Thema: Anzahl und Intervalle zwischen je 7 benachbarten Primzahlen (7-Tupel) Number and intervals between 7 adjacent prime numbers (7-tuples) Main index: http://www.fermatquotient.com Erste 3 Primzahl-Septupel vor dem jeweiligen Intervall zum Septupel-Nachbarn. First 3 prime number septuples before the respective interval to the septuple neighbor. Variant A: p-10,p-8,p-4,p-2,p+2,p+8,p+10 ==> Factor 1 Variant B: p-10,p-8,p-2,p+2,p+4,p+8,p+10 ==> Factor 1 Septuple: Sum of factors = 2 Beginn: 21±10 Interval, Septuple±10, Septuple±10, Septuple±10 210 Var. A, 1683059174212311, 5943810258381891, 6332966627448741 210 Var. B, 482046424357029, 559709659502889, 1207091067810159 252 B d A, 17647294011492789, 55838023630923639, 105741067122115299 378 A d B, 406811307719901, 17582181333275181, 17691220108068591 420 Var. A, 90835427794196211, 296103968781395901, 353733139865140161 420 Var. B, 9720062593485699, 104888800062160569, 155192285409165459 462 B d A, 1720453355589969, 6773003917073289, 12982542547415259 588 A d B, 16383435184207581, 43759609182682701, 60321619630938771 672 B d A, 837929050280469, 2171658514540389, 4133652387982299 798 A d B, 4280913340011, 1000352647061691, 3679617443628021 840 Var. A, 30953001413566701, 63092767785840261, 64014514550625261 840 Var. B, 6848950728286359, 44724658416019659, 69951846381557679 882 B d A, 18950874690315309, 67206509075066739, 123818440653605439 1050 Var. A, 1708375741704891, 5434104173519511, 72370275720334851 1050 Var. B, 327410564148129, 915913487184339, 31440745343274729 1092 B d A, 19890881923989069, 63409009469992179, 83051880776826339 1260 Var. A, 11625112228710981, 13265231981581041, 19907694943138821 1260 Var. B, 8947464344025669, 10964551369012059, 13169707954438959 1428 A d B, 2156179048144611, 2291308515644001, 36125670859094901 1470 Var. A, 3850642559356131, 29770649377761831, 35166557081248131 1470 Var. B, 1779674695574769, 81681552881694759, 81951823844865429 1638 A d B, 639947328993441, 2448438386471721, 3001630993296831 1722 B d A, 540378033904989, 1007882168480649, 1955786183739129 1848 A d B, 471888767172081, 16011953142167241, 133262802607348491 1890 Var. A, 82043502277544661, 87641922931855761, 145028784770353491 1890 Var. B, 128798991518020929, 371092247740844889, 378796828836942159 1932 B d A, 122863128449439, 1841527951374489, 5111631965702409 2058 A d B, 470675044955241, 1627002539467431, 5217475571663481 2100 Var. A, 242283063932091, 38078416419461061, 61338965767987551 2100 Var. B, 11474855714928249, 46250666479341219, 56681459661559089 2310 Var. A, 2298056785355331, 40947963037864701, 98113759682327571 2310 Var. B, 9920226178034319, 19148675872761309, 42973293724876389 ... https://www.pzktupel.de/smarchive.php 53.9719483001296523960730291062 = Septuple Prime Constant = (Hardy-Littlewood Constant C_7)*35^6/(2^22*3) Integral: 53.9719...*dX/[Ln(X)]^7 von X = 8 bis X = 1E+15 ==> 1166417.093070733410 Näherung ; Approximation: Li_7(X) = X/[Ln(X)-1-3.6/Ln(X)-67/Ln(X)^2]^7*53.972*2 ; X > 1E+08 Error = [pi_7(X)-Li_7(X)]/Sqrt[Li_7(X)] ====================================================== Septuples: p-10,p-8,p-4,p-2,p+2,p+8,p+10 and p-10,p-8,p-2,p+2,p+4,p+8,p+10 X Li_7(X) pi_7(X) Error delta_7(X) ====================================================== 1E+08 29.55 33 +0.63 3.45 1E+09 108.02 103 -0.48 -5.02 1E+10 471.78 473 +0.06 1.22 1E+11 2292.33 2335 +0.89 42.67 1E+12 11992.59 11969 -0.22 -23.59 1E+13 66447.04 66461 +0.05 13.96 1E+14 385942.45 385809 -0.21 -133.45 5E+14 1352088.65 1352546 +0.39 457.35 1E+15 2332834.19 2334123 +0.84 1288.81 2E+15 4037457.24 4040202 +1.37 2744.76 5E+15 8375143.79 8373748 -0.48 -1395.79 1E+16 14592456.97 14592876 +0.11 419.03 1.5E+16 20217614.50 20213369 -0.94 -4245.50 2E+16 25493864.87 25490333 -0.70 -3531.87 2.5E+16 30526905.89 30522399 -0.82 -4506.89 3E+16 35375360.80 35372426 -0.49 -2934.80 4E+16 44654823.94 44649382 -0.81 -5441.94 5E+16 53514137.87 53503829 -1.41 -10308.87 6E+16 62054293.33 62042511 -1.50 -11782.33 7E+16 70339075.24 70330308 -1.05 -8767.24 8E+16 78412183.10 78406621 -0.63 -5562.1 9E+16 86305293.47 86298212 -0.76 -7091.47 1E+17 94042342.48 94033362 -0.93 -8980.48 1E+50 4.28949438229E+37 1E+100 3.24457266938E+85 ====================================================== Maximale Intervalle von Sextupel-Primzahlen der Differenz 20 Maximum intervals of sextuple prime numbers of difference 20 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 - 20/2)]/Ln(letzte Primzahl - 20/2)/2.5}^7 K = {1-Ln[Ln(last prime number - 20/2)]/Ln(last prime number - 20/2)/2.5}^7 relativ = (Rekord-Intervall)/K/Ln(letzte Primzahl - 20/2)^7 Rekord-Intervall, 1. Primzahl, letzte Primzahl der gefundenen Intervall, relativ Record interval, 1st prime number, last prime number of the found interval, relative 5628, 11, 5659, 0.04541 83160, 5639, 88819, 0.06454 119028, 165701, 284749, 0.04065 341880, 284729, 626629, 0.06992 5414220, 1146779, 6561019, 0.28382 13773480, 25658441, 39431941, 0.29179 22813770, 45002591, 67816381, 0.37441 30440088, 93625991, 124066099, 0.37935 33424818, 327864611, 361289449, 0.26151 34863108, 367438061, 402301189, 0.26065 39869718, 575226131, 615095869, 0.24978 66303048, 1228244651, 1294547719, 0.30742 70350000, 1519658669, 1590008689, 0.30074 109212852, 1590008669, 1699221541, 0.45486 131165160, 3008352029, 3139517209, 0.43098 161786520, 8436205691, 8597992231, 0.36547 228283230, 13835037311, 14063320561, 0.43206 257159490, 15863572589, 16120732099, 0.46371 289262610, 19763247389, 20052510019, 0.48303 350304570, 47827797749, 48178102339, 0.43276 439508622, 81720304649, 82159813291, 0.45434 561983940, 130121377481, 130683361441, 0.49912 571529112, 338490796739, 339062325871, 0.37482 777665070, 378184596059, 378962261149, 0.49263 815371200, 632131550579, 632946921799, 0.44106 908352732, 811003946099, 811912298851, 0.4556 1063368810, 819221051261, 820284420091, 0.5317 1211707350, 1330453426439, 1331665133809, 0.52409 1234420782, 2219662600019, 2220897020821, 0.4594 1261947498, 3618415983641, 3619677931159, 0.40789 1468079298, 4064130673901, 4065598753219, 0.45903 1887408222, 4683800652899, 4685688061141, 0.5668 1947769572, 15457675215119, 15459622984711, 0.41984 2217999168, 17343772341011, 17345990340199, 0.46337 2478988722, 17668104903719, 17670583892461, 0.5153 3484084380, 18865153070831, 18868637155231, 0.71148 3523349172, 57075938362349, 57079461711541, 0.53627 3668832300, 66634278495131, 66637947327451, 0.53638 3677810220, 99373821578399, 99377499388639, 0.48505 3704425998, 141387806832731, 141391511258749, 0.44658 3747710778, 146258465290571, 146262213001369, 0.44794 3974542320, 147400207111679, 147404181654019, 0.47412 4296334602, 178458970788809, 178463267123431, 0.48837 5343753870, 185072465945441, 185077809699331, 0.6019 5372585022, 393208334708459, 393213707293501, 0.50187 5842028388, 503820405016541, 503826247044949, 0.51362 6549026022, 566548557083219, 566555106109261, 0.55957 6810221880, 782195366635589, 782202176857489, 0.53826 7247706060, 1235175843036251, 1235183090742331, 0.51362 7561482180, 1574993028808469, 1575000590290669, 0.50593 8270089212, 2017317988864679, 2017326258953911, 0.52211 8425586190, 2175705123427931, 2175713549014141, 0.52262 9564343950, 2710640296617551, 2710649860961521, 0.56368 9848718180, 3417906881673281, 3417916730391481, 0.55015 11447130828, 3510982467857981, 3510993914988829, 0.63549 12557720952, 4534734118206359, 4534746675927331, 0.65741 16763556012, 8025455431960649, 8025472195516681, 0.77105 * 20038024650, 38515499255977811, 38515519294002481, 0.65256 20067414990, 75716044254111629, 75716064321526639, 0.56566 22899799902, 83041301081141309, 83041323980941231, 0.63302 15.04.2025 Richard Fischer