### Maximal gaps between prime decuplets

© 2011 by Alexei Kourbatov, JavaScripter.net/math
Main article: Maximal gaps between prime k-tuples

Prime decuplets (10-tuples) are the densest permissible clusters of 10 consecutive primes. There are two types of prime decuplets:

• {p+0,2,6,8,12,18,20,26,30,32} (OEIS A027569, A202281, A202282)
• {p+0,2,6,12,14,20,24,26,30,32} (OEIS A027570, A202361 A202362).

The observed maximal gaps between prime decuplets near p are at most log p times the average gap.
The approximate size of a maximal gap that ends at p is given by the following empirical formula:

E(max g10(p))  =  a(log(p/a) − 0.2)  =  O(log11p)
where a = 0.00059 log10p is the average gap, as predicted by the Hardy-Littlewood k-tuple conjecture.

Maximal gaps between prime decuplets of each type are listed below.

#### Maximal gaps between prime decuplets {p+0,2,6,8,12,18,20,26,30,32} up to 6 × 1015

```   1st 10-tuple:    2nd 10-tuple:     Gap g10(p):
11      33081664151     33081664140
33081664151      83122625471     50040961320
83122625471     294920291201    211797665730
294920291201     573459229151    278538937950
730121110331    1044815397161    314694286830
1291458592421    1738278660731    446820068310
4700094892301    5289415841441    589320949140
6218504101541    7353767766461   1135263664920
7908189600581    9062538296081   1154348695500
10527733922591   11808683662661   1280949740070
21939572224301   23280376374371   1340804150070
23960929422161   25419097742651   1458168320490
30491978649941   32031885520751   1539906870810
46950720918371   48809302182911   1858581264540
84254447788781   86844628716731   2590180927950
118565337622001  121748202896051   3182865274050
124788318636251  129737394812561   4949076176310
235474768767851  241194271107521   5719502339670
513639397576421  520113801406931   6474403830510
826551814786541  836728470696731  10176655910190
1811660166341561 1822356561339311  10696394997750
3336445098184811 3351240113757071  14795015572260
4860936813525251 4883001500251991  22064686726740
```

#### Maximal gaps between prime decuplets {p+0,2,6,12,14,20,24,26,30,32} up to 6 × 1015

```   1st 10-tuple:    2nd 10-tuple:     Gap g10(p):
9853497737      21956291867     12102794130
22741837817     164444511587    141702673770
242360943257     666413245007    424052301750
1418575498577    2118274828907    699699330330
4396774576277    5111078123507    714303547230
8639103445097    9378647660507    739544215410
11105292314087   12728490626207   1623198312120
12728490626207   15420024060797   2691533434590
119057768524127  123265617079457   4207848555330
226608256438997  231544331258477   4936074819480
581653272077387  587540846737697   5887574660310
896217252921227  902779907026157   6562654104930
987041423819807  994246494727457   7205070907650
1408999953009347 1417129014533357   8129061524010
1419018243046487 1427380791698987   8362548652500
2189095026865907 2198836733614877   9741706748970
2274112300607657 2284079627820227   9967327212570
2369286971460107 2380971400045937  11684428585830
4498231210741967 4513269435966647  15038225224680
5320814086365287 5337031951168427  16217864803140
5406353435580257 5424929217415217  18575781834960
```
The ratio g10(p)/log11p is never greater than 0.00059, i.e. maximal gap sizes are less than log p times the average gap, where p is the prime at the end of the gap.