mersenneforum.org QUADRUPLET PRIMES AND THEIR GAPS
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 2019-12-27, 09:15 #1 rudy235     Jun 2015 Vallejo, CA/. 24×61 Posts QUADRUPLET PRIMES AND THEIR GAPS Here is a list of first instances For the Quadruplets (notice how different it is from the list of First Differences between Twin Primes) But first a review GAPS between primes P≥3 Minumum gasp is 2 Maximum gap Until Dec 2019 is the gap of 5,103,138 after a PRP of 216,849 digits. GAP between Twins p,p+2 to q,q+2 these gaps between p and q are always of the form 6k Minimum gap (expressed as k) 1 Maximum gap yet found 518546 As to the gap between Quadruplet primes we define the number k as the distance between the first element of the QP (or alternatively of the barycenter) divided by 30 The barycenter –or Pivot– is the central number (it ends in 5) For instance the baricenter of the quadruplet {821, 823, 827, 829 is 825 A159910 FIRST OCCURRENCE Code: Gap After p= 1 1006301 rudy235 2 NONE 3 11 rudy235 4 1022381 rudy235 5 NONE 6 3512051 rudy235 7 1871 rudy235 8 632081 rudy235 9 NONE 10 71121831 rudy235 11 15731 rudy235 12 NONE 13 1481 rudy235 14 1155611 rudy235 15 1068251 rudy235 16 NONE 17 18911 rudy235 18 284741 rudy235 19 NONE 20 12390011 rudy235 21 191 rudy235 22 821 rudy235 23 NONE 24 3837131 rudy235 25 875261 rudy235 26 NONE 27 854921 rudy235 28 10865291 rudy235 29 18041 rudy235 30 NONE 31 958541 rudy235 32 680291 rudy235 33 NONE 34 299471 rudy235 35 1063961 rudy235 36 3995441 rudy235 37 NONE 38 4404551 39 5733525 40 NONE The largest gap I have found is a gap of 1897 between the Quadruplet of barycenter 3,741,165 and that of 3,798,075 3'798,075-3,741,165= 56910 . 56,910/30 = 1897
 2019-12-27, 12:06 #2 mart_r     Dec 2008 you know...around... 27E16 Posts You might want to have a look at this paper, which lists record gaps between prime quadruplets up to 10^15: https://arxiv.org/abs/1309.4053
2019-12-27, 14:53   #3
rudy235

Jun 2015
Vallejo, CA/.

24×61 Posts

Quote:
 Originally Posted by mart_r You might want to have a look at this paper, which lists record gaps between prime quadruplets up to 10^15: https://arxiv.org/abs/1309.4053
Thank you very much. Very interesting read.

Here is a more complete table of First Occurrences. (It is easy to prove GAPS 2 mod 7 and 5 mod 7 do not exist.

There were a couple of mistakes in the previous table This is the corrected version
Note gaps like 2,5,9,12,16,19,23,26,30,33,37,40,44,47,51,54,58,61,65,68,72,75,79,82,86,89,93,96,100… and all 2 mod7 or 5 mod cannot exist.

Code:
GAP    After Quadruple
1	1006301

3	11

4	1246241

6	3512051

7	1871

8	632081

10	1121831

11	15731

13	1481

14	1155611

15	1068251

17	18911

18	284741

20	12390011

21	191

22	821

24	3837131

25	875261

27	854921

28	10865291

29	18041

31	958541

32	680291

34	299471

35	1063961

36	663581

38	165701

39	2081

41    Unknown

42	21011

43	97841

45	79691

46	2223671

48	1022501

49	3436241

50	300491

52	294311

53	19421

55     Unknown

56	81041

57	463451

59	442571

60	419051

62	1902611

63	734471

64	6054281

66	16061

67	6332861

69	10037381

70	2246141

71	266681

73	3461

74	855731

76	67211

77	358901

78	7159511

80	2761721

81	77261

83	661091

84	1006331

85	257861

87	467471

88	13001

90	389561

91	69491

92	116531

94	2846861

95	594821

97	119291

98	1003361

99	11718821

101	22271

102	1230371

104	31721

105	2728541

106	1830341

108	1065011

109	5154761

111	654161

112	833711

113   Unknown

115	1627601

116	1912451

118	9585881

119	9431

120	295871

122	240041

123	5248091

125	22692611

126	5651

127	1023941

129	6880871

130	243701

132	2875331

133	51341

134	1798631

136	8557721

137	1588751

139	5201291

140	559211

141	62981

143	2541941

144	166841

146	397751

147	6556601

148	8827541

150	5234681

151	1122131

153	4575281

154	495611

155	354251

157	741751

158	2418671

160  Unknown

161	768191

162	13179581

164	1616801

165	1954361

167  Unknown

168	72221

169	19842671

171	1324571

172   Unknown

174   Unkown

175	2413421

176	2002331

178	14912741

179	978071

181  Unknown
182	340931
183  Unknown
185	14877671
186   Unknown
188	1791731
189	2136131
190	1508621
192	195731
193	1904471
195	6187451
196	1157831
197 Unknown
199	4734671
200	823721

The largest gap I've found is 5261  after 21872441

 2019-12-27, 16:46 #4 rudy235     Jun 2015 Vallejo, CA/. 24×61 Posts More First occurrences Code: GAP After Quadruple 202 12021671 203 82721 204 13597271 206 16728071 207 4326401 209 260411 210 4609301 211 1749491 213 420851 214 25301 216 Unknown 217 334421 218 7267451 220 1352201 221 563411 223 109841 224 4138241 225 8101781 227 6918341 228 1093991 230 18468251 231 470081 232 15010571 234 412031 235 1711811 237 9541991 238 1577291 239 7328771 241 187631 242 4032401 244 Unknown 245 6692261 246 12382631 248 3727721 249 2704901 251 Unknown 252 43781 253 4370081 255 55331 256 1360781 258 10872131 259 4396781 260 9291641 262 9725951 263 4405691 265 Unknown 266 217361 267 13056311 269 2596661 270 814061 272 4353311 273 444341 274 14235671 276 326141 277 946661 279 5417381 280 881471 281 157271 283 2007911 284 11138711 286 11192801 287 938051 288 Unknown 290 135461 291 101111 294 1146791 295 9138131 297 585911 298 34841 300 5318891 301 88811 302 Unknown 304 3215741 305 19451981 307 5037911 308 6474821 309 402761 311 2604731 312 10606361 314 2159231 315 6631901 316 9617981 318 2294051 321 597671 322 539501 323 12600821 325 1514321 326 Unknown 328 13680671 329 247991 330 7465481 332 5841461 333 705161 335 549161 336 20612231 337 Unknown 339 Unknown 340 14974571 342 2065571 343 9154961 344 Unknown 346 500231 347 Unknown 349 681251 350 1819841 351 5425751 353 Unknown 354 803441 356 Unknown 357 1246361 358 Unknown 360 7659011 361 7422221 363 5158031 364 452531 365 4144961 367 Unknown 368 8183891 370 9280541 371 2362271 372 Unknown 374 1126661 375 1780481 377 4174601 378 Unknown 379 Unknown 381 21633371 382 1659101 384 8658101 385 6715991 386 Unknown 388 Unknown 389 2050331 391 20598701 392 12027731 393 Unknown 395 11477231 396 4098461 396 5450111 398 12953651 399 3328901 400 Unknown
 2020-01-01, 17:27 #5 Bobby Jacobs     May 2018 3248 Posts The value of the gap of 4 should be 1022381.
2020-01-01, 18:14   #6
rudy235

Jun 2015
Vallejo, CA/.

3D016 Posts

Quote:
 Originally Posted by Bobby Jacobs The value of the gap of 4 should be 1022381.
You are correct. (1246241 is the second instance only.)

Can you find the first occurrence on 41? I have searched up to 5,001,000 to no avail.

 2020-01-01, 20:54 #7 rudy235     Jun 2015 Vallejo, CA/. 11110100002 Posts I have found the first occurrences of many of the missing gaps. Code: GAP after prime 41 181773281 55 72506501 113 51633731 160 78139751 167 46187831 172 45290801 174 23482001 181 33113651 I believe all gaps (except the obvious 2mod 7 and 5mod 7) are represented although a few that are 1 mod 7 and 6 mod 7 can sometimes be underrepresented.
2020-01-02, 15:02   #8
Thomas11

Feb 2003

22·32·53 Posts

Here is a C program based on my twingaps program (itself based on fastsieve by TOS). A Windows binary is included as well as a batch file for running it.

In order to save the results you should redirect the output into a text file, e.g.:
Code:
run.bat >> logfile.txt
See the batch file for the parameter settings.

BTW.: It runs pretty fast. Up to 1 billion (1e9) in just 1 second...

And a list of largest gaps found up to 1e9:
Code:
 10045 948653801
10074 785474531
10164 736611521
10409 540053531
10415 783970331
10515 511502861
10528 204335771
10780 898511981
11133 737220431
11141 987996431
11200 644979191
12417 643002791
13184 628246181
Attached Files

2020-01-02, 16:35   #9
Thomas11

Feb 2003

22·32·53 Posts

Here are the first occurrences of gaps between quadruplets searched up to 1e12.

Largest gap: 85025 342611795411

Smallest unknown gap: 21764

I stopped my search at 1e12, so you may continue from there...
Attached Files
 gaplist_1e12.zip (196.2 KB, 144 views)

2020-01-02, 21:25   #10
rudy235

Jun 2015
Vallejo, CA/.

24·61 Posts

Quote:
 Originally Posted by Thomas11 Here are the first occurrences of gaps between quadruplets searched up to 1e12. Largest gap: 85025 342611795411 Smallest unknown gap: 21764 Yes, it does coincide with what I believe are the most difficult gaps to find +1 or -1 mod7 I stopped my search at 1e12, so you may continue from there...
I'll leave it like that, thanks!

2020-01-03, 13:19   #11
robert44444uk

Jun 2003
Oxford, UK

22·3·7·23 Posts

Quote:
 Originally Posted by Thomas11 Here are the first occurrences of gaps between quadruplets searched up to 1e12. Largest gap: 85025 342611795411 Smallest unknown gap: 21764 I stopped my search at 1e12, so you may continue from there...
21764 1206449526011

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