Thema: Allgemeine Repunitpaar-Primzahlen (B^N+1)/(B+1) Main table of content: http://www.fermatquotient.com/ Siehe auch unter: https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/ 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://oeis.org/ --> 701, 1709, 2617, 3539, 5807 Search (for example) Repunitpaar (Basis B=10) Primzahlen Mittlere Anzahl Primzahlen pro Basis bis N: (ln(N)+m*ln(2)*ln(ln(N))+1/sqrt(N)-1.6)*e^C/ln(B) m = 1 für die Nichtsonderfälle m = 2 für die Basen 9, 25, 36, 49, 100, 121, 144, 169, 196, 225, 289, 361, 400, 441, 484, 529, 576 m = 3 für die Basen 16, 81, 625, 1296, 2401, 10000, 14641, 20736 m = 4 für die Basen 256, 6561, 390625, 10^8 m = 5 für die Basis 65536, 3^16 und m = 6 für die Basis 2^32 C = 0.57721566490... e^C = 1.78107241799... Basis Exponent N 2 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, (?) 13347311, 13372531, 15135397 3 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, 1896463, 2533963 5 5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, 1856147 6 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, 1313371 7 3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, 1178033 10 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, 1600787 11 5, 7, 179, 229, 439, 557, 6113, 223999, 327001, 2264611 12 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, 495953 13 3, 11, 17, 19, 919, 1151, 2791, 9323, 56333, 1199467 14 7, 53, 503, 1229, 22637, 1091401, 1385203 15 3, 7, 29, 1091, 2423, 54449, 67489, 551927 17 7, 17, 23, 47, 967, 6653, 8297, 41221, 113621, 233689, 348259 18 3, 7, 23, 73, 733, 941, 1097, 1933, 4651, 481147 19 17, 37, 157, 163, 631, 7351, 26183, 30713, 41201, 77951, 476929 20 5, 79, 89, 709, 797, 1163, 6971, 140053, 177967, 393257 21 3, 5, 7, 13, 37, 347, 17597, 59183, 80761, 210599, 394579 22 3, 5, 13, 43, 79, 101, 107, 227, 353, 7393, 50287 23 11, 13, 67, 109, 331, 587, 24071, 29881, 44053 24 7, 11, 19, 2207, 2477, 4951 26 11, 109, 227, 277, 347, 857, 2297, 9043 28 3, 19, 373, 419, 491, 1031, 83497, 223381 29 7, 112153, 151153 30 139, 173, 547, 829, 2087, 2719, 3109, 10159, 56543, 80599 31 109, 461, 1061, 50777 33 5, 67, 157, 12211, 313553 34 3, 294277 35 11, 13, 79, 127, 503, 617, 709, 857, 1499, 3823, 135623, 280979 37 5, 7, 2707, 163193 38 5, 167, 1063, 1597, 2749, 3373, 13691, 83891, 131591 39 3, 13, 149, 15377 40 53, 67, 1217, 5867, 6143, 11681, 29959 41 17, 691, 113749 42 3, 709, 1637, 17911, 127609, 172663 43 5, 7, 19, 251, 277, 383, 503, 3019, 4517, 9967, 29573 44 7, 41233 45 103, 157, 37159 46 7, 23, 59, 71, 107, 223, 331, 2207, 6841, 94841 47 5, 19, 23, 79, 1783, 7681 48 5, 17, 131, 84589 50 1153, 26903, 56597 51 3, 149, 3253 52 7, 163, 197, 223, 467, 5281, 52901, 85259 53 21943, 24697, 158341 54 7, 19, 67, 197, 991, 99563, 128189, 164839 55 3, 5, 179, 229, 1129, 1321, 2251, 15061, 299087 56 37, 107, 1063, 4019 57 53, 227, 18211, 20231, 22973, 87719, 111119 58 3, 17, 1447, 11003 59 17, 43, 991, 33613, 203309 60 3, 937, 1667, 3917, 18077, 31393, 119083 61 7, 41, 359, 17657 62 11, 29, 167, 313, 16567, 38699, 170539, 286483 63 3, 37, 41, 2131, 4027, 22283, 51439, 102103, 188147, 238481 65 19, 31 66 7, 17, 211, 643, 28921, 58741, 63079, 67349 67 3, 2347, 2909, 3203, 203431, 239053 68 757, 773, 71713 69 11, 211, 239, 389, 503, 4649, 24847 70 3, 61, 97, 13399, 42737 71 5, 37, 5351, 7499, 68539, 77761 72 3, 7, 79, 277, 3119 73 7, 39181, 280697 74 13, 31, 37, 109, 17383, 167311 75 5, 83, 6211 76 3, 5, 191, 269, 23557, 165947 77 37, 317 78 3, 7, 31, 661, 4217 79 3, 107, 457, 491, 2011 80 5, 13, 227, 439, 191953, 192133, 228419 82 293, 1279, 97151 83 19, 31, 37, 43, 421, 547, 3037, 8839 84 7, 13, 139, 359, 971, 1087, 3527 85 167, 3533, 48677, 138647 86 7, 17, 397, 7159, 103471, 123677 87 7, 467, 43189 88 709, 1373, 61751, 208739 89 13, 59, 137, 1103, 4423, 82609, 101363 90 3, 47 91 3, 11, 43, 397, 21529, 37507, 61879 92 37, 59, 113 93 89, 571, 601, 3877 94 71, 307, 613, 1787, 3793, 10391 95 43, 93377, 127583 96 37, 103, 131, 263, 32369 97 (>500000) 98 19, 101, 78797, 114859, 189619 99 7, 37, 41, 71, 357779 101 7, 229, 91463, 166849 102 3 103 104 673, 839, 1031, 24877, 28201 105 11, 149, 1187, 1627 106 3, 7, 19, 23, 31, 3989 107 103, 983, 18049, 28703 108 13, 223, 15731 109 59, 79, 811 110 23, 101, 17041 111 3, 5, 23, 53, 383, 2039, 12109 112 3 113 114 7, 13, 1801, 12487 115 7, 31, 293 116 113, 1481, 2089, 16889 117 271 118 3, 23, 109, 2357 119 29, 53, 797, 11491 120 3, 31, 43, 263, 4919 122 293, 3877, 12889, 22277 123 29, 739 124 16427 126 5, 13, 47, 163, 239, 4523 127 317, 1061, 23887 129 17, 227, 1753 130 467 131 5, 101, 3389, 3581 132 3, 101, 157, 1303 133 5, 7, 17, 59, 79, 157 134 13, 1171, 6733 135 5, 7, 2671, 11953 136 5, 7, 23, 59, 199, 2053, 6067 137 101, 241, 353, 1999, 21851 138 103, 577, 10781 139 3, 17, 47, 2683, 2719, 26437 140 59, 29819 141 5, 1471 142 3, 7537 143 7, 17, 19, 47, 103, 4423, 18287 145 7, 23, 281, 24229 146 17, 1439, 11027 147 11, 151, 6599 148 3, 7, 31, 43, 163, 317, 1933, 5669, 11789, 19289, 22171 149 17, 769 150 6883, 15139 151 3, 367, 3203, 7993, 10273, 14437 152 13, 19 153 13, 1063, 5749 154 3, 29, 263, 601, 619, 809, 1217, 2267 155 5, 22679 156 3, 1301, 25933 157 5, 157, 809, 1861, 2203 158 5, 769, 5023 159 283, 449, 1949, 7457, 23369, 29303 160 11, 37, 1907, 10487 Durchsucht bis mindestens N = 32803 Sonderfälle bis B = 625 mit etwas höherer Primwahrscheinlichkeit: [Erwartung bis 32801] Basis Exponent N 9 3, 59, 223, 547, 773, 1009, 1823, 3803 [9.8], 49223, 193247, 703393, 860029 16 3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317, 30197 [8.8], 1025393 25 3, 7, 23, 29, 59, 1249, 1709, 1823, 1931, 3433, 8863 [6.7], 43201, 78707 36 31, 191, 257, 367, 3061 [6.0], 110503 49 7, 19, 37, 83, 1481, 12527, 20149 [5.5] 81 3, 5, 701, 829, 1031, 1033, 7229, 19463 [5.5], 370421 100 3, 293, 461, 11867 [4.7], 90089 121 5, 13, 97, 1499, 11321 [4.5], 320483 144 3, 23, 41, 317, 3371 [4.3], 45259, 119671 169 3, 7, 109, 21943 [4.2] 196 43, 1049, 5441, 18089 [4.1] 225 383, 1277 [4.0] 256 5, 13, 23029 [4.9], 50627, 51479, 72337 289 3, 179, 181, 683 [3.8] 361 5, 23, 223, 4441 [3.6] 400 263 [3.6] 441 101, 197 [3.5] 484 257 [3.5] 529 587, 683, 25693 [3.4] 576 379, 461, 1861, 28307 [3.4] 625 3, 7, 11, 31, 67, 9173, 17737, 26267 [3.8] Durchsucht bis mindestens N = 32803 Spezielle Ausnahmen: Basis Exponent N 4 3 keine weiteren, weil (4^N+1) = {2^N-2^[(N+1)/2]+1} * {2^N+2^[(N+1)/2]+1} 324 keine, weil (324^N+1)/325 = {18^N-6*18^[(N-1)/2]+1}/25 * {18^N+6*18^[(N-1)/2]+1}/13 oder (324^N+1)/325 = {18^N+6*18^[(N-1)/2]+1}/25 * {18^N-6*18^[(N-1)/2]+1}/13 2500 none, because (2500^N+1)/2501 = {50^N-10*50^[(N-1)/2]+1}/61 * {50^N+10*50^[(N-1)/2]+1}/41 or (2500^N+1)/2501 = {50^N+10*50^[(N-1)/2]+1}/61 * {50^N-10*50^[(N-1)/2]+1}/41 5184 keine, weil (5184^N+1)/5185 = {72^N-12*72^[(N-1)/2]+1}/85 * {72^N+12*72^[(N-1)/2]+1}/61 oder (5184^N+1)/5185 = {72^N+12*72^[(N-1)/2]+1}/85 * {72^N-12*72^[(N-1)/2]+1}/61 9604 none, because (9604^N+1)/9605 = {98^N-14*98^[(N-1)/2]+1}/113 * {98^N+14*98^[(N-1)/2]+1}/85 or (9604^N+1)/9605 = {98^N+14*98^[(N-1)/2]+1}/113 * {98^N-14*98^[(N-1)/2]+1}/85 Weitere: Basis Exponent N N 3, 5, 17, 157 (>24077) ==> (N^N+1)/(N+1) N^2 3, 7, 29, 41, 43, 61, 577 (>16417) ==> (N^2N+1)/(N^2+1) 197 = 14^2+1 31, 37, 101, 163 (>32803) 255 = 2^8-1 7, 59, 179, 263, 4283, 15527 (>32803) 257 = 2^8+1 5, 47, 2909, 8747, 25537 (>32803) 359 = 2^19-2 7, 17, 59 (>32803) 360 = 2^19-1 41, 43, 167, 1987 (>32803) 401 = 20^2+1 7, 41, 227 (>32803) 1296 = 6^4 3, 2153, 3517 (>24077) 1297 = 6^4+1 311, 3833 (>16417) 2401 = 7^4 37, 3583, 8059 (>16417) 6561 = 3^8 19, 29, 11213 (>16417) 8191 = 2^13-1 (>16417) 10000 = 10^4 3, 283, 1087 (>16417) 14641 = 11^4 13, 211 (>16417) 20736 = 12^4 7, 593 (>16417) 65536 = 2^16 239 (>16417) 65537 = 2^16+1 5 (>16417) 390625 = 5^8 3 (>16417) 3^16 (>16417) 10^8 3 (>16417) 2^32 3, 13619 (>16417) Weitere Sonderfälle bis B = 1024: Basis Exponent N 216, 343, 729 3 keine weiteren 128 7 keine weiteren 8, 27, 32, 64, 125 keine 243, 512, 1000, 1024 keine Statistik ohne Sonderfälle: [Erwartung] Exponent N = 3 liefert 39 Primzahlen Exponent N = 5 und 7 liefert 66 Primzahlen Exponent N = 11 und 13 liefert 31 Primzahlen Exponent N zwischen 16 und 32 liefert 61 Primzahlen [51.6] Exponent N zwischen 32 und 64 liefert 49 Primzahlen [51.1] Exponent N zwischen 64 und 128 liefert 51 Primzahlen [50.8] Exponent N zwischen 128 und 256 liefert 51 Primzahlen [50.6] Exponent N zwischen 256 und 512 liefert 56 Primzahlen [50.4] Exponent N zwischen 512 und 1024 liefert 47 Primzahlen [50.1] Exponent N zwischen 1024 und 2048 liefert 52 Primzahlen [49.9] Exponent N zwischen 2048 und 4096 liefert 49 Primzahlen [49.7] Exponent N zwischen 4096 und 8192 liefert 35 Primzahlen [49.5] Exponent N zwischen 8192 und 16384 liefert 38 Primzahlen [49.3] Exponent N zwischen 16384 und 32768 liefert 54 Primzahlen [49.2] Mittlere Primzahlen Anzahl pro N-Faktor 2 ohne Sonderfälle Arp2 = e^C*(ln(2*N)-ln(N))*(1/ln(2)+1/ln(3)+1/ln(5)+...+1/ln(160)) Arp2 = 1.781072418*ln(2)*37.37567946 = 46.14197034 (vereinfacht für sehr grosse N) 30.03.2022 Richard Fischer