Jun 2003
Oxford, UK
3·5^{4} Posts

Twin Prime Constellations
rudy235 has pm'd me on the topic of twin prime constellations and it got me thinking that a lot of work has been carried out on ktuple permissable patterns and prime constellations that fit these patterns.
To advance this idea requires us to define subgroups of ktuple permissible patterns that contain a preponderance of twins.
For example from Tony Forbes paper at https://sites.google.com/site/anthon...attredirects=0
we can see a pattern of 6 positions twins in a width of 81 integers ( I don't know if this is the most dense)
Quote:
k=20 s=80 B={0 2 8 12 14 18 24 30 32 38 42 44 50 54 60 68 72 74 78 80}
1 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109
The only other known example of this pattern is 3941119827895253385301920029 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (28 digits, 24 June 2014, Raanan Chermoni & Jaroslaw Wroblewski)
k=20 s=80 B={0 2 6 8 12 20 26 30 36 38 42 48 50 56 62 66 68 72 78 80}
The smallest known example of this pattern is 14374153072440029138813893241 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (29 digits, October 6, 2014, Raanan Chermoni & Jaroslaw Wroblewski)

rudy235 points out there are Hargrave Primes, where there are 5 twin positions in a width of 39, presumably 0,2, 6,8,18,20,30,32,36,38
I seems to me that we can easily find first instance twin prime constellations that fit known ktuple positions up to perhaps 10length. To do so we need to generate the ktuples with leading numbers of twin prime positions, but I don't know where to source the list of all possible efficient ktuples. The leading work of Engelsma does not actually list the ktuples see http://www.opertech.com/primes/ktuples.html
Then maybe this work has already been carried out, in which case my Google skills aren't what they used to be.
Last fiddled with by robert44444uk on 20190430 at 10:50
