Thema: Allgemeine Repunit-Primzahlen (B^N-1)/(B-1) Main table of content: http://www.fermatquotient.com/ Siehe auch unter: http://www.primenumbers.net/Henri/us/MersFermus.htm http://www.primenumbers.net/prptop/prptop.php?page=1#haut --> Search by form --> (x^y-1)/z http://primes.utm.edu/top20/page.php?id=4 http://oeis.org/ --> 19, 23, 317, 1031, 49081 Search (for example) Mersenne'sche (Basis B=2) Primzahlen Repunit (Basis B=10) Primzahlen Mittlere Anzahl Primzahlen pro Basis bis N: [ln(N)+ln(2)*ln(ln(N))+1/sqrt(N)-1.1]*e^C/ln(B) C = 0.57721566490... e^C = 1.78107241799... Basis Exponent N 2 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, (?) 74207281, (?) 77232917, (?) 82589933 3 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117 5 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, 3300593, 4939471, 5154509 6 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, 1365019, 3360347 7 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699 10 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, 5794777, 8177207 11 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, 1868983 12 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543 13 5, 7, 137, 283, 883, 991, 1021, 1193, 3671, 18743, 31751, 101089, 1503503 14 3, 7, 19, 31, 41, 2687, 19697, 59693, 67421, 441697 15 3, 43, 73, 487, 2579, 8741, 37441, 89009, 505117, 639833 17 3, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509, 1990523 18 2, 25667, 28807, 142031, 157051, 180181, 414269, 1270141 19 19, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359 20 3, 11, 17, 1487, 31013, 48859, 61403, 472709, 984349 21 3, 11, 17, 43, 271, 156217, 328129 22 2, 5, 79, 101, 359, 857, 4463, 9029, 27823 23 5, 3181, 61441, 91943, 121949, 221411 24 3, 5, 19, 53, 71, 653, 661, 10343, 49307, 115597, 152783 26 7, 43, 347, 12421, 12473, 26717 28 2, 5, 17, 457, 1423, 115877 29 5, 151, 3719, 49211, 77237 30 2, 5, 11, 163, 569, 1789, 8447, 72871, 78857, 82883 31 7, 17, 31, 5581, 9973, 54493, 101111, 535571 33 3, 197, 3581, 6871, 183661 34 13, 1493, 5851, 6379, 125101 35 313, 1297, 568453 37 13, 71, 181, 251, 463, 521, 7321, 36473, 48157, 87421, 168527, 249341 38 3, 7, 401, 449, 109037 39 349, 631, 4493, 16633, 36341 40 2, 5, 7, 19, 23, 29, 541, 751, 1277 41 3, 83, 269, 409, 1759, 11731 42 2, 1319, 337081 43 5, 13, 6277, 26777, 27299, 40031, 44773, 194119 44 5, 31, 167, 100511 45 19, 53, 167, 3319, 11257, 34351, 216551 46 2, 7, 19, 67, 211, 433, 2437, 2719, 19531 47 127, 18013, 39623 48 19, 269, 349, 383, 1303, 15031, 200443 50 3, 5, 127, 139, 347, 661, 2203, 6521, 210319 51 4229, 35227 52 2, 103, 257, 4229, 6599, 264697 53 11, 31, 41, 1571, 25771, 181981 54 3, 389, 16481, 18371, 82471 55 17, 41, 47, 151, 839, 2267, 3323, 3631, 5657, 35543, 98419, 179573 56 7, 157, 2083, 2389, 57787 57 3, 17, 109, 151, 211, 661, 16963, 22037, 275669 58 2, 41, 2333, 67853 59 3, 13, 479, 12251, 169777 60 2, 7, 11, 53, 173 61 7, 37, 107, 769 62 3, 5, 17, 47, 163, 173, 757, 4567, 9221, 10889 63 5, 3067, 38609, 195893 65 19, 29, 631, 375017 66 2, 3, 7, 19, 19973 67 19, 367, 1487, 3347, 4451, 10391, 13411, 167449 68 5, 7, 107, 149, 2767 69 3, 61, 2371, 3557, 8293, 106397, 178093 70 2, 29, 59, 541, 761, 1013, 11621, 27631 71 3, 31, 41, 157, 1583, 31079, 55079, 72043, 150697, 172259 72 2, 7, 13, 109, 227, 273149 73 5, 7, 35401, 110603 74 5, 191, 3257, 31267 75 3, 19, 47, 73, 739, 13163, 15607, 93307 76 41, 157, 439, 593, 3371, 3413, 4549, 157649 77 3, 5, 37, 15361, 200657 78 2, 3, 101, 257, 1949, 67141, 245183 79 5, 109, 149, 659, 28621 80 3, 7 82 2, 23, 31, 41, 7607, 12967 83 5, 2713 84 17, 3917 85 5, 19, 2111, 259159, 275881 86 11, 43, 113, 509, 1069, 2909, 4327, 40583 87 7, 17, 121487, 257689 88 2, 61, 577, 3727, 22811, 40751 89 3, 7, 43, 47, 71, 109, 571, 11971, 50069 90 3, 19, 97, 5209, 159463 91 4421, 20149 92 439, 13001, 22669, 44491, 271639 93 7, 4903 94 5, 13, 37, 1789, 3581, 170371, 241993 95 7, 523, 9283, 10487, 11483 96 2, 3343, 46831 97 17, 37, 1693, 187393 98 13, 47, 2801 99 3, 5, 37, 47, 383, 5563 101 3, 337, 677, 1181, 6599 102 2, 59, 673, 25087 103 19, 313, 1549, 41183 104 97, 263, 5437 105 3, 19, 389, 2687, 4783 106 2, 149 107 17, 24251 108 2, 449, 2477 109 17, 1193, 13679, 27061 110 3, 5, 13, 691, 1721, 3313, 11827 111 3, 337 112 2, 79, 107, 701, 1697, 5657 113 23, 37, 6563 114 29, 43, 73, 89, 569, 709 115 7, 241, 1409, 2341, 2539, 7673, 12539, 16879 116 59, 2503 117 3, 5, 19, 31 118 5, 163, 193 119 3, 19, 827, 2243, 3821 120 5, 373, 1693 122 5, 7, 67, 3803 123 43, 563, 1693, 4877, 22741 124 599, 18367, 28591 126 2, 7, 37, 59, 127, 20947 127 5, 23, 31, 167, 5281, 8969, 23297, 165601 129 5, 17, 109, 8447 130 2, 37 131 3, 31, 263 132 47, 71, 3343 133 13, 599, 991, 1181, 3083, 14827 134 5, 37, 353, 2843, 21379 135 1171, 15227 136 2, 227, 293, 4133 137 11, 19, 1009, 2939 138 2, 3, 61, 13679 139 163, 173, 3821 140 79, 577, 1721 141 3, 23, 173, 3217 142 1231, 6133 143 3, 5 145 5, 31 146 7, 83, 857, 21961 147 3, 17, 19, 37, 163, 571, 983, 3697 148 2, 1201 149 7, 13, 17, 317, 3251 150 2, 3, 3389 151 13, 29, 127, 4831, 5051, 13249, 18251 152 270217 153 3, 5099 154 5, 8161 155 3, 61, 449, 2087 156 2, 7, 199, 5591 157 17, 107, 2791, 39047, 53819, 90239 158 7, 79, 109, 4003, 6151, 10453 159 13, 89, 577, 1433, 9643 160 7, 17, 151, 1487, 3989, 20773 Durchsucht bis mindestens N = 32803 Weitere: Basis Exponent N N 2, 3, 19, 31, 7547 (>24077) ==> (N^N-1)/(N-1) 170 = 13^2+1 17, 23, 79, 1237, 19843 (>32803) 197 = 14^2+1 31, 47, 283, 11719 (>32803) 217 = 6^3+1 281, 821 (>32803) 226 = 15^2+1 2, 127, 619, 7043 (>32803) 244 = 3^5+1 3331, 5099 (>32803) 255 = 2^8-1 5, 151, 701 (>32803) 257 = 2^8+1 23, 59, 487, 967, 5657 (>32803) 290 = 17^2+1 3, 7, 31963 (>32803) 344 = 7^3+1 3, 23 (>32803) 359 = 19^2-2 5, 59, 101, 383, 74189, 98327 360 = 19^2-1 2609 (>32803) 401 = 20^2+1 127, 199, 6551 (>32803) 496 = 2^4*31 47, 67 (>24077) 513 = 2^9+1 17, 2663, 6883 (>24077) 577 = 24^2+1 109, 139, 227 (>24077) 626 = 5^4+1 3, 11, 61, 1249, 21379 (>24077) 730 = 3^6+1 13 (>24077) 1001 = 10^3+1 3, 1787 (>20051) 1025 = 2^10+1 13, 83 (>20051) 1297 = 6^4+1 5, 7, 29, 2423 (>20051) 1601 = 40^2+1 11 (>16417) 2188 = 3^7+1 7, 3011, 12437 (>16417) 2402 = 7^4+1 3, 19 (>16417) 3126 = 5^5+1 11, 2749, 14431, 14983 (>16417) 4097 = 2^12+1 7, 37, 3673, 8311 (>16417) 6562 = 3^8+1 2, 701 (>16417) 7777 = 6^5+1 5 (>16417) 8128 = 2^6*127 13, 151, 1009, 15859 (>16417) 8191 = 2^13-1 (>16417) 10001 = 10^4+1 11, 569 (>16417) 65537 = 2^16+1 7, 11 (>16417) 100001 = 10^5+1 31, 53 (>16417) 1000001 = 10^6+1 11, 277 (>16417) Sonderfälle bis B = 360: Basis Exponent N 4, 16, 36, 100, 196, 256 2 keine weiteren 8, 27 3 keine weiteren 128 7 keine weiteren 9, 25, 32, 49, 64 ,81 keine 121, 125, 144, 169, 216 keine 225, 243, 289, 324, 343 keine Statistik ohne Sonderfälle: [Erwartung] Exponent N = 2 liefert 32 Primzahlen Exponent N = 3 liefert 43 Primzahlen Exponent N = 5 und 7 liefert 69 Primzahlen Exponent N = 11 und 13 liefert 25 Primzahlen Exponent N zwischen 16 und 32 liefert 66 Primzahlen [51.6] Exponent N zwischen 32 und 64 liefert 48 Primzahlen [51.1] Exponent N zwischen 64 und 128 liefert 50 Primzahlen [50.8] Exponent N zwischen 128 und 256 liefert 41 Primzahlen [50.6] Exponent N zwischen 256 und 512 liefert 46 Primzahlen [50.4] Exponent N zwischen 512 und 1024 liefert 50 Primzahlen [50.1] Exponent N zwischen 1024 und 2048 liefert 42 Primzahlen [49.9] Exponent N zwischen 2048 und 4096 liefert 58 Primzahlen [49.7] Exponent N zwischen 4096 und 8192 liefert 46 Primzahlen [49.5] Exponent N zwischen 8192 und 16384 liefert 51 Primzahlen [49.3] Exponent N zwischen 16384 und 32768 liefert 46 Primzahlen [49.2] Mittlere Primzahlen Anzahl pro N-Faktor 2 ohne Sonderfälle Am2 = e^C*(ln(2*N)-ln(N))*(1/ln(2)+1/ln(3)+1/ln(5)+...+1/ln(160)) Am2 = 1.781072418*ln(2)*37.37567946 = 46.14197034 (vereinfacht für sehr grosse N) 03.04.2024 Richard Fischer