Thema: Anzahl und Intervalle zwischen je 8 benachbarten Primzahlen (8-Tupel) Number and intervals between 8 adjacent prime numbers (8-tuples) Main index: http://www.fermatquotient.com Erste 3 Primzahl-Octupel vor dem jeweiligen Intervall zum Octupel-Nachbarn. First 3 prime number octuples before the respective interval to the octuple neighbor. Variant A: p-13,p-11,p-7,p-5,p-1,p+5,p+7,p+13 ==> Factor 6/5 Variant B: p-13,p-11,p-7,p-1,p+1,p+7,p+11,p+13 ==> Factor 16/5 Variant C: p-13,p-7,p-5,p+1,p+5,p+7,p+11,p+13 ==> Factor 6/5 Octuples: Sum of factors = 28/5 = 5.6 Begin: 24±13 Interval, Octuple±13, Octuple±13, Octuple±13 210 Var. B, 420 Var. A, 7387829334225124914, 12072258109083857814 420 Var. B, 839127400597710450, 1818743886123408690, 1947915980437935720 420 Var. C, 1365135293377079676 630 Var. B, 5550473314704358980, 22964705179106205870 840 Var. A, 840 Var. C, 18263588245075895796, 24940796541060310446 1050 Var. A, 7787352707025024444, 13582163042760806574 1050 Var. B, 1050 Var. C, 572763847597837536 1260 Var. A, 7358539621385904234, 18909038612035855104, 20774438314353866964 1260 Var. B, 8975703536148172920, 10599861784151947260 1260 Var. C, 9809980019049219006, 13750566157734815406, 23652735826929748536 1470 Var. A, 18980921214089879124 1470 Var. C, 1908642756881197866, 17545069342775462016, 23602213840366415706 1680 Var. B, 1075876674163445250, 20939888359076744010 1890 Var. A, 2582241812968029684, 15925383974910482544 1890 Var. B, 5565038852145834000, 11091508947596020050 1890 Var. C, 2100 Var. B, 2310 Var. A, 122495334048108024, 1534436133978118374 2310 Var. B, 133580450164085790, 5419376890650597870, 12764101955245776000 2310 Var. C, 1729345218739815366 ... https://www.pzktupel.de/smarchive.php 148.551628663785371162800656735 = Basis Octuple Prime Constant = (Hardy-Littlewood Constant C_8)*35^7/(2^25*3) Integral: 148.5516...*dX/[Ln(X)]^8 von X = 9 bis X = 1E+15 ==> 96698.383242724800 Näherung ; Approximation: Li_8(X) = X/[Ln(X)-1-4/Ln(X)-87/Ln(X)^2]^8*148.552*5.6 ; X > 1E+09 Error = [pi_8(X)-Li_8(X)]/Sqrt[Li_8(X)] ==================================================== Variants A and B and C X SLi_8(X) Spi_8(X) Error delta_8(X) ==================================================== 1E+08 19.76 19 -0.2 -0.76 1E+09 50.26 47 -0.46 -3.26 1E+10 176.85 183 +0.46 6.15 1E+11 750.40 724 -0.96 -26.40 1E+12 3542.33 3660 +1.98 117.67 1E+13 17969.69 18123 +1.14 153.31 1E+14 96389.13 96521 +0.42 131.87 5E+14 320655.05 320178 -0.84 -477.05 1E+15 541510.95 541066 -0.60 -444.95 2E+15 917755.69 916640 -1.16 -1115.69 5E+15 1852996.56 1851344 -1.21 -1652.56 1E+16 3164801.21 3164872 +0.04 70.79 1.5E+16 4334724.43 4334559 -0.08 -165.43 2E+16 5422067.57 5421145 -0.40 -922.57 3E+16 7439485.54 7437259 -0.82 -2226.54 5E+16 11097703.24 11091683 -1.81 -6020.24 7E+16 14454579.82 14446427 -2.14 -8152.82 1E+17 19141637.84 19134374 -1.66 -7263.84 1.5E+17 26364375.98 26356333 -1.57 -8042.98 2E+17 33106043.39 33094502 -2.01 -11541.39 2.5E+17 39513670.58 39503610 -1.60 -10060.58 3E+17 45668518.63 45664923 -0.53 -3595.63 1E+50 2.89845542383E+36 1E+100 1.09085802350E+84 ==================================================== Maximale Intervalle von Octupel-Primzahlen der Differenz 26 Maximum intervals of octuple prime numbers of difference 26 Die relativen Werte habe ich so definiert, dass ich auch bei riesigen Primzahlen maximal, nur Werte wenig über 1 erwarte. I defined the relative values in such a way that even with huge prime numbers maximum, only expect values slightly above 1. Da sich für kleinere Primzahlen verhältnismässig kleine Werte ergeben, habe ich den Korrektur-Faktor K eingeführt. Since smaller prime numbers result in relatively small values, I have that Correction factor K introduced. K = {1-Ln[Ln(letzte Primzahl - 26/2)]/Ln(letzte Primzahl - 26/2)/2}^8 K = {1-Ln[Ln(last prime number - 26/2)]/Ln(last prime number - 26/2)/2}^8 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 26/2)^8 Rekord-Intervall, 1. Primzahl, letzte Primzahl des gefundenen Intervalls, relativ Record interval, 1st prime number, last prime number of the found interval, relative 1260, 17, 1303, 0.07995 87516, 1277, 88819, 0.06213 171576, 113147, 284749, 0.04774 570990, 284723, 855739, 0.07094 1433874, 1146773, 2580673, 0.08463 3980346, 2580647, 6561019, 0.13077 9199098, 6560993, 15760117, 0.17996 32549946, 25658441, 58208413, 0.30906 34106370, 100658627, 134765023, 0.20933 35630232, 182403491, 218033749, 0.17184 52428180, 311725847, 364154053, 0.19687 106510800, 408936947, 515447773, 0.33885 209256864, 701679047, 910935937, 0.51056 213077286, 3734403131, 3947480443, 0.27137 217647306, 4149404087, 4367051419, 0.26548 232215690, 4723119677, 4955335393, 0.26845 310686756, 4955335367, 5266022149, 0.35004 473025162, 7602979451, 8076004639, 0.44565 547025694, 14253395063, 14800420783, 0.40234 775720266, 37654490171, 38430210463, 0.39136 894089904, 53262070223, 54156160153, 0.39537 1063249896, 57440594201, 58503844123, 0.45655 1081594866, 91192669481, 92274264373, 0.39116 1082118180, 149440186787, 150522304993, 0.3266 1110726168, 246062159243, 247172885437, 0.28008 1126329600, 251117697977, 252244027603, 0.28195 1316367996, 252486203927, 253802571949, 0.3288 1461989400, 257617642967, 259079632393, 0.36249 1374300114, 264794984543, 266169284683, 0.33747 1574210814, 334897899227, 336472110067, 0.35559 2486301930, 369422358287, 371908660243, 0.54207 2927478378, 807979592663, 810907071067, 0.4866 2934040566, 1021851148931, 1024785189523, 0.45023 2941710654, 1494723868487, 1497665579167, 0.39714 4597912350, 1993523182577, 1998121094953, 0.56383 4685092920, 3612589540517, 3617274633463, 0.47283 4930582182, 7289418138461, 7294348720669, 0.39735 5280401016, 7578818435387, 7584098836429, 0.42033 5349481830, 8043493895603, 8048843377459, 0.41789 5388430104, 8568575395397, 8573963825527, 0.41263 5538305166, 11402613244487, 11408151549679, 0.38778 5892124194, 11532561230147, 11538453354367, 0.41109 7786181340, 11647384527197, 11655170708563, 0.54154 8244845748, 14072152489253, 14080397335027, 0.54072 8774673600, 16277198070431, 16285972744057, 0.55016 10655025414, 22740914300957, 22751569326397, 0.60297 13490361786, 48498392765087, 48511883126899, 0.60781 15253039824, 100128830897987, 100144083937837, 0.55538 17119512894, 138148915489217, 138166035002137, 0.56797 19820279376, 171551546835287, 171571367114689, 0.618 21846721056, 287273520255077, 287295366976159, 0.58862 28800949320, 444601660361687, 444630461311033, 0.68691 31624909464, 1028386184626727, 1028417809536217, 0.59949 33903266040, 1381373849943533, 1381407753209599, 0.59359 35635543650, 2090985893294147, 2091021528837823, 0.55866 46208484270, 2338886920593257, 2338933129077553, 0.70327 51016528290, 6581160482088323, 6581211498616639, 0.59323 65960132856, 9128356772086457, 9128422732219339, 0.70556 71953721280, 18526614551898137, 18526686505619443, 0.64414 72744540336, 30870755736394571, 30870828480934933, 0.57394 73963926276, 33481065603534161, 33481139567460463, 0.57205 74163801024, 37976697503774987, 37976771667576037, 0.55618 79216551966, 39526576135794257, 39526655352346249, 0.5883 93934089108, 39602227417865243, 39602321351954377, 0.69728 95235546996, 90859562272189637, 90859657507736659, 0.57853 95333663460, 97973843782917917, 97973939116581403, 0.56881 101887816380, 117147321330488147, 117147423218304553, 0.58263 124040578404, 147105047422641017, 147105171463219447, 0.67212 126318465372, 154826578527161531, 154826704845626929, 0.67627 147537453426, 196468577567144411, 196468725104597863, 0.74691 * 02.05.2025 Richard Fischer