u306e\u4e8c\u4e57\u304cn \u3092<\/i>\u5272\u3089\u306a\u3044\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002<\/p>

n \u306f<\/i>\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u3067\u3042\u308a\u3001 p \u306f<\/i>n<\/i>\u306e\u7d20\u56e0\u6570\u3067\u3042\u308b\u304b\u3001\u307e\u305f\u306fp<\/i>\u306e\u539f\u59cb\u6839\u3092\u6301\u3061\u307e\u3059<\/i>( p<\/i>\u3092\u4f34\u3046\u7d20\u6570<\/i>\u3068\u306a\u308a\u307e\u3059)\u3002

$$ {a^{n} \\equiv a \\pmod{n}} $$<\/div>\u305d\u308c\u3067
$$ {a^{n} \\equiv a \\pmod{p}} $$<\/div> p \u306f<\/i>n \u3092<\/i>\u9664\u7b97\u3059\u308b\u305f\u3081\u3001\u3064\u307e\u308a\u518d\u3073
$$ {a^{n-1} \\equiv 1 \\pmod{p}} $$<\/div> a \u306f p \u3068\u7d20\u6570\u306a\u306e\u3067\u3002\u3053\u308c\u304b\u3089\u3001 n<\/i> -1 \u306f\u6cd5<\/i>p<\/i>\u306e\u6b21<\/a><\/span>\u6570\u306e\u500d\u6570\u3001\u3064\u307e\u308ap<\/i> -1 \u306e\u500d\u6570\u3067\u3042\u308b\u3068\u63a8\u6e2c\u3057\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4efb\u610f\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570n<\/i>\u306b\u3064\u3044\u3066\u3001\u305d\u306e\u7d20\u56e0\u6570p<\/i>\u306e\u305d\u308c\u305e\u308c\u306f\u3001 p<\/i> -1 \u304cn<\/i> -1 \u3092\u9664\u7b97\u3059\u308b\u3053\u3068\u3092\u691c\u8a3c\u3057\u307e\u3059\u3002<\/p>

\u9006\u306b\u3001 n \u304c<\/i>\u3059\u3079\u3066\u306e\u7570\u306a\u308b\u7d20\u6570p<\/i> 1<\/sub> \u3001 p<\/i> 2<\/sub> \u3001… p<\/i> k<\/i><\/sub>\u306e\u7a4d\u3067\u3042\u308a\u3001\u6570\u5024p<\/i> 1<\/sub> -1\u3001 p<\/i> 2<\/sub> -1\u3001… \u306f\u3059\u3079\u3066n<\/i> -1 \u3092\u9664\u7b97\u3059\u308b\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002\u6b21\u306b\u3001\u3059\u3079\u3066\u306e\u6574\u6570a<\/i>\u3068\u3059\u3079\u3066\u306ep<\/i> i<\/i><\/sub>\u306b\u3064\u3044\u3066\u3001 a n<\/i> \u22611 (mod p<\/i> i<\/i><\/sub> -1) \u3068\u306a\u308b\u305f\u3081\u3001 a<\/i> \u2261 a<\/i> p<\/i> i<\/i><\/sub><\/sup> \u2261 a<\/i> 2 p<\/i> i<\/i><\/sub> -1<\/sup> \u2261 a<\/i> 3 p<\/i> i<\/i><\/sub> -2<\/sup> \u2261…\u2261 a<\/i> n<\/i><\/sup> ( mod p<\/i> i<\/i><\/sub> )\u3002\u6570\u5024a<\/i> n \u306f\u3001<\/i><\/sup>\u5404p<\/i> i<\/i><\/sub>\u306e\u30e2\u30b8\u30e5\u30ed<\/i>\u306b\u4e00\u81f4\u3059\u308b\u305f\u3081\u3001\u305d\u306e\u7a4dn<\/i>\u306e\u30e2\u30b8\u30e5\u30ed\u306b\u3082\u4e00\u81f4\u3057\u307e\u3059\u3002\u3053\u308c\u306f\u4efb\u610f\u306e\u6574\u6570a<\/i>\u306b\u5f53\u3066\u306f\u307e\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001 n \u306f<\/i>\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306b\u306a\u308a\u307e\u3059\u3002<\/p>

\u3053\u308c\u3067\u30b3\u30fc\u30bb\u30eb\u30c8\u306e\u5b9a\u7406\u306e\u8a3c\u660e\u304c\u5b8c\u4e86\u3057\u307e\u3057\u305f\u3002<\/p>

\u30b3\u30fc\u30bb\u30eb\u30c8\u306e\u5b9a\u7406\u306e\u7d50\u679c:<\/p>

n \u3092<\/i>\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u3068\u3057\u307e\u3059\u3002\u3053\u308c\u306f\u3044\u304f\u3064\u304b\u306e\u7570\u306a\u308b\u7d20\u6570\u3067\u5272\u308a\u5207\u308c\u308b\u306e\u3067\u3001\u305d\u306e\u3046\u3061\u306e\u5c11\u306a\u304f\u3068\u3082 1 \u3064\u306f 2 \u3068\u306f\u7570\u306a\u308a\u307e\u3059\u3002\u3053\u306e<\/i>\u5947\u6570\u306en p<\/i>\u306e\u7d20\u7d04\u6570\u3092 p \u3068\u547c\u3073\u307e\u3057\u3087\u3046\u3002 p<\/i> -1 \u306f\u5076\u6570\u306a\u306e\u3067\u3001\u305d\u306e\u500d\u6570n<\/i> -1 \u3082\u5076\u6570\u3067\u3059\u3002\u3053\u308c\u306f\u3001\u3059\u3079\u3066\u306e\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u304c\u5947\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002<\/p>

p \u304c<\/i>\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570n<\/i>\u306e\u7d20\u56e0\u6570\u3067\u3042\u308b\u5834\u5408\u3001 p<\/i> -1 \u3092\u6cd5\u3068\u3057\u3066p<\/i> \u22611\u2261 n<\/i>\u304c\u5f97\u3089\u308c\u3001\u3057\u305f\u304c\u3063\u3066 1\u2261( n<\/i> \/ p<\/i> ) p<\/i> \u2261( n<\/i> \/ p<\/i> )1= n<\/i> \/ p<\/i>\u3068\u306a\u308a\u307e\u3059\u3002\u8a00\u3044\u63db\u3048\u308c\u3070\u3001 p \u304c<\/i>\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306e\u7d20\u56e0\u6570\u3067\u3042\u308b\u5834\u5408\u3001\u4ed6\u306e\u7d20\u56e0\u6570\u306e\u7a4d\u306f 1 \u3092\u6cd5\u3068\u3059\u308bp<\/i> -1 \u306b\u4e00\u81f4\u3057\u307e\u3059\u3002<\/p>

\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f\u30012 \u3064\u306e\u7d20\u6570p<\/i>\u3068q<\/i>\u306e\u7a4d\u306b\u306a\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u305d\u306e\u5834\u5408\u30012 \u3064\u306e\u6570p<\/i> -1 \u3068q<\/i> -1 \u306f\u305d\u308c\u305e\u308c\u4ed6\u65b9\u3092\u9664\u7b97\u3057\u3001\u7b49\u3057\u304f\u306a\u308b\u304b\u3089\u3067\u3059\u3002<\/p>

\u3057\u305f\u304c\u3063\u3066\u3001\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570\u306f\u3001\u5c11\u306a\u304f\u3068\u3082 3 \u3064\u306e\u7570\u306a\u308b\u5947\u6570\u7d20\u6570\u306e\u7a4d\u306b\u306a\u308a\u307e\u3059\u3002<\/p><\/div>

\n\"\u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. \u0639\u062f\u062f \u0643\u0627\u0631\u0645\u064a\u0643\u0627\u0626\u064a\u0644 \u2013 arabe<\/a><\/li>
  2. Nombres de Carmichael \u2013 catalan<\/a><\/li>
  3. Carmichaelovo \u010d\u00edslo \u2013 tch\u00e8que<\/a><\/li>
  4. Carmichael-Zahl \u2013 allemand<\/a><\/li>
  5. Carmichael number \u2013 anglais<\/a><\/li>
  6. Nombro de Carmichael \u2013 esp\u00e9ranto<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u30ab\u30fc\u30de\u30a4\u30b1\u30eb\u6570 – \u5b9a\u7fa9\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n