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 rudy235 2019-12-27 09:15

Here is a list of first instances For the Quadruplets (notice how different it is from the list of First Differences between Twin Primes)

[B]But first a review[/B]
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 [B]825[/B]

[OEIS]A159910[/OEIS]

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
[/code]

The largest gap I have found is a gap of [B]1897[/B] between the Quadruplet of barycenter 3,741,165 and that of 3,798,075

3'798,075-3,741,165= 56910 . 56,910/30 = [B]1897[/B]

 mart_r 2019-12-27 12:06

You might want to have a look at this paper, which lists record gaps between prime quadruplets up to 10^15:
[URL="https://arxiv.org/abs/1309.4053"]https://arxiv.org/abs/1309.4053[/URL]

 rudy235 2019-12-27 14:53

[QUOTE=mart_r;533621]You might want to have a look at this paper, which lists record gaps between prime quadruplets up to 10^15:
[URL="https://arxiv.org/abs/1309.4053"]https://arxiv.org/abs/1309.4053[/URL][/QUOTE]

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]
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
[/code]

 rudy235 2019-12-27 16:46

More First occurrences

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[/code]

 Bobby Jacobs 2020-01-01 17:27

The value of the gap of 4 should be 1022381.

 rudy235 2020-01-01 18:14

[QUOTE=Bobby Jacobs;533917]The value of the gap of 4 should be 1022381.[/QUOTE]

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. :bangheadonwall:

 rudy235 2020-01-01 20:54

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
[/code]

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.

 Thomas11 2020-01-02 15:02

1 Attachment(s)
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[/CODE]

See the batch file for the parameter settings.

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

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
[/CODE]

 Thomas11 2020-01-02 16:35

1 Attachment(s)
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...

 rudy235 2020-01-02 21:25

[QUOTE=Thomas11;534012]Here are the first occurrences of gaps between quadruplets searched up to 1e12.

Largest gap: 85025 342611795411

Smallest unknown gap: 21764 [COLOR="Red"]Yes, it does coincide with what I believe are the most difficult gaps to find +1 or -1 mod7[/COLOR]

I stopped my search at 1e12, so you may continue from there...[/QUOTE] [COLOR="red"]I'll leave it like that, thanks![/COLOR]

 robert44444uk 2020-01-03 13:19

[QUOTE=Thomas11;534012]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...[/QUOTE]

21764 1206449526011

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