Thema: Anzahl und maximale Luecken (Intervallen) von Primzahlen-Zwillingen Topic: Number and maximum gaps (intervals) of twin primes Main index: http://www.fermatquotient.com/ Siehe auch unter / See too: http://oeis.org/ 3424506 , A113274 , A113275 , A036061 , A036062 https://pzktupel.de/RecordGaps/GAP02.php 1.32032363169373914785562422003 = Twin Prime Constant Integral: 1.3203236...*dX/[Ln(X)]^2 von X = 3 bis X = 1E+15 ==> 1177208491859.000339093331 Näherung ; Approximation: Li_2(X) = X/(lnX-1-1.5/lnX-7/(lnX)^2)^2*1.32032 ; X>100 Error = [pi_2(X)-Li_2(X)]/Sqrt[Li_2(X)] ============================================================= Twin: p-1, p+1 X Li_2(X) pi_2(X) Error delta_2(X) ============================================================= 1E+03 44.11 35 -1.37 -9.11 1E+04 212.53 205 -0.52 -7.53 1E+05 1247.03 1224 -0.65 -23.03 1E+06 8246.35 8169 -0.85 -77.35 1E+07 58752.14 58980 +0.94 227.86 1E+08 440366.11 440312 -0.08 -54.11 1E+09 3425306.47 3424506 -0.43 -800.47 1E+10 27411414.85 27412679 +0.24 1264.15 1E+11 224368863.00 224376048 +0.48 7185.00 1E+12 1870559865.14 1870585220 +0.59 25354.86 1E+13 15834598303.38 15834664872 +0.53 66568.62 1E+14 135780264892.8 135780321665 +0.15 56772.2 5E+14 613789681909.4 613790177314 +0.63 495404.6 1E+15 1177208491859.0 1177209242304 +0.69 750445.0 1.5E+15 1723753824138.7 1723754585354 +0.58 761215.3 2E+15 2259757691020.1 2259758303674 +0.41 612653.9 2.5E+15 2788122444353.9 2788122612616 +0.10 168262.1 3E+15 3310516903420.2 3310517800844 +0.49 897423.8 4E+15 4341400642441.9 4341401630211 +0.47 987769.1 5E+15 5357873955331.7 5357875276068 +0.57 1320736.3 6E+15 6363080679933.7 6363082106477 +0.57 1426543.3 7E+15 7359060980845.9 7359062903160 +0.71 1922314.1 8E+15 8347243739264.7 8347246232893 +0.86 2493628.3 9E+15 9328684162138.1 9328687670269 +1.15 3508130.9 1E+16 10304192554494 10304195697298 +0.98 3142804 1.2E+16 12239856619857 12239858939771 +0.66 2319914 1.5E+16 15111534853341 15111540068618 +1.34 5215277 2E+16 19831843368786 19831847025792 +0.82 3657006 1E+17 90948833260988 90948839353159 +0.63 6092171 1E+18 808675901493605 808675888577436 -0.45 -12916169 1E+25 4.130654449164E+21 1E+50 1.013885487622E+46 1E+100 2.512201969745E+95 ============================================================= Primzahlen-Zwillinge ±1 ; Twin prime numbers ±1 relativ = Rekordluecke * T / {ln(letzte Primzahl - 1)-ln[ln(letzte Primzahl - 1)]/3}^3 relative = (Record gap) * T / {ln(last prime number - 1)-ln[ln(last prime number - 1)]/3}^3 T = 1.32032363169... = Twin Prime Constant Rekord-Luecke, 1. Primzahl, letzte Primzahl der gefundenen Luecke, relativ Record gap, 1. prime number, last prime number of the found gap, relative 2, 3, 7, 0.64789 Merit: 1.019 = 2*T/(ln 5)^2 6, 5, 13, 0.76308 Merit: 2.092 = 6*T/(ln 7)^2 12, 17, 31, 0.59085 18, 41, 61, 0.49913 30, 71, 103, 0.56866 Merit: 2.152 = 30*T/(ln 73)^2 36, 311, 349, 0.32600 72, 347, 421, 0.59024 Merit: 2.773 = 72*T/(ln 349)^2 150, 659, 811, 0.88854 Merit: 4.697 = 150*T/(ln 661)^2 168, 2381, 2551, 0.60499 210, 5879, 6091, 0.54302 282, 13397, 13681, 0.55153 372, 18539, 18913, 0.65498 Merit: 5.085 = 372*T/(ln 18541)^2 498, 24419, 24919, 0.80396 Merit: 6.442 = 498*T/(ln 24421)^2 630, 62297, 62929, 0.77265 Merit: 6.825 = 630*T/(ln 62299)^2 924, 187907, 188833, 0.84188 Merit: 8.273 = 924*T/(ln 187909)^2 930, 687521, 688453, 0.61735 1008, 688451, 689461, 0.66890 1452, 850349, 851803, 0.91764 Merit: 10.284 = 1452*T/(ln 850351)^2 1512, 2868959, 2870473, 0.73219 1530, 4869911, 4871443, 0.66435 1722, 9923987, 9925711, 0.64945 1902, 14656517, 14658421, 0.66582 2190, 17382479, 17384671, 0.74246 Merit: 10.404 = 2190*T/(ln 17382481)^2 2256, 30752231, 30754489, 0.68875 2832, 32822369, 32825203, 0.85451 Merit: 12.484 = 2832*T/(ln 32822371)^2 2868, 96894041, 96896911, 0.71651 3012, 136283429, 136286443, 0.71069 3102, 234966929, 234970033, 0.66955 3180, 248641037, 248644219, 0.68015 3480, 255949949, 255953431, 0.74086 3804, 390817727, 390821533, 0.75722 Merit: 12.832 = 3804*T/(ln 390817729)^2 4770, 698542487, 698547259, 0.86794 Merit: 15.186 = 4770*T/(ln 698542489)^2 5292, 2466641069, 2466646363, 0.79911 6030, 4289385521, 4289391553, 0.84196 Merit: 16.184 = 6030*T/(ln 4289385523)^2 6282, 19181736269, 19181742553, 0.71641 6474, 24215097497, 24215103973, 0.71626 6552, 24857578817, 24857585371, 0.72244 6648, 40253418059, 40253424709, 0.68917 7050, 42441715487, 42441722539, 0.72596 7980, 43725662621, 43725670603, 0.81864 Merit: 17.551 = 7980*T/(ln 43725662623)^2 8040, 65095731749, 65095739791, 0.78469 8994, 134037421667, 134037430663, 0.80352 Merit: 18.090 = 8994*T/(ln 134037421669)^2 9312, 198311685749, 198311695063, 0.79381 Merit: 18.169 = 9312*T/(ln 198311685751)^2 9318, 223093059731, 223093069051, 0.78331 10200, 353503437239, 353503447441, 0.81242 Merit: 19.046 = 10200*T/(ln 353503437241)^2 10338, 484797803249, 484797813589, 0.79392 10668, 638432376191, 638432386861, 0.79391 Merit: 19.063 = 10668*T/(ln 638432376193)^2 10710, 784468515221, 784468525933, 0.77867 11388, 794623899269, 794623910659, 0.82676 Merit: 20.026 = 11388*T/(ln 794623899271)^2 11982, 1246446371789, 1246446383773, 0.82720 Merit: 20.395 = 11982*T/(ln 1246446371791)^2 12138, 1344856591289, 1344856603429, 0.83095 Merit: 20.548 = 12138*T/(ln 1344856591291)^2 12288, 1496875686461, 1496875698751, 0.83134 Merit: 20.643 = 12288*T/(ln 1496875686463)^2 12630, 2156652267611, 2156652280243, 0.82102 Merit: 20.676 = 12630*T/(ln 2156652267613)^2 13050, 2435613754109, 2435613767161, 0.83721 Merit: 21.181 = 13050*T/(ln 2435613754111)^2 14262, 4491437003327, 4491437017591, 0.85695 Merit: 22.186 = 14262*T/(ln 4491437003329)^2 14436, 13104143169251, 13104143183689, 0.77604 14952, 14437327538267, 14437327553221, 0.79588 15396, 18306891187511, 18306891202909, 0.80003 15720, 18853633225211, 18853633240933, 0.81445 Merit: 22.213 = 15720*T/(ln 18853633225213)^2 16362, 23275487664899, 23275487681263, 0.82995 Merit: 22.805 = 16362*T/(ln 23275487664901)^2 16422, 23634280586867, 23634280603291, 0.83172 Merit: 22.866 = 16422*T/(ln 23634280586869)^2 16590, 38533601831027, 38533601847619, 0.80042 16896, 43697538391391, 43697538408289, 0.80517 17082, 56484333976919, 56484333994003, 0.79388 18384, 74668675816277, 74668675834663, 0.83160 Merit: 23.787 = 18384*T/(ln 74668675816279)^2 19746, 116741875898981, 116741875918729, 0.85580 Merit: 24.849 = 19746*T/(ln 116741875898983)^2 19992, 136391104728629, 136391104748623, 0.85377 Merit: 24.919 = 19992*T/(ln 136391104728631)^2 20532, 221346439666109, 221346439686643, 0.83786 21930, 353971046703347, 353971046725279, 0.85687 Merit: 25.800 = 21930*T/(ln 353971046703349)^2 22548, 450811253543219, 450811253565769, 0.86173 Merit: 26.148 = 22548*T/(ln 450811253543221)^2 23358, 742914612256169, 742914612279529, 0.85321 Merit: 26.303 = 23358*T/(ln 742914612256171)^2 23382, 1121784847637957, 1121784847661341, 0.82323 25230, 1149418981410179, 1149418981435411, 0.88638 Merit: 27.701 = 25230*T/(ln 1149418981410181)^2 26268, 2543288406389231, 2543288406415501, 0.86076 28842, 2797282815481499, 2797282815510343, 0.93735 * Merit: 30.102 = 28842*T/(ln 2797282815481501)^2 31302, 12914226879316517, 12914226879347821, 0.89380 31482, 16155559543324757, 16155559543356241, 0.88246 31512, 37338553629118097, 37338553629149611, 0.82508 31920, 37962553054417547, 37962553054449469, 0.83464 33042, 38617975949216087, 38617975949249131, 0.86279 34692, 52000545890760149, 52000545890794843, 0.88454 Merit: 30.918 = 34692*T/(ln 52000545890760151)^2 35640, 70478530884377381, 70478530884413023, 0.88702 Merit: 31.267 = 35640*T/(ln 70478530884377383)^2 04.11.2025 Richard Fischer