Thema: Anzahl und Maximalintervalle zwischen 9 bis 16 benachbarten Primzahlen (9- bis 16-Tupel) Number and maximum intervals between 9 to 16 adjacent prime numbers (9- to 16-tuples) Main index: http://www.fermatquotient.com https://www.pzktupel.de/smarchive.php https://pzktupel.de/ktpatt_hl.php http://www.plouffe.fr/simon/constants/twinprime.txt 4 nonuple variants: Variant A: p,p+2,p+6,p+8,p+12,p+18,p+20,p+26,p+30 ==> Factor 15/8 Variant B: p,p+2,p+6,p+12,p+14,p+20,p+24,p+26,p+30 ==> Factor 15/4 Variant C: p,p+4,p+6,p+10,p+16,p+18,p+24,p+28,p+30 ==> Factor 15/4 Variant D: p,p+4,p+10,p+12,p+18,p+22,p+24,p+28,p+30 ==> Factor 15/8 Nonuples: Sum of factors = 45/4 = 11.25 Begin: 26±15 336.034326749231865528620127461 = Basis Nonuple Prime Constant = (Hardy-Littlewood Constant C_9)*35^8/(2^28*3) Integral: 336.0343...*dX/[Ln(X)]^9 von X = 10 bis X = 1E+15 ==> 6601.3927180239 Näherung ; Approximation: Li_9(X) = X/[Ln(X)-1-4.4/Ln(X)-110/Ln(X)^2]^9*336.034*11.25 ; X > 1E+10 Error = [pi_9(X)-Li_9(X)]/Sqrt[Li_9(X)] ==================================================== 9-tuple variants A; B; C and D X SLi_9(X) Spi_9(X) Error delta_9(X) ==================================================== 1E+09 19.92 17 -0.65 -2.92 1E+10 45.92 47 +0.16 1.08 1E+11 152.53 135 -1.42 -17.53 1E+12 626.63 615 -0.46 -11.63 1E+13 2881.63 2812 -1.30 -69.63 1E+14 14235.85 14337 +0.85 101.15 1E+15 74265.67 74522 +0.94 256.33 5E+15 242144.84 242292 +0.30 147.16 1E+16 405356.74 406147 +1.24 790.26 2E+16 680969.15 682029 +1.28 1059.85 5E+16 1358892.38 1358454 -0.38 -438.38 1E+17 2300336.27 2300170 -0.11 -166.27 1.5E+17 3134301.76 3134125 -0.10 -176.76 2E+17 3906027.63 3905162 -0.44 -865.63 3E+17 5331434.21 5330219 -0.53 -1215.21 5E+17 7900845.88 7898004 -1.01 -2841.88 7E+17 10246159.61 10243456 -0.84 -2703.61 1E+18 13506307.8 13505546 -0.21 -761.8 1E+25 6.48215781563E+12 1E+50 1.15496814990E+35 1E+100 2.16267512736E+82 ==================================================== Maximale Intervalle von Nunupel-Primzahlen der Differenz 30 Maximum intervals of nonuple prime numbers of difference 30 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 - 30/2)]/Ln(letzte Primzahl - 30/2)/1.5}^9 K = {1-Ln[Ln(last prime number - 30/2)]/Ln(last prime number - 30/2)/1.5}^9 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 30/2)^9 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, 1307, 0.10126 87512, 1277, 88819, 0.04166 742562, 113147, 855739, 0.04861 73410540, 855709, 74266279, 0.22406 108137242, 74266249, 182403521, 0.19599 156659940, 252277007, 408936977, 0.1816 284008468, 626927443, 910935941, 0.21523 472686632, 1645175087, 2117861749, 0.23312 481120226, 2966003057, 3447123313, 0.18663 500357134, 3447123283, 3947480447, 0.18172 1343310936, 4422726013, 5766036979, 0.40677 2389429356, 14005112893, 16394542279, 0.4453 2637580464, 37045175329, 39682755823, 0.33166 3250520282, 43813839521, 47064359833, 0.37949 3962460392, 67406579477, 71369039899, 0.38682 4707167634, 88431318949, 93138486613, 0.4105 5148354116, 158532152477, 163680506623, 0.35499 6908925298, 242360943259, 249269868587, 0.40119 7701391886, 414847098557, 422548490473, 0.36193 9041208546, 428943544633, 437984753209, 0.4189 9536948190, 589711716673, 599248664893, 0.39055 9828700136, 758040039917, 767868740083, 0.36544 13498923900, 800169804433, 813668728363, 0.49076 14719738106, 1372112527127, 1386832265263, 0.43627 16400779866, 1615939173461, 1632339953357, 0.45704 17528075670, 3514108139599, 3531636215299, 0.36664 23405225336, 3722619926147, 3746025151513, 0.47912 27409381864, 4296421864723, 4323831246617, 0.53247 30118945346, 4840688423927, 4870807369303, 0.56034 38096251624, 11315324779453, 11353421031107, 0.52392 57562459706, 13314078841157, 13371641300893, 0.74746 57959411008, 17398943650789, 17456903061827, 0.68588 74883631680, 32526257436703, 32601141068413, 0.71523 84357057956, 95021243797847, 95105600855833, 0.56372 86058916020, 140296582478863, 140382641394913, 0.50658 95905127996, 146233211267237, 146329116395263, 0.55701 111719386740, 148144260133603, 148255979520373, 0.64612 123432161582, 218673584680907, 218797016842519, 0.62996 158295956488, 279107563691629, 279265859648147, 0.74756 164745675570, 843169173326083, 843333919001683, 0.55138 215762599292, 1271611524407657, 1271827287006979, 0.63722 227673778770, 1300553951681837, 1300781625460637, 0.66784 240928000890, 1384126286714447, 1384367214715367, 0.69354 253614532078, 2372587985018803, 2372841599550911, 0.62127 319366872452, 2687175252475781, 2687494619348263, 0.75398 323241175532, 4175044694359151, 4175367935534713, 0.67037 367526500374, 6799837631044877, 6800205157545281, 0.66159 370562154532, 9299090625091429, 9299461187245991, 0.60975 384872623852, 12600969913714309, 12601354786338191, 0.58085 465256314600, 14438784002628889, 14439249258943519, 0.67565 474211801928, 20789829792326711, 20790304004128669, 0.62165 586532044982, 28046303417652251, 28046889949697263, 0.70743 700657586528, 41753460869629121, 41754161527215679, 0.7573 740602618776, 65064824596761943, 65065565199380749, 0.70932 833417154988, 76600495767490813, 76601329184645831, 0.76376 * 911984670874, 119045264661971353, 119046176646642257, 0.74256 951258249694, 200247199019847583, 200248150278097307, 0.67489 953547390454, 220226263685972623, 220227217233363107, 0.65982 1078637630066, 298885888961914337, 298886967599544433, 0.68912 1100107551206, 380125562112202847, 380126662219754083, 0.66032 1136074393648, 587495472519475759, 587496608593869437, 0.60964 1173672728910, 604024239868391803, 604025413541120743, 0.62536 1277911417226, 616637940900558737, 616639218811975993, 0.67731 1355134611184, 764566845810980893, 764568200945592107, 0.67993 1418937207874, 926406940615421113, 926408359552629017, 0.67811 2 decuple variants: Variant A: p,p+2,p+6,p+8,p+12,p+18,p+20,p+26,p+30,p+32 ==> Factor 10/3 Variant B: p,p+2,p+6,p+12,p+14,p+20,p+24,p+26,p+30,p+32 ==> Factor 10/3 Decuples: Sum of factors = 20/3 = 6.666666666666... Begin: 27±16 http://oeis.org/A257127 511.422282058995855969309000575 = Basis Decuple Prime Constant = (Hardy-Littlewood Constant C_10)*35^9/(2^31*3) Integral: 511.4222...*dX/[Ln(X)]^10 von X = 11 bis X = 1E+15 ==> 304.1245648036 Näherung ; Approximation: Li_10(X) = X/[Ln(X)-1-4.8/Ln(X)-136/Ln(X)^2]^10*511.42*20/3 ; X > 1E+11 Error = [pi_10(X)-Li_10(X)]/Sqrt[Li_10(X)] ==================================================== 10-tuple variants A and B X SLi_10(X) Spi_10(X) Error delta_10(X) ==================================================== 1E+11 7.87 6 -0.7 -1.87 1E+12 23.86 21 -0.59 -2.86 1E+13 93.84 102 +0.84 8.16 1E+14 420.24 459 +1.89 38.76 1E+15 2027.50 2144 +2.59 116.5 1E+16 10323.33 10400 +0.75 76.67 2E+16 17002.29 17214 +1.62 211.71 5E+16 33073.50 33358 +1.56 284.5 1E+17 54941.71 55383 +1.88 441.29 2E+17 91584.97 92295 +2.35 710.03 5E+17 180881.23 181803 +2.17 921.77 1E+18 303795.89 304561 +1.39 765.11 2E+18 511790.62 512703 +1.28 912.38 5E+18 1024407.93 1025081 +0.67 673.07 1E+19 1737311.55 1738502 +0.90 1190.45 1.5E+19 2369294.25 2370869 +1.02 1574.75 2E+19 2954332.44 2955884 +0.90 1551.56 2.5E+19 3506986.17 3508900 +1.02 1913.83 1E+25 103761101767 1E+50 9.13471853102E+32 1E+100 8.50951649513E+79 ==================================================== Maximale Intervalle von Decupel-Primzahlen der Differenz 32 Maximum intervals of decuple prime numbers of difference 32 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 - 32/2)]/Ln(letzte Primzahl - 32/2)*3/4}^10 K = {1-Ln[Ln(last prime number - 32/2)]/Ln(last prime number - 32/2)*3/4}^10 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 32/2)^10 Rekord-Intervall, 1. Primzahl, letzte Primzahl des gefundenen Intervalls, relativ Record interval, 1st prime number, last prime number of the found interval, relative 9853497726, 11, 9853497769, 0.11483 12102794130, 9853497737, 21956291899, 0.09404 50040961320, 33081664151, 83122625503, 0.20438 81321886116, 83122625471, 164444511619, 0.24217 278538937950, 294920291201, 573459229183, 0.47544 319703162154, 1418575498577, 1738278660763, 0.34034 374567507580, 3699811791647, 4074379299259, 0.28108 410983231206, 4700094892301, 5111078123539, 0.28151 563272830120, 6390526086797, 6953798916949, 0.34124 703391348574, 11105292314087, 11808683662693, 0.34607 728748575256, 20933920839101, 21662669414389, 0.28379 1458168320490, 23960929422161, 25419097742683, 0.53433 2690434371924, 119057768524127, 121748202896083, 0.55252 3036596580204, 279787876907237, 282824473487473, 0.46214 3141561078810, 285539202502637, 288680763581479, 0.47469 3362913297504, 555343131317057, 558706044614593, 0.40407 3407968063500, 829367805582287, 832775773645819, 0.35731 4358939581920, 872322923004341, 876681862586293, 0.44912 4660763832426, 898119143193731, 902779907026189, 0.47547 5093857694046, 1558072653622601, 1563166511316679, 0.43217 5282087928546, 1677786567870221, 1683068655798799, 0.43726 6011287324614, 1888531104078317, 1894542391402963, 0.47846 6734090115324, 3049075116536147, 3055809206651503, 0.45801 13040253984450, 4499162039650121, 4512202293634603, 0.78149 13279237325970, 15167297271417647, 15180576508743649, 0.54147 13574096572626, 21367378474780181, 21380952571352839, 0.49765 15412220654094, 24115852659843797, 24131264880497923, 0.54433 16665755852130, 34469249821245971, 34485915577098133, 0.52758 22726140274950, 37423405897013021, 37446132037288003, 0.7016 23094517277106, 75468950570776571, 75492045088053709, 0.57712 24796207364040, 84255638044289837, 84280434251653909, 0.59962 30228433744740, 124641828165146417, 124672056598891189, 0.65093 31470119943990, 241668103762660007, 241699573882604029, 0.55847 34045404413370, 387343710445857701, 387377755850271103, 0.52744 40689122583690, 410007925926634571, 410048615049218293, 0.6202 41310834926130, 556788093874787891, 556829404709714053, 0.57715 71685014583576, 704427611935812881, 704499296950396489, 0.93706 * 86401604779140, 3406851435021026801, 3406937836625805973, 0.73053 103494438996324, 4155648846458506907, 4155752340897503263, 0.82927 134488362002610, 17149680820650628487, 17149815309012631129, 0.73982 2 variants 11-tuple: Variant A: p,p+2,p+6,p+8,p+12,p+18,p+20,p+26,p+30,p+32,p+36 ==> Factor 7/3 Variant B: p,p+4,p+6,p+10,p+16,p+18,p+24,p+28,p+30,p+34,p+36 ==> Factor 7/3 11-tuples: Sum of factors = 14/3 = 4.666666666666... Begin: 29±18 1312.31971129864326745583635097 = Basis 11-Tuple Prime Constant = (Hardy-Littlewood Constant C_11)*77^10/(2^45*3*5) Integral: 1312.3197...*dX/[Ln(X)]^11 von X = 12 bis X = 1E+15 ==> 23.8755994822 Näherung ; Approximation: Li_11(X) = X/[Ln(X)-1-5.2/Ln(X)-165/Ln(X)^2]^11*1312.32*14/3 ; X > 1E+12 Error = [pi_11(X)-Li_11(X)]/Sqrt[Li_11(X)] =================================================== 11-tuple variants A and B X SLi_11(X) Spi_11(X) Error delta_11(X) =================================================== 1E+12 2.66 1 -1.0 -1.66 1E+13 6.99 7 +0.0 0.01 1E+14 25.68 28 +0.46 2.32 1E+15 111.42 111 -0.04 -0.42 1E+16 525.52 500 -1.11 -25.52 1E+17 2618.40 2672 +1.05 53.6 5E+17 8259.41 8350 +1.00 90.59 1E+18 13627.41 13723 +0.82 95.59 2E+18 22560.47 22713 +1.02 152.53 5E+18 44149.80 44162 +0.06 12.20 1E+19 73633.09 73810 +0.65 176.91 1.5E+19 99454.79 99465 +0.03 10.21 2E+19 123174.16 123092 -0.23 -82.16 3E+19 166653.16 166415 -0.58 -238.16 5E+19 244244.25 244189 -0.11 -55.25 7E+19 314435.15 314284 -0.27 -151.15 1E+20 411272.56 411551 +0.43 278.44 1E+25 3309573014 1E+50 1.43902139951E+31 1E+100 6.66858112527E+77 =================================================== Maximale Intervalle von 11-Tupel-Primzahlen der Differenz 36 Maximum intervals of 11-tuple prime numbers of difference 36 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 - 36/2)]/Ln(letzte Primzahl - 36/2)/1.25}^11 K = {1-Ln[Ln(last prime number - 36/2)]/Ln(last prime number - 36/2)/1.25}^11 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 36/2)^11 Rekord-Intervall, 1. Primzahl, letzte Primzahl des gefundenen Intervalls, relativ Record interval, 1st prime number, last prime number of the found interval, relative 1418575498562, 11, 1418575498609, 0.12539 2278499747370, 2118274828903, 4396774576309, 0.12092 2619544322010, 7908189600581, 10527733922627, 0.09508 15258482588312, 12640876669691, 27899359258039, 0.36735 16210151708788, 44058461657443, 60268613366267, 0.28479 18526365283078, 90793299817453, 109319665100567, 0.25651 37800253134992, 109319665100531, 147119918235559, 0.46554 53035067750012, 871195311482381, 924230379232429, 0.32413 104231611129800, 1603667122451473, 1707898733581309, 0.50843 111506881985462, 2742581831609351, 2854088713594849, 0.45186 125863514143558, 6841762142313793, 6967625656457387, 0.37193 137264297702338, 7051376006538943, 7188640304241317, 0.40123 140448397140088, 7346196928594603, 7486645325734727, 0.40476 140794049020742, 8272719280170461, 8413513329191239, 0.3896 228762951130560, 8925341992111393, 9154104943241989, 0.61477 246120503865390, 26581766875458911, 26827887379324337, 0.45818 249645651134612, 39138758504100371, 39388404155235019, 0.40871 263383971626160, 52058240670234493, 52321624641860689, 0.39245 329284247509408, 80650970626364833, 80980254873874277, 0.42509 368187504364680, 96053268432136901, 96421455936501617, 0.44905 411813906614612, 109074964758309281, 109486778664923929, 0.48197 435006956365018, 132162559063209523, 132597566019574577, 0.47857 503573757664168, 178457083524684043, 178960657282348247, 0.50318 579793658619058, 206891153251867933, 207470946910487027, 0.55265 770184586807020, 312722792069213411, 313492976656020467, 0.64414 919577625131282, 1595913076808345411, 1596832654433476729, 0.46532 1211254433560978, 1631014489696996483, 1632225744130557497, 0.60886 1315841914413180, 3785767016105978983, 3787082858020392199, 0.51422 1714493702235930, 5345808009229341733, 5347522502931577699, 0.6052 2302197605527050, 6641913431366968603, 6644215628972495689, 0.7626 * 2427586726658940, 23390752778933020751, 23393180365659679727, 0.55977 2978644494681240, 27823469332598375321, 27826447977093056597, 0.65393 3047594326322700, 43513737912970579333, 43516785507296902069, 0.59007 3262851568166190, 58735695811402421081, 58738958662970587307, 0.58111 3486561427946788, 81537818084553018883, 81541304645980965707, 0.56709 3852769846124430, 81823622191212516463, 81827474961058640929, 0.62605 4166798473523070, 94105480082102838043, 94109646880576361149, 0.65153 2 variants 12-tuple: Variant A: p,p+2,p+6,p+8,p+12,p+18,p+20,p+26,p+30,p+32,p+36,p+42 ==> Factor 21/5 Variant B: p,p+6,p+10,p+12,p+16,p+22,p+24,p+30,p+34,p+36,p+40,p+42 ==> Factor 21/5 12-tuples: Sum of factors = 42/5 = 8.4 Begin: 32±21 2364.59896330592750706739036716 = Basis 12-Tuple Prime Constant = (Hardy-Littlewood Constant C_12)*77^11/(2^49*3*5) Integral: 2364.598...*dX/[Ln(X)]^12 von X = 13 bis X = 1E+15 ==> 1.4149602253 Näherung ; Approximation: Li_12(X) = X/[Ln(X)-1-5.6/Ln(X)-196/Ln(X)^2]^12*2364.6*8.4 ; X > 3E+13 Error = [pi_12(X)-Li_12(X)]/Sqrt[Li_12(X)] =================================================== 12-tuples variants A and B X SLi_12(X) Spi_12(X) Error delta_12(X) =================================================== 1E+14 3.62 5 +0.7 1.38 1E+15 11.89 17 +1.48 5.11 1E+16 49.22 53 +0.54 3.78 1E+17 226.53 253 +1.76 26.47 1E+18 1106.13 1185 +2.37 78.87 5E+18 3440.75 3515 +1.26 74.25 1E+19 5642.47 5759 +1.55 116.53 2E+19 9283.92 9417 +1.38 133.08 5E+19 18019.36 18115 +0.71 95.64 7E+19 23018.99 23125 +0.70 106.01 1E+20 29864.39 30054 +1.10 189.61 1.4E+20 38205.66 38239 +0.17 33.34 1E+25 190708153.1 1E+50 4.09367697270E+29 1E+100 9.43633846551E+75 =================================================== Maximale Intervalle von 12-Tupel-Primzahlen der Differenz 42 Maximum intervals of 12-tuple prime numbers of difference 42 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 - 42/2)]/Ln(letzte Primzahl - 42/2)*6/7}^12 K = {1-Ln[Ln(last prime number - 42/2)]/Ln(last prime number - 42/2)*6/7}^12 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 42/2)^12 Rekord-Intervall, 1. Primzahl, letzte Primzahl des gefundenen Intervalls, relativ Record interval, 1st prime number, last prime number of the found interval, relative 1418575498556, 11, 1418575498609, 0.01772 26480783759430, 1418575498567, 27899359258039, 0.08025 41614642273050, 34460918582317, 76075560855409, 0.08081 110385055740960, 76075560855367, 186460616596369, 0.14577 167275369395476, 437163765888581, 604439135284099, 0.1354 252707529286024, 727417501795057, 980125031081123, 0.16813 294228747907984, 1539765965257747, 1833994713165773, 0.15246 579332222653526, 3843547642594391, 4422879865247959, 0.21294 1178234306003644, 6308411019731047, 7486645325734733, 0.35416 1308138316048410, 11140102475962687, 12448240792011139, 0.32469 1316489363200054, 12448240792011097, 13764730155211193, 0.31472 2165010689410800, 14727257011031407, 16892267700442249, 0.47963 2685175550882936, 106920298210935941, 109605473761818919, 0.30264 3393857853620850, 124568430037136201, 127962287890757093, 0.36225 3442769694870660, 142977538681261357, 146420308376132059, 0.35053 4539207862808850, 216812607870223207, 221351815733032099, 0.40029 6489192166738050, 485616539165071387, 492105731331809479, 0.43532 7310737867282080, 842617217622945607, 849927955490227729, 0.40806 11529242833092600, 1035624071743706561, 1047153314576799203, 0.60028 13627266591097886, 2844738245030787221, 2858365511621885149, 0.51025 14140332007226314, 4081819125074525677, 4095959457081752033, 0.47141 14281634215463726, 4614246116693368271, 4628527750908832039, 0.4578 15801916932897480, 4837830340027454501, 4853632256960352023, 0.49889 16391422592873516, 6900205088923091831, 6916596511515965389, 0.46225 17372520188668260, 11473517342470328261, 11490889862658996563, 0.41742 20577802367145870, 12908144983107039551, 12928722785474185463, 0.47651 21562758888249810, 22539011692840407607, 22560574451728657459, 0.42001 29440648392325170, 22972168192963624477, 23001608841355949689, 0.57005 34220713277621366, 34557112838351783411, 34591333551629404819, 0.58457 34429768587069240, 63015703346062757021, 63050133114649826303, 0.49018 43643596540302716, 80685261082594283231, 80728904679134585989, 0.57686 49064452667579854, 104907885468383778007, 104956949921051357903, 0.59959 54642888354104820, 118170152255853853271, 118224795144207958133, 0.64452 * 6 variants 13-tuple: Variant A: p,p+2,p+6,p+8,p+12,p+18,p+20,p+26,p+30,p+32,p+36,p+42,p+48 ==> Factor 33/10 Variant B: p,p+2,p+8,p+14,p+18,p+20,p+24,p+30,p+32,p+38,p+42,p+44,p+48 ==> Factor 77/20 Variant C: p,p+2,p+12,p+14,p+18,p+20,p+24,p+30,p+32,p+38,p+42,p+44,p+48 ==> Factor 77/24 Variant D: p,p+4,p+6,p+10,p+16,p+18,p+24,p+28,p+30,p+34,p+36,p+46,p+48 ==> Factor 77/24 Variant E: p,p+4,p+6,p+10,p+16,p+18,p+24,p+28,p+30,p+34,p+40,p+46,p+48 ==> Factor 77/20 Variant F: p,p+6,p+12,p+16,p+18,p+22,p+28,p+30,p+36,p+40,p+42,p+46,p+48 ==> Factor 33/10 13-tuples: Sum of factors = 1243/60 = 20.716666666666... Begin: 35±24 7820.6000302445688588042613158 = Basis 13-Tuple Prime Constant = (Hardy-Littlewood Constant C_13)*1001^12/(2^79*3^14*5) Integral: 7820.600...*dX/[Ln(X)]^13 von X = 14 bis X = 1E+20 ==> 263.633808882 Näherung ; Approximation: Li_13(X) = X/[Ln(X)-1-6/Ln(X)-230/Ln(X)^2]^13*7820.6*1243/60 ; X > 1E+15 Error = [pi_13(X)-Li_13(X)]/Sqrt[Li_13(X)] =================================================== 13-tuple variants A; B; C; D; E and F X SLi_13(X) Spi_13(X) Error delta_13(X) =================================================== 1E+15 4.91 4 -0.4 -0.91 1E+16 13.38 12 -0.38 -1.38 1E+17 51.17 52 +0.12 0.83 1E+18 227.96 212 -1.06 -15.96 1E+19 1090.63 1000 -2.74 -90.63 5E+19 3348.98 3250 -1.71 -98.98 1E+20 5461.61 5256 -2.78 -205.61 2E+20 8936.37 8748 -1.99 -188.37 3E+20 11936.84 11777 -1.46 -159.84 5E+20 17216.30 16990 -1.72 -226.3 7E+20 21932.69 21694 -1.61 -238.69 1E+21 28371.82 28224 -0.88 -147.82 1.2E+21 32371.50 32284 -0.49 -87.50 1E+25 27652789.0 1E+50 2.92910885579E+28 1E+100 3.35827075525E+74 =================================================== Maximale Intervalle von 13-Tupel-Primzahlen der Differenz 48 Maximum intervals of 13-tuple prime numbers of difference 48 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 - 48/2)]/Ln(letzte Primzahl - 48/2)*12/13}^13 K = {1-Ln[Ln(last prime number - 48/2)]/Ln(last prime number - 48/2)*12/13}^13 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 48/2)^13 Rekord-Intervall, 1. Primzahl, letzte Primzahl des gefundenen Intervalls, relativ Record interval, 1st prime number, last prime number of the found interval, relative 10527733922566, 13, 10527733922627, 0.01689 175932882673742, 10527733922579, 186460616596369, 0.07071 1521438116984952, 186460616596321, 1707898733581321, 0.22893 1558691309879550, 1707898733581273, 3266590043460871, 0.17807 1889917252092588, 5693002600430263, 7582919852522899, 0.15216 2137943784035778, 7933248530182091, 10071192314217917, 0.15327 4003966287265830, 11987120084474369, 15991086371740247, 0.23808 4484629613279982, 15991086371740199, 20475715985020229, 0.24152 6117880346386770, 22443709342850669, 28561589689237487, 0.28867 6754736905115430, 33502273017038711, 40257009922154189, 0.27844 8033936431400882, 40257009922154141, 48290946353555071, 0.30844 10644108551235120, 104814760374339133, 115458868925574301, 0.29207 13456478149022102, 197053322268438509, 210509800417460659, 0.29428 29017043184746748, 272039012072134243, 301056055256881039, 0.55528 32832293265683730, 1047475863642863773, 1080308156908547551, 0.39395 44660207284402710, 1443771785192736719, 1488431992477139477, 0.47779 46306309576887080, 1714532248479805871, 1760838558056692999, 0.46664 49399108065501796, 3270495071809685953, 3319894179875187797, 0.39814 50091363404193150, 4223113186660803269, 4273204550064996467, 0.36972 61375960786358116, 4345452296465505343, 4406828257251863507, 0.44819 92239236683463870, 4619280148472746693, 4711519385156210611, 0.65813 * 93263356963666890, 12304484935179522959, 12397748292143189897, 0.47769 103444317228890536, 13417585713157422733, 13521030030386313317, 0.51452 116895417753906610, 19142318145795018241, 19259213563548924899, 0.51617 117532514082713730, 47369052318358130593, 47486584832440844371, 0.38473 130046758556844372, 53916947360351371169, 54046994118908215589, 0.40802 179230748794005090, 55828528810609601983, 56007759559403607121, 0.55582 205378412253903826, 123821308453977238183, 124026686866231142057, 0.49227 226082435436947806, 124260428516214805933, 124486510951651753787, 0.54125 229993366994554998, 175809592423376487461, 176039585790371042507, 0.49285 302624248968928590, 241746935156728164179, 242049559405697092817, 0.58613 346666274971826042, 349121577868484646059, 349468244143456472149, 0.59806 394146128902980300, 739887265568046055693, 740281411696949036041, 0.53831 505205471751015688, 1018275345691180997653, 1018780551162932013389, 0.62543 2 variants 14-tuple: Variant A: p,p+2,p+6,p+8,p+12,p+18,p+20,p+26,p+30,p+32,p+36,p+42,p+48,p+50 ==> Factor 20/9 Variant B: p,p+2,p+8,p+14,p+18,p+20,p+24,p+30,p+32,p+38,p+42,p+44,p+48,p+50 ==> Factor 20/9 14-tuples: Sum of factors = 40/9 = 4.444444444444... Begin: 36±25 22938.9086323254268456557745047 = Basis 14-Tuple Prime Constant = (Hardy-Littlewood Constant C_14)*1001^13/(2^85*3^15*5) Integral: 22938.908...*dX/[Ln(X)]^14 von X = 15 bis X = 1E+20 ==> 17.425661107 Näherung ; Approximation: Li_14(X) = X/[Ln(X)-1-6.4/Ln(X)-267/Ln(X)^2]^14*22938.9*40/9 ; X>1E+16 Error = [pi_14(X)-Li_14(X)]/Sqrt[Li_14(X)] =================================================== 14-tuple variants A and B X SLi_14(X) Spi_14(X) Error delta_14(X) =================================================== 1E+18 3.93 4 +0.0 0.07 1E+19 16.59 15 -0.39 -1.59 1E+20 77.45 78 +0.06 0.55 5E+20 234.81 263 +1.84 28.19 1E+21 380.94 394 +0.67 13.06 2E+21 620.10 623 +0.12 2.90 3E+21 825.80 821 -0.17 -4.80 5E+21 1186.52 1143 -1.26 -43.52 7E+21 1507.77 1438 -1.80 -69.77 1E+22 1945.22 1810 -3.07 -135.22 1E+25 309505.72 1E+50 1.61699342459E+26 1E+100 9.22028620754E+71 =================================================== Maximale Intervalle von 14-Tupel-Primzahlen der Differenz 50 Maximum intervals of 14-tuple prime numbers of difference 50 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 - 50/2)]/Ln(letzte Primzahl - 50/2)}^14 K = {1-Ln[Ln(last prime number - 50/2)]/Ln(last prime number - 50/2)}^14 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 50/2)^14 Rekord-Intervall, 1. Primzahl, letzte Primzahl des gefundenen Intervalls, relativ Record interval, 1st prime number, last prime number of the found interval, relative 21817283854511250, 11, 21817283854511311, 0.02378 57470521611732948, 21817283854511261, 79287805466244259, 0.03634 761974641103906512, 79287805466244209, 841262446570150771, 0.18646 1675784608444106160, 1006587882969594041, 2682372491413700251, 0.26267 2294504145248125092, 2714623996387988519, 5009128141636113661, 0.28443 2762595421478512338, 7993822923596334941, 10756418345074847329, 0.25822 3250311817059572500, 27367669288651556699, 30617981105711113999, 0.20813 3820498055844159690, 35863507147601267969, 39684005203445427709, 0.22306 4453862600174808102, 50079448898342988959, 54533311498517797111, 0.23239 5212642265415765180, 55603189618237916771, 60815831883653682001, 0.26174 8596064239562449368, 98733575252292966101, 107329639491855415519, 0.35396 9696176181714212190, 294883369734311433191, 304579545916025645431, 0.27919 15947853408687129930, 473029310348949789551, 488977163757636919531, 0.39144 19022094759222825552, 958585542764963318039, 977607637524186143641, 0.37072 20424171342813878640, 1342822630273778658839, 1363246801616592537529, 0.35677 28797753387111208500, 1392392425078137978329, 1421190178465249186879, 0.49622 38643589780604278470, 2125478663401345427009, 2164122253181949705529, 0.58049 * 43358796938205859110, 3322258534990494654389, 3365617331928700513549, 0.56465 53311036642542268308, 7947079609610035209431, 8000390646252577477789, 0.52672 4 variants 15-tuple: Variant A: p,p+2,p+6,p+8,p+12,p+18,p+20,p+26,p+30,p+32,p+36,p+42,p+48,p+50,p+56 ==> Factor 27/8 Variant B: p,p+2,p+6,p+12,p+14,p+20,p+24,p+26,p+30,p+36,p+42,p+44,p+50,p+54,p+56 ==> Factor 12 Variant C: p,p+2,p+6,p+12,p+14,p+20,p+26,p+30,p+32,p+36,p+42,p+44,p+50,p+54,p+56 ==> Factor 12 Variant D: p,p+6,p+8,p+14,p+20,p+24,p+26,p+30,p+36,p+38,p+44,p+48,p+50,p+54,p+56 ==> Factor 27/8 15-tuples: Sum of factors = 123/4 = 30.75 Begin: 39±28 http://oeis.org/A257169 55651.4625534991443965782327095 = Basis 15-Tuple Prime Constant = (Hardy-Littlewood Constant C_15)*1001^14/(2^91*3^16*5) Integral: 55651.462...*dX/[Ln(X)]^15 von X = 16 bis X = 1E+20 ==> 0.9966428653 Näherung ; Approximation: Li_15(X) = X/[Ln(X)-1-6.8/Ln(X)-306/Ln(X)^2]^15*55651.5*30.75 ; X>3E+17 Error = [pi_15(X)-Li_15(X)]/Sqrt[Li_15(X)] =================================================== 15-tuple variants A; B; C and D X SLi_15(X) Spi_15(X) Error delta_15(X) =================================================== 1E+19 8.04 8 -0.0 -0.04 1E+20 30.65 39 +1.51 8.35 1E+21 137.91 142 +0.35 4.09 5E+21 412.16 419 +0.34 6.84 1E+22 665.15 676 +0.42 10.85 1E+25 92464.00 1E+50 2.38136957302E+25 1E+100 6.75279810091E+70 =================================================== Maximale Intervalle von 15-Tupel-Primzahlen der Differenz 56 Maximum intervals of 15-tuple prime numbers of difference 56 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 - 56/2)]/Ln(letzte Primzahl - 56/2)}^15 K = {1-Ln[Ln(last prime number - 56/2)]/Ln(last prime number - 56/2)}^15 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 56/2)^15 Rekord-Intervall, 1. Primzahl, letzte Primzahl des gefundenen Intervalls, relativ Record interval, 1st prime number, last prime number of the found interval, relative 1158722981124148350, 17, 1158722981124148423, 0.111 5229964613725866210, 3172206835341609797, 8402171449067476063, 0.22573 8176842109164158340, 17905159760365247387, 26082001869529405783, 0.22753 13202593450794337590, 50035247022695224217, 63237840473489561863, 0.26275 14352561778062217620, 90798766993022298227, 105151328771084515903, 0.23635 14793438845198541066, 151477098804870766217, 166270537650069307339, 0.20577 24414869006900201214, 166270537650069307283, 190685406656969508553, 0.323 26370658297525951794, 470807023319315838887, 497177681616841790737, 0.24672 44903964137948468280, 524567351879416758917, 569471316017365227253, 0.40023 51656273486207498550, 823491664526499042287, 875147938012706540893, 0.39528 54428723707573302360, 1210392827233162795877, 1264821550940736098293, 0.36587 63565395868366297710, 1343339881857931111367, 1406905277726297409133, 0.41165 67699545660476651550, 3191426144896963821137, 3259125690557440336693, 0.32767 84170188184334529290, 3940794344489079457307, 4024964532673413986653, 0.37896 109688713092436935996, 6439563382863366440801, 6549252095955803376853, 0.41845 155621007510816489816, 8667033223088203698851, 8822654230599020188723, 0.53687 * 2 variants 16-tuple: Variant A: p,p+2,p+6,p+12,p+14,p+20,p+26,p+30,p+32,p+36,p+42,p+44,p+50,p+54,p+56,p+60 ==> Factor 320/39 Variant B: p,p+4,p+6,p+10,p+16,p+18,p+24,p+28,p+30,p+34,p+40,p+46,p+48,p+54,p+58,p+60 ==> Factor 320/39 16-tuples: Sum of factors = 640/39 = 16.410256410256... Begin: 43±30 91555.1112261441955926805744818 = Basis 16-Tuple Prime Constant = (Hardy-Littlewood Constant C_16)*1001^15/(2^97*3^17*5) Integral: 91555.111...*dX/[Ln(X)]^16 von X = 17 bis X = 1E+20 ==> 0.0564891032 Näherung ; Approximation: Li_16(X) = X/[Ln(X)-1-7.2/Ln(X)-348/Ln(X)^2]^16*91555*640/39 ; X>1E+19 Error = [pi_16(X)-Li_16(X)]/Sqrt[Li_16(X)] =================================================== 16-tuple variants A and B X SLi_16(X) Spi_16(X) Error delta_16(X) =================================================== 1E+21 2.91 4 +0.6 1.09 1E+22 12.21 18 +1.66 5.79 5E+22 35.73 34 -0.29 -1.73 1E+23 57.30 67 +1.28 9.70 1E+25 1446.09 1E+50 1.83457598783E+23 1E+100 2.58688717533E+68 =================================================== Maximale Intervalle von 16-Tupel-Primzahlen der Differenz 60 Maximum intervals of 16-tuple prime numbers of difference 60 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 - 60/2)]/Ln(letzte Primzahl - 60/2)}^16 K = {1-Ln[Ln(last prime number - 60/2)]/Ln(last prime number - 60/2)}^16 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 60/2)^16 Rekord-Intervall, 1. Primzahl, letzte Primzahl des gefundenen Intervalls, relativ Record interval, 1st prime number, last prime number of the found interval, relative 47710850533373130094, 13, 47710850533373130167, 0.02233 299998600213146604770, 47710850533373130107, 347709450746519734937, 0.0645 348165435428733176186, 347709450746519734877, 695874886175252911123, 0.0575 403763689947799307194, 695874886175252911063, 1099638576123052218317, 0.05616 507535834538971258020, 1750052554011927712483, 2257588388550898970563, 0.05407 619971225945186215100, 2639154464612254121537, 3259125690557440336697, 0.05773 865454633082729209520, 3789227751026345304613, 4654682384109074514193, 0.07079 4020477691727936425054, 5022156579757255625623, 9042634271485192050737, 0.25898 * 4359840411512173491836, 9239395687646993061197, 13599236099159166553093, 0.24288 5057095292650401360360, 15571053758048293307807, 20628149050698694668227, 0.24321 5203231804946016501660, 42421183685552747462323, 47624415490498763964043, 0.18696 10.06.2025 Richard Fischer