GAPS BETWEEN PRIME PAIRS (Twin Primes)

[mart_r]

Originally Posted by Cybertronic

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

The number of prime quadruplets below floor(1/0.0000000145553806...) = 68703115 is 3593, a prime number
(The number of twins below 68703115 is 316154, which is twice a prime number)

A follow-up to my last post, with new results for d <= 40140 and p <= 2^32:

d	gap	start p		end p		expected *	CSG equivalent
1762	6960	1161678559	1161685519	21.09182930188	1.010477281538
2038	8352	3399730699	3399739051	22.89401673778	1.043151898744
2274	3658	1413479839	1413483497	21.75970227694	1.032767057624
2298	2592	147452491	147455083	19.34694712807	1.028598703340
3118	5862	355526911	355532773	19.96518857500	1.014021164210
3156	2274	44527843	44530117	19.35971567320	1.099253952479
3244	4080	36970783	36974863	17.74018815706	1.018045181811 <--,
3336	2010	36908593	36910603	17.48273441990	1.003369315590 <--- two more close pairs of large gaps between prime pairs
3352	7590	2015459527	2015467117	21.83314802908	1.019092141745
4308	4120	1967146393	1967150513	23.75669768056	1.110133740498
4958	7452	1156278491	1156285943	22.59288528354	1.082632161384
5268	1932	26241749	26243681	17.48192552737	1.023352233093
5956	8316	4135644811	4135653127	22.39367935080	1.011325003037
5988	1170	2111873		2113043		14.56426313296	0.999964651670 <--,
6186	1170	2108627		2109797		14.56724110104	1.000271362864 <--- almost! (a "pair" with the same gap size)
8058	1698	11929993	11931691	16.88654042023	1.036303552135
8706	2484	120816623	120819107	18.93987624421	1.017735007855
9402	3170	531079049	531082219	20.73915457948	1.032290060559
9456	2990	248908601	248911591	21.12516413285	1.092721146265
11166	2280	60171863	60174143	18.76363472517	1.047497114274
11562	2532	145171997	145174529	18.93039860728	1.007285521121
13088	4716	65502293	65507009	19.22288869689	1.068072862061
13228	6870	902281663	902288533	21.33245913439	1.034529351060
13652	7644	1199210237	1199217881	23.09422062982	1.104725625964
15018	2210	60433031	60435241	18.17870410891	1.014595772052
15328	1140	84691		85831		11.50932842677	1.006427165567
15516	1470	5348467		5349937		16.16984617259	1.043626189605
15648	378	13963		14341		10.13729921538	1.021603335340
15896	2406	872561		874967		16.95102545560	1.238243207583
16312	1650	343309		344959		13.35524857321	1.045672756308
16466	4674	62836073	62840747	19.13990801287	1.065921487373
16672	1758	395671		397429		13.92421312300	1.078465754917
16746	684	181943		182627		12.22081026433	1.005233041585
16794	826	292777		293603		13.70313351858	1.086137014502
16998	2806	281482603	281485409	19.57528393577	1.006150797484
17002	1830	249367		251197		15.55518527304	1.248093263741
17088	2650	159132943	159135593	19.62040252199	1.038923897927
17154	960	699539		700499		13.96961383632	1.037016792282
18658	708	6883		7591		10.35180363178	1.089349429976
18774	108	29		137		6.748101033837	1.038018970509
18834	74	5		79		5.851143833900	1.012488514288
19212	482	18587		19069		12.26093058881	1.203390103281
19442	690	6761		7451		10.08467467408	1.061031055151
20184	1456	4112293		4113749		16.57109856838	1.087905236808
20478	1602	9754559		9756161		16.33144543551	1.014729962847
20938	1032	42901		43933		11.52429918604	1.059843498463
21768	1568	3099461		3101029		18.52446245212	1.239053892194
21828	930	333721		334651		15.10433029615	1.184555846849
21842	1884	654257		656141		13.83449510824	1.031725877864
22048	3054	6512419		6515473		16.37721824769	1.043723568068
22416	2088	31927223	31929311	18.46655665218	1.068706475315
22558	6240	351469309	351475549	21.27734655303	1.081293787165
22898	4164	31679309	31683473	18.43002440549	1.067070974545
22922	6468	590298791	590305259	20.93691154140	1.036677620818
23208	3134	599428499	599431633	20.25871504465	1.002335590794
23382	110	17		127		7.074837810889	1.104137585131
23416	480	1117		1597		8.692406656100	1.018003418944
23626	6300	267882487	267888787	22.08735710987	1.138164816631
24162	420	18701		19121		10.55465763290	1.029642319673
24292	852	22159		23011		10.43316257399	1.004763769442
24558	756	235673		236429		12.93914931677	1.041699589642
24744	1470	3680647		3682117		16.97464760025	1.122502422489
25446	108	23		131		6.707750384676	1.028722563228
25812	1590	9674299		9675889		16.22519473489	1.008628228505
25986	580	84247		84827		11.62498811280	1.012790416813
27282	572	75797		76369		11.63922702274	1.021722596222
27338	1380	163781		165161		12.47135016068	1.031787989783
27822	288	2731		3019		9.243367265687	1.019046199773
28092	1030	452077		453107		15.96389492358	1.223022600008
29644	420	367		787		8.495575299546	1.051536487278
29712	360	8167		8527		9.982101327209	1.022884304191
29814	636	96827		97463		12.44452286877	1.071232206603
30098	1608	242873		244481		13.67138122962	1.097072604775
30474	1634	5071169		5072803		18.09430755654	1.171741278585
30676	2484	850273		852757		17.54441544105	1.283194117163
30972	1408	2407549		2408957		17.20406604016	1.170280929502
30978	532	36979		37511		11.99148878100	1.107895111181
31302	3630	1240560649	1240564279	21.86309502754	1.044140532664
31392	84	5		89		6.138468182511	1.021163244521
31442	2040	870059		872099		14.36062809275	1.048587969436
31468	6078	411001081	411007159	20.39923187780	1.028490242412
31588	840	18541		19381		10.39599283298	1.006512367962
31706	8790	3744949433	3744958223	23.88367215439	1.083470312124
31938	670	97651		98321		13.07427977053	1.123916542684
33486	236	311		547		9.884036518720	1.244773443725
33546	1746	12843377	12845123	17.20571050683	1.051069905367
33676	360	181		541		7.816480592812	1.002363934711
34134	2908	357343489	357346397	19.79818902654	1.005277129931
34754	2928	6495569		6498497		15.70478393888	1.000972657281
35016	476	18911		19387		11.69563013172	1.128175764017
35586	136	31		167		7.694281164267	1.126230872317
35634	3320	651206033	651209353	21.28616752192	1.048870753092
35982	3066	468196241	468199307	20.31272956552	1.017445507418
36422	1002	39839		40841		11.09026209108	1.015403498957
36732	938	677021		677959		13.68691926449	1.017423917871
37234	6624	729263629	729270253	20.99993151625	1.029026259344
37342	1638	272971		274609		13.65730842497	1.085301814006
37438	6162	525993679	525999841	20.17614949499	1.004746230335
37668	232	709		941		8.646014361882	1.027128406567
37866	574	40213		40787		12.67812553272	1.159501805293
38596	3492	12915061	12918553	17.19348111838	1.049949493238
38916	1010	1022981		1023991		13.88897046743	1.002280310133
39022	1860	360049		361909		14.87771200938	1.157997311841
39138	252	739		991		9.292135209243	1.098039123274
39176	2196	1538027		1540223		14.25992188223	1.000044563941
39342	3218	415550909	415554127	21.57679427007	1.087256881390
39546	450	18691		19141		10.99353584670	1.057412992489
39648	2246	23749343	23751589	20.56093732056	1.210610096643
39960	700	161977		162677		12.61078705744	1.041619429921
40028	2070	1118219		1120289		14.05263856621	1.007654026079
40134	1508	2275859		2277367		18.56161861346	1.267266342915
* expected average number of prime pairs p+{0,d} in the gap, or "merit" equivalent



bonus: record CSG equivalent from a search to d<=80000 / p<=20000:
59544	320	227		547		12.98009662424	1.608743120775