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Old 2021-11-04, 21:55   #1
mart_r
 
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Dec 2008
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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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