mersenneforum.org Gaps between non-consecutive primes
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 2021-11-04, 21:55 #1 mart_r     Dec 2008 you know...around... 2×73 Posts 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?
 2021-11-07, 13:34 #2 Bobby Jacobs     May 2018 3·79 Posts For each k, what are the first few gaps with record CSG ratio? This is very interesting.
 2021-11-09, 18:10 #3 mart_r     Dec 2008 you know...around... 12568 Posts 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.
 2021-11-14, 17:20 #4 Bobby Jacobs     May 2018 3×79 Posts 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.
2021-11-17, 13:51   #5
mart_r

Dec 2008
you know...around...

12568 Posts

Quote:
 Originally Posted by Bobby Jacobs 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...

 2021-11-23, 01:33 #6 Bobby Jacobs     May 2018 111011012 Posts 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!
 2021-11-29, 11:54 #7 mart_r     Dec 2008 you know...around... 2·73 Posts 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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