Twin Prime Constellations

[danaj]

Originally Posted by henryzz

I was correct about this taking a while. I have searched 11#*x + 1451 with x upto 1e14 in about 3 hours with no results(best was upto 48). Started a run upto 1e15.
I think I am being a victim here of the numbers getting larger and less likely to be prime as I search deeper.

$ 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

There are no valid residues...

[henryzz]

Originally Posted by danaj

$ 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

There are no valid residues...

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

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]

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]

Originally Posted by mart_r

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?

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]

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]

Originally Posted by henryzz

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

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]

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

I found all dense twin patterns up to 26 twins today. Here's a small overview, just in case someone is interested:

#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  ?)

[robert44444uk]

Originally Posted by mart_r

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

I found all dense twin patterns up to 26 twins today. Here's a small overview, just in case someone is interested:

#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  ?)

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]

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]

Originally Posted by mart_r

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

I found all dense twin patterns up to 26 twins today. Here's a small overview, just in case someone is interested:

#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  ?)

What do the patterns look like?

[mart_r]

Originally Posted by Bobby Jacobs

What do the patterns look like?

All right, don't panic
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.