new record gaps of prime k-tuplets

[Cybertronic]

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

[Cybertronic]

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

[Cybertronic]

Searching continue with prime triplets , pattern d=0,2,6: starting at 10^15
Rank 73: GAP=355650; 1039463608480391, 1039463608836041; merit=24.58911

[Cybertronic]

Rank 74, GAP=392370 [1440653814746297, 1440653815138667] merit=26.37393
Rank 75, GAP=402456 [1516877715570671, 1516877715973127] merit=26.93236


upper bound: 1.71e15

[Cybertronic]

[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 )

[Cybertronic]

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

[Cybertronic]

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

[Cybertronic]

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.

[Cybertronic]

Prime septuplets

Both pattern up to 5e16 complete:
https://pzktupel.de/RecordGaps/GAP07.php

[Cybertronic]

Septuplets complete up to 10^17 !
https://pzktupel.de/RecordGaps/GAP07.php

[mart_r]

The Cramér-Shanks-Granville analogue would also be interesting to consider, in this table the records for k=1..7 (not pattern-specific):

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.