• It is known that prime twins, quadruplets, sextuplets and octuplets may be symmetrical: 6k + {1} 15k + {2; 4} 15k + {2; 4; 8} 30k + {1; 7; 11; 13},
Message 1 of 4 , May 31, 2010
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It is known that prime twins, quadruplets, sextuplets and octuplets may be symmetrical:
6k + {1}
15k + {2; 4}
15k + {2; 4; 8}
30k + {1; 7; 11; 13}, http://www.research.att.com/~njas/sequences/A145315

It seems that there are no more symmetrical patterns for the densest k-tuplets:
http://www.opertech.com/primes/k-tuples.html

But there are some symmetrical admissible patterns which are quite dense, but not the densest, e.g.
210k + {1; 11; 13; 17; 19}, http://www.research.att.com/~njas/sequences/A178565

Some of them are noted at the bottom of that page:
http://www.opertech.com/primes/modexample.html

The prime decuplet mentioned above consists of two prime quadruplets surrounding a twin prime pair; it has a width of 38 while the densest decuplets have 32. This cluster had already raised people's interest before:
http://www.trnicely.net/dense/dense1.html

But unfortunately there's no information about symmetrical dense prime clusters of more than 12 terms. So I propose a symmetrical cluster of 18 primes:

30030k + {1; 17; 19; 23; 29; 31; 37; 41; 43}

It consists of two prime octuplets surrounding the central twin prime pair, both octuplets being also symmetrical. The cluster is quite dense (its width is 86, while the densest ones have 70). Obviously, k should be always + 8 (mod 17) and + 7 (mod 19).

I just tried to find with APSieve some k's which would give all 18 terms prime but failed. That's not surprising since these clusters seem to be quite rare. But it would be an interesting to challenge to find some of them, as it was some years ago for 210k + {1; 11; 13; 17; 19}.

Best regards,

Andrey

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• All the plus signs in my previous message should be considered as +/- symbols (they actually was, but yahoo! stole the underlines).
Message 2 of 4 , May 31, 2010
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All the plus signs in my previous message should be considered as +/- symbols (they actually was, but yahoo! stole the underlines).
• ... In http://www.math.ethz.ch/~waldvoge/Projects/clprimes03.pdf Jörg Waldvogel and Peter Leikauf studied the pattern of 16 primes n +/- {17; 19; 23; 29; 31;
Message 3 of 4 , May 31, 2010
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Andrey Kulsha wrote:
> I propose a symmetrical cluster of 18 primes:
>
> 30030k + {1; 17; 19; 23; 29; 31; 37; 41; 43}

In http://www.math.ethz.ch/~waldvoge/Projects/clprimes03.pdf
Jörg Waldvogel and Peter Leikauf studied the pattern of 16 primes
n +/- {17; 19; 23; 29; 31; 37; 41; 43}

An exhaustive search to 5*10^22 found 94 occurrences with n-43
listed on page 14. There are 6 cases where either n-1 or n+1 is
also prime but none where they both are.
The first case is n-43 = 3741636047391669917447 where n-1 is prime.
n-47 also happens to be prime, but not n+1.

--
Jens Kruse Andersen
• You may want to take a look at: youtube.com/watch?v=ZbC4k5o6kzc Using that method until you get seeds that meet your 30030k + {1; 17; 19; 23; 29; 31; 37;
Message 4 of 4 , Jun 1, 2010
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You may want to take a look at: youtube.com/watch?v=ZbC4k5o6kzc

Using that method until you get "seeds" that meet your "30030k + {1; 17; 19; 23; 29; 31; 37; 41; 43}" criteria would then let you jump directly to all relative primes that have the same pattern pattern.
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