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In der Unterhaltungsmathematik ist eine einzigartige Primzahl oder einzigartige periodische Primzahl (vom englischen unique prime oder unique period prime) eine Primzahl , für welche gilt:
Einzigartige Primzahlen wurden erstmals im Jahr 1980 von Samuel Yates untersucht.[1]
Perioden- länge |
Primzahl | Perioden- länge |
Primzahl | Perioden- länge |
Primzahl | Perioden- länge |
Primzahl | Perioden- länge |
Primzahl |
---|---|---|---|---|---|---|---|---|---|
1 | 3 | 21 | 43, 1933, 10838689 | 41 | 83, 1231, 538987, 201763709900322803748657942361 | 61 | 733, 4637, 329401, 974293, 1360682471, 106007173861643, 7061709990156159479 | 81 | 163, 9397, 2462401, 676421558270641, 130654897808007778425046117 |
2 | 11 | 22 | 23, 4093, 8779 | 42 | 127, 2689, 459691 | 62 | 909090909090909090909090909091 | 82 | 2670502781396266997, 3404193829806058997303 |
3 | 37 | 23 | 11111111111111111111111 | 43 | 173, 1527791, 1963506722254397, 2140992015395526641 | 63 | 10837, 23311, 45613, 45121231, 1921436048294281 | 83 | 3367147378267, 9512538508624154373682136329, 346895716385857804544741137394505425384477 |
4 | 101 | 24 | 99990001 | 44 | 89, 1052788969, 1056689261 | 64 | 19841, 976193, 6187457, 834427406578561 | 84 | 226549, 4458192223320340849 |
5 | 41, 271 | 25 | 21401, 25601, 182521213001 | 45 | 238681, 4185502830133110721 | 65 | 162503518711, 5538396997364024056286510640780600481 | 85 | 262533041, 8119594779271, 4222100119405530170179331190291488789678081 |
6 | 7, 13 | 26 | 859, 1058313049 | 46 | 47, 139, 2531, 549797184491917 | 66 | 599144041, 183411838171 | 86 | 57009401, 2182600451, 7306116556571817748755241 |
7 | 239, 4649 | 27 | 757, 440334654777631 | 47 | 35121409, 316362908763458525001406154038726382279 | 67 | 493121, 79863595778924342083, 28213380943176667001263153660999177245677 | 87 | 4003, 72559, 310170251658029759045157793237339498342763245483 |
8 | 73, 137 | 28 | 29, 281, 121499449 | 48 | 9999999900000001 | 68 | 28559389, 1491383821, 2324557465671829 | 88 | 617, 16205834846012967584927082656402106953 |
9 | 333667 | 29 | 3191, 16763, 43037, 62003, 77843839397 | 49 | 505885997, 1976730144598190963568023014679333 | 69 | 277, 203864078068831, 1595352086329224644348978893 | 89 | 497867, 103733951, 104984505733, 5078554966026315671444089, 403513310222809053284932818475878953159 |
10 | 9091 | 30 | 211, 241, 2161 | 50 | 251, 5051, 78875943472201 | 70 | 4147571, 265212793249617641 | 90 | 29611, 3762091, 8985695684401 |
11 | 21649, 513239 | 31 | 2791, 6943319, 57336415063790604359 | 51 | 613, 210631, 52986961, 13168164561429877 | 71 | 241573142393627673576957439049, 45994811347886846310221728895223034301839 | 91 | 547, 14197, 17837, 4262077, 43442141653, 316877365766624209, 110742186470530054291318013 |
12 | 9901 | 32 | 353, 449, 641, 1409, 69857 | 52 | 521, 1900381976777332243781 | 72 | 3169, 98641, 3199044596370769 | 92 | 1289, 18371524594609, 4181003300071669867932658901 |
13 | 53, 79, 265371653 | 33 | 67, 1344628210313298373 | 53 | 107, 1659431, 1325815267337711173, 47198858799491425660200071 | 73 | 12171337159, 1855193842151350117, 49207341634646326934001739482502131487446637 | 93 | 900900900900900900900900900900990990990990990990990990990991 |
14 | 909091 | 34 | 103, 4013, 21993833369 | 54 | 70541929, 14175966169 | 74 | 7253, 422650073734453, 296557347313446299 | 94 | 6299, 4855067598095567, 297262705009139006771611927 |
15 | 31, 2906161 | 35 | 71, 123551, 102598800232111471 | 55 | 1321, 62921, 83251631, 1300635692678058358830121 | 75 | 151, 4201, 15763985553739191709164170940063151 | 95 | 191, 59281, 63841, 1289981231950849543985493631, 965194617121640791456070347951751 |
16 | 17, 5882353 | 36 | 999999000001 | 56 | 7841, 127522001020150503761 | 76 | 722817036322379041, 1369778187490592461 | 96 | 97, 206209, 66554101249, 75118313082913 |
17 | 2071723, 5363222357 | 37 | 2028119, 247629013, 2212394296770203368013 | 57 | 21319, 10749631, 3931123022305129377976519 | 77 | 5237, 42043, 29920507, 136614668576002329371496447555915740910181043 | 97 | 12004721, 846035731396919233767211537899097169, 109399846855370537540339266842070119107662296580348039 |
18 | 19, 52579 | 38 | 909090909090909091 | 58 | 59, 154083204930662557781201849 | 78 | 157, 6397, 216451, 388847808493 | 98 | 197, 5076141624365532994918781726395939035533 |
19 | 1111111111111111111 | 39 | 900900900900990990990991 | 59 | 2559647034361, 4340876285657460212144534289928559826755746751 | 79 | 317, 6163, 10271, 307627, 49172195536083790769, 3660574762725521461527140564875080461079917 | 99 | 199, 397, 34849, 362853724342990469324766235474268869786311886053883 |
20 | 3541, 27961 | 40 | 1676321, 5964848081 | 60 | 61, 4188901, 39526741 | 80 | 5070721, 19721061166646717498359681 | 100 | 60101, 7019801, 14103673319201, 1680588011350901 |
Einzigartige Primzahlen sind von der Basis abhängig, mit der gezählt wird. In den oberen Abschnitten wurden einzigartige Primzahlen zur Basis , also im Dezimalsystem betrachtet. In diesem Abschnitt werden einzigartige Primzahlen im Dualsystem, also mit Basis , behandelt.
Eine Primzahl ist eine einzigartige Primzahl zur Basis b=2, genau dann, wenn gilt:
Eine Primzahl ist eine einzigartige Primzahl zur Basis b, genau dann, wenn gilt:
Es folgt eine Auflistung von Primzahlen , für die der Bruch bei gegebener Basis die Periodenlänge besitzt. Einzigartige Primzahlen werden in gelben Zellen geschrieben:
Periodenlänge n Basis b |
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | es gibt keine | 2 | 3 | 2 | 5 | 2, 3 | 7 | 2 | 3 | 2, 5 | 11 | 2, 3 | 13 | 2, 7 | 3, 5 | 2 | 17 | 2, 3 | 19 | 2, 5 | 3, 7 | 2, 11 | 23 |
2 | 3 | es gibt keine | 5 | 3 | 7 | es gibt keine | 3 | 5 | 11 | 3 | 13 | 7 | 3, 5 | es gibt keine | 17 | 3 | 19 | 5 | 3, 7 | 11 | 23 | 3 | 5 |
3 | 7 | 13 | 7 | 31 | 43 | 19 | 73 | 7, 13 | 37 | 7, 19 | 157 | 61 | 211 | 241 | 7, 13 | 307 | 7 | 127 | 421 | 463 | 13 | 7, 79 | 601 |
4 | 5 | 5 | 17 | 13 | 37 | 5 | 5, 13 | 41 | 101 | 61 | 5, 29 | 5, 17 | 197 | 113 | 257 | 5, 29 | 5, 13 | 181 | 401 | 13, 17 | 5, 97 | 5, 53 | 577 |
5 | 31 | 11 | 11, 31 | 11, 71 | 311 | 2801 | 31, 151 | 11, 61 | 41, 271 | 3221 | 22621 | 30941 | 11, 3761 | 11, 4931 | 11, 31, 41 | 88741 | 41, 2711 | 151, 911 | 11, 61, 251 | 40841 | 245411 | 292561 | 346201 |
6 | es gibt keine | 7 | 13 | 7 | 31 | 43 | 19 | 73 | 7, 13 | 37 | 7, 19 | 157 | 61 | 211 | 241 | 7, 13 | 307 | 7 | 127 | 421 | 463 | 13 | 7, 79 |
7 | 127 | 1093 | 43, 127 | 19531 | 55987 | 29, 4733 | 127, 337 | 547, 1093 | 239, 4649 | 43, 45319 | 659, 4943 | 5229043 | 8108731 | 1743463 | 29, 43, 113, 127 | 25646167 | 449, 80207 | 701, 70841 | 29, 71, 32719 | 43, 631, 3319 | 16968421 | 29, 5336717 | 29, 239, 28771 |
8 | 17 | 41 | 257 | 313 | 1297 | 1201 | 17, 241 | 17, 193 | 73, 137 | 7321 | 89, 233 | 14281 | 41, 937 | 17, 1489 | 65537 | 41761 | 113, 929 | 17, 3833 | 160001 | 97241 | 73, 3209 | 139921 | 331777 |
9 | 73 | 757 | 19, 73 | 19, 829 | 19, 2467 | 37, 1063 | 262657 | 19, 37, 757 | 333667 | 1772893 | 37, 80749 | 1609669 | 397, 18973 | 541, 21061 | 19, 37, 73, 109 | 19, 1270657 | 991, 34327 | 523, 29989 | 64008001 | 85775383 | 127, 297613 | 19, 7792003 | 19, 2017, 4987 |
10 | 11 | 61 | 41 | 521 | 11, 101 | 11, 191 | 11, 331 | 1181 | 9091 | 13421 | 19141 | 11, 2411 | 71, 101 | 31, 1531 | 61681 | 11, 71, 101 | 11, 9041 | 11, 2251 | 152381 | 185641 | 224071 | 31, 41, 211 | 11, 5791 |
11 | 23, 89 | 23, 3851 | 23, 89, 683 | 12207031 | 23, 3154757 | 1123, 293459 | 23, 89, 599479 | 23, 67, 661, 3851 | 21649, 513239 | 15797, 1806113 | 23, 266981089 | 23, 419, 859, 18041 | 67, 4027, 1154539 | 67, 463, 2333, 8537 | 23, 89, 397, 683, 2113 | 2141993519227 | 23, 199, 16127, 51217 | 104281, 62060021 | 10778947368421 | 17513875027111 | 67, 353, 1176469537 | 3937230404603 | 67, 7349, 134367047 |
12 | 13 | 73 | 241 | 601 | 13, 97 | 13, 181 | 37, 109 | 6481 | 9901 | 13, 1117 | 20593 | 28393 | 37, 1033 | 13, 3877 | 97, 673 | 83233 | 229, 457 | 13, 769 | 13, 12277 | 61, 3181 | 157, 1489 | 37, 7549 | 13, 73, 349 |
13 | 8191 | 797161 | 2731, 8191 | 305175781 | 3433, 760891 | 16148168401 | 79, 8191, 121369 | 398581, 797161 | 53, 79, 265371653 | 1093, 3158528101 | 477517, 20369233 | 53, 264031, 1803647 | 157, 29914249171 | 53, 157483, 16655159 | 53, 157, 1613, 2731, 8191 | 212057, 2919196853 | 79, 521, 29759719289 | 599, 29251, 133338869 | 3121, 142559, 9690539 | 79, 189437, 516094151 | 79, 2003, 85107437663 | 47691619, 480393499 | 53, 6553, 15913, 6895253 |
14 | 43 | 547 | 29, 113 | 29, 449 | 29, 197 | 113, 911 | 43, 5419 | 29, 16493 | 909091 | 1623931 | 211, 13063 | 29, 22079 | 7027567 | 10678711 | 15790321 | 22796593 | 32222107 | 197, 226871 | 827, 10529 | 81867661 | 29, 43, 86969 | 71, 673, 2969 | 183458857 |
15 | 151 | 4561 | 151, 331 | 181, 1741 | 1171, 1201 | 31, 159871 | 631, 23311 | 31, 271, 4561 | 31, 2906161 | 195019441 | 61, 661, 9781 | 4651, 161971 | 31, 2851, 15511 | 61, 39225301 | 61, 151, 331, 1321 | 6566760001 | 31, 601, 558721 | 31, 211, 2460181 | 31, 3001, 261451 | 211, 9391, 18181 | 61, 858794191 | 74912328481 | 241, 17881, 24481 |
16 | 257 | 17, 193 | 65537 | 17, 11489 | 17, 98801 | 17, 169553 | 97, 257, 673 | 21523361 | 17, 5882353 | 17, 6304673 | 17, 97, 260753 | 407865361 | 17, 5393, 16097 | 7121, 179953 | 641, 6700417 | 18913, 184417 | 97, 113607841 | 15073, 563377 | 17, 1505882353 | 62897, 300673 | 17, 3227992561 | 17, 3697, 623009 | 17, 2801, 2311681 |
17 | 131071 | 1871, 34511 | 43691, 131071 | 409, 466344409 | 239, 409, 1123, 30839 | 14009, 2767631689 | 103, 2143, 11119, 131071 | 103, 307, 1021, 1871, 34511 | 2071723, 5363222357 | 50544702849929377 | 2693651, 74876782031 | 103, 443, 15798461357509 | 103, 22771730193675277 | 1045002649, 6734509609 | 137, 953, 26317, 43691, 131071 | 10949, 1749233, 2699538733 | 7563707819165039903 | 3044803, 99995282631947 | 689852631578947368421 | 1502097124754084594737 | 239, 74729519, 176634767651 | 103, 62246266355102810647 | 307, 120574031, 341563234253 |
18 | 19 | 19, 37 | 37, 109 | 5167 | 46441 | 117307 | 87211 | 530713 | 19, 52579 | 590077 | 1657, 1801 | 19, 271, 937 | 19, 132049 | 19, 739, 811 | 433, 38737 | 1423, 5653 | 73, 465841 | 199, 236377 | 307, 69481 | 19, 37, 199, 613 | 19, 5966803 | 163, 271, 1117 | 127, 199, 7561 |
19 | 524287 | 1597, 363889 | 174763, 524287 | 191, 6271, 3981071 | 191, 638073026189 | 419, 4534166740403 | 32377, 524287, 1212847 | 1597, 2851, 101917, 363889 | 1111111111111111111 | 6115909044841454629 | 29043636306420266077 | 12865927, 9468940004449 | 459715689149916492091 | 4272113, 370649274902657 | 229, 457, 174763, 524287, 525313 | 229, 1103, 202607147, 291973723 | 6841, 6089884909802812423 | 109912203092239643840221 | 75368484119, 192696104561 | 12061389013, 54921106624003 | 45943, 341203, 97404596002423 | 2129, 63877469, 24939218613613 | 7282588256957615350925401 |
20 | 41 | 1181 | 61681 | 41, 9161 | 241, 6781 | 281, 4021 | 41, 61, 1321 | 42521761 | 3541, 27961 | 212601841 | 85403261 | 421, 601, 641 | 1061, 1383881 | 19421, 131381 | 4278255361 | 21881, 63541 | 15101, 145501 | 16936647121 | 41, 2801, 222361 | 41, 920421641 | 181, 401, 150901 | 61, 941, 272341 | 61, 1801385941 |
21 | 337 | 368089 | 337, 5419 | 379, 519499 | 1822428931 | 11898664849 | 92737, 649657 | 43, 2269, 368089 | 43, 1933, 10838689 | 1723, 8527, 27763 | 8177824843189 | 43, 337, 547, 2714377 | 43, 547, 2239000891 | 43, 2817034275427 | 337, 1429, 5419, 14449 | 43, 13567, 940143709 | 156107192084257 | 30640261, 68443621 | 460951, 8442733531 | 4789, 6427, 227633407 | 12271836836138419 | 43, 170689, 408030421 | 43, 10426753, 78066619 |
22 | 683 | 67, 661 | 397, 2113 | 23, 67, 5281 | 51828151 | 23, 10746341 | 67, 683, 20857 | 5501, 570461 | 23, 4093, 8779 | 23, 89, 199, 58367 | 57154490053 | 128011456717 | 23, 11737870057 | 23, 23504771357 | 353, 2931542417 | 23, 947, 87415373 | 536801, 6301307 | 23, 253239693257 | 23, 424016563147 | 23, 6073, 10362529 | 89, 285451051007 | 39700406579747 | 60867245726761 |
23 | 47, 178481 | 47, 1001523179 | 47, 178481, 2796203 | 8971, 332207361361 | 47, 139, 3221, 7505944891 | 47, 3083, 31479823396757 | 47, 178481, 10052678938039 | 47, 1001523179, 23535794707 | 11111111111111111111111 | 829, 28878847, 3740221981231 | 47, 39891250417, 321218438243 | 1381, 2519545342349331183143 | 47, 461, 2347, 10627, 2249861, 14525237 | 829, 31741, 3046462151831565769 | 47, 277, 1013, 1657, 30269, 178481, 2796203 | 47, 26552618219228090162977481 | 47, 599, 7468009, 20801237997245359 | 277, 2347, 16497763013, 1335495402823 | 691, 1381, 46266279097921483078651 | 47, 19597, 139870566115103282847737 | 4463, 1323064018651, 60575166785239 | 461, 1289, 831603031789, 1920647391913 | 47, 124799, 304751, 58769065453824529 |
24 | 241 | 6481 | 97, 673 | 390001 | 1678321 | 73, 193, 409 | 433, 38737 | 97, 577, 769 | 99990001 | 10657, 20113 | 193, 2227777 | 815702161 | 1475750641 | 2562840001 | 193, 22253377 | 73, 1321, 72337 | 11019855601 | 4297, 3952393 | 31177, 821113 | 73, 518118697 | 191353, 286777 | 937, 83575993 | 97, 1134793633 |
Es folgt eine Auflistung der Periodenlängen von Bruchzahlen der Form mit den ersten 34 Primzahlen zu verschiedensten Basen . Wenn die Primzahl ein Teiler der Basis ist, endet die Dezimalbruchentwicklung, die Periodenlänge beträgt somit . Ist die Primzahl eine einzigartige Primzahl zur Basis , so wird die Periodenlänge in einer gelben Zelle geschrieben:
Primzahl Basis |
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||||
3 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | ||||||||
5 | 4 | 4 | 2 | 1 | 4 | 4 | 2 | 1 | 4 | 4 | 2 | 1 | 4 | 4 | 2 | 1 | 4 | 4 | 2 | ||||
7 | 3 | 6 | 3 | 6 | 2 | 1 | 3 | 6 | 3 | 6 | 2 | 1 | 3 | 6 | 3 | 6 | 2 | 1 | 3 | 6 | |||
11 | 10 | 5 | 5 | 5 | 10 | 10 | 10 | 5 | 2 | 1 | 10 | 5 | 5 | 5 | 10 | 10 | 10 | 5 | 2 | 1 | 10 | ||
13 | 12 | 3 | 6 | 4 | 12 | 12 | 4 | 3 | 6 | 12 | 2 | 1 | 12 | 3 | 6 | 4 | 12 | 12 | 4 | 3 | 6 | 12 | |
17 | 8 | 16 | 4 | 16 | 16 | 16 | 8 | 8 | 16 | 16 | 16 | 4 | 16 | 8 | 2 | 1 | 8 | 16 | 4 | 16 | 16 | 16 | |
19 | 18 | 18 | 9 | 9 | 9 | 3 | 6 | 9 | 18 | 3 | 6 | 18 | 18 | 18 | 9 | 9 | 2 | 1 | 18 | 18 | 9 | 9 | |
23 | 11 | 11 | 11 | 22 | 11 | 22 | 11 | 11 | 22 | 22 | 11 | 11 | 22 | 22 | 11 | 22 | 11 | 22 | 22 | 22 | 2 | 1 | |
29 | 28 | 28 | 14 | 14 | 14 | 7 | 28 | 14 | 28 | 28 | 4 | 14 | 28 | 28 | 7 | 4 | 28 | 28 | 7 | 28 | 14 | 7 | 7 |
31 | 5 | 30 | 5 | 3 | 6 | 15 | 5 | 15 | 15 | 30 | 30 | 30 | 15 | 10 | 5 | 30 | 15 | 15 | 15 | 30 | 30 | 10 | 30 |
37 | 36 | 18 | 18 | 36 | 4 | 9 | 12 | 9 | 3 | 6 | 9 | 36 | 12 | 36 | 9 | 36 | 36 | 36 | 36 | 18 | 36 | 12 | 36 |
41 | 20 | 8 | 10 | 20 | 40 | 40 | 20 | 4 | 5 | 40 | 40 | 40 | 8 | 40 | 5 | 40 | 5 | 40 | 20 | 20 | 40 | 10 | 40 |
43 | 14 | 42 | 7 | 42 | 3 | 6 | 14 | 21 | 21 | 7 | 42 | 21 | 21 | 21 | 7 | 21 | 42 | 42 | 42 | 7 | 14 | 21 | 21 |
47 | 23 | 23 | 23 | 46 | 23 | 23 | 23 | 23 | 46 | 46 | 23 | 46 | 23 | 46 | 23 | 23 | 23 | 46 | 46 | 23 | 46 | 46 | 23 |
53 | 52 | 52 | 26 | 52 | 26 | 26 | 52 | 26 | 13 | 26 | 52 | 13 | 52 | 13 | 13 | 26 | 52 | 52 | 52 | 52 | 52 | 4 | 13 |
59 | 58 | 29 | 29 | 29 | 58 | 29 | 58 | 29 | 58 | 58 | 29 | 58 | 58 | 29 | 29 | 29 | 58 | 29 | 29 | 29 | 29 | 58 | 58 |
61 | 60 | 10 | 30 | 30 | 60 | 60 | 20 | 5 | 60 | 4 | 15 | 3 | 6 | 15 | 15 | 60 | 60 | 30 | 5 | 12 | 15 | 20 | 20 |
67 | 66 | 22 | 33 | 22 | 33 | 66 | 22 | 11 | 33 | 66 | 66 | 66 | 11 | 11 | 33 | 33 | 66 | 33 | 66 | 33 | 11 | 33 | 11 |
71 | 35 | 35 | 35 | 5 | 35 | 70 | 35 | 35 | 35 | 70 | 35 | 70 | 10 | 35 | 35 | 10 | 35 | 35 | 7 | 70 | 70 | 14 | 35 |
73 | 9 | 12 | 9 | 72 | 36 | 24 | 3 | 6 | 8 | 72 | 36 | 72 | 72 | 72 | 9 | 24 | 18 | 36 | 72 | 24 | 8 | 36 | 12 |
79 | 39 | 78 | 39 | 39 | 78 | 78 | 13 | 39 | 13 | 39 | 26 | 39 | 26 | 26 | 39 | 26 | 13 | 39 | 39 | 13 | 13 | 3 | 6 |
83 | 82 | 41 | 41 | 82 | 82 | 41 | 82 | 41 | 41 | 41 | 41 | 82 | 82 | 82 | 41 | 41 | 82 | 82 | 82 | 41 | 82 | 41 | 82 |
89 | 11 | 88 | 11 | 44 | 88 | 88 | 11 | 44 | 44 | 22 | 8 | 88 | 88 | 88 | 11 | 44 | 44 | 88 | 44 | 44 | 22 | 88 | 88 |
97 | 48 | 48 | 24 | 96 | 12 | 96 | 16 | 24 | 96 | 48 | 16 | 96 | 96 | 96 | 12 | 96 | 16 | 32 | 32 | 96 | 4 | 96 | 24 |
101 | 100 | 100 | 50 | 25 | 10 | 100 | 100 | 50 | 4 | 100 | 100 | 50 | 10 | 100 | 25 | 10 | 100 | 25 | 50 | 50 | 50 | 50 | 25 |
103 | 51 | 34 | 51 | 102 | 102 | 51 | 17 | 17 | 34 | 102 | 102 | 17 | 17 | 51 | 51 | 51 | 51 | 51 | 102 | 102 | 34 | 17 | 34 |
107 | 106 | 53 | 53 | 106 | 106 | 106 | 106 | 53 | 53 | 53 | 53 | 53 | 53 | 106 | 53 | 106 | 106 | 53 | 106 | 106 | 106 | 53 | 106 |
109 | 36 | 27 | 18 | 27 | 108 | 27 | 12 | 27 | 108 | 108 | 54 | 108 | 108 | 27 | 9 | 36 | 108 | 36 | 54 | 27 | 27 | 36 | 108 |
113 | 28 | 112 | 14 | 112 | 112 | 14 | 28 | 56 | 112 | 56 | 112 | 56 | 28 | 4 | 7 | 112 | 8 | 112 | 112 | 112 | 56 | 112 | 112 |
127 | 7 | 126 | 7 | 42 | 126 | 126 | 7 | 63 | 42 | 63 | 126 | 63 | 126 | 63 | 7 | 63 | 63 | 3 | 6 | 63 | 9 | 126 | 18 |
131 | 130 | 65 | 65 | 65 | 130 | 65 | 130 | 65 | 130 | 65 | 65 | 65 | 130 | 65 | 65 | 130 | 26 | 26 | 65 | 65 | 130 | 130 | 26 |
137 | 68 | 136 | 34 | 136 | 136 | 68 | 68 | 68 | 8 | 68 | 136 | 136 | 34 | 34 | 17 | 68 | 34 | 68 | 136 | 136 | 34 | 136 | 136 |
139 | 138 | 138 | 69 | 69 | 23 | 69 | 46 | 69 | 46 | 69 | 138 | 69 | 46 | 138 | 69 | 138 | 138 | 138 | 69 | 138 | 138 | 46 | 69 |
Nun folgt eine Tabelle, der man die kleinsten Periodenlängen (bis inklusive ) entnehmen kann, für die der Bruch mit eine einzigartige Länge hat. Es gibt somit keine andere Primzahl zur gegebenen Basis mit der gleichen Periodenlänge. Außerdem wird jeweils auch die dazugehörige einzigartige Primzahl angegeben, deren Bruch diese Periodenlänge hat.
Basis | die kleinsten Periodenlängen von einzigartigen Primzahlen zur Basis | OEIS-Folge |
---|---|---|
die dazugehörigen einzigartigen Primzahlen mit diesen Periodenlängen | OEIS-Folge | |
2 | 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 342, 345, 366, 374, 382, 398, 410, 414, 425, 447, 471, 507, 521, 550, 567, 579, 590, 600, … | Folge A161508 in OEIS |
3, 7, 5, 31, 127, 17, 73, 11, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 241, 2731, 262657, 331, 2147483647, 65537, 599479, 43691, 174763, 61681, 5419, 2796203, 4432676798593, 87211, 15790321, 2305843009213693951, 715827883, 145295143558111, 10052678938039, 581283643249112959, 22366891, 4278255361, 9520972806333758431, 2932031007403, 618970019642690137449562111, … | Folge A161509 in OEIS | |
3 | 1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 20, 21, 24, 26, 32, 33, 36, 40, 46, 60, 63, 64, 70, 71, 72, 86, 103, 108, 128, 130, 132, 143, 145, 154, 161, 236, 255, 261, 276, 279, 287, 304, 364, 430, 464, 513, 528, 541, 562, … | |
2, 13, 5, 11, 7, 1093, 41, 757, 61, 73, 797161, 547, 4561, 1181, 368089, 6481, 398581, 21523361, 2413941289, 530713, 42521761, 23535794707, 47763361, 144542918285300809, 926510094425921, 374857981681, 3754733257489862401973357979128773, 282429005041, 82064241848634269407, 6957596529882152968992225251835887181478451547013, … | ||
4 | 1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, 596, … | |
3, 5, 7, 17, 13, 257, 41, 241, 65537, 61681, 15790321, 4278255361, 4562284561, 291280009243618888211558641, 18446744069414584321, 78919881726271091143763623681, 84179842077657862011867889681, 20988936657440586486151264256610222593863921, 84159375948762099254554456081, 1461503031127477825099979369543473122548042956801, … | ||
5 | 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 18, 24, 28, 47, 48, 49, 56, 57, 88, 90, 92, 108, 110, 116, 120, 127, 134, 141, 149, 161, 171, 181, 198, 202, 206, 236, 248, 288, 357, 384, 420, 458, 500, 530, 536, … | |
2, 3, 31, 13, 7, 19531, 313, 521, 12207031, 601, 305175781, 5167, 390001, 234750601, 177635683940025046467781066894531, 152587500001, 227376585863531112677002031251, 59509429687890001, 11735415506748076408140121, 9080418348371887359375390001, 60081451169922001, 5465713352000770660547109750601, 14551915228363037109375001, … | ||
6 | 1, 2, 3, 4, 5, 6, 7, 8, 18, 21, 22, 24, 29, 30, 42, 50, 62, 71, 86, 90, 94, 118, 124, 127, 129, 144, 154, 186, 192, 214, 271, 354, 360, 411, 480, 509, 558, 575, … | |
5, 7, 43, 37, 311, 31, 55987, 1297, 46441, 1822428931, 51828151, 1678321, 7369130657357778596659, 1950271, 2527867231, 3655688315536801, 189491931189200021056951, 3546245297457217493590449191748546458005595187661976371, 412482688627178079807598675848631, 4760317816590150361, … | ||
7 | 3, 4, 5, 6, 8, 13, 18, 21, 28, 30, 34, 36, 46, 48, 50, 54, 55, 58, 63, 76, 84, 94, 105, 122, 131, 148, 149, 224, 280, 288, 296, 332, 352, 456, 528, 531, 581, … | |
19, 5, 2801, 43, 1201, 16148168401, 117307, 11898664849, 13564461457, 6568801, 29078814248401, 13841169553, 3421093417510114543, 33232924804801, 79787519018560501, 1628413557556843, 5457586804596062091175455674392801, 402488219476647465854701, 2643999917660728787808396988849, 2598696228942460402343442913969, 195489390796456327201, … | ||
8 | 1, 2, 3, 6, 9, 18, 30, 42, 78, 87, 114, 138, 189, 303, 318, 330, 408, 462, 504, 561, … | |
7, 3, 73, 19, 262657, 87211, 18837001, 77158673929, 5302306226370307681801, 328017025014102923449988663752960080886511412965881, 19177458387940268116349766612211, 6113142872404227834840443898241613032969, 34175792320105064276509600649933535697253970335472049142780400956425111741139140798213387072831489, … | ||
9 | 1, 2, 4, 6, 10, 12, 16, 18, 20, 30, 32, 36, 54, 64, 66, 118, 138, 152, 182, 232, 264, 336, 340, 380, 414, 446, 492, 540, … | |
2, 5, 41, 73, 1181, 6481, 21523361, 530713, 42521761, 47763361, 926510094425921, 282429005041, 150094634909578633, 1716841910146256242328924544641, 13490012358249728401, 19966781110160346782368664772328944885905284750420567849, 1076050302914923449767311155851656076154481, … | ||
10 | 1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385, 586, 597, … | Folge A007498 in OEIS |
3, 11, 37, 101, 333667, 9091, 9901, 909091, 1111111111111111111, 11111111111111111111111, 99990001, 999999000001, 909090909090909091, 900900900900990990990991, 9999999900000001, 909090909090909090909090909091, 900900900900900900900900900900990990990990990990990990990991, 9090909090909090909090909090909090909090909090909091, 100009999999899989999000000010001, 909090909090909090909090909090909090909090909090909090909090909091, 10000099999999989999899999000000000100001, 999999999999990000000000000099999999999999000000000000009999999999999900000000000001, 142857157142857142856999999985714285714285857142857142855714285571428571428572857143, … | Folge A007615 in OEIS | |
11 | 2, 4, 5, 6, 8, 9, 10, 14, 15, 17, 18, 19, 20, 27, 36, 42, 45, 52, 60, 73, 91, 104, 139, 205, 234, 246, 318, 358, 388, 403, 458, 552, … | |
3, 61, 3221, 37, 7321, 1772893, 13421, 1623931, 195019441, 50544702849929377, 590077, 6115909044841454629, 212601841, 5559917315850179173, 3138426605161, 3421169496361, 9842332430037465033595921, 9768997162071483134919121, 46329453543600481, 1051153199500053598403188407217590190707671147285551702341089650185945215953, … | ||
12 | 1, 2, 3, 5, 10, 12, 19, 20, 21, 22, 56, 60, 63, 70, 80, 84, 92, 97, 109, 111, 123, 164, 189, 218, 276, 317, 353, 364, 386, 405, 456, 511, … | |
11, 13, 157, 22621, 19141, 20593, 29043636306420266077, 85403261, 8177824843189, 57154490053, 79493013628273739882868481, 186168115009253521, 708391688852136898302887193094373767489, 86121235964912696227980301, 34182189107670005092862256297738241, 80048881834094656438235281, 302669957628317561107372328495588758678132736113, … | ||
13 | 2, 3, 5, 6, 7, 8, 9, 12, 16, 22, 24, 28, 33, 34, 38, 78, 80, 102, 137, 140, 147, 224, 230, 283, 304, 341, 360, 372, 384, 418, 420, 436, 483, 568, 570, 594, … | |
7, 61, 30941, 157, 5229043, 14281, 1609669, 28393, 407865361, 128011456717, 815702161, 23161037562937, 17551032119981679046729, 617886851384381281, 104422877883960436477, 584288727345658049575114801, 442779263234039928595359287744639041, 476622264829847630603684799705499201, … | ||
14 | 1, 3, 4, 6, 7, 14, 19, 24, 31, 33, 35, 36, 41, 55, 60, 106, 114, 129, 152, 153, 172, 222, 265, 286, 400, 448, 560, 584, … | |
13, 211, 197, 61, 8108731, 7027567, 459715689149916492091, 1475750641, 26063080998214179685167270877966651, 77720275181800334933851, 2984619585279628795345143571, 56693904845761, 7538867501749984216983927242653776257689563451, 590942011471566261212035041517359275008998041, 2189065053896955781, … | ||
15 | 3, 4, 6, 7, 14, 24, 43, 54, 58, 73, 85, 93, 102, 184, 220, 221, 228, 232, 247, 291, 305, 486, 487, 505, 551, 552, 590, … | |
241, 113, 211, 1743463, 10678711, 2562840001, 26656068987980386414408582952871386493955339704241, 1477891879996957031251, 798962746803683694452047348022461, 5111329463430071646630167819950683399621676569261698373582346123709742512021745954241, … | ||
16 | 2, 4, 6, 8, 10, 14, 20, 30, 46, 48, 52, 70, 74, 78, 150, 178, 204, 298, 306, 346, 366, 378, 400, 476, 498, 502, … | |
17, 257, 241, 65537, 61681, 15790321, 4278255361, 4562284561, 291280009243618888211558641, 18446744069414584321, 78919881726271091143763623681, 84179842077657862011867889681, 20988936657440586486151264256610222593863921, 84159375948762099254554456081, 1461503031127477825099979369543473122548042956801, … | ||
17 | 1, 2, 3, 5, 7, 8, 11, 12, 14, 15, 34, 42, 46, 47, 48, 50, 71, 77, 94, 110, 114, 147, 154, 176, 228, 235, 258, 275, 338, 350, 419, 450, 480, 515, 589, … | |
2, 3, 307, 88741, 25646167, 41761, 2141993519227, 83233, 22796593, 6566760001, 45957792327018709121, 88109799136087, 1109309383381084655697725873, 423622795798733187216959754496018087627393990881167960767, 48661191868691111041, 4064228544226537005066401, 143798195172461138521036839345269251740737334259640879028155379795667047030720519999127, … | ||
18 | 1, 2, 3, 6, 14, 17, 21, 24, 30, 33, 38, 45, 46, 72, 78, 114, 146, 168, 288, 414, 440, 448, … | |
17, 19, 7, 307, 32222107, 7563707819165039903, 156107192084257, 11019855601, 11630180251, 12042065697120681040605799, 1961870762757168078553, 1338029376807245057016053427001, 3913037558632733048069409307, 1338258845052393545608356556801, 1412364383703504438982118048251, 1633867441076854816741224240423374163767714299, … | ||
19 | 2, 3, 4, 6, 19, 20, 31, 34, 47, 56, 59, 61, 70, 74, 91, 92, 96, 98, 107, 120, 145, 156, 168, 242, 276, 314, 326, 337, 387, 565, … | |
5, 127, 181, 7, 109912203092239643840221, 16936647121, 243270318891483838103593381595151809701, 274019342889240109297, 70169234660105574400577005075855017842743056666917902427141, 4898725341275828472027787456561, 155306613932666028670208812450645212905178047040045530562317564121001023821, … | ||
20 | 1, 3, 4, 6, 8, 9, 10, 11, 17, 30, 98, 100, 110, 126, 154, 158, 160, 168, 178, 182, 228, 266, 270, 280, 340, 416, 480, 574, … | |
19, 421, 401, 127, 160001, 64008001, 152381, 10778947368421, 689852631578947368421, 26876632021, 628292358238289452269193508271835428805485714102857143, 10995116277758926258176000104857599999989760000000001, 11544868483876542417134734645670674427914239932800021, 68728066670457765494784262143995903488000008001, … | ||
21 | 2, 3, 5, 6, 8, 9, 10, 11, 14, 17, 26, 43, 64, 74, 81, 104, 192, 271, 321, 335, 348, 404, 437, 445, 516, … | |
11, 463, 40841, 421, 97241, 85775383, 185641, 17513875027111, 81867661, 1502097124754084594737, 7021471715414521, 35842614220783025524408588074144786493150233831596714503, 1023263388750334684164671319051311082339521, 379919184478057330357419845346252603881265273961, … | ||
22 | 2, 3, 5, 6, 7, 10, 21, 25, 26, 69, 79, 86, 93, 100, 101, 154, 158, 161, 171, 202, 214, 294, 354, 359, 424, 454, … | |
23, 13, 245411, 463, 16968421, 224071, 12271836836138419, 705429635566498619547944801, 12296089473177511, 111284674149221479321933039375712865638979268198676574953779, 536009503964613991286957683005287140685493324523698332668846831791767500295593367113600925667991227216067, … | ||
23 | 2, 5, 6, 8, 11, 15, 22, 26, 39, 42, 45, 54, 56, 132, 134, 145, 147, 196, 212, 218, 252, 343, 580, … | |
3, 292561, 13, 139921, 3937230404603, 74912328481, 39700406579747, 21001515080686141, 459408054528299360264076035007841, 22865554874031409, 480211292412647894626919619228001, 1081383636631149044212969, 480249047846803230704957710381921, 2950758285992728866481208896744379674128936788494711201, … | ||
24 | 1, 2, 3, 4, 5, 8, 14, 19, 22, 38, 45, 53, 54, 70, 71, 117, 140, 144, 169, 186, 192, 195, 196, 430, … | |
23, 5, 601, 577, 346201, 331777, 183458857, 7282588256957615350925401, 60867245726761, 6699981196401006122851369, 1333639297121560770726162830707201, 615840114784814774501200690134862345946783236130283731411280186824640601, 6979147079581739570429953, 1389307926104143220565076487602201, … |
Die beiden Primzahlen und nennt man bi-einzigartige Primzahlen (vom englischen bi-unique prime), wenn gilt:
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Analog zu den bi-einzigartigen Primzahlen kann man auch tri-einzigartige Primzahlen definieren:
Die drei Primzahlen nennt man tri-einzigartige Primzahlen (vom englischen tri-unique prime), wenn gilt:
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Die Primzahlen nennt man n-einzigartige Primzahlen (vom englischen n-unique prime), wenn gilt:
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