20201206, 00:06  #1 
Dec 2020
7 Posts 
Twin prime gap bounds
I'm working with prime and twin gap bounds and it appears that there may be an approach to establishing maximum bounds on the distance to the next pair (as per convention, a pair is denoted by the first member). It was possible to verify the results with the first 100k pairs but that's getting close to the limit of the capacity. We need to verify the maximum known relative gap between consecutive pairs i.e. (t_{next}  t) / t. According to the results, the ratio is bounded by a specific number so any data on large twin gaps or references to such would be much appreciated and I checked the related threads as well. I'll gladly share the details if the results are confirmed.

20201208, 17:52  #3 
Dec 2008
you know...around...
1010010001_{2} Posts 
I'm still pondering about updating the twin prime gap sequences (there are six of them that I know of, so while submitting it all, my mailbox will get flooded...). I just can't decide how to go about the credits. Could I just give a link to the twin prime gap search thread?
@ CRGreathouse: you're an editor of OEIS, right? What if I just post the bfiles here, would that be okay (and perhaps less work for everyone)? Knowing myself, I probably spend hours checking whether I filled the forms right if I submit it the regular way. 
20201209, 14:46  #4 
Dec 2020
7_{10} Posts 
@CRGreathouse, thank you this was helpful. From these results it appears that the relative gap tends to decrease with the prime order and this is also in a good agreement with our results.

20201211, 12:44  #5 
Dec 2020
7 Posts 
twin prime distribution
Is it acceptable to post directly related research here for information / comments / feedback?

20201211, 20:19  #6  
Dec 2008
you know...around...
1010010001_{2} Posts 
Quote:
Here's the most recent data on twin prime gaps: https://www.mersenneforum.org/showpo...&postcount=223 And the search goes on... 

20201213, 15:06  #7 
Dec 2020
7 Posts 
Twin gap estimates
Some recent results on estimates of twin prime gaps. Appreciate comments and feedback.

20201214, 06:48  #8  
Aug 2006
3×1,993 Posts 
Quote:


20201217, 21:09  #9  
Dec 2008
you know...around...
3^{2}×73 Posts 
Quote:
P.S.: Here's a tidbit for further study: Code:
Maximal gap between twin coprimes mod p# p# gap #inst. first instance after 3# 6 1 6 ± 1 5# 12 2 18 ± 1 7# 30 2 72 ± 1 11# 42 4 900 ± 1 13# 66 12 732 ± 1 17# 108 20 702 ± 1 19# 150 20 660 ± 1 23# 204 4 76166568 ± 1 29# 258 2 1205437110 ± 1 31# 348 4 8813641452 ± 1 37# 528 2 544899485412 ± 1 41# 540 16 14416696724700 ± 1 43# 618 8 830330079152052 ± 1 47# 702 (?) 53# 816 (?) 59# 912+ (?) 

20201218, 04:10  #10 
Aug 2006
3·1,993 Posts 
If you'd like me to look anything over, post the Anumber here or PM me once it's up. (If you're happy with how it looks, submit for review at the same time; if not, leave it in /draft while you have me look.)

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