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 fuer die Nichtsonderfaelle m = 2 fuer die Basen 9, 25, 36, 49, 100, 121, 144, 169, 196, 225, 289, 361, 400, 441, 484, 529, 576 m = 3 fuer die Basen 16, 81, 625, 1296, 2401, 10000, 14641, 20736 m = 4 fuer die Basen 256, 6561, 390625, 10^8 m = 5 fuer die Basis 65536, 3^16 und m = 6 fuer 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, 2674381, 7034611 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, 1522841 15, 3, 7, 29, 1091, 2423, 54449, 67489, 551927, 1841911, 1848811 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, 463511 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, 821497 26, 11, 109, 227, 277, 347, 857, 2297, 9043, 501409 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, 735439 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, 294947 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, 379703 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, (>800000) 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 161, 31, 331, 1483 162, 3, 1823, 7703, 15377 163, 3, 11, 31, 661, 1999, 4079, 6917 164, 7, 103, 541, 1109 165, 3, 5, 383 166, 17, 5437, 14747 167, 17, 59, 1301, 3167 168, 3, 31, 1741, 2099 170, 7 171, 13, 149, 257, 4967 172, 37, 283, 647, 4483, 5417 173, 7, 59, 569, 2647 174, 3, 3191 175, 31627 176, 5, 31, 269, 479, 599, 809, 1307, 17707 177, 3, 5, 19, 419 178, 61, 167, 227 179, 827, 5011, 8867 180, 5, 13, 7369, 8101 181, 449, 2687, 4877, 20407 182, 1487, 8081 183, 11, 16363 184, 19, 79, 149, 7283 185, 11 186, 187, 188, 22037 189, 3, 31, 71, 8123 190, 3, 19, 1153 191, 479, 1163 192, 109, 197, 587, 727, 1997, 2441 193, 3, 11, 67, 3253 194, 19, 31 195, 3, 13, 19, 43, 89, 1087, 1949, 2939 197, 31, 37, 101, 163 (>32803) 198, 37, 151, 937 199, 313, 2579, 5387 200, 7, 277 Durchsucht bis mindestens N = 24917 Sonderfaelle bis B = 625 mit etwas hoeherer 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, 1145393 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) 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) 496 = 2^4*31 , 3, 787, 9059 (>24077) 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) 7777 = 6^5+1 , 677 (>16417) 8128 = 2^6*127 , 5, 71, 709 (>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 Sonderfaelle 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 Sonderfaelle: [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 Sonderfaelle 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 fuer sehr grosse N) 29.03.2025 Richard Fischer