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-   -   Twin prime gap bounds (https://www.mersenneforum.org/showthread.php?t=26278)

gnostique 2020-12-06 00:06

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[SUB]next[/SUB] - 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.

CRGreathouse 2020-12-06 06:54

I think you're looking for [url=https://oeis.org/A113275]A113275[/url] and [url=https://oeis.org/A113274]A113274[/url].

mart_r 2020-12-08 17:52

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...:smile:). 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 b-files 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.

gnostique 2020-12-09 14:46

@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.

gnostique 2020-12-11 12:44

twin prime distribution
 
Is it acceptable to post directly related research here for information / comments / feedback?

mart_r 2020-12-11 20:19

[QUOTE=gnostique;565936]Is it acceptable to post directly related research here for information / comments / feedback?[/QUOTE]
I guess so. It's what I do most of the time.
Here's the most recent data on twin prime gaps:
[URL]https://www.mersenneforum.org/showpost.php?p=563186&postcount=223[/URL]
And the search goes on...

gnostique 2020-12-13 15:06

Twin gap estimates
 
Some recent results on [URL="https://www.cambridge.org/engage/coe/article-details/5fce49501ad9a000182972f3"]estimates of twin prime gaps[/URL]. Appreciate comments and feedback.

CRGreathouse 2020-12-14 06:48

[QUOTE=mart_r;565680]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...:smile:). 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 b-files 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.[/QUOTE]

Yes, I am an OEIS Editor-in-Chief. If you need help submitting something, feel free to PM me.

mart_r 2020-12-17 21:09

[QUOTE=CRGreathouse;566132]Yes, I am an OEIS Editor-in-Chief. If you need help submitting something, feel free to PM me.[/QUOTE]
That doesn't quite answer all of my questions, but since sending files via PM isn't allowed (AFAICT) I'm going to make a regular submission in the next couple of days. Or weeks. When I'm in the right mood to do so.:smile:


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+ (?)[/CODE]Not yet sure about 47#+. But before I try to go any further and spending time that I currently don't have, I'm curious to know whether anyone has already investigated this.

CRGreathouse 2020-12-18 04:10

If you'd like me to look anything over, post the A-number 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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