In the last few days I dug my fangs into gaps between primes p

_{n} and p

_{n+k} (with k=1 these are the usual well-known prime gaps, for k=2 see A

144103, for k=3 see A

339943, for k=4 see A

339944).

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*10

^{12}:

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 CramÃ©r-Shanks-Granville ratio. Only a quick spreadsheet formula, this could probably use some fine tuning

^{1)}, 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(p

_{n+k})-Gram(p

_{n})-k+1 where Gram(x) is Gram's version of Riemann's pi(x) approximation, and CSG = M

^{2}/gap - pending negotiations...

Calculations will have reached p ~ 7*10

^{12} by tomorrow, and additionally for k=2 with p ~ 16*10

^{12}. Not terribly fast, I admit.

Does anybody know of any further work on this topic?