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Old 2019-12-27, 09:15   #1
rudy235
 
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Default 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
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Old 2019-12-27, 12:06   #2
mart_r
 
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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
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Old 2019-12-27, 14:53   #3
rudy235
 
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Quote:
Originally Posted by mart_r View Post
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
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Old 2019-12-27, 16:46   #4
rudy235
 
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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
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Old 2020-01-01, 17:27   #5
Bobby Jacobs
 
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The value of the gap of 4 should be 1022381.
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Old 2020-01-01, 18:14   #6
rudy235
 
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Quote:
Originally Posted by Bobby Jacobs View Post
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.
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Old 2020-01-01, 20:54   #7
rudy235
 
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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.
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Old 2020-01-02, 15:02   #8
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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
File Type: zip quadgaps.zip (1.34 MB, 86 views)
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Old 2020-01-02, 16:35   #9
Thomas11
 
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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
File Type: zip gaplist_1e12.zip (196.2 KB, 93 views)
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Old 2020-01-02, 21:25   #10
rudy235
 
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Quote:
Originally Posted by Thomas11 View Post
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!
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Old 2020-01-03, 13:19   #11
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Quote:
Originally Posted by Thomas11 View Post
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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