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Primi diedri (II)

Rappresentazione dei numeri 

Sono talvolta chiamati “primi diedri” i numeri primi che restano primi, ma non necessariamente uguali a se stessi, se su un visore a 7 segmenti sono letti al contrario o riflessi specularmente.

Se si ruota di 180° un visore a 7 segmenti (ossia se lo si legge dalla parte opposta):

  • le cifre 0, 1, 2, 5 e 8 restano invariate;

  • le cifre 6 e 9 si scambiano.

Se invece lo si riflette specularmente:

  • le cifre 0, 1 e 8 restano invariate;

  • le cifre 2 e 5 si scambiano.

Quindi un primo diedro può contenere solo le cifre 0, 1, 2, 5 e 8.

 

Normalmente i primi diedri sono considerati solo in base 10, però siccome molte calcolatrici possono convertire i numeri nelle basi 2, 8 e 16, ho preferito considerare anche queste.

La tabella seguente mostra i primi diedri minori di 10000 in base 2 e minori di 109 nelle basi 8, 10 e 16.

Base

Primi diedri

2

112 = 3, 1012 = 5, 1112 = 7, 10112 = 11, 11012 = 13, 100012 = 17, 101112 = 23, 111012 = 29, 111112 = 31, 1001012 = 37, 1010012 = 41, 1010112 = 43, 1011112 = 47, 1101012 = 53, 1111012 = 61, 10000112 = 67, 10001112 = 71, 10010012 = 73, 10100112 = 83, 11000012 = 97, 11001012 = 101, 11010112 = 107, 11100012 = 113, 11111112 = 127, 100000112 = 131, 100101112 = 151, 101000112 = 163, 101001112 = 167, 101011012 = 173, 101101012 = 181, 110000012 = 193, 110001012 = 197, 110001112 = 199, 110111112 = 223, 111000112 = 227, 111001012 = 229, 111010012 = 233, 111110112 = 251, 1000000012 = 257, 1000001112 = 263, 1000011012 = 269, 1000101012 = 277, 1000110112 = 283, 1001100112 = 307, 1001110012 = 313, 1010010112 = 331, 1010100012 = 337, 1010111012 = 349, 1011000012 = 353, 1011001112 = 359, 1011101012 = 373, 1011111112 = 383, 1100110012 = 409, 1101001012 = 421, 1101011112 = 431, 1101100012 = 433, 1101110112 = 443, 1110000012 = 449, 1110011012 = 461, 1110011112 = 463, 1110111112 = 479, 1111001112 = 487, 1111010112 = 491, 1111101112 = 503, 1111111012 = 509, 10000010012 = 521, 10001110112 = 571, 10010000012 = 577, 10010101112 = 599, 10010110012 = 601, 10011010012 = 617, 10011010112 = 619, 10011101112 = 631, 10100000112 = 643, 10100011012 = 653, 10100101012 = 661, 10101001012 = 677, 10101010112 = 683, 10101100112 = 691, 10101111012 = 701, 10110001012 = 709, 10110101112 = 727, 10111000112 = 739, 10111101012 = 757, 11000001012 = 773, 11000111012 = 797, 11001101012 = 821, 11001101112 = 823, 11001110112 = 827, 11010001112 = 839, 11010101012 = 853, 11010110012 = 857, 11011100012 = 881, 11011100112 = 883, 11100010112 = 907, 11100011112 = 911, 11101010012 = 937, 11101011012 = 941, 11101100112 = 947, 11101110012 = 953, 11110001112 = 967, 100000010012 = 1033, 100000110012 = 1049, 100001111112 = 1087, 100010000112 = 1091, 100010001012 = 1093, 100010101012 = 1109, 100010111012 = 1117, 100011010012 = 1129, 100100000012 = 1153, 100100100112 = 1171, 100100111012 = 1181, 100101010012 = 1193, 100101100012 = 1201, 100110000012 = 1217, 100110011012 = 1229, 100111010112 = 1259, 101000010112 = 1291, 101000100012 = 1297, 101000101112 = 1303, 101000110112 = 1307, 101010100012 = 1361, 101010101112 = 1367, 101010111012 = 1373, 101100011112 = 1423, 101100100112 = 1427, 101100110012 = 1433, 101100111112 = 1439, 101101010112 = 1451, 101101011012 = 1453, 101101111112 = 1471, 101110010012 = 1481, 101110010112 = 1483, 101110011112 = 1487, 101110100012 = 1489, 101110101012 = 1493, 101111110112 = 1531, 110000100012 = 1553, 110000111112 = 1567, 110001000112 = 1571, 110001010112 = 1579, 110010001112 = 1607, 110010010012 = 1609, 110010011012 = 1613, 110010100112 = 1619, 110010110112 = 1627, 110100001012 = 1669, 110100111012 = 1693, 110101000112 = 1699, 110101011012 = 1709, 110101110012 = 1721, 110110001012 = 1733, 110110100112 = 1747, 110110111112 = 1759, 110111110112 = 1787, 110111111012 = 1789, 111000100112 = 1811, 111001001112 = 1831, 111010001012 = 1861, 111010101012 = 1877, 111010101112 = 1879, 111100011012 = 1933, 111100111012 = 1949, 111100111112 = 1951, 111110000112 = 1987, 111110011012 = 1997, 111110011112 = 1999, 111110110112 = 2011, 111111000012 = 2017, 111111011012 = 2029, 1000000101012 = 2069, 1000001000012 = 2081, 1000001000112 = 2083, 1000010000012 = 2113, 1000010111112 = 2143, 1000011100012 = 2161, 1000100000112 = 2179, 1000101011012 = 2221, 1000101111112 = 2239, 1000110000112 = 2243, 1000111000012 = 2273, 1000111010012 = 2281, 1000111101012 = 2293, 1001000001112 = 2311, 1001000111012 = 2333, 1001001001012 = 2341, 1001010011012 = 2381, 1001010011112 = 2383, 1001010101012 = 2389, 1001010110012 = 2393, 1001011010112 = 2411, 1001011100012 = 2417, 1001101010012 = 2473, 1001111010112 = 2539, 1001111101112 = 2551, 1010010010012 = 2633, 1010011101012 = 2677, 1010011110112 = 2683, 1010100000012 = 2689, 1010100100112 = 2707, 1010100101112 = 2711, 1010100111112 = 2719, 1010101010012 = 2729, 1010101010112 = 2731, 1010111001012 = 2789, 1010111001112 = 2791, 1010111100012 = 2801, 1011000110112 = 2843, 1011001010012 = 2857, 1011010100012 = 2897, 1011100010012 = 2953, 1011100100112 = 2963, 1011110010112 = 3019, 1011111110112 = 3067, 1100000010112 = 3083, 1100000100012 = 3089, 1100001011112 = 3119, 1100001100012 = 3121, 1100010000012 = 3137, 1100010110112 = 3163, 1100010111112 = 3167, 1100011100112 = 3187, 1100100101012 = 3221, 1100100111012 = 3229, 1100101110112 = 3259, 1100111000112 = 3299, 1100111110112 = 3323, 1101000000112 = 3331, 1101000011112 = 3343, 1101001111012 = 3389, 1101010101012 = 3413, 1101011010012 = 3433, 1101011110012 = 3449, 1101100011012 = 3469, 1101101000112 = 3491, 1101101101112 = 3511, 1101110001112 = 3527, 1101110100112 = 3539, 1101111001012 = 3557, 1101111100112 = 3571, 1101111111012 = 3581, 1101111111112 = 3583, 1110000010012 = 3593, 1110000101112 = 3607, 1110001011112 = 3631, 1110001110112 = 3643, 1110011101012 = 3701, 1110100001112 = 3719, 1110100011112 = 3727, 1110100101012 = 3733, 1110110110112 = 3803, 1110111110012 = 3833, 1111000010112 = 3851, 1111000101112 = 3863, 1111001010012 = 3881, 1111010000112 = 3907, 1111010001112 = 3911, 1111011111112 = 3967, 1111100101012 = 3989, 1111101000012 = 4001, 1111101000112 = 4003, 1111110100012 = 4049, 1111111011112 = 4079, 1111111110112 = 4091, 10000000011112 = 4111, 10000000111112 = 4127, 10000001110012 = 4153, 10000011010012 = 4201, 10000100001012 = 4229, 10000101001012 = 4261, 10000101011112 = 4271, 10000101100012 = 4273, 10000111001112 = 4327, 10000111111012 = 4349, 10001000010112 = 4363, 10001000101012 = 4373, 10001001110012 = 4409, 10001010111112 = 4447, 10001011000112 = 4451, 10001011010012 = 4457, 10001100110112 = 4507, 10001101000012 = 4513, 10001101001012 = 4517, 10001101010112 = 4523, 10001110000112 = 4547, 10001110001012 = 4549, 10001110101112 = 4567, 10010000011012 = 4621, 10010000111012 = 4637, 10010001010112 = 4651, 10010001101112 = 4663, 10010010100112 = 4691, 10010010111112 = 4703, 10010011111012 = 4733, 10010101100112 = 4787, 10010110000012 = 4801, 10010110011012 = 4813, 10010110100012 = 4817, 10010111111012 = 4861, 10011000001112 = 4871, 10011000011012 = 4877, 10011000110012 = 4889, 10011001001112 = 4903, 10011001101112 = 4919, 10011010001012 = 4933, 10011011011012 = 4973, 10011011110112 = 4987, 10011100000012 = 4993, 10011100100012 = 5009, 10011100111112 = 5023, 10011111011012 = 5101, 10011111110012 = 5113, 10100000110112 = 5147, 10100001000012 = 5153, 10100010001012 = 5189, 10100010011012 = 5197, 10100010110012 = 5209, 10100011011112 = 5231, 10100011100012 = 5233, 10100011101012 = 5237, 10100100011012 = 5261, 10100101000012 = 5281, 10100101100012 = 5297, 10100101101112 = 5303, 10100101111012 = 5309, 10100110101012 = 5333, 10100111000112 = 5347, 10101000100012 = 5393, 10101001101112 = 5431, 10101001111012 = 5437, 10101010000112 = 5443, 10101011001012 = 5477, 10101011010112 = 5483, 10101011111012 = 5501, 10101100000112 = 5507, 10101100101112 = 5527, 10101101101012 = 5557, 10101110001012 = 5573, 10101110011012 = 5581, 10101110101112 = 5591, 10101111101112 = 5623, 10110000010012 = 5641, 10110000110012 = 5657, 10110001001012 = 5669, 10110001111012 = 5693, 10110010001012 = 5701, 10110010011112 = 5711, 10110011010012 = 5737, 10110011101012 = 5749, 10110101011112 = 5807, 10110110011112 = 5839, 10110110110012 = 5849, 10110111011012 = 5869, 10110111110012 = 5881, 10111000010012 = 5897, 10111001000112 = 5923, 10111010111012 = 5981, 10111100011012 = 6029, 10111100101012 = 6037, 10111101001012 = 6053, 10111110010012 = 6089, 10111110010112 = 6091, 10111110101012 = 6101, 10111111000012 = 6113, 10111111010012 = 6121, 10111111100112 = 6131, 11000001101012 = 6197, 11000001110112 = 6203, 11000010000112 = 6211, 11000010101012 = 6229, 11000011100012 = 6257, 11000100110112 = 6299, 11000100111012 = 6301, 11000101001112 = 6311, 11000101100112 = 6323, 11000110100012 = 6353, 11000110101112 = 6359, 11000111001012 = 6373, 11000111010112 = 6379, 11001001100112 = 6451, 11001010010012 = 6473, 11001100100112 = 6547, 11001101000112 = 6563, 11001101010012 = 6569, 11001110001112 = 6599, 11001110110112 = 6619, 11001111111012 = 6653, 11010000100012 = 6673, 11010011111012 = 6781, 11010100001112 = 6791, 11010100010012 = 6793, 11010101010112 = 6827, 11010101100012 = 6833, 11010110101012 = 6869, 11010111000112 = 6883, 11010111111112 = 6911, 11011000001012 = 6917, 11011001000112 = 6947, 11011001100012 = 6961, 11011011100112 = 7027, 11011100000112 = 7043, 11011101111112 = 7103, 11011110101112 = 7127, 11011110110012 = 7129, 11100000110012 = 7193, 11100001010112 = 7211, 11100011100112 = 7283, 11100100110012 = 7321, 11100101000112 = 7331, 11100101101112 = 7351, 11100111000012 = 7393, 11101001101012 = 7477, 11101001111112 = 7487, 11101011000112 = 7523, 11101011100012 = 7537, 11101011101012 = 7541, 11101011110112 = 7547, 11101100010012 = 7561, 11101100101012 = 7573, 11101100110012 = 7577, 11101101001012 = 7589, 11101101001112 = 7591, 11101101101112 = 7607, 11101111101012 = 7669, 11110000000012 = 7681, 11110001011112 = 7727, 11110010011012 = 7757, 11110010011112 = 7759, 11110011011012 = 7789, 11110100011112 = 7823, 11110101000012 = 7841, 11110101011012 = 7853, 11110110001012 = 7877, 11110111011112 = 7919, 11111000000012 = 7937, 11111001110012 = 7993, 11111010010012 = 8009, 11111010100012 = 8017, 11111100101112 = 8087, 11111101110112 = 8123, 11111111010112 = 8171, 11111111111112 = 8191, 100000000111012 = 8221, 100000001001112 = 8231, 100000001010012 = 8233, 100000010011012 = 8269, 100000010111112 = 8287, 100000011010012 = 8297, 100000011101112 = 8311, 100000011111012 = 8317, 100000100010012 = 8329, 100000101000012 = 8353, 100000101010112 = 8363, 100000110001012 = 8389, 100000111011112 = 8431, 100001010000012 = 8513, 100001010010012 = 8521, 100001010111112 = 8543, 100001011100112 = 8563, 100001100001012 = 8581, 100001110001112 = 8647, 100001110101112 = 8663, 100001111010012 = 8681, 100010000011112 = 8719, 100010000110112 = 8731, 100010001001012 = 8741, 100010010010112 = 8779, 100010011000112 = 8803, 100010011001112 = 8807, 100010011111112 = 8831, 100010100001012 = 8837, 100010101111012 = 8893, 100010110110112 = 8923, 100010111101112 = 8951, 100011001010012 = 9001, 100011001101012 = 9013, 100011011000112 = 9059, 100011100000112 = 9091, 100011100011112 = 9103, 100011101011012 = 9133, 100011110101012 = 9173, 100011111100112 = 9203, 100100001111012 = 9277, 100100010000012 = 9281, 100100100001012 = 9349, 100100101000012 = 9377, 100100101110112 = 9403, 100100110010112 = 9419, 100100110110012 = 9433, 100100110111112 = 9439, 100100111110112 = 9467, 100101000000012 = 9473, 100101000100112 = 9491, 100101000110012 = 9497, 100101001001112 = 9511, 100101001100012 = 9521, 100101001111012 = 9533, 100101010000112 = 9539, 100101100000012 = 9601, 100101100011012 = 9613, 100101110011012 = 9677, 100101111000012 = 9697, 100110000010112 = 9739, 100110001010012 = 9769, 100110010100112 = 9811, 100110100000112 = 9859, 100110100110112 = 9883, 100110110010012 = 9929

8

28 = 2, 58 = 5, 1118 = 73, 10118 = 521, 11018 = 577, 1010118 = 33289, 1011118 = 33353, 1012018 = 33409, 1015018 = 33601, 1021018 = 33857, 1051018 = 35393, 1101018 = 36929, 1111018 = 37441, 10010018 = 262657, 11010118 = 295433, 100010118 = 2097673, 110001118 = 2359369, 110100018 = 2363393, 111000118 = 2392073, 112125518 = 2430313, 112252518 = 2435753, 115152218 = 2529937, 115525218 = 2544977, 122515118 = 2708297, 125255118 = 2796361, 152522118 = 3495049, 155212118 = 3580553, 1001001018 = 16810049, 1010010018 = 17039873, 1010010118 = 17039881, 1022022018 = 17368193, 1022152218 = 17373841, 1022555518 = 17390441, 1055055018 = 18254657, 1055125518 = 18257257, 1055222218 = 18261137, 1100200118 = 18882569, 1100500118 = 18894857, 1101001018 = 18907201, 1101112118 = 18911881, 1101115118 = 18912073, 1120105118 = 19403081, 1121110118 = 19436041, 1125152118 = 19569289, 1150102118 = 20189321, 1151110118 = 20222473, 1152125118 = 20256073, 1200000218 = 20971537, 1210102518 = 21237929, 1212012218 = 21299857, 1220005518 = 21496169, 1221021218 = 21529681, 1221512218 = 21549713, 1222025518 = 21562729, 1222255018 = 21572417, 1225055518 = 21662569, 1225122018 = 21664897, 1250101518 = 22286441, 1251115218 = 22319953, 1255022518 = 22447273, 1255055218 = 22448977, 1500000518 = 27263017, 1510105218 = 27529553, 1515015518 = 27689833, 1520101218 = 27791441, 1521112518 = 27824809, 1522022518 = 27853993, 1522055218 = 27855697, 1550002218 = 28573841, 1551051518 = 28609129, 1551215518 = 28615529, 1552022218 = 28640401, 1552155018 = 28646209, 1555052218 = 28740241, 1555522018 = 28759169, 10000111018 = 134222401, 10111000018 = 136609793, 10111212118 = 136618633, 10111515118 = 136631113, 10121505018 = 136892737, 10151202018 = 137666689, 10202151018 = 138484289, 10210222018 = 138683521, 10222012018 = 139002497, 10255005518 = 139886953, 10505121018 = 144872513, 10510555018 = 144988993, 10522002218 = 145293457, 10555015018 = 146178881, 11001010118 = 151028233, 11010100118 = 151261193, 11022522218 = 151606417, 11055255518 = 152480617, 11212111018 = 155521601, 11212200218 = 155525137, 11252010518 = 156566057, 11515111018 = 161911361, 11515500518 = 161927209, 11525010218 = 162169361, 12000511218 = 167793233, 12002212118 = 167846537, 12010525118 = 168056137, 12020525218 = 168318289, 12050212218 = 169091729, 12110122218 = 170136721, 12115000218 = 170295313, 12200225018 = 171976001, 12212050218 = 172296721, 12221011218 = 172524113, 12225220118 = 172663817, 12525020218 = 178947089, 15000211518 = 218112617, 15005515118 = 218288969, 15010252118 = 218376841, 15020515518 = 218649449, 15050252518 = 219425449, 15110155518 = 220470121, 15112000518 = 220528681, 15252050518 = 223676969, 15500552018 = 228612737, 15515020518 = 229016617, 15551011518 = 229933673, 15552550118 = 229988873

10

2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081, 188011, 188801, 1008001, 1022201, 1028011, 1055501, 1058011, 1082801, 1085801, 1088081, 1108201, 1108501, 1110881, 1120121, 1120211, 1150151, 1150511, 1180811, 1181881, 1208021, 1210151, 1210211, 1221221, 1250521, 1280821, 1281281, 1508051, 1510121, 1510511, 1520251, 1551551, 1580851, 1581581, 1802081, 1805081, 1808801, 1821821, 1851851, 1880111, 1880881, 1881811, 1881881, 10001081, 10001801, 10011101, 10080011, 10101181, 10111001, 10111081, 10128011, 10158011, 10180801, 10188811, 10808101, 10810001, 10888081, 10888181, 11008001, 11008111, 11010281, 11010581, 11018881, 11082101, 11085101, 11088881, 11180011, 11810081, 11888101, 12158521, 12252551, 12552251, 12585121, 12815581, 15128251, 15225521, 15282151, 15525221, 15812281, 18001811, 18010001, 18011101, 18088801, 18110101, 18111881, 18188801, 18201011, 18221851, 18288581, 18501011, 18551821, 18588281, 18811181, 18881011, 18888011, 100080181, 100100111, 100111001, 100252511, 100525211, 100808101, 100810811, 100818881, 100888001, 101018081, 101088881, 101108881, 101180081, 101218021, 101218801, 101518051, 101518801, 101808001, 108000811, 108081881, 108088181, 108101801, 108108881, 108110081, 108121801, 108151801, 108180811, 108181111, 108225151, 108282011, 108552121, 108585011, 108810011, 108810181, 108812101, 108815101, 108820021, 108850051, 108881111, 110018801, 110100811, 110111011, 110222251, 110225221, 110251181, 110282801, 110521181, 110552551, 110555521, 110585801, 110822111, 110855111, 110880811, 110881811, 111001001, 111010111, 111020111, 111050111, 111118811, 111181111, 111181801, 111188801, 111228011, 111558011, 111800881, 111828121, 111858151, 112050121, 112111211, 112251121, 112525001, 112828151, 115020151, 115111511, 115252001, 115521151, 115858121, 118000801, 118001011, 118008881, 118018001, 118081801, 118088011, 118188011, 118252811, 118525811, 118800181, 118811111, 118818811, 118880821, 118880851, 120028801, 120080251, 120155821, 120801521, 120808021, 120808081, 120812101, 121050211, 121058081, 121080121, 121080221, 121152211, 121255801, 121281821, 121282121, 121555121, 121828111, 121858511, 122008181, 122080121, 122105821, 122505881, 122522011, 122858221, 125025281, 125051821, 125058181, 125080051, 125108021, 125225851, 125555011, 128088811, 128150521, 128182121, 128255251, 128285851, 128501221, 128525821, 128551021, 128802181, 128810081, 150058801, 150080521, 150122851, 150801251, 150808051, 150808081, 150815101, 151020511, 151028081, 151080151, 151080551, 151125511, 151222151, 151522801, 151581851, 151585151, 151828211, 151858111, 152021851, 152028181, 152052581, 152080021, 152108051, 152222011, 152552821, 155008181, 155080151, 155102851, 155202881, 155255011, 155828551, 158088811, 158120251, 158185151, 158201551, 158221051, 158252851, 158522521, 158582821, 158805181, 158810081, 180011801, 180018821, 180018851, 180020881, 180050881, 180081101, 180101081, 180215081, 180512081, 180808021, 180808051, 180810101, 180811181, 180820151, 180850121, 181008811, 181010881, 181018801, 181080001, 181111181, 181118081, 181125011, 181152011, 181208821, 181508851, 181800221, 181800551, 181811881, 181820251, 181850521, 181880801, 181888181, 182108581, 182520521, 182801581, 182888581, 185108281, 185250251, 185801281, 185888281, 188008111, 188010181, 188010881, 188020081, 188050081, 188118181, 188180801, 188202551, 188505221, 188800811, 188801101, 188801801, 188818001, 188880101, 188888881

16

216 = 2, 516 = 5, 1116 = 17, 10116 = 257, 1000116 = 65537, 1018116 = 65921, 1108116 = 69761, 1212116 = 74017, 1515116 = 86353, 1801116 = 98321, 1810116 = 98561, 1818116 = 98689, 100018116 = 16777601, 108880116 = 17336321, 125052116 = 19203361, 152025116 = 22151761, 181000116 = 25231361, 181818116 = 25264513, 1000180116 = 268441601, 1000188116 = 268441729, 1000818116 = 268468609, 1080081116 = 276826129, 1081000116 = 276889601, 1081818116 = 276922753, 1088001116 = 277348369, 1088182116 = 277354529, 1088185116 = 277354577, 1100880116 = 285247489, 1101118116 = 285282689, 1108881116 = 285771793, 1180080116 = 293603329, 1181881116 = 293701649, 1182085116 = 293734481, 1185082116 = 293931041, 1188181116 = 294131729, 1188801116 = 294158353, 1200052116 = 301991201, 1215885116 = 303401041, 1250002116 = 307232801, 1280581116 = 310401041, 1281880116 = 310478849, 1288215116 = 310911313, 1500025116 = 352322129, 1512882116 = 353536033, 1520005116 = 354418769, 1580281116 = 360720401, 1581880116 = 360810497, 1588512116 = 361255201, 1808088116 = 403179649, 1811101116 = 403771409, 1818000116 = 404226049, 1818180116 = 404232193, 1880808116 = 411074689, 1881000116 = 411107329

 

Tutti i primi pluriunitari sono diedri, come pure tutti i primi palindromi che si rappresentano usando solo le cifre 0, 1 e 8.

 

Il minimo primo diedro che contenga tutte le cifre possibili è:

  • 1012 = 5 in base 2;

  • 112125518 = 2430313 in base 8;

  • 108225151 in base 10;

  • 1182085116 = 293734481 in base 16.

 

Il massimo primo diedro noto è Primo diedro palindromo, che è palindromo e ha 180055 cifre (Darren Bedwell, 2009).

Vedi anche

Primi (numeri).

Contattami

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