View Single Post
Old 2019-04-30, 09:30   #1
robert44444uk
 
robert44444uk's Avatar
 
Jun 2003
Oxford, UK

3×54 Posts
Default 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 k-tuple permissable patterns and prime constellations that fit these patterns.

To advance this idea requires us to define sub-groups of k-tuple 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 k-tuple positions up to perhaps 10-length. To do so we need to generate the k-tuples with leading numbers of twin prime positions, but I don't know where to source the list of all possible efficient k-tuples. The leading work of Engelsma does not actually list the k-tuples see http://www.opertech.com/primes/k-tuples.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 2019-04-30 at 10:50
robert44444uk is offline   Reply With Quote