Thema: Fermatquotient B^(P-1) == 1 (mod P^2) Fermatquotienten mit extremen Erst-Basen Main table of content: http://www.fermatquotient.com/ 26.11.2006: Die Primzahl 36300023 hat 9 Fermatqotienten-Basen, die kleiner als sie selbst sind! 16.02.2007: Die Primzahl 45051187 hat ebenfalls 9 Fermatqotienten-Basen! 02.03.2009: Kleinste Basen der Primzahlen: 11950711691, 23147324413, 87249620417, 103053130679, 405697846751, 588024812497, 604807523183, 663840051067, 719867822369 10.03.2009: Kleinste Basen der Primzahlen: 1219715607577, 1251922253819, 1822333408543, 2549536629329 28.03.2009: Kleinste Basen der Primzahlen: 5744393199187, 6683052151409 09.04.2009: Kleinste Basen der Primzahlen: 7036306088681, 9809862296159 15.04.2009: Kleinste Basen der Primzahl: 312679516579, with 4 prime bases! 21.06.2009: Kleinste Basen der Primzahl: 13368932516573 14.08.2009: Kleinste Basen der Primzahl: 14438374183481 29.08.2009: Kleinste Basen der Primzahl: 14802884210119 05.10.2009: Kleinste Basen der Primzahl: 13842663034709 und je erste 10 statt je 6 08.02.2010: Kleinste Basen der Primzahl: 18088417183289 22.03.2010: Die Primzahl 89518199 hat 9 Fermatqotienten-Basen, die kleiner als sie selbst sind! 19.04.2010: Die Primzahl 105408031 hat 10 Fermatqotienten-Basen, die kleiner als sie selbst sind! 12.07.2010: Basen-Statistik zugefügt 23.08.2010: Die Primzahl 156673243 hat 10 Fermatqotienten-Basen, die kleiner als sie selbst sind! (06.09.2010: Updates in all probability 21 dates) Fermatquotienten mit relativ vielen Basen kleiner als Primzahl Gegenwärtig durchsucht bis zur Primzahl 160523431 Anzahl (je erste 10) ; P: Primzahl^2 ; B: Basen Quanity (10 piece each) ; P: (prime number)^2 ; B: bases 1 P: 29 B: 14 1 P: 37 B: 18 1 P: 43 B: 19 1 P: 59 B: 53 1 P: 79 B: 31 1 P: 97 B: 53 1 P: 103 B: 43 1 P: 109 B: 96 1 P: 113 B: 68 1 P: 137 B: 19 2 P: 11 B: 3, 9 2 P: 71 B: 11, 26 2 P: 127 B: 38, 62 2 P: 131 B: 58, 111 2 P: 163 B: 65, 84 2 P: 211 B: 165, 182 2 P: 229 B: 44, 209 2 P: 313 B: 104, 213 2 P: 347 B: 75, 156 2 P: 349 B: 223, 317 3 P: 269 B: 171, 180, 207 3 P: 331 B: 18, 71, 324 3 P: 571 B: 262, 422, 516 3 P: 863 B: 13, 169, 434 3 P: 883 B: 644, 657, 754 3 P: 907 B: 127, 761, 771 3 P: 1097 B: 425, 579, 776 3 P: 1103 B: 284, 793, 1054 3 P: 1291 B: 62, 1148, 1286 3 P: 1579 B: 603, 906, 1235 4 P: 487 B: 10, 100, 175, 307 4 P: 1163 B: 78, 170, 241, 618 4 P: 2203 B: 184, 209, 767, 2037 4 P: 2731 B: 1534, 1577, 1872, 2290 4 P: 3373 B: 527, 1238, 1686, 2739 4 P: 4391 B: 2336, 2489, 2837, 3751 4 P: 5261 B: 1827, 2037, 2115, 2268 4 P: 6343 B: 80, 3754, 4701, 5033 4 P: 6451 B: 31, 961, 1204, 3933 4 P: 6569 B: 2496, 3154, 3667, 5050 5 P: 653 B: 84, 120, 197, 287, 410 5 P: 4909 B: 2189, 2234, 2406, 3266, 4649 5 P: 7723 B: 352, 3066, 3503, 4172, 6223 5 P: 9811 B: 1216, 1637, 3565, 4871, 7765 5 P: 13691 B: 83, 6889, 8677, 9345, 11689 5 P: 15413 B: 7134, 8387, 9628, 10535, 12309 5 P: 18133 B: 1997, 6375, 10565, 13098, 15160 5 P: 18223 B: 1814, 6143, 10532, 12472, 13141 5 P: 21061 B: 1654, 5302, 8769, 16116, 18104 5 P: 22147 B: 3752, 5113, 5271, 9336, 21673 6 P: 5107 B: 560, 1209, 1779, 2621, 4295, 4361 6 P: 20771 B: 5, 25, 125, 625, 3125, 15625 6 P: 51427 B: 10364, 14795, 26183, 28411, 34111, 39159 6 P: 52517 B: 13425, 18243, 34196, 38462, 39362, 51787 6 P: 61417 B: 12947, 15631, 17144, 20287, 41739, 51605 6 P: 66161 B: 6, 36, 216, 1296, 7776, 46656 6 P: 116731 B: 5999, 21399, 32127, 61099, 69145, 115067 6 P: 119359 B: 25627, 26486, 43165, 57879, 78988, 98633 6 P: 128657 B: 28237, 62334, 85135, 120099, 123891, 125137 6 P: 140741 B: 44757, 53828, 63099, 107890, 133072, 137002 7 P: 103291 B: 14866, 27419, 39660, 80408, 92041, 96106, 98404 7 P: 491531 B: 7, 49, 343, 2401, 16807, 117649, 397783 7 P: 534851 B: 6, 36, 216, 1296, 7776, 46656, 279936 7 P: 804367 B: 100933, 434207, 586707, 654355, 677456, 750045, 751958 7 P: 997961 B: 196462, 324572, 505976, 517837, 612235, 636080, 990873 7 P: 1026899 B: 102185, 131429, 289746, 430862, 556539, 895649, 980494 7 P: 1062427 B: 279412, 384570, 393638, 534085, 651095, 731789, 1033542 7 P: 1457389 B: 50917, 329577, 356210, 727539, 790181, 1058766, 1209683 7 P: 1550513 B: 440322, 445055, 468710, 628196, 742517, 774770, 1534043 7 P: 2327629 B: 648296, 995385, 1127223, 1410269, 1727926, 1757491, 1789865 Weitere Primzahlen (Fermatquotienten) mit 7 Basen sind: 2602307, 3093121, 3137257, 3181481, 3412741, 3497381, 3720179, 3814253, 4087301, 4234057, 4891973, 5063087, 5131237, 5194789, 5736611, 6253349, 6903191, 6906469, 6945047, 7401091, 7584587, 7587761, 8410849, 8426149, 8556277, 8603407, 8792137, 8852881, 8905003, 9052181, 9238967, 9377747, 10051439, 10161163, 10265779, 10889981, 11333923, 11895029, 12082663, 12494231, 12740789, 12854069, 12928753, 13066759, 13501673, 14091871, 14169577, 14905193, 15574847, 15694313, 15824917, 16528333, 16833799, 17209631, 17723311, 18256801, 18560981, 18795613, 19160381, 19588727, 19825459, 20017693, 20057549, 20311547, 20418721, 20511011, 20657999, 20856881, 21064567, 21215561, 21607249, 22011503, 22303817, 22316779, 22383353, 22665749, 22787857, 22947751, 23038759, 23094283, 23664149, 23804531, 24351169, 24615401, 24663281, 24851413, 25339499, 25941961, 26299907, 26675569, 26687867, 26873377, 27266441, 27489997, 27759799, 27823571, 27979309, 28271879, 28281391, 28373369, 28465057, 28484563, 28504001, 28556707, 28828333, 29136719, 29486581, 29671819, 29689403, 29891839, 30200539, 30705131, 31211629, 31404341, 31587707, 31784483, 32427233, 32456219, 32540003, 32833861, 33304703, 33374741, 33968993, 34542007, 34983787, 35294093, 35347993, 36105163, 36683107, 36764993, 37101539, 37119289, 38056037, 38516659, 38780717, 38967407, 39026167, 39070891, 39132649, 39166157, 39180023, 39574429, 39856559, 40241209, 40361353, 40362169, 40510069, 40515247, 40812379, 40999193, 41106721, 41173339, 41447353, 41457401, 41501347, 41513221, 41617621, 41868389, 42132611, 42278009, 42478213, 42623447, 42713417, 43026883, 43090547, 43343507, 43675633, 44058011, 44146579, 44615399, 44885063, 45025511, 45341603, 45569753, 46280483, 47783297, 47907511, 47923483, 47934571, 48017831, 48798467, 49305643, 49569269, 49580171, 49717079, 50072917, 50202797, 50887061, 51232579, 51451513, 51566153, 51675301, 51810569, 51999853, 52117487, 52133267, 52238023, 52609703, 52745359, 52754953, 53013391, 53127929, 53394023, 53977991, 54940159, 55187317, 55347871, 56491249, 56598313, 56695909, 56776859, 56993569, 57214331, 57404261, 58007233, 58224827, 58427407, 58554179, 58610177, 58954463, 59137961, 59183741, 59298287, 59421217, 59756401, 60226889, 60322921, 60347099, 60386897, 60916459, 60920989, 61016173, 61076531, 61408189, 61878017, 62022607, 62799419, 62904929, 63061489, 63264533, 64418461, 64537309, 64605451, 64663343, 64714861, 65146657, 65442931, 65479639, 65628611, 65674633, 65944103, 66132779, 66308707, 66907129, 67037921, 67080281, 67391239, 67453039, 67579339, 67747151, 68034451, 68075213, 68373247, 68392829, 68547653, 69095393, 69172657, 69225881, 69248411, 69378091, 69730637, 69957361, 69967543, 70343927, 70521169, 70687121, 70699117, 70897061, 70898783, 71131703, 71763973, 72081301, 72726307, 73018333, 73233451, 73388701, 73777541, 74130011, 74569301, 74670889, 74748743, 74822927, 74862061, 74935051, 75230569, 75387457, 75430423, 75489179, 75839521, 76028417, 76062991, 76138327, 76232347, 76536833, 76608761, 76821763, 77153591, 77176721, 77510113, 77634449, 78302827, 78793577, 78959011, 79572799, 80073781, 80120111, 80497957, 80634079, 80825977, 80949347, 81426439, 81498563, 81983171, 82124359, 82250227, 82322111, 82995919, 83218657, 83261473, 83745191, 84319073, 84340397, 85592069, 85773679, 85783939, 85871801, 86361953, 86485837, 86623079, 86726723, 86759261, 86818009, 87085631, 87153259, 87313211, 87371761, 87956243, 88635109, 88850413, 89406287, 90160243, 90178819, 90558929, 90671233, 90892871, 90958349, 90981281, 91637869, 91672957, 91712681, 91841821, 92077927, 92281873, 92760671, 93380887, 93588557, 93919219, 94287043, 94462229, 94498549, 94607899, 94722779, 95477689, 95478793, 95931397, 96308999, 96309091, 96690131, 96855169, 96927907, 97226029, 97337809, 97765831, 97809259, 97810561, 98009917, 98495717, 98542757, 98562757, 98844979, 98900953, 98960503, 99290879, 99500831, 99509299, 99815851, 100109491, 100153133, 101278313, 101444521, 101687947, 101862643, 101878697, 101894953, 102248479, 102648913, 102686027, 103065227, 103246841, 103313729, 103683869, 103723331, 103743077, 103828861, 103836853, 103841207, 104075497, 104088157, 104698651, 105303823, 105518321, 105579421, 105653897, 105813941, 106071751, 106380011, 106417579, 106809161, 106893019, 107744743, 107879311, 107958073, 108012497, 108049631, 108134353, 108206473, 109133503, 109152863, 109163921, 109292129, 109529621, 109544293, 109617229, 109628809, 109720657, 110354747, 110523233, 111059951, 111088301, 111120409, 111381301, 111419831, 111688777, 112091069, 112249259, 112265891, 112398617, 112473089, 112667801, 112670267, 113109863, 113465623, 113617507, 113628143, 113842627, 114181349, 114417271, 114784231, 114850481, 114851327, 114875483, 116027137, 116395093, 116631521, 116811271, 117191989, 117323867, 117815167, 117851161, 119045119, 119085301, 119576591, 119655251, 119677037, 119709749, 120120263, 120188027, 120894901, 121462811, 121774109, 121911113, 121975507, 123167393, 123356803, 123502241, 123602243, 123983309, 124584259, 124594619, 124739561, 124773343, 125743601, 126037139, 126899189, 127134019, 127224499, 127513681, 127518113, 127648097, 127760011, 128377031, 128558813, 128896973, 129240061, 129799613, 130032571, 130116509, 130165703, 130242941, 130261631, 130420987, 130769929, 130786871, 131020433, 131180011, 131261149, 131957107, 132074087, 132243847, 132315329, 132447307, 132802211, 133257353, 133263821, 133272977, 133339873, 133546321, 133764331, 133820801, 133827299, 134143631, 134510993, 134626103, 134931557, 134981867, 135368339, 135630997, 135847501, 135881513, 135959293, 136113613, 137291989, 137606731, 137723983, 138068347, 138266203, 139670371, 139784327, 139956559, 140085629, 140124679, 140202943, 140234747, 140289631, 140720689, 140755063, 140866487, 141342431, 141473147, 141680269, 142524223, 142834903, 142929133, 143460277, 143585203, 143604449, 143673461, 144251881, 144329587, 144334733, 144455419, 144929077, 145437623, 146761613, 146863267, 147883067, 147903073, 148035049, 148041689, 148063991, 148564387, 148603669, 148954951, 149001947, 149475197, 149742773, 149909797, 150245717, 150791177, 150966689, 151059877, 151158143, 151847701, 152147267, 152241767, 152405777, 152518783, 152959991, 153101917, 153165409, 153253913, 153258031, 153599969, 153601619, 153902789, 154031159, 154307737, 154381039, 154931041, 155387677, 155870857, 156239387, 156353807, 156736337, 157578959, 157651199, 157674721, 157705829, 157710257, 157972421, 158023897, 158422499, 158802541, 159369031, 160305227, 160425319 8 P: 40487 B: 5, 25, 125, 625, 3125, 4492, 15625, 22460 8 P: 728471 B: 36709, 159316, 241830, 288664, 418571, 443653, 653451, 679977 8 P: 1491769 B: 774740, 792387, 948605, 1145688, 1161654, 1175740, 1353311, 1400304 8 P: 2519453 B: 125989, 136464, 339163, 370740, 778051, 1087929, 1674962, 2514772 8 P: 6528499 B: 638520, 1163806, 2145319, 2960253, 4269303, 4989666, 5413294, 6055130 8 P: 7242839 B: 166139, 2711128, 3721813, 3735285, 4116272, 4520251, 5355280, 6366530 8 P: 9566399 B: 1918479, 2000014, 3917085, 4395789, 5446003, 5824932, 7427860, 8265504 8 P: 9624319 B: 953800, 2022515, 2754065, 4145078, 5124174, 8736191, 8996536, 9460124 8 P: 10338851 B: 175078, 1936742, 4930344, 6701194, 6720019, 7820944, 7960429, 8395977 8 P: 10346797 B: 1904393, 2816351, 4070823, 4696371, 4716163, 5386008, 7312639, 9759581 Weitere Primzahlen (Fermatquotienten) mit 8 Basen sind: 11518099, 13655093, 15799639, 19012271, 19158743, 19864981, 20336149, 23927983, 29923547, 37011409, 37952771, 39885761, 40506199, 41455903, 42613481, 43203379, 46364233, 48038609, 52708961, 53375789, 54478547, 60340993, 63559147, 65595493, 72437801, 73493447, 74106007, 75959833, 78095233, 83154949, 85802543, 87377911, 88568741, 89624021, 90959347, 96963829, 98312021, 100157639, 105007769, 106295089, 108016009, 108776201, 110330771, 111087001, 111261571, 115667471, 116733871, 116951689, 121820869, 124280327, 124602659, 126036767, 126857153, 131081183, 132510661, 134079059, 136640213, 138088963, 139654709, 140813479, 144333913, 145333973, 145345927, 149247661, 156577013, 158797937 9 P: 2544079 B: 112676, 795574, 1094810, 1220551, 1392775, 1400707, 1528862, 2051627, 2203851 9 P: 3152573 B: 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 1693042 9 P: 36300023 B: 992025, 6574670, 7150109, 8294877, 9644532, 12420134, 23490327, 25306848, 32644915 9 P: 45051187 B: 2141207, 2953730, 6331316, 13429349, 14871517, 28881712, 29318609, 29703953, 35122297 9 P: 89518199 B: 1320874, 5728201, 16885161, 21320343, 35065948, 56361288, 62069867, 70946800, 89267728 9 P: 120616039 B: 5888278, 12468053, 31802854, 31940787, 54665641, 76598539, 103217290, 111951502, 117320929 9 P: 145946191 B: 17480341, 17633875, 27788219, 87708805, 97507162, 123858102, 127611842, 141707008, 144869956 9 P: 158559257 B: 25819732, 40750113, 46996750, 50287321, 89989957, 101198107, 109135844, 147792763, 153061086 10 P: 1093 B: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 10 P: 105408031 B: 5664036, 5724231, 20438091, 21712498, 53476247, 62708745, 67245243, 77022748, 83311882, 98762172 10 P: 156673243 B: 6833263, 35182832, 43713519, 54989512, 56697192, 73509268, 83170964, 84248066, 94731012, 115865120 11 P: 3511 B: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048 11 P: 53471161: 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125 12 P: 1006003 B: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441 Weitere bekannte Primzahlen > 5E+07 mit Erst-Basis < 152 und mit mindestens 7 Basen sind: Anzahl ; P: Primzahl^2 ; B: Basen Quanity ; P: (prime number)^2 ; B: bases 7 P: 56598313 B: 10, 100, 1000, 10000, 100000, 1000000, 10000000 (100000000, 197713319, 207037044, 234454221, 346417103, 415537484, 498728726, 546662487, 590085073, 621192973, 648416139, 749343279) 7 P: 63061489 B: 19, 361, 6859, 130321, 2476099, 15941914, 47045881 (103289908, 264669157, 302896366, 319387512, 327327061, 348858675, 383681229, 434373013, 456426426, 586893208, 662549451, 742690181) 7 P: 486999673 B: 26, 676, 17576, 456976, 11881376, 175898620, 308915776 (671434989, 765575411, 1751404322, 2260672534, 2499865453, 3554404145, 3623757338, 3638074818, 4573364120, 4877917266, 5308767373, 5403135367) 7 P: 492366587 B: 22, 484, 10648, 234256, 5153632, 113379904, 271278503 (968000364, 1806749569, 2494357888, 2713433298, 2765267357, 3828183014, 3956728024, 4383396359, 5411314423, 5814179437, 5968127066, 6138094866) 7 P: 2243046959 B: 5231, 27363361, 145089403, 316581637, 344503074, 669443510, 927771514 (3851083943, 7762307536, 10507152124, 11368932316, 13595379036, 15783351768, 17870484250, 20003833259, 21127788118, 23002863965, 27768797268, 29756677768) 7 P: 9566295763 B: 60, 3600, 216000, 12960000, 777600000, 5024104003, 7360763268 (13047944972, 18991111122, 27230527988, 28299403585, 34952458539, 40996831246, 44503642869, 46656000000, 50362559102, 93216911503, 109149242033, 110900750248) 7 P: 9639595369 B: 33, 1089, 35937, 1185921, 39135393, 1291467969, 9438122555 (27176292678, 40999141175, 42251984362, 42618442977, 51238605888, 53162215759, 56927107153, 70210277746, 72606478698, 90562355471, 111409290066, 111810630128) 7 P: 23147324413 B: 135, 18225, 2460375, 332150625, 351450254, 15589679287, 19401993332 (44840334375, 47445784290, 53057506707, 53509380337, 103635375207, 109992538152, 118631522265, 120250218254, 176056347055, 198511658476, 220382782123, 221460618308) 7 P: 76704103313 B: 97, 9409, 912673, 88529281, 8587340257, 56066217318, 60029996975 (151626351426, 179352503420, 236917183271, 307130546780, 676683036343, 724595128250, 742007798629, 768882094629, 821292221161, 832972004929, 855845286731, 860824033177) 7 P: 87249620417 B: 136, 18496, 2515456, 342102016, 46525874176, 51726266154, 58682607166 (228879393843, 249057499631, 355765387515, 460946047826, 497744483808, 687876490858, 708432532765, 796370342546, 810781081708, 1462447650965, 1545436777231, 1660088839105) 7 P: 103053130679 B: 150, 22500, 3375000, 506250000, 35783913210, 75937500000, 87533014073 (482798464958, 513595631827, 574848578489, 589320479937, 611198218625, 668121855968, 759925972396, 858950252739, 1061395940649, 1072355035046, 1214834729021, 1574933563922) 7 P: 312679516579 B: 691*, 477481, 329939371, 64907851831*, 77658868919*, 227988105361, 235522640761* ; * = prime (454488760701, 796565938594, 1362216471274, 2111784844448, 2848371412392, 2902433166749, 2921553332500, 3458514187688, 3851768125641, 3903086847245, 4255891721244, 4573751228192) 7 P: 405697846751 B: 113, 12769, 1442897, 163047361, 18424351793, 73932100785, 188062244682 (432019080594, 547617437402, 680971250695, 1403731225126, 1561101360620, 1918967153837, 2081951752609, 2616670876348, 3985041434219, 4075301855097, 4384858643767, 4435712515456) 7 P: 588024812497 B: 66, 4356, 287496, 18974736, 1252332576, 82653950016, 407916174766 (1303089130535, 3059505491138, 4136324791239, 4217173419460, 5232510962341, 5455160701056, 5795952078262, 6159016646257, 7502205110881, 7670368575298, 8557557134041, 8667209198246) 7 P: 604807523183 B: 87, 7569, 658503, 57289761, 4984209207, 322994504180, 433626201009 (823636962293, 1078005869514, 1111021561394, 2383311314419, 2490087152053, 3951057707782, 4280019221819, 4657609681126, 4665869722362, 4727436774610, 6382557539316, 7671170104099) 7 P: 663840051067 B: 84, 7056, 592704, 49787136, 4182119424, 339683704554, 351298031616 (1442288261569, 1620601235502, 3874827183621, 3918419743513, 4270207230614, 7090821423989, 7230772109930, 8367752562157, 8985842251225, 9190103984866, 9794766452975, 10154099282616) 7 P: 719867822369 B: 42, 1764, 74088, 3111696, 130691232, 5489031744, 230539333248 (1118390129936, 1189994108652, 1922088952027, 2152913927397, 2202653330129, 3713404602578, 3726351799246, 3763465498162, 4515355192852, 5244359172109, 5452802478422, 5776244462755) 7 P: 1251922253819 B: 74, 5476, 405224, 29986576, 2219006624, 41768194945, 164206490176 (1739027378366, 3090846425930, 6444907635753, 6587914632942, 8492940125028, 9170176939270, 10208481212657, 12151280273024, 12872583803813, 13459658905770, 15962019806864, 16307584342586) 7 P: 13842663034709 B: 360, 129600, 46656000, 16796160000, 6046617600000, 6319022671788, 13616429970784 (23856816437019, 61573683834221, 64163427287775, 72509997902409, 76279136260071, 86874262182939, 88856493944103, 92293958672195, 95704852695665, 107254535672230, 109517232348259, 141999509449780) 7 P: 18088417183289 B: 59, 3481, 205379, 12117361, 714924299, 42180533641, 2488651484819 (24994140702987, 43407189793036, 93641040305598, 121192954574404, 129060931553590, 146830437604321, 148925022838234, 156061505457865, 166200822266365, 167441459753109, 192347622227833, 193866328971328) 8 P: 1228488439 B: 52, 2704, 140608, 6747236, 7311616, 102109169, 350856272, 380204032 (2067855823, 2202746865, 2306896042, 5309676788, 6181110212, 6420783289, 6534805740, 7314187281, 10631535346, 11485298795, 12827564884, 12954283112) 8 P: 11950711691 B: 104, 10816, 1124864, 116985856, 241104353, 444145810, 2859511419, 2940201032 (12166529024, 25074852712, 30473934937, 46191164240, 46503913440, 46816987261, 70582215023, 71075186793, 73871267711, 74495290500, 100857220246, 101140719089) 8 P: 15546404183 B: 23, 529, 12167, 279841, 6436343, 148035889, 3404825447, 5840778226 (23545155743, 27148818850, 53543136724, 61332751773, 78310985281, 91877961403, 105000814231, 117043883775, 119017540367, 134337899198, 148233831210, 170861510393) 8 P: 94727075783 B: 30, 900, 27000, 810000, 24300000, 729000000, 21870000000, 42986456628 (151119266445, 156197456434, 353665585264, 417395626558, 656100000000, 859944470279, 996745043353, 1000158170385, 1061646187872, 1098129621247, 1187434411188, 1289593698840) 8 P: 115755260963 B: 56, 3136, 175616, 9834496, 550731776, 30840979456, 83686593107, 114532627710 (217437185670, 474950399746, 636868839407, 741785119621, 894011925149, 1071150435875, 1078318539231, 1104815768579, 1303348847289, 1558645732272, 1659017930964, 1727094849536) 8 P: 1219715607577 B: 110, 12100, 1331000, 146410000, 16105100000, 347914584477, 650390161812, 685274994103 (1771561000000, 2753500825153, 4622704292966, 5626400698597, 6341433153298, 9113815358192, 9922188190895, 10096979594292, 13298481247828, 13382232420524, 16667562083365, 17831302144814) 8 P: 1822333408543 B: 139, 19321, 2685619, 373301041, 51888844699, 386389357290, 974183630554, 1425799713522 (7212549413161, 7292351518558, 13835803241071, 16937109289961, 18571426512689, 18918746792366, 19195698757214, 20770571524572, 21676801118247, 23134625775494, 25288344448181, 26093453346616) 8 P: 5744393199187 B: 78, 6084, 474552, 37015056, 2887174368, 225199600704, 2161608938045, 3322314150128 (6295807132508, 8273858755864, 9389479829433, 17565568854912, 25669287991378, 26295664596137, 29203703107165, 29369660982332, 32032762700985, 32443535474746, 33957846154475, 35090139939130) 8 P: 6683052151409 B: 60, 3600, 216000, 12960000, 777600000, 46656000000, 2264013798892, 2799360000000 (7418808149138, 23635158874552, 26846999553861, 29781009957747, 37380372495867, 46423601275929, 53524742520117, 55070755128895, 70472458009066, 73700201263456, 80757744718613, 83245408129138) 8 P: 7036306088681 B: 40, 1600, 64000, 2560000, 102400000, 4096000000, 163840000000, 6553600000000 (27544039483644, 33518469499724, 34680139054879, 37497429479835, 45332275525404, 53656928867447, 67153106122459, 67249906614251, 68297828743769, 76336595927863, 82076590355685, 82296382814022) 8 P: 14438374183481 B: 110, 12100, 1331000, 146410000, 16105100000, 1771561000000, 2704058032875, 6078809438104 (41684805907977, 43173838942680, 50616964188244, 56993421931097, 62634044602867, 67764920513601, 75136186438956, 75429689261419, 82090414639282, 88195330679738, 88716895412501, 121755351683904) 8 P: 14802884210119 B: 51, 2601, 132651, 6765201, 345025251, 17596287801, 897410677851, 9984769265225 (23638761309169, 38857502043365, 41016688965971, 45767944570401, 47832280488875, 53459930603691, 67866903916664, 73021433796296, 76101066607003, 79076528375873, 111106546191407, 111703890725142) 9 P: 7596952219 B: 14, 196, 2744, 38416, 537824, 7529536, 105413504, 1475789056, 3148358644 (14308423759, 14819717576, 17370538894, 18573599281, 19913607683, 20661046784, 24318824912, 32439828682, 34759085204, 44077021016, 62935491863, 80492048293) 9 P: 2549536629329 B: 23, 529, 12167, 279841, 6436343, 148035889, 3404825447, 78310985281, 1801152661463 (3068274133735, 3577432562697, 5198139050319, 6065172460992, 10986689254242, 14061537545947, 16229475376810, 18304451025511, 19892565027224, 19925360452639, 23421003705632, 23677983036119) 9 P: 13368932516573 B: 43, 1849, 79507, 3418801, 147008443, 6321363049, 271818611107, 8147024755195, 11688200277601 (34019929751467, 34424133914649, 36460104261962, 43727748357650, 75933088226453, 84688209718879, 106110522047822, 118369885589276, 130601306229756, 133071568674797, 151914491268076, 193877128333111) 10 P: 119327070011 B: 15, 225, 3375, 50625, 759375, 11390625, 170859375, 2562890625, 38443359375, 55995107732 (139361386385, 228397789113, 261086004684, 487360026007, 576650390625, 713765539280, 839926615980, 1084643723474, 1272103668557, 1390463126829, 1413657853278, 1423288157972) 10 P: 478225523351 B: 17, 289, 4913, 83521, 1419857, 24137569, 410338673, 6975757441, 118587876497, 423329209746 (1195441552422, 2015993900449, 2227246108447, 2367505224618, 3437685430734, 3726419794976, 3781367525475, 4852143038698, 5075622063616, 5079946164589, 5341907228809, 6601290986332) 11 P: 53471161 B: 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125 (171358553, 237207222, 244140625, 277361309, 292659296, 309469159, 324973313, 355862547, 397792654, 416113118, 456442499, 554878728) 11 P: 9809862296159: 22, 484, 10648, 234256, 5153632, 113379904, 2494357888, 54875873536, 1207269217792, 4376017644649, 6065078091938 (14455084927962, 18565159688747, 26559922791424, 30760473665861, 39500629439869, 41811115853113, 45173883702470, 76094369902758, 96272388182278, 114247234921481, 119885644812686, 121814886187067) 14 P: 1645333507 B: 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 1317772341 (2806442861, 4728352314, 6103515625, 6588861705, 9721957398, 10602260038, 10639387189, 11298528672, 12369415874, 13015969356, 13063745546, 14032214305) 16 P: 6692367337 B: 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 4961139411, 6103515625, 6462265338 (9269835902, 9758066546, 16367316223, 20737395363, 24805697055, 27620871478, 30517578125, 32311326690, 37276920658, 41477984997, 46349179510, 48790332730) 16 P: 188748146801 B: 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625 (361970431381, 762939453125, 993257029491, 1125215380306, 1427298772747, 1526322632992, 1661315355562, 1796655913688, 1809852156905, 1888241169258, 1914123678467, 2143520565704) Statistik der Anzahl Basen < Primzahl P von bis ; Anzahl Primzahlen ; 6 Basen ; 7 Basen ; 8 Basen ; >8 Basen from until ; quanity of prime number ; 6 bases ; 7 bases ; 8 bases ; >8 bases (Erwartungswert mit Poisson-Verteilung, Ist) ; (estimate with Poisson distribution, actual) 101 9973 ; 1204 ; (0.6, 1) ; (0.09, 0) ; (0.01, 0) ; (0.001, 2) 10007 99991 ; 8363 ; (4.3, 5) ; (0.6, 0) ; (0.08, 1) ; (0.01, 0) 100003 999983 ; 68906 ; (35.2, 53) ; (5.0, 5) ; (0.6, 1) ; (0.08, 0) 1000003 9999991 ; 586081 ; (299, 327) ; (42.8, 37) ; (5.3, 6) ; (0.7, 3) 10000019 99999989 ; 5096876 ; (2604, 2571) ; (372, 364) ; (46.5, 39) ; (5.7, 4) 100000007 199999991 ; 5317482 ; (2717, ?) ; (388, ?) ; (48.5, ?) ; (6.0, ?) bis April 2011 Fermatquotienten mit rekordgrosser Erst-Basis Gegenwärtig durchsucht bis zur Primzahl 160523431 B: Basis ; P: Primzahl^2 ; Basis/Primzahl B: first base ; P: (prime number)^2 ; base/(prime number) B: 5 P: 2 B/P: 2.5 B: 8 P: 3 B/P: 2.66667 B: 18 P: 7^3 B/P: 2.57143 B: 19 P: 13 B/P: 1.46154 B: 38 P: 17 B/P: 2.23529 B: 115 P: 31 B/P: 3.70968 B: 338 P: 53 B/P: 6.37736 B: 532 P: 179 B/P: 2.97207 B: 747 P: 271 B/P: 2.75646 B: 1260 P: 311 B/P: 4.05145 B: 1449 P: 503 B/P: 2.88072 B: 2069 P: 569 B/P: 3.63620 B: 2788 P: 587 B/P: 4.74957 B: 3325 P: 1231 B/P: 2.70106 B: 4298 P: 1307 B/P: 3.28845 B: 5455 P: 1543 B/P: 3.53532 B: 10675 P: 1931 B/P: 5.52822 B: 10957 P: 2647 B/P: 4.13940 B: 14854 P: 2711 B/P: 5.47916 B: 16936 P: 3089 B/P: 5.48268 B: 19847 P: 3917 B/P: 5.06689 B: 31153 P: 4919 B/P: 6.3332 B: 32850 P: 5879 B/P: 5.58768 B: 45329 P: 6491 B/P: 6.98336 B: 50961 P: 8933 B/P: 5.70480 B: 57177 P: 9137 B/P: 6.25774 B: 77749 P: 11437 B/P: 6.79802 B: 83384 P: 13411 B/P: 6.21758 B: 108682 P: 14431 B/P: 7.53115 B: 136471 P: 16657 B/P: 8.19301 B: 163250 P: 21599 B/P: 7.55822 B: 178557 P: 26053 B/P: 6.85361 B: 295176 P: 29129 B/P: 10.13341 B: 298670 P: 57367 B/P: 5.20630 B: 321224 P: 58481 B/P: 5.49279 B: 331097 P: 62071 B/P: 5.33417 B: 346715 P: 62971 B/P: 5.50595 B: 439577 P: 68351 B/P: 6.43117 B: 937745 P: 70639 B/P: 13.27517 B: 1162517 P: 109721 B/P: 10.59521 B: 1406492 P: 156967 B/P: 8.96043 B: 1510208 P: 193811 B/P: 7.79217 B: 1882268 P: 216211 B/P: 8.7057 B: 1889213 P: 282713 B/P: 6.68244 B: 1972849 P: 285763 B/P: 6.90379 B: 2115511 P: 293149 B/P: 7.2165 B: 2244610 P: 315493 B/P: 7.11461 B: 2722074 P: 324757 B/P: 8.38188 B: 3174837 P: 371971 B/P: 8.53517 B: 3210430 P: 405829 B/P: 7.91079 B: 3515694 P: 458057 B/P: 7.67523 B: 4110444 P: 466243 B/P: 8.8161 B: 4135780 P: 509797 B/P: 8.11260 B: 4629075 P: 540269 B/P: 8.56809 B: 4874069 P: 569369 B/P: 8.56047 B: 4878548 P: 592393 B/P: 8.23532 B: 7072356 P: 646873 B/P: 10.93314 B: 7250295 P: 762899 B/P: 9.50361 B: 7544995 P: 794011 B/P: 9.50238 B: 7628020 P: 822581 B/P: 9.27328 B: 8859002 P: 984761 B/P: 8.99609 B: 9803338 P: 1108619 B/P: 8.84284 B: 10643795 P: 1162729 B/P: 9.15415 B: 10837063 P: 1183561 B/P: 9.15632 B: 12210491 P: 1338391 B/P: 9.12326 B: 12406756 P: 1461851 B/P: 8.48702 B: 12653477 P: 1480093 B/P: 8.54911 B: 13026434 P: 1562593 B/P: 8.33642 B: 13038299 P: 1622639 B/P: 8.03524 B: 13893955 P: 1630427 B/P: 8.52167 B: 17404169 P: 1635761 B/P: 10.6398 B: 17839448 P: 1988023 B/P: 8.97346 B: 21942152 P: 2061673 B/P: 10.64289 B: 22553727 P: 2113667 B/P: 10.67043 B: 23301925 P: 2404859 B/P: 9.68952 B: 23423860 P: 2598199 B/P: 9.01542 B: 23563154 P: 2650093 B/P: 8.89144 B: 25053557 P: 2819497 B/P: 8.88583 B: 28565195 P: 2821831 B/P: 10.12293 B: 30243125 P: 3364741 B/P: 8.98825 B: 31427133 P: 3415757 B/P: 9.20063 B: 36449372 P: 3536101 B/P: 10.30779 B: 37972677 P: 3998707 B/P: 9.49624 B: 40775927 P: 4053941 B/P: 10.05834 B: 41933834 P: 4065197 B/P: 10.31533 B: 42145057 P: 4509623 B/P: 9.34558 B: 42540939 P: 4515169 B/P: 9.42178 B: 45840287 P: 4805513 B/P: 9.5391 B: 55280391 P: 4976459 B/P: 11.10838 B: 68164170 P: 5595599 B/P: 12.18175 B: 70901672 P: 6047579 B/P: 11.72398 B: 76671798 P: 6902627 B/P: 11.10763 B: 76950757 P: 7601621 B/P: 10.12294 B: 86236947 P: 7864243 B/P: 10.9657 B: 89860260 P: 8731103 B/P: 10.29197 B: 91521057 P: 8901461 B/P: 10.28158 B: 99152674 P: 9430367 B/P: 10.51419 B: 108325082 P: 9650183 B/P: 11.22518 B: 123957828 P: 9781313 B/P: 12.67292 B: 153826602 P: 12166013 B/P: 12.64396 B: 155511778 P: 13012171 B/P: 11.95126 B: 212705716 P: 13369823 B/P: 15.90939 B: 267298932 P: 17957369 B/P: 14.88519 B: 283140559 P: 22413529 B/P: 12.63257 B: 321739035 P: 25532977 B/P: 12.60092 B: 322809614 P: 28348709 B/P: 11.3871 B: 345356467 P: 29064239 B/P: 11.88252 B: 368239040 P: 31507243 B/P: 11.68744 B: 381412125 P: 32057897 B/P: 11.8976 B: 384607752 P: 33016103 B/P: 11.6491 B: 388574544 P: 34077767 B/P: 11.40258 B: 412340994 P: 35580619 B/P: 11.58892 B: 439940432 P: 39705781 B/P: 11.08001 B: 451143978 P: 40355039 B/P: 11.17937 B: 507237777 P: 41788871 B/P: 12.13811 B: 533719723 P: 42469709 B/P: 12.56707 B: 690600360 P: 43059437 B/P: 16.03831 B: 769737572 P: 49339691 B/P: 15.60078 B: 777934986 P: 51843101 B/P: 15.00556 B: 813548642 P: 58288957 B/P: 13.95717 B: 843876057 P: 61369967 B/P: 13.75064 B: 874033697 P: 68652029 B/P: 12.73136 B: 885932533 P: 69270571 B/P: 12.78945 B: 937294832 P: 71279981 B/P: 13.14948 B: 950168997 P: 79023661 B/P: 12.02385 B: 998841342 P: 81383747 B/P: 12.27323 B: 1051419042 P: 81596041 B/P: 12.88566 B: 1148613243 P: 87910153 B/P: 13.06576 B: 1183925803 P: 94112797 B/P: 12.57986 B: 1199868631 P: 95158873 B/P: 12.60911 B: 1248949421 P: 98416079 B/P: 12.6905 B: 1511084510 P: 108248177 B/P: 13.95945 B: 1580634438 P: 120040931 B/P: 13.16746 B: 2039759423 P: 125689423 B/P: 16.22857 B: 2267295935 P: 157974923 B/P: 14.35225 Weitere Primzahlen (Fermatquotienten) mit einer Erstbasis B/P > 10 sind: 5108027, 5924851, 5967277, 6022609, 6231529, 6413809, 8083919, 8269477, 10038443, 10061911, 11269367, 13037681, 13157033, 13731511, 13842923, 13880579, 14340589, 14421821, 14793379, 15281471, 15518039, 16084457, 17062301, 17441309, 17464963, 18196109, 18669061, 18673139, 19018367, 19303309, 19705901, 19828693, 19875467, 20331917, 20434951, 21009719, 21049949, 21147869, 21236867, 21605057, 21684881, 22218377, 22251631, 22975567, 23132113, 23273693, 23891639, 24207049, 24605719, 24806027, 25612651, 25879459, 25992097, 26612851, 27266951, 27489437, 28018043, 28321537, 28993697, 29111359, 29113789, 29175833, 29590481, 30693367, 30780901, 31390091, 31823713, 31920631, 32073563, 32537053, 33002231, 33403129, 33641389, 34231699, 34634683, 34734613, 34896167, 34946017, 36602827, 36681131, 37780529, 37798289, 38189533, 38625569, 38727427, 39732029, 40024091, 40140643, 40170071, 41888767, 42332419, 42618679, 43580767, 43710731, 44062913, 44218651, 44539511, 45004481, 45004783, 45071087, 45185633, 45384839, 45971851, 46118531, 46167269, 46477283, 46539413, 46656851, 46992587, 47160859, 47413207, 47621303, 48408889, 48540743, 48889721, 49202597, 49321763, 49384523, 49583579, 49633943, 50046767, 51226811, 51375461, 51561551, 52011679, 52137023, 52257187, 52533497, 53549009, 54127807, 55573403, 55879163, 55899979, 56577503, 56714219, 56721031, 56900659, 57919703, 58489553, 58819729, 58860367, 59010011, 59138897, 59241529, 59302501, 59579873, 59814647, 60095417, 60566837, 60635639, 60882329, 61225499, 61367261, 61824991, 62271647, 62404079, 63137593, 63647407, 63715601, 63956357, 64570349, 64666207, 64787579, 64807201, 64895737, 65002037, 65137859, 65771161, 66071737, 66284951, 66646291, 67068403, 67534063, 67753171, 67932269, 68185877, 68609029, 68829757, 68869553, 69316547, 69441089, 70573753, 70861943, 71187587, 71292931, 71441417, 71671991, 73402237, 74226781, 74555633, 74680259, 74918777, 75113119, 75548069, 75717877, 76726051, 77441491, 78305543, 78372521, 79648963, 80080751, 80213369, 80994899, 81345421, 81855517, 82834517, 82932497, 83175733, 84167231, 84674791, 85146773, 85847851, 85873759, 86162569, 86360783, 86508271, 86585141, 86646499, 87952343, 88498639, 88708681, 88973723, 89223131, 90472073, 90678421, 91455163, 91852757, 91882589, 92000941, 92028929, 92331727, 92568079, 92635883, 93717713, 95287519, 95494381, 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135203693, 135551879, 136004111, 136368581, 136920947, 137692909, 138090751, 138677387, 139348129, 139780159, 140917597, 143876347, 144198833, 144358733, 145148033, 145536247, 146166109, 146269741, 146415803, 146479031, 146823277, 146984521, 147207961, 147221153, 148563137, 148817069, 148831231, 149434561, 150247693, 150467201, 151752973, 151992601, 152977819, 153792389, 154504631, 155404201, 155836607, 156078383, 157170617, 157685399, 158362723, 158375011, 158488403, 158979389, 160223137, 160416049 06.09.2010 Richard Fischer