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Old 2021-11-04, 21:55   #1
mart_r
 
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Default Gaps between non-consecutive primes

In the last few days I dug my fangs into gaps between primes pn and pn+k (with k=1 these are the usual well-known prime gaps, for k=2 see A144103, for k=3 see A339943, for k=4 see A339944).

This can be seen as part of the effort to further improve the amount of empirical data related to prime gaps.
Recently I found the paper https://arxiv.org/abs/2011.14210 (Abhimanyu Kumar, Anuraag Saxena: Insulated primes), which makes some predictions regarding k=2, but is based on quite limited empirical study.

Here's a tidbit of data of especially large gaps for k=1..19 and p<6*1012:

Code:
 k  CSG_max *            p_n          p_n+k
 1  0.7975364  2614941710599  2614941711251
 2  0.8304000  5061226833427  5061226834187
 3  0.8585345  5396566668539  5396566669381
 4  0.8729716     4974522893     4974523453 (largest CSG_max thus far)
 5  0.8486459   137753857961   137753858707
 6  0.8358987  5550170010173  5550170011159
 7  0.8396098  3766107590057  3766107591083
 8  0.8663070    11878096933    11878097723
 9  0.8521843  1745499026867  1745499027983
10  0.8589305  5995661470529  5995661471797
11  0.8467931  5995661470481  5995661471797
12  0.8347906  5995661470529  5995661471893
13  0.8439277  5995661470529  5995661471977
14  0.8312816  5995661470481  5995661471977
15  0.7987377  5995661470471  5995661471977
16  0.7901341  5568288566663  5568288568217
17  0.7632862   396016668869   396016670261
18  0.7476038   396016668833   396016670261
19  0.7560424   968269822189   968269823761
* A version of the Cramér-Shanks-Granville ratio. Only a quick spreadsheet formula, this could probably use some fine tuning1), but for the time being, in this table
\(CSG = \Large \frac{gap}{(\log \frac{p_n+p_{n+k}}{2} +k-1)^2}\)

1) I'd prefer something like M (the "merit") = Gram(pn+k)-Gram(pn)-k+1 where Gram(x) is Gram's version of Riemann's pi(x) approximation, and CSG = M2/gap - pending negotiations...

Calculations will have reached p ~ 7*1012 by tomorrow, and additionally for k=2 with p ~ 16*1012. Not terribly fast, I admit.


Does anybody know of any further work on this topic?
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Old 2021-11-07, 13:34   #2
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For each k, what are the first few gaps with record CSG ratio? This is very interesting.
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Old 2021-11-09, 18:10   #3
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Greetings Bobby,

I'd like to run these numbers through Pari again before posting more inconsistent/approximate numbers. The formula with the term "-k+1" (see 1) from previous post) is only working properly when CSG = max(0,M)2/gap since M can be negative (because of the aforementioned term). Working out details like these takes me inordinately long...

Good news is, for p = 8,281,634,108,677 and k = 19, I get a CSG > 1 with the rough-and-ready version of the fine-tuned formula: gap = 1812, M = gap/log(p+gap/2)-18 ~ 42.918 (there are 60.918 primes on average in a range of 1812 integers, i.e. 42.918 more than the 18 that are actually between the bounding primes), and CSG = M2/gap ~ 1.0165. With p that large, there won't be much of a difference anymore when using Gram(x) in the calculation of CSG.
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Old 2021-11-14, 17:20   #4
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When n=3, a big gap seems like 35617, 35671, 35677, 35729. There is a gap of 54 between 35617 and 35671, which is big for numbers of that size. After the gap of 6 between 35671 and 35677, there is another big gap of 52 between 35677 and 35729. Therefore, the 3-gap between 35617 and 35729 is a surprisingly large prime gap.
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Old 2021-11-17, 13:51   #5
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Quote:
Originally Posted by Bobby Jacobs View Post
When n=3, a big gap seems like 35617, 35671, 35677, 35729.
It doesn't only seem like a big gap, it's listed in A339943 as a(56), since 56=(35729-35617)/2.
I have a lot of data ready for submission, it just takes me longer to actually submit it, my schedule is pretty clogged at the moment...
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Old 2021-11-23, 01:33   #6
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I looked at A115401, the record gaps between primes 3 apart, and it turns out that the gap of 112 between 35617 and 35729 is very big. The sequence starts out smoothly. After the initial 5, every even number from 8 to 36 is in the sequence. There are not many even numbers missing up to 68. Then, it jumps to 78, 84, and a really big leap to 112. That corresponds to the 35617, 35729 gap. It is an enormous prime gap!
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Old 2021-11-29, 11:54   #7
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I have uploaded some data for posterity, differences d between primes pn and pn+k for k <= 130 and d <= 740, see 2nd link for A086153.
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