GAPS BETWEEN PRIME PAIRS (Twin Primes)

[Cybertronic]

Okay, after reaching 1e17 , I will send OEIS a new b-file.

[Cybertronic]

Update 9.3e16-9.4e16

4880 93127653532737407
4731 93539476885612781
5122 93635219969125277
4884 93679515356569097
4801 93973499868673391
4992 93993640519365989

[Cybertronic]

Update 9.4e16-9.4989e16

5116 94185634582223201
5171 94215570739195151
4875 94365062134750259
4601 94375388522968151
5183 94704401239516151
4782 94737856215568919

[Cybertronic]

Update: 9.5e16 - 9.6e16

5371 95186107022160071
4915 95278213555666277
5012 95476085403701459
5055 95697220892225297
4944 95782102758939407
4594 95847907877481257
5307 95847991444578179

[Cybertronic]

Update: 9.6e16 - 9.7e16

5063 96559616615931281
4832 96790930844389337

[Cybertronic]

Update 9.7e16 - 9.8e16

4815 97379587218246209
4783 97716423712646021

[Cybertronic]

A just for fun table....

gaps found below record gap / record gap

4162 / 5217 =>79.77%
4251 / 5247 =>81,0%
4576 / 5252 => 87,1%
4583 / 5320 => 86,14%
4589 / 5507 => 83,33%
4742 / 5782 => 82,01%
4878 / 5940 => 82,12%
4989 / 5940 => 83,8% <- status now
Next: record gap ~6100 ?

[Cybertronic]

9.8 - 9.9e16

4872 98178542270123129
5152 98370851264622917
4633 98378002461666329
5056 98590143800904911

[Cybertronic]

Searching up to 10^17 complete (k<=1.66666...e16)
https://pzktupel.de/RecordGaps/GAP02.php

Latest open gaps found:
4861 99530388665966831
4951 99896889053870801

I stop here....

[mart_r]

Ever looking for irregularities in the prime number distribution, as a generalization to gaps between twin primes, I considered gaps between consecutive prime pairs {p, p+d} for even d > 2. (So e. g. with d = 4, we're looking at gaps between cousin primes.) Specifically, I'm once again focusing on exceptionally large gaps, similar to "CSG > 1" between regular prime gaps, in this case the comparative measure for such a large gap (which may be exceeded only finitely often for fixed d) is \(\large \sqrt{\frac{2 c_2}{(\frac{2 c_2}{gap} \cdot \int_{start\:p}^{end\:p} \frac{\text{d} t}{\log t \cdot \log (t+d)})^3}}\) if d mod 3 <> 0, and \(\large \sqrt{\frac{4 c_2}{(\frac{4 c_2}{gap} \cdot \int_{start\:p}^{end\:p} \frac{\text{d} t}{\log t \cdot \log (t+d)})^3}}\) if d mod 3 = 0, where \(c_2\) is the twin prime constant 0.6601618... .

Here are some exceptionally large gaps (d<=15000, p<10000000):

d	gap	start p	end p	expected *	CSG equivalent
96	792	287501	288293	13.23514749702	1.052871552102
1038	106	283	389	6.666569300312	1.028834379845
1374	46	7	53	5.274758801878	1.099182477006
1396	240	337	577	6.895143207451	1.017112792979
1758	1404	4968829	4970233	15.59424518190	1.011364690747
1926	1568	8944073	8945641	16.16043348677	1.009604514475
2154	46	7	53	4.971179604627	1.005669014240
2448	450	26981	27431	11.30113354517	1.102101408721
2648	750	17033	17783	10.23718319847	1.040878128217
2804	1056	60923	61979	11.42236266670	1.033860065575
2972	600	8147	8747	9.378221048165	1.020386899216
3102	320	10211	10531	9.610917222022	1.024983961588
3288	302	4289	4591	10.60689435095	1.223273864676
3428	396	1583	1979	8.165543494411	1.020445590697
3558	1188	1586161	1587349	15.38765793633	1.077693761422 <--
3642	162	647	809	7.746986607389	1.042525062759    } interesting!
3888	1110	1586191	1587301	14.37715965943	1.006915341704 <--
4192	798	22699	23497	10.26649948449	1.013425762962
4402	2436	1638061	1640497	15.70403218868	1.097331698777
4596	1376	3705347	3706723	15.88090710355	1.049901472763
4602	486	13921	14407	13.64510715424	1.406993512761 <-- largest known CSG equivalent
5228	2046	1054043	1056089	14.03892506112	1.012063519611
6186	1170	2108627	2109797	14.56724110104	1.000271362864
5204	198	29	227	6.539921041165	1.034393931617
6304	258	19	277	8.235396562418	1.280491570678
6328	480	1789	2269	9.219128862252	1.111921201591
6662	288	179	467	7.498612534161	1.053015898018
6852	280	4651	4931	9.319135901047	1.046235027387
7928	810	8009	8819	12.19835757557	1.302774748457
8094	1068	1348619	1349687	14.14935522055	1.002222508440
8604	316	2887	3203	11.11155044769	1.282222094500
10674	2100	8037439	8039539	21.93374343273	1.379444700673
11224	1668	203659	205327	14.66397442453	1.196573334288
12332	1422	187409	188831	12.66287435404	1.039940399644
13012	2010	912511	914521	14.07346079495	1.024856675921
13428	948	717191	718139	13.74991956012	1.019039155011
14268	776	282713	283489	12.95211437279	1.029733755338
14986	1140	84391	85531	11.51930604377	1.007736180365
* expected average number of prime pairs p+{0,d} in the gap, or "merit" equivalent

Eventually, this data shall serve as another yardstick for the validity of the Hardy-Littlewood heuristic.

[Cybertronic]

The lowest known merit for twins is: merit = 0,0000000145553806...
It is a consequence of the largest known prime quadruplet with 10132 digits.