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-   -   Twin Prime Constellations (https://www.mersenneforum.org/showthread.php?t=24363)

 danaj 2019-05-01 16:31

I think I am being a victim here of the numbers getting larger and less likely to be prime as I search deeper.[/QUOTE]

[code]\$ perl -Mntheory=:all -E 'prime_set_config(verbose=>2); say for sieve_prime_cluster(0,1e14,2,6,8,18,20,30,32,36,38,48,50,60,62,78,80);'
cluster sieve found 1 residues mod 30
cluster sieve found 1 residues mod 210
cluster sieve using 0 residues mod 2310[/code]

There are no valid residues...

 henryzz 2019-05-01 22:08

[QUOTE=danaj;515413][code]\$ perl -Mntheory=:all -E 'prime_set_config(verbose=>2); say for sieve_prime_cluster(0,1e14,2,6,8,18,20,30,32,36,38,48,50,60,62,78,80);'
cluster sieve found 1 residues mod 30
cluster sieve found 1 residues mod 210
cluster sieve using 0 residues mod 2310[/code]

There are no valid residues...[/QUOTE]

And that's what I get for blindly continuing to search 2310*x+1451 without testing for factors <=11. :redface:

The best I can now find checking more carefully is: -24 -22 0 2 6 8 18 20 30 32 36 38 48 50 60 62

 mart_r 2019-05-02 17:04

During my lunch break today, and just for fun, I calculated some of the different possibilities for dense twin prime patterns:

2 Twins:
p+{0,2,6,8}

3 Twins:
p+{0,2,6,8,18,20}
p+{0,2,12,14,18,20}

4 Twins:
p+{0,2,12,14,24,26,30,32}
p+{0,2,12,14,18,20,30,32}
p+{0,2,6,8,18,20,30,32}

5 Twins:
p+{0,2,6,8,18,20,30,32,36,38}

6 Twins:
p+{0,2,12,14,18,20,30,32,42,44,48,50}
p+{0,2,6,8,18,20,30,32,36,38,48,50}

7 Twins:
p+{0,2,12,14,24,26,30,32,42,44,54,56,60,62}
p+{0,2,6,8,18,20,30,32,36,38,48,50,60,62}

8 Twins:
p+{0,2,12,14,24,26,30,32,42,44,54,56,60,62,84,86}
p+{0,2,12,14,24,26,30,32,42,44,54,56,72,74,84,86}
p+{0,2,24,26,30,32,42,44,54,56,60,62,72,74,84,86}
p+{0,2,12,14,30,32,42,44,54,56,60,62,72,74,84,86}

9 Twins:
p+{0,2,12,14,24,26,30,32,42,44,54,56,60,62,84,86,102,104}
p+{0,2,12,14,18,20,42,44,48,50,60,62,72,74,90,92,102,104}
p+{0,2,18,20,42,44,48,50,60,62,72,74,78,80,90,92,102,104}
p+{0,2,12,14,30,32,42,44,54,56,60,62,84,86,90,92,102,104}

10 Twins:
p+{0,2,12,14,30,32,42,44,54,56,60,62,84,86,90,92,102,104,132,134}
p+{0,2,18,20,30,32,42,44,48,50,72,74,78,80,90,92,120,122,132,134}
p+{0,2,12,14,30,32,42,44,54,56,72,74,84,86,90,92,114,116,132,134}
p+{0,2,12,14,30,32,42,44,54,56,72,74,90,92,102,104,114,116,132,134}
p+{0,2,18,20,30,32,42,44,60,62,72,74,78,80,102,104,120,122,132,134}
p+{0,2,12,14,30,32,54,56,60,62,72,74,90,92,102,104,114,116,132,134}
p+{0,2,18,20,30,32,42,44,60,62,78,80,90,92,102,104,120,122,132,134}
p+{0,2,18,20,42,44,48,50,60,62,78,80,90,92,102,104,120,122,132,134}
p+{0,2,12,14,42,44,54,56,60,62,84,86,90,92,102,104,114,116,132,134}
p+{0,2,30,32,42,44,48,50,72,74,78,80,90,92,102,104,120,122,132,134}

11 Twins:
p+{0,2,12,14,30,32,42,44,54,56,60,62,84,86,90,92,102,104,132,134,144,146}
p+{0,2,12,14,30,32,42,44,54,56,72,74,84,86,90,92,114,116,132,134,144,146}
p+{0,2,12,14,30,32,42,44,54,56,72,74,90,92,102,104,114,116,132,134,144,146} (note the symmetric pattern here)
p+{0,2,12,14,30,32,54,56,60,62,72,74,90,92,102,104,114,116,132,134,144,146}
p+{0,2,12,14,42,44,54,56,60,62,84,86,90,92,102,104,114,116,132,134,144,146}

Does anyone need more patterns?

 robert44444uk 2019-05-03 08:46

[QUOTE=mart_r;515514]During my lunch break today, and just for fun, I calculated some of the different possibilities for dense twin prime patterns:

2 Twins:
p+{0,2,6,8}

Does anyone need more patterns?[/QUOTE]

Well Done mart_r, Extremely useful. I'm not sure we need more than this given we are already needing to locate 22 primes - that is probably at the limit of what can be discovered right now.

 henryzz 2019-05-03 10:12

I have failed so far to find 8 twins. I have searched mart_r's third type of 8 twins upto 2310*1e16.

I will attempt the others upto the same bound and then tap out.

 danaj 2019-05-03 11:28

[QUOTE=henryzz;515612]I have failed so far to find 8 twins. I have searched mart_r's third type of 8 twins upto 2310*1e16. [/quote]I ran to 10^20 last night without finding any. I just spent a few minutes and tested the other three types to 10^17 without finding any.

I did a simple look at the 3rd type of 11 twins to 10^19 without finding any.

 mart_r 2019-05-03 19:13

And here I was hoping to be fast enough to post the list of patterns before someone else does :smile:

I found all dense twin patterns up to 26 twins today. Here's a small overview, just in case someone is interested:
[CODE]#tw width #patterns
2 8 1
3 20 2
4 32 3
5 38 1
6 50 2
7 62 2
8 86 4
9 104 4
10 134 10
11 146 5
12 170 4
13 182 2
14 212 18
15 224 2
16 254 34
17 266 2
18 290 8
19 302 6
20 338 8
21 350 2
22 380 12
23 392 2
24 410 4
25 422 2
26 440 2
(27 >=470 ?)
[/CODE]

 robert44444uk 2019-05-04 06:34

[QUOTE=mart_r;515666]And here I was hoping to be fast enough to post the list of patterns before someone else does :smile:

I found all dense twin patterns up to 26 twins today. Here's a small overview, just in case someone is interested:
[CODE]#tw width #patterns
2 8 1
3 20 2
4 32 3
5 38 1
6 50 2
7 62 2
8 86 4
9 104 4
10 134 10
11 146 5
12 170 4
13 182 2
14 212 18
15 224 2
16 254 34
17 266 2
18 290 8
19 302 6
20 338 8
21 350 2
22 380 12
23 392 2
24 410 4
25 422 2
26 440 2
(27 >=470 ?)
[/CODE][/QUOTE]

This is good work! Even if we haven't the computer resources to explore these upper reaches it is worth posting the detail for posterity. I found it annoying that the detail on longer prime constellations was not readily available on the web.

 henryzz 2019-05-08 19:40

1667+4108566605770697*(4290)+{0,2,12,14,24,26,30,32,42,44,54,56,72,74,84,86} are all prime.

The 1st and 3rd types for 8 twins are searched upto 2310e16
The 2nd and 4th types for 8 twins are searched upto 4290e16.

 Bobby Jacobs 2019-06-01 22:30

[QUOTE=mart_r;515666]And here I was hoping to be fast enough to post the list of patterns before someone else does :smile:

I found all dense twin patterns up to 26 twins today. Here's a small overview, just in case someone is interested:
[CODE]#tw width #patterns
2 8 1
3 20 2
4 32 3
5 38 1
6 50 2
7 62 2
8 86 4
9 104 4
10 134 10
11 146 5
12 170 4
13 182 2
14 212 18
15 224 2
16 254 34
17 266 2
18 290 8
19 302 6
20 338 8
21 350 2
22 380 12
23 392 2
24 410 4
25 422 2
26 440 2
(27 >=470 ?)
[/CODE][/QUOTE]

What do the patterns look like?

 mart_r 2019-06-09 12:51

1 Attachment(s)
[QUOTE=Bobby Jacobs;518298]What do the patterns look like?[/QUOTE]

All right, don't panic :smile:
I've attached a file with patterns for up to 31 twins. Had to re-write my program and I'm actually not 100% sure if it works correctly, for example I found 82 (!) different dense patterns for 29 twins.

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