Prvočísla/8191
8191 je prvočíslo. Z hlediska délky periody převrácené hodnoty má relativně dosti výjimečné vlastnosti: je unikátním prvočíslem (v číselné soustavě o základu 91 o délce period převrácené hodnoty l = 6 a také jedním z jediných dvou repunitovým prvočíslem s délkou > 2, vyskytujícím se zároveň ve dvou nemocninových číselných soustavách: ve dvojkové soustavě (l = 13 a zároveň v devadesátkové soustavě (l = 3); (dalším je prvočíslo 31). Z tohoto důvodu tomuto prvočíslu věnuji větší pozornost.
Prvočíselný (nejen) rozklad kořene
[editovat]Kořen tohoto prvočísla k = 8190. Prvočíselné w:faktory tohoto kořene jsou: 2, 3, 3, 5, 7 a 13. Zároveň je tento kořen součinem dvou po sobě jdoucích čísel a to 90 a 91, což je typická vlastnost repunitů o délce 3, zmenšených o 1. Zároveň je tento kořen roven 2^13 - 2, což je opět typická vlastnost kořenů Mersenových prvočísel. Z rozboru prvočíselného rozkladu kořene vyplývá, že v různých soustavách se mohou vyskytovat tyto délky period převrácené hodnoty tohoto prvočísla: 0 (pro z = 8191 + n*8191), 1 (pro z = 8192 + n*8191 - dále již následující možnosti "+ n*8191" nebudu uvádět, jsou samozřejmé), 2 (pro z = 8190), 3 (pro z = 90 a pro z = 8100), ...další v tabulce (celkový počet* je míněno menších z než 8193 a větších, než 1):
| l | z | celkový počet* |
|---|---|---|
| 0 | 8191 | 1 |
| 1 | 8192 | 1 |
| 2 | 8190 | 1 |
| 3 | 90, 8100 | 2 |
| 5 | 1904, 3002, 4794, 6681 | 4 |
| 6 | 91, 8101 | 2 |
| 7 | 1378, 3085, 6247, 6763, 7474, 7816 | 6 |
| 9 | 1477, 1874, 2723, 4840, 6128, 7531 | 6 |
| 10 | 1510, 3397, 5189, 6287 | 4 |
| 13 | 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096 | 12 |
| 14 | 375, 717, 1428, 1944, 5106, 6813 | 6 |
| 15 | 3347, 5312, 5528, 6060, 6354, 6938, 7540, 8068 | 8 |
| 18 | 660, 2063, 3351, 5468, 6317, 6714 | 6 |
| 21 | 998, 1155, 1361, 2536, 4893, 5242, 5658, 5950, 7083, 7205, 7347, 7910 | 12 |
| 26 | 4095, 6143, 7167, 7679, 7926, 8063, 8127, 8159, 8175, 8183, 8187, 8189 | 12 |
| 30 | 123, 651, 1253, 1837, 2131, 2663, 2879, 4844 | 8 |
| 35 | 301, 500, 893, 956, 1071, 1458, 1799, 1822, 1844, 2047, 2329, 2592, 2729, 2922, 3062, 4186, 4270, 4295, 4608, 4735, 5228, 5340, 6808, 7925 | 24 |
| 39 | 45, 180, 360, 720, 1277, 1440, 2025, 2059, 2367, 2554, 2880, 3329, 4050, 4118, 4734, 5108, 5125, ... | 24 |
| 42 | 281, 844, 986, 1108, 2241, 2533, 2949, 3298, 5655, 6830, 7036, 7193 | 12 |
| 45 | 152, 485, 902, 2550, 2623, 2695, 3714, 3728, 4346, 4706, 4774, 5011, 5489, 5799, ... | 24 |
| 63 | 52, 515, 816, 1676, 1684, 1770, 1939, 2046, 2207, 2281, 2349, 2385, 2499, 2704, 2750, 3133, 3402, 3459, ... | 36 |
| 65 | 119, 238, 413, 476, 826, 952, 1397, 1501, 1652, 1859, 2151, 2397, 2423, 2647, 2794, 2985, 3304, 3591, 3718, ... | 48 |
| 70 | 266, 1383, 2851, 2963, 3456, 3583, 3896, 3921, 4005, 5129, 5269, 5462, 5599, 5862, 6144, 6347, 6369, 6392, 6733, 7120, 7235, 7298, 7691, 7890 | 24 |
| 78 | 182, 364, 728, 1456, 1533, 2431, 2912, 3066, 3083, 3457, 4073, 4141, 4862, 5311, 5637, 5824, 6132, 6166, 6751, 6914, ... | 24 |
| 90 | 172, 311, 730, 1154, 1469, 1571, 2028, 2143, 2314, 2318, 2392, 2702, 3180, 3417, 3485, ... | 24 |
| 91 | 107, 214, 415, 428, 555, 573, 689, 830, 830, 856, 977, 1110, 1146, 1491, 1629, 1660, 1712, 1725, 1954, 1987, ... | 72 |
| 105 | 111, 160, 164, 218, 633, 647, 831, 868, 1027, 1573, 1667, 2116, 2140, 2323, 2517, 2988, 3105, 3238, 3633, 3646, ... | 48 |
| 117 | 383, 605, 663, 766, 853, 937, 1141, 1210, 1326, 1489, 1532, 1706, 2199, 22282, 2333, 2417, 2420, 2613, 2631, ... | 72 |
| 126 | 279, 334, 537, 793, 1556, 1858, 2126, 2370, 2796, 2933, 2947, 3118, 3400, 3511, 4069, 4253, 4438, 4520, 4732, ... | 36 |
| 130 | 557, 575, 693, 721, 755, 1009, 1114, 1150, 1386, 1442, 1583, 2018, 2187, 2221, 2228, 2300, 2603, 2772, , ... | 48 |
| 182 | 243, 357, 486, 714, 750, 972, 1291, 1343, 1500, 1551, 1675, 1909, 2021, 2137, 2227, 2525, 2553, 2582, 2679, 2686, ... | 72 |
| 195 | 83, 166, 319, 332, 621, 638, 664, 691, 843, 859, 1242, 1276, 1328, 1382, 1421, 1515, 1541, 1579, 1686, 1718, 1745, ... | 96 |
| 210 | 46, 120, 148, 367, 670, 677, 1540, 1587, 1605, 1622, 1902, 1910, 1982, 2609, ... | 48 |
| 234 | 61, 122, 165, 244, 330, 488, 695, 749, 941, 976, 1285, 1320, 1367, 1390, ... | 72 |
| 273 | 201, 303, 317, 402, 427, 499, 597, 606, 634, 804, 854, 947, 981, 1049, 1194, 1212, 1268, 1439, 1505, 1567, 1595, 1608, ... | 144 |
| 315 | 127, 129, 145, 210, 259, 286, 337, 355, 372, 459, 475, 490, 495, 523, 566, 716, 761, 771, 836, 880, 883, 907, 940, 1004, 1142, ... | 144 |
| 390 | 246, 333, 492, 666, 709, 984, 1211, 1302, 1319, 1332, 1418, 1447, 1497, 1653, 1669, 1833, 1875, 1968, 2027, 2225, 2361, ... | 96 |
| 455 | 9, 18, 36, 72, 81, 125, 144, 162, 181, 231, 239, 250, 288, 324, 349, 362, 377, 399, 461, 462, 478, 517, 576, 602, 648, 698, ... | 288 |
| 546 | 163, 207, 211, 277, 326, 343, 379, 414, 422, 493, 554, 562, 615, 652, 673, 686, 857, 773, 801, 828, 1059, 1124, 1230, 1304, ... | 144 |
| 585 | 19, 38, 76, 79, 157, 158, 189, 233, 304, 314, 316, 361, 378, 391, 451, 466, 469, 501, 608, 628, 632, 659, 722, 756, 782, 855, ... | 288 |
| 630 | 39, 114, 253, 280, 287, 496, 612, 614, 627, 705, 753, 767, 814, 863, 875, 946, 958, 1018, 1028, ... | 144 |
| 819 | 13, 26, 49, 51, 53, 98, 102, 104, 106, 169, 196, 204, 208, 212, 219, 331, 338, 392, 408, 416, 419, 421, 423, 424, 438, 585, 629, ... | 432 |
| 910 | 3, 6, 12, 24, 27, 48, 54, 77, 96, 108, 133, 154, 185, 191, 192, 216, 308, 321, 370, 382, 384, 432, 445, 455, 487, 497, 532, 543, ... | 288 |
| 1170 | 43, 86, 87, 137, 155, 174, 175, 274, 299, 310, 329, 344, 348, 350, 365, 381, 431, 457, 507, 548, 563, 577, 598, 620, 622, 658, ... | 288 |
| 1365 | 5, 10, 20, 25, 40, 41, 50, 69, 80, 82, 100, 109, 138, 200, 205, 217, 222, 223, 225, 263, 267, 273, 276, 320, 328, 345, 353, 369, 400, ... | 576 |
| 1638 | 7, 14, 28, 56, 112, 167, 187, 213, 215, 224, 285, 293, 313, 315, 323, 371, 374, 389, 401, 425, 426, 430, 435, 448, 558, 565, 570, ... | 432 |
| 2730 | 15, 23, 30, 37, 60, 67, 74, 75, 89, 92, 101, 115, 134, 135, 150, 151, 178, 184, 199, 202, 230, 240, 249, 268, 270, 296, 300, 302, 327, ... | 576 |
| 4095 | 11, 21, 22, 29, 42, 44, 47, 55, 58, 65, 71, 84, 88, 93, 94, 95, 97, 99, 103, 105, 110, 116, 117, 121, 130, 131, 142, 143, 149, 161, ... | 1728 |
| 8190 | 17, 31, 33, 34, 35, 57, 59, 62, 63, 66, 68, 70, 73, 78, 85, 113, 118, 124, 126, 132, 136, 139, 140, 141, 146, 147, 153, 156, 159, ... | 1728 |
Délky p.h. podle soustav
[editovat]| z | zápis p 8191 v dané soustavě | délka p.h. (l) |
|---|---|---|
| 2 | 1111111111111 | 13 |
| 3 | 102020101 | 910 |
| 4 | 1333333 | 13 |
| 5 | 230231 | 1365 |
| 6 | 101531 | 910 |
| 7 | 32611 | 1638 |
| 8 | 17777 | 13 |
| 9 | 12211 | 455 |
| 10 | 8191 | 1365 |
| 11 (χ) | 6177 | 4095 |
| 12 | 48A7 | 910 |
| 13 | 3961 | 819 |
| 14 | 2DB1 | 1638 |
| 15 | 2661 | 2730 |
| 16 | 1FFF | 13 |
| 17 | 1B5E | 8190 (max.) |
| 18 | 1751 (01:07:05:01) | 455 |
| 19 | 01:03:13:02 | 585 |
| 20 | 01:00:09:11 | 1365 |
| 21 | 18:12:01 | 4095 |
| 22 | 16:20:07 | 4095 |
| 23 | 15:11:03 | 2730 |
| 24 | 14:05:07 | 910 |
| 25 | 13:02:16 | 1365 |
| 26 | 12:03:01 | 819 |
| 27 | 11:06:10 | 910 |
| 28 | 10:12:15 | 1638 |
| 29 | 09:21:13 | 4095 |
| 30 | 09:03:01 | 2730 |
| 31 | 08:16:07 | 8190 |
| 32 | 07:31:31 | 13 |
| 33 | 07:17:07 | 8190 |
| 34 | 07:02:31 | 8190 |
| 35 | 06:24:01 | 8190 |
| 36 | 06:11:19 | 455 |
| 37 | 05:36:14 | 2730 |
| 38 | 05:25:21 | 585 |
| 39 | 05:15:01 | 630 |
| 40 | 05:04:31 | 1365 |
| 41 | 04:35:32 | 1365 |
| 42 | 04:27:01 | 4095 |
| 43 | 04:18:21 | 1170 |
| 44 | 04:10:07 | 4095 |
| 45 | 04:02:01 | 39 |
| 46 | 03:40:03 | 210 |
| 47 | 03:33:13 | 4095 |
| 48 | 03:26:31 | 910 |
| 49 | 03:20:08 | 819 |
| 50 | 03:13:41 | 1365 |
| 51 | 03:07:31 | 819 |
| 52 | 03:01:27 | 63 |
| 53 | 02:48:29 | 819 |
| 54 | 02:43:37 | 910 |
| 55 | 02:38:51 | 4095 |
| 56 | 02:34:15 | 1638 |
| 57 | 02:29:40 | 8190 |
| 58 | 02:25:13 | 4095 |
| 59 | 02:20:49 | 8190 |
| 60 | 02:16:31 | 2730 |
| 61 | 02:12:17 | 234 |
| 62 | 02:08:07 | 8190 |
| 63 | 02:04:01 | 8190 |
| 64 | 01:63:63 | 13 |
| 65 | 01:61:01 | 4095 |
| 66 | 01:58:07 | 8190 |
| 67 | 01:55:17 | 2730 |
| 68 | 01:52:31 | 8190 |
| 69 | 01:49:49 | 1365 |
| 70 | 01:47:01 | 8190 |
| 71 | 01:44:26 | 4095 |
| 72 | 01:41:55 | 455 |
| 73 | 01:39:15 | 8190 |
| 74 | 01:36:51 | 2730 |
| 75 | 01:34:16 | 2730 |
| 76 | 01:31:59 | 585 |
| 77 | 01:29:29 | 910 |
| 78 | 01:27:01 | 8190 |
| 79 | 01:24:54 | 585 |
| 80 | 01:22:31 | 1365 |
| 81 | 01:20:10 | 455 |
| 82 | 01:17:73 | 1365 |
| 83 | 01:15:57 | 195 |
| 84 | 01:13:43 | 4095 |
| 85 | 01:11:31 | 8190 |
| 86 | 01:09:21 | 1170 |
| 87 | 01:07:13 | 1170 |
| 88 | 01:05:07 | 4095 |
| 89 | 01:03:03 | 2730 |
| 90 | 01:01:01 | 3 |
| 91 | 90:01 | 6 |
| 92 | 89:03 | 2730 |
| 93 | 88:07 | 4095 |
| 94 | 87:13 | 4095 |
| 95 | 86:21 | 4095 |
| 96 | 85:31 | 910 |
| 97 | 84:43 | 4095 |
| 98 | 83:57 | 819 |
| 99 | 82:73 | 4095 |
| 100 | 81:91 | 1365 |
| 101 | 081:010 | 2730 |
| 102 | 080:031 | 819 |
| 103 | 079:054 | 4095 |
| 104 | 078:079 | 819 |
| 105 | 078:001 | 4095 |
| 106 | 077:029 | 819 |
| 107 | 076:059 | 91 |
| 108 | 075:091 | 910 |
| 109 | 075:016 | 1365 |
| 110 | 074:051 | 4095 |
| 111 | 073:088 | 105 |
| 112 | 073:015 | 1638 |
| 113 | 072:055 | 8190 |
| 114 | 071:097 | 630 |
| 115 | 071:026 | 2730 |
| 116 | 070:071 | 4095 |
| 117 | 070:001 | 4095 |
| 118 | 069:049 | 8190 |
| 119 | 068:099 | 65 |
| 120 | 068:031 | 210 |
| 121 | 067:084 | 4095 |
| 122 | 067:017 | 234 |
| 123 | 066:073 | 30 |
| 124 | 066:007 | 8190 |
| 125 | 065:066 | 455 |
| 126 | 065:001 | 8190 |
| 127 | 064:063 | 315 |
| 128 | 063:127 | 13 |
| 129 | 063:064 | 315 |
| ... | ... | ... |