mersenneforum.org new record gaps of prime k-tuplets
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 2022-11-10, 09:37 #1 Cybertronic     Jan 2007 Germany 27116 Posts new record gaps of prime k-tuplets prime sextuplets range 10^15 to 10^16 done , computing time 8h 10 new ranks found incl. new merit record. https://pzktupel.de/RecordGaps/GAP06.php Last fiddled with by Cybertronic on 2022-11-10 at 09:38
 2022-11-11, 16:40 #2 Cybertronic     Jan 2007 Germany 10011100012 Posts gap-update for 5- and 7-tuplets 5-tuplets: pattern 0,2,6,8,12 increased range : 10^15 -> 3*10^15 6 new gaps records 7-tuplets: both pattern increased range : 10^15 -> 10^16 16 new gaps records ( 11 and 5 ) , time to compute 6h see: https://pzktupel.de/Rectables.php
 2022-11-19, 10:59 #3 Cybertronic     Jan 2007 Germany 62510 Posts update Searching continue with prime triplets , pattern d=0,2,6: starting at 10^15 Rank 73: GAP=355650; 1039463608480391, 1039463608836041; merit=24.58911 Last fiddled with by Cybertronic on 2022-11-19 at 11:01
 2022-11-20, 07:22 #4 Cybertronic     Jan 2007 Germany 54 Posts triplets [d=0,2,6] Rank 74, GAP=392370 [1440653814746297, 1440653815138667] merit=26.37393 Rank 75, GAP=402456 [1516877715570671, 1516877715973127] merit=26.93236 upper bound: 1.71e15 Last fiddled with by Cybertronic on 2022-11-20 at 08:17
 2022-11-22, 06:27 #5 Cybertronic     Jan 2007 Germany 54 Posts status triplets [d=0,2,6] : no new record ( up to 2.80e15 ) [d=0,4,6] : Rank 80: GAP=401286 ; [1413850739732107, 1413850740133393]; merit=27.01683 ( up to 1.83e15 ) Last fiddled with by Cybertronic on 2022-11-22 at 06:30
 2022-11-23, 12:18 #6 Cybertronic     Jan 2007 Germany 54 Posts status 5-tuplets Prime quintuplet, pattern: d=0,4,6,10,12 RANK 72, GAP= 112167180, [7836574258165897, 7836574370333077] merit=17.30974 RANK 73, GAP= 113994300, [8140099858614037, 8140099972608337] merit=17.50065 RANK 74, GAP= 123697560, [8232448271331367, 8232448395028927] merit=18.96111 Both pattern up to 10^16 complete. reference: https://pzktupel.de/RecordGaps/GAP05.php Last fiddled with by Cybertronic on 2022-11-23 at 12:18
 2022-11-24, 09:28 #7 Cybertronic     Jan 2007 Germany 62510 Posts status sextuplet Prime sextuplet, pattern: d=0,4,6,10,12,16 Rank 67 Gap=2865745680 [21050295686721937, 21050298552467617] merit=17.58384 Rank 68 Gap=2974185270 [23026103227984387, 23026106202169657] merit=17.99002 Up to 2.5e16 complete Last fiddled with by Cybertronic on 2022-11-24 at 09:28
 2022-11-24, 10:59 #8 Cybertronic     Jan 2007 Germany 54 Posts status septuplets Prime septuplet, pattern: d=0,2,6,8,12,18,20 RANK 48, GAP=26886742260, [13725091836704861, 13725118723447121]; merit=14.83678 RANK 49, GAP=28638152340, [15773525045646281, 15773553683798621]; merit=15.39525 RANK 50, GAP=38182265940, [19076409136724111, 19076447318990051]; merit=19.80826 Both pattern up to 2*10e16 complete. Last fiddled with by Cybertronic on 2022-11-24 at 11:01
 2022-11-24, 19:58 #9 Cybertronic     Jan 2007 Germany 54 Posts Prime septuplets Both pattern up to 5e16 complete: https://pzktupel.de/RecordGaps/GAP07.php Last fiddled with by Cybertronic on 2022-11-24 at 19:58
 2022-11-25, 20:50 #10 Cybertronic     Jan 2007 Germany 54 Posts Septuplets complete up to 10^17 ! https://pzktupel.de/RecordGaps/GAP07.php
 2022-11-25, 21:50 #11 mart_r     Dec 2008 you know...around... 35516 Posts The Cramér-Shanks-Granville analogue would also be interesting to consider, in this table the records for k=1..7 (not pattern-specific): Code: k gap gap start gap end CSG 1 1132 1693182318746371 1693182318747503 0.92063859 2 28842 2797282815481499 2797282815510341 0.84634708 3 38340 148770907 148809247 0.87388424 (!) 4 2550750 342611795411 342614346161 0.80114297 5 107597490 991851356676277 991851464273767 0.64307883 6 2650231920 6654545231260177 6654547881492097 0.53794859 7 38182265940 19076409136724111 19076447318990051 0.52840023 The CSG analogue here is equal to $$G_k \cdot \int_{gap\:start}^{gap\:end} \frac{\text{d}t}{log^{k+1}t}$$, with $$G_k$$ being the respective Hardy-Littlewood constant.

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