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Old 2019-04-16, 18:20   #1
rudy235
 
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Jun 2015
Vallejo, CA/.

22·239 Posts
Default GAPS BETWEEN PRIME PAIRS (Twin Primes)

As we all know the twin primes are
{3,5} {5,7} {11.13} {17,19} {29,31} {41,43} {59,61} {71,73} {101,103} {107,109} {137,139} … A077800

Chris Caldwell has a link to the first 10k Twin primes first element of twin primes
Except for the first pair all the primes p, p+2 are the form 6k+1 and 6k-1

So we can adopt the convention of denoting a twin pair of primes by simple using the number k

Thus k=58 represents the twin primes 347, 349 or 6*58-1 and 6*58+1

Then we can create a sequence of all k's and have a good shorthand for all the pairs of twin primes (except for the aforementioned pair {3,5}

This sequence is A002822 1,2,3,5,7,10,12,17,18,23,25,30,32,33,38,40,45,47,
52,58,70,72,77,87,95,100,103,107,110,135,137,138,
143,147,170,172,175,177,182,192,205,213,215,217,
220,238,242,247,248,268,270,278,283,287,298,312,
313,322,325

With the exception of the first, all of the members of this sequence are congruent to (0, 2 or 3 mod 5

So in a comparison to the "gap between primes" we now can establish gaps between contiguous pairs of primes. (of the form 6k +/-1

The first gap of 1 appears at the start as we can see here

Code:
k         Gap   
1                     1
2                     1
3                     2
5                     2
7                     3
10                    2
12                    5
17                    1
18                    5
23                    2
25                    5
30                    2
32                    1
33                    5
38                    2
40                    5
45                    2
47                    5
52                    6
58                   12
70                    2
72                    5
77                   10
87                    8
95                    5
100                   3
103                   4
107                   3
110                  25
135                   2
137                   1
138                   5
143                   4  
147                  23
170                   2
172                   3
175                   2
177                   5
182                  10
192                  13
205                   8
213                   2
215                   2
217                   3
220                  18 
238                   4
242                   5
247                   1
248                  20
268                   2
270                   8
278                   5
283                   4
287                  11
298                  14
312                   1
313                   9
322                   3
325                   8
333                   5
338
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Old 2019-04-16, 19:01   #2
rudy235
 
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Jun 2015
Vallejo, CA/.

22×239 Posts
Default

We can see a few things on this list of gaps.

First of all that all numbers seem to be represented ( we have at least 1 to 6 then 8 to 14 and 18, 20,23,25 )

What does a gap of 1 mean and are they infinite of those?

A gap of 1 between two pairs of twin primes represents a prime quadruplet For instance the gap of 1 after element 247 represents 6*247-1 and 6*247+1 which is a twin prime pair {1481,1483} and with the next closest pair of primes {1487, 1489} make a quadruplet.

As quadruplet primes are theorized to be infinite the gaps of 1 would also be infinite.

Are all gaps represented? I believe so but, of course, this is an open question. I do not see any reason why a gap of 15 or of 7 might not exist and if someone with time and resourses makes a run up to twin primes under 100,000 I am confident that they should appear a few times. (I have only searched primes ≤ 2100 which is a paltry seach)

Code:
k         Gap   
1                     1  (first time)
2                     1
3                     2  (first time)
5                     2
7                     3  (first time)
10                    2
12                    5  (first time)
17                    1
18                    5
23                    2
25                    5
30                    2
32                    1
33                    5
38                    2
40                    5
45                    2
47                    5
52                    6 (first time)
58                   12 (first time)
70                    2
72                    5
77                   10 (first time)
87                    8 (first time)
95                    5
100                   3 
103                   4 (first time)
107                   3
110                  25 (first time)
135                   2
137                   1
138                   5
143                   4  
147                  23 (first time)
170                   2
172                   3
175                   2
177                   5
182                  10
192                  13 (first time)
205                   8
213                   2
215                   2
217                   3
220                  18 (first time)
238                   4
242                   5
247                   1
248                  20 (first time)
268                   2
270                   8
278                   5
283                   4
287                  11 (first time)
298                  14 (first time)
312                   1
313                   9 (first time)
322                   3
325                   8
333                   5
338
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Old 2019-04-16, 22:29   #3
ATH
Einyen
 
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Dec 2003
Denmark

B2716 Posts
Default

First occurrence gaps up to gap=1023.


Code:
gap	k (before gap)
1	1
2	3
3	7
4	103
5	12
6	52
7	378
8	87
9	313
10	77
11	287
12	58
13	192
14	298
15	597
16	357
17	1075
18	220
19	3563
20	248
21	2042
22	800
23	147
24	3843
25	110
26	3257
27	2063
28	397
29	6458
30	1755
31	6227
32	1438
33	1507
34	5638
35	980
36	13372
37	2560
38	7637
39	6018
40	2438
41	6332
42	6088
43	14542
44	11833
45	2478
46	6692
47	2233
48	9105
49	6808
50	8432
51	23277
52	7968
53	6585
54	23133
55	13815
56	25347
57	13953
58	7462
59	35798
60	31972
61	25587
62	3090
63	17475
64	90423
65	9002
66	72942
67	19033
68	17850
69	32378
70	4377
71	12952
72	23693
73	48785
74	37058
75	14845
76	58077
77	35368
78	77697
79	18308
80	55143
81	33397
82	74400
83	4070
84	24168
85	20478
86	138187
87	80868
88	22890
89	56523
90	55632
91	37942
92	81448
93	44660
94	213103
95	97545
96	354662
97	27620
98	27977
99	383148
100	44905
101	125472
102	20013
103	76967
104	293123
105	10383
106	241847
107	91303
108	89567
109	91103
110	140420
111	129022
112	149863
113	130757
114	239678
115	126070
116	334862
117	100658
118	429182
119	304243
120	140795
121	614567
122	322018
123	199937
124	535353
125	73238
126	148897
127	135550
128	410312
129	217368
130	37355
131	264492
132	150433
133	187047
134	376973
135	500720
136	828307
137	156343
138	311532
139	767163
140	505115
141	569097
142	240345
143	41995
144	989058
145	116403
146	407412
147	308028
148	802070
149	989443
150	282408
151	707262
152	277258
153	234965
154	31318
155	114587
156	607952
157	292908
158	839762
159	410923
160	154445
161	518717
162	793900
163	526890
164	443443
165	437057
166	705787
167	632180
168	114742
169	1571558
170	1074442
171	2040992
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192	1052370
193	485555
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196	1402747
197	682880
198	1188775
199	2004258
200	340200
201	1511522
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203	1335645
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205	1722322
206	1372987
207	3142030
208	546497
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217	901498
218	1416445
219	1886258
220	1793428
221	3411077
222	1580488
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240	924093
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305	8106710
306	2681627
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313	6400455
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315	4702603
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318	16896975
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411	22425307
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413	47602420
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417	68602238
418	41778520
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445	12612057
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448	21431515
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885	2306786122
886	3287184092
887	1031407750
888	5187146730
889	8226414758
890	3048037043
891	4781139942
892	4455278398
893	5321666817
894	4568127788
895	6559094538
896	3352338792
897	4322805470
898	2179079075
899	2248681003
900	2238954123
901	4346895982
902	7060275863
903	1223175910
904	11060909463
905	1574161108
906	4349096642
907	2526316303
908	5763562942
909	9224381248
910	3187343453
911	4162545082
912	3920873523
913	6284266240
914	7887840663
915	3964391842
916	4023472667
917	4967220493
918	2785190895
919	4516769603
920	1654347340
921	6886816167
922	1522788600
923	1731506367
924	4163553853
925	1507820423
926	6972833117
927	2528311860
928	6198848495
929	5310531548
930	2689512325
931	2719152782
932	6172499860
933	4207178020
934	3005762743
935	4422668477
936	18462023822
937	2615806835
938	4397130915
939	8493754803
940	1299446533
941	8527231937
942	4945354223
943	3912129900
944	13643349213
945	5531673495
946	11543632172
947	4015179735
948	12862817032
949	9407061128
950	5657024718
951	3749254647
952	7884111378
953	3780605152
954	4441086328
955	9549957132
956	9626553492
957	3765668163
958	5365459852
959	14906845753
960	4077390530
961	856359122
962	8640331193
963	1863989655
964	10792231053
965	3157455605
966	6129872887
967	9973337810
968	5706917330
969	15473299583
970	3670338742
971	7514667242
972	14374380998
973	5016435072
974	17508006308
975	1065472893
976	4204219947
977	16162465793
978	8987211702
979	9417392003
980	6262281762
981	13519426337
982	10207490848
983	11779435180
984	16686552823
985	13121206348
986	5917519772
987	7103990803
988	3770789082
989	10920336278
990	6159537563
991	3553869007
992	7971882450
993	16539204720
994	3923891423
995	11323476390
996	2829945767
997	8108352970
998	15359688132
999	18263276708
1000	4718578212
1001	2811629627
1002	11024680285
1003	6835356575
1004	19374221148
1005	714897587
1006	14257463852
1007	4267983675
1008	10764699650
1009	18724975378
1010	4959775740
1011	6904565602
1012	6009418280
1013	6582996530
1014	11991672908
1015	7504060620
1016	12499155547
1017	8955975688
1018	5986929792
1019	7242983443
1020	13688528333
1021	12937575437
1022	6003407428
1023	8561645130

Last fiddled with by ATH on 2019-04-16 at 22:32
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Old 2019-04-16, 23:22   #4
rudy235
 
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Jun 2015
Vallejo, CA/.

22×239 Posts
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Quote:
Originally Posted by ATH View Post
First occurrence gaps up to gap=1023.


Code:
gap	k (before gap)
1	1
2	3
3	7
4	103
5	12
6	52
7	378
8	87
9	313
10	77
11	287
12	58
13	192
14	298
15	597
16	357
17	1075
18	220
19	3563
20	248
21	2042
22	800
23	147
24	3843
25	110
26	3257
27	2063
28	397
29	6458
30	1755
…
…

1000	4718578212
1001	2811629627
1002	11024680285
1003	6835356575
1004	19374221148
1005	714897587
1006	14257463852
1007	4267983675
1008	10764699650
1009	18724975378
1010	4959775740
1011	6904565602
1012	6009418280
1013	6582996530
1014	11991672908
1015	7504060620
1016	12499155547
1017	8955975688
1018	5986929792
1019	7242983443
1020	13688528333
1021	12937575437
1022	6003407428
1023	8561645130
Thank very much ATH for that.

I could only modestly go until k≤ 2700 because I was doing it in EXCEL.

I could find all the first occurrences from 1-18 and the next few ones up to 30 with the exception of 19, 24, 26 and 29

Seems clear from what you have done that it can be safely assumed (but of course not easy to prove) that ALL gaps are going to be present.

Last fiddled with by rudy235 on 2019-04-17 at 00:14 Reason: added color
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Old 2019-04-17, 01:03   #5
retina
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Quote:
Originally Posted by rudy235 View Post
... I was doing it in EXCEL.
If you are serious about doing this kind of stuff then download and learn to use some dedicated computing software.
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Old 2019-04-17, 01:09   #6
ATH
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Dec 2003
Denmark

5×571 Posts
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Continued up to k=200B. Here are consecutive first occurrence gaps 1024 up to 1358, as well as the other nonconsecutive first occurrence gaps 1360 to 1898.

Also added the maximal gap list in order of occurrence.

Code:
gap	k
1024	30370627668
1025	16507782197
1026	12133392337
1027	8527092578
1028	5400391645
1029	8780665563
1030	5007391888
1031	8309499172
1032	3740360183
1033	12579042990
1034	7490905648
1035	4354724050
1036	12790338547
1037	14585762935
1038	12918899037
1039	10528753153
1040	8643398725
1041	22228983707
1042	18421312110
1043	4277863287
1044	22247337703
1045	9117393532
1046	11812293782
1047	3196956045
1048	7919321767
1049	10991916183
1050	9865733768
1051	22563333397
1052	22070444965
1053	16686868527
1054	12996465758
1055	8472371157
1056	31366197567
1057	9491799368
1058	8659623317
1059	11055610988
1060	13235724187
1061	41915068062
1062	18498428755
1063	9023952725
1064	9653598588
1065	5298861998
1066	12036778707
1067	14334530700
1068	39987891870
1069	22736738038
1070	4416179072
1071	12539957742
1072	25946320698
1073	14636179420
1074	12432480398
1075	7762655897
1076	19372935412
1077	14255597088
1078	16521687485
1079	4035849583
1080	24070972425
1081	21562226522
1082	14943718163
1083	13624623332
1084	18383401348
1085	15837017163
1086	43177754272
1087	18253515870
1088	17472321867
1089	11557132158
1090	8186246962
1091	41500930727
1092	4142929803
1093	9490473660
1094	21439776823
1095	40213458707
1096	28895725292
1097	21386758558
1098	25137783535
1099	29311613548
1100	14524563518
1101	12875454587
1102	10942894720
1103	11202556685
1104	16615629933
1105	16606921495
1106	20860243742
1107	17500043025
1108	6708903010
1109	19821577948
1110	12852163973
1111	8412207522
1112	25186770675
1113	14333623160
1114	23078677218
1115	9254604950
1116	15134152567
1117	24189828365
1118	29646936667
1119	34226827423
1120	18463125062
1121	58663327587
1122	10133636193
1123	26227925140
1124	42383800163
1125	38347022838
1126	20497482947
1127	8185647088
1128	29451224835
1129	29579425773
1130	16298537952
1131	37485937562
1132	7578097340
1133	25913710302
1134	14524565688
1135	20792431795
1136	28809229977
1137	13582827433
1138	37975916165
1139	61756317343
1140	15892671560
1141	10439154257
1142	25262591533
1143	28593081655
1144	48826150303
1145	41705622962
1146	25794434892
1147	16293399008
1148	8582270267
1149	8525721438
1150	16251349042
1151	45316370542
1152	21034892753
1153	15728686990
1154	17504175188
1155	24735441112
1156	33174606362
1157	27616548440
1158	42035688247
1159	23092251588
1160	14998583343
1161	56801711312
1162	29812173248
1163	17756220570
1164	16797436063
1165	14435182932
1166	15804331282
1167	33766997653
1168	29664641207
1169	51537427428
1170	25320667690
1171	21420242172
1172	39668112358
1173	13791132407
1174	18761657808
1175	7073619248
1176	30537616247
1177	23153721783
1178	15554271252
1179	57841170578
1180	13255405793
1181	31234475137
1182	30924553110
1183	26168680170
1184	49319201213
1185	23812067153
1186	41936084087
1187	8939732550
1188	14330487105
1189	44674906188
1190	8031358767
1191	10603108182
1192	18835412365
1193	34746153190
1194	16560031848
1195	9391637273
1196	14528709892
1197	18242511845
1198	25046740252
1199	33650418018
1200	18359485235
1201	33051842117
1202	66002847468
1203	26782423792
1204	40847394758
1205	15928146832
1206	64999749057
1207	9988457248
1208	19986121662
1209	53500919883
1210	29408935490
1211	32056558362
1212	46819768958
1213	23359512832
1214	20331890558
1215	36293533640
1216	68006592497
1217	48101859580
1218	35070286585
1219	10774224213
1220	36726742647
1221	25678337692
1222	8695593693
1223	43949266767
1224	90705803338
1225	25905710483
1226	35530520182
1227	50758082210
1228	43848563047
1229	44351452008
1230	26404449530
1231	101208274662
1232	22556325743
1233	37269319510
1234	45484451658
1235	41601982865
1236	20875795512
1237	56829323030
1238	32867985517
1239	33904468903
1240	37951687547
1241	23524711222
1242	51497620853
1243	64872205377
1244	65789117053
1245	57057845338
1246	40884738972
1247	36050767643
1248	53024192545
1249	48725636383
1250	28543693757
1251	64133691577
1252	45976631095
1253	56015688972
1254	100761544543
1255	22544518132
1256	38237309287
1257	37685943368
1258	20020326737
1259	89817312058
1260	40437430637
1261	81952221312
1262	34386364680
1263	64312175652
1264	41031871478
1265	7595970438
1266	66001896202
1267	102711625448
1268	52055075942
1269	60464331573
1270	26072298478
1271	94947179337
1272	37482239365
1273	52338755872
1274	31605073873
1275	37883577230
1276	19485934097
1277	10846539718
1278	53318021160
1279	80634404163
1280	40551655827
1281	31260613627
1282	51115333138
1283	69732506812
1284	70023017393
1285	12347431502
1286	41906763817
1287	29873248415
1288	29314359620
1289	44402436488
1290	43944263470
1291	60328695492
1292	35124127965
1293	44978931772
1294	78981727158
1295	66748779720
1296	136174101612
1297	83379921555
1298	9634521157
1299	29706714558
1300	59173429653
1301	64935700882
1302	77686997383
1303	68044441195
1304	40831407493
1305	24030385738
1306	37042953262
1307	92308689725
1308	91890048212
1309	69089723273
1310	71131302125
1311	65562906117
1312	50485273458
1313	48624178687
1314	83852500748
1315	48882377632
1316	75146429732
1317	36692181453
1318	53992589912
1319	186438133443
1320	51173468143
1321	30530362357
1322	94707449170
1323	42450011142
1324	70051577718
1325	40320006480
1326	78720103417
1327	31520502065
1328	34445495270
1329	143446677043
1330	7287610437
1331	86046387072
1332	51772611128
1333	104162957980
1334	44438479298
1335	46412332407
1336	92479857902
1337	51780274385
1338	97017192822
1339	79715880988
1340	10849288625
1341	82083852107
1342	132223116333
1343	25083224815
1344	62994611978
1345	78773730823
1346	103895222522
1347	66824446040
1348	93470563142
1349	46032970723
1350	110940120422
1351	147555423337
1352	44028349955
1353	72388630452
1354	176010608498
1355	35415187398
1356	73721641492
1357	39997802863
1358	68875590515



1360	120701590522
1361	103656076897
1362	129780127055
1363	45464706810
1365	27432631065
1367	66664704428
1368	79675959260
1369	105795508798
1370	22751684835
1371	150422884902
1372	73987959848
1373	91101657665
1374	74619205678
1375	106433204497
1376	109109662112
1377	140926519020
1378	158222902962
1379	59513006413
1380	83535481318
1381	81120569787
1383	82037154155
1384	91105669713
1385	35954269187
1386	36654125322
1387	82436707565
1388	52106714982
1390	134783026287
1391	91527303572
1392	160241217295
1393	78441568602
1394	107218832948
1395	93204583073
1397	71529209675
1398	89717150497
1399	167763335213
1400	78769799252
1401	113406055667
1402	83591984703
1404	115946907303
1405	60066856568
1406	157757582267
1408	75688827280
1409	131711929323
1410	199574686315
1411	48278160327
1412	157757709578
1413	32135929095
1414	179075794238
1415	49037845735
1416	64167955752
1417	117577979190
1418	108513597027
1419	77078300443
1420	74951308542
1421	42119304362
1422	101735004478
1423	142152878497
1425	88942037140
1426	114261540757
1428	100389566412
1429	175835907578
1430	48978765967
1431	105371512887
1432	89851341625
1434	197709087693
1435	110562730142
1436	34440671017
1437	141185526520
1438	167850568830
1439	94595234198
1440	34516209527
1441	195969981917
1442	136888972125
1443	134581666552
1444	159728123338
1445	38840382823
1447	57301290370
1448	77920988750
1449	102332614223
1450	108516370937
1451	64095115322
1452	151527991675
1453	145272707550
1454	174799142838
1455	81631078743
1456	195778814802
1460	64584093742
1462	88475830910
1463	69741598787
1465	135767046975
1467	85661945075
1468	63139736062
1470	106208976653
1471	156171984272
1472	138299223690
1473	184783689402
1475	67410836280
1477	185821439738
1478	96139203857
1482	176427443750
1483	48773094382
1484	136319467498
1485	169809420105
1486	150516739082
1489	130741136258
1490	88340661855
1492	121639858400
1493	66924164457
1495	69782047618
1496	174973906777
1499	22339570278
1500	172823544343
1502	128837925765
1503	181674927455
1505	160823433270
1507	176020578323
1510	163775464003
1515	155390806498
1516	85283577452
1517	196413282468
1518	184159612907
1520	159460255028
1523	147488057610
1525	98616660978
1526	56105289977
1527	194013931730
1529	137914593968
1530	164942161335
1531	82643395422
1536	151913191462
1538	119496454527
1540	139666887840
1542	181392272663
1543	180056667807
1544	146578135928
1545	70464152903
1546	168996745347
1547	160429275275
1548	142976782372
1549	170014476063
1550	174553795390
1552	33051947625
1553	37182176622
1555	119208146283
1556	107282165062
1557	143282458305
1558	164872223950
1560	62578713270
1561	176158811742
1562	118027566675
1565	136243056022
1575	103750795712
1577	185624758875
1580	184298726117
1583	129919858890
1585	96643867775
1587	144666194755
1590	116320889387
1595	177702480225
1603	121584681445
1605	185545828858
1610	109354338662
1613	128518326722
1618	151521188780
1620	111423209912
1631	117959421867
1633	103293294022
1635	101323871015
1643	158850706422
1646	197557993677
1649	158176828393
1658	166226961612
1659	165136905488
1660	133864375453
1664	131415761768
1672	161263436148
1687	156290145345
1700	58917239540
1710	89470349442
1723	80799633875
1778	106405396032
1785	130744752537
1810	133135498198
1815	169866686215
1890	151287342070
1898	132437316545
Code:
maxgap	k
1       1
2       3
3       7
5       12
6       52
12      58
25      110
28      397
35      980
47      2233
62      3090
83      4070
105     10383
154     31318
155     114587
168     114742
242     141725
252     478160
255     811652
287     1653998
317     2442753
365     2897080
376     5125372
472     5470395
478     16149007
502     22713905
517     39161155
530     41440173
580     42658325
634     65136288
795     116423748
882     411106845
1005    714897587
1047    3196956045
1079    4035849583
1092    4142929803
1108    6708903010
1175    7073619248
1330    7287610437
1340    10849288625
1499    22339570278
1552    33051947625
1553    37182176622
1700    58917239540
1723    80799633875
1778    106405396032
1785    130744752537
1898    132437316545
ATH is offline   Reply With Quote
Old 2019-04-17, 08:21   #7
robert44444uk
 
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Jun 2003
Oxford, UK

187510 Posts
Default

These gaps do look interesting.

If ATH shared his program, we could co-ordinate this to higher levels.

The values of k of the first instance gaps are much smaller than those for prime gaps, even when the latter are divided by two.

I wonder whether it is possible to hack Robert G.'s prime gap program to get a massive speed up?

Last fiddled with by robert44444uk on 2019-04-17 at 08:21
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Old 2019-04-17, 13:53   #8
Thomas11
 
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Feb 2003

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A quick and easy solution would be using NewPGen (sieving for twin primes of the type k*6^1+/-1) and then using some Perl script or similar tool to generate and update the gap list.

In a Linux environment the following bash script would do the job:

Code:
kmin=1
kmax=100010000
kstep=100000000
kfinal=1000010000

while [ $kmax -le $kfinal ]
do
  ./newpgen -v -t=2 -base=6 -n=1 -kmin=$kmin -kmax=$kmax -wp=6k.txt
  ./update_gaplist.pl 6k.txt
  kmin=`expr $kmin + $kstep`
  kmax=`expr $kmax + $kstep`
done
The necessary Perl script can be found in the attached ZIP file.

The range up to k=1B takes less than a minute. But note that the sieve time increases with increasing k (roughly with the square root of k).

(We use a little overlap between consecutive ranges in order to make sure that we do not loose any gaps between the intervals.)
Attached Files
File Type: zip twingaps.zip (822 Bytes, 35 views)

Last fiddled with by Thomas11 on 2019-04-17 at 14:00
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Old 2019-04-17, 17:59   #9
Dr Sardonicus
 
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Feb 2017
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Earlier, this Forum was treated to a link about Jumping champions, having to do with the differences between consecutive primes. Using the table of the first 100k (not 10k) twin primes (I dropped the "3" and just looked at the 99999 remaining ones) I found that if p = 6*k - 1 and p' = 6*k' - 1 are consecutive, then

the largest value of g = k' - k which occurs is 365.

I decided to compile a list of gap values g = k' - k from g = 1 to g = 365. I list the first 65 of these. It looks like, at this stage, g = 5 is somewhat favored [pairs of twin primes differing by 6*5 = 30], with g = 7 a close second and g = 2 a somewhat distant third.

[1377, 3482, 2465, 1660, 4804, 1386, 4342, 2368, 1488, 3352, 1637, 2608, 2476, 1630, 3195, 1272, 2007, 1968, 1121, 2345, 1283, 2026, 2183, 548, 2214, 1201, 1281, 2087, 658, 2459, 579, 1170, 1670, 616, 2545, 409, 1359, 969, 520, 1872, 372, 1152, 732, 592, 988, 333, 773, 774, 467, 798, 505, 578, 519, 290, 920, 370, 460, 606, 197, 595, 329, 536, 611, 149, 807]

If someone is aware of "jumping champions" heuristics for twin primes, please share.

Last fiddled with by Dr Sardonicus on 2019-04-17 at 18:00 Reason: finxig spoty
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Old 2019-04-17, 18:27   #10
rudy235
 
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Quote:
Originally Posted by robert44444uk View Post
These gaps do look interesting.

The values of k of the first instance gaps are much smaller than those for prime gaps, even when the latter are divided by two.
Yes, Robert I did notice that. While it took years and years of search to get to the first kilogap by B. Nyman in 2001 (in case of the primes) which is really a gap of 566 if divided by 2, in this case, it was relatively easy for ATH to get to 1023.

Just to add one other fact. As the gaps of 1 are _by definition_ Quadruplet Primes, we can make an accurate estimation of how many gaps of 1 they are for a particular level.


So for instance π (1012) is 37,607,912,018 while π_4(1012) is 8,398,278
In other words at that level (1012) for every 4478 gaps one of those is equal to 1 . At the 1016 level the ratio goes down to 1 for every 11,002 gaps

T. R . Nicely has tabulated the Quad primes up to 1.p*1016 here
Code:
     x         π_4 (x)          π(x)
======================
   1e07         899     664,579
   1e08        4768     5,761,455   
   1e09       28388     50,847,534      
   1e10      180529     455,052,511   
   1e11     1209318     4,118,054,813 
   1e12     8398278     37,607,912,018
   1e13    60070590     346,065,536,839
   1e14   441296836     3,204,941,750,802
   2e14   807947960     6,270,424,651,315
   3e14  1151928827     9,287,441,600,280
   4e14  1482125418     12,273,824,155,491 
   5e14  1802539207     15,237,833,654,620 
   6e14  2115416076     18,184,255,291,570
   7e14  2422194981     21,116,208,911,023
   8e14  2723839871     24,035,890,368,161
   9e14  3021126140     26,944,926,466,221
   1e15  3314576487     29,844,570,422,669
 1.1e15  3604646822     32,735,816,605,908 
 1.2e15  3891706125     35,619,471,693,548
 1.3e15  4175985018     38,496,205,973,965
 1.4e15  4457782901     41,366,582,391,891
 1.5e15  4737286827     44,231,080,178,273
 1.6e15  5014641832     47,090,114,439,072
 1.7e15  5290057283     49,944,045,778,207  
 1.8e15  5563600032     52,793,190,012,734
 1.9e15  5835422569     55,637,829,945,151
 2.0e15  6105617289     58,478,215,681,891  
 3.0e15  8734892736     86,688,602,810,119
 4.0e15 11265509044     114,630,988,904,000 
 5.0e15 13725978764     142,377,417,196,364  
 6.0e15 16132120984     169,969,662,554,551  
 7.0e15 18494314750     197,434,994,078,331
 8.0e15 20819642284     224,792,606,318,600 
 9.0e15 23113346779     252,056,733,453,928
 1.0e16 25379433651     279,238,341,033,925
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Old 2019-04-18, 00:42   #11
rudy235
 
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CORRECTION:

T. R . Nicely has tabulated the Quad primes up to 1*1016 here
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