20211231, 11:18  #12 
Dec 2008
you know...around...
1354_{8} Posts 
New Year's Eve consolidation
Some data for maximal gaps in the file in close proximity. In the next update, I'll include the data for p=2, I promise.
There are three primes (well, actually, 54 primes:) that await discovery, for k=16 / d=76 k=17 / d=82 k=18 / d=84 And possibly feasible for J. Wroblewski and R. Chermoni: k=19 / d=86 and d=88 k=20 / d=90 and d=92 As a byproduct, a puzzle: Given x, find the next three consecutive primes >= x. Denote the two gaps between them g_{1} and g_{2}, and let g_{1} >= g_{2}. Let r = g_{1}/g_{2}. As x becomes larger, the geometric mean r_{gm} of values of r also become larger. Find an asymptotic function f(x) ~ r_{gm}. 
20220103, 20:42  #13 
Dec 2008
you know...around...
2EC_{16} Posts 
Staking claims
A measurably unusual scarcity of primes appears between 6,215,409,275,042 and 6,215,409,279,556  there are only 83 primes inbetween, just a little over half as many as expected on average, and the associate CSG value is 1.0944363.
The year starts off pretty well. 
20220109, 14:14  #14 
May 2018
103_{16} Posts 
This conversation does not make sense to me. It first seems like Robert is agreeing with you that you are being cocky. However, your response acts like he is not agreeing. Then, in the next post, he says that you are not cocky. What is going on?

20220109, 20:37  #15 
Dec 2008
you know...around...
2^{2}×11×17 Posts 
Building mountains out of molehills, I guess
I thought Robert was making fun of me ... one word responses can be confusing, maybe it was a misunderstanding on my part. Those language barriers... Any suggestions on whether I should rather continue to search larger primes for k<=109, or to look at larger values of k? 
20220110, 10:05  #16 
Jun 2003
Oxford, UK
2,039 Posts 
Hi all
I did a bit of searching around the average merit of 100 gaps in the range of 101 primes, and my best performance is (I've checked up to 9.675e11): Gap=4354 Average merit=1.622541804, from prime 450867605017 to 450867609371 My method takes the average gap to be [g/ln(p1)+g/ln(p101)]/(2*100) where g is, in this case 4354 At the other end of the spectrum, the following range: Gap=1554 Average Merit=0.584366417 from prime 354120798439 to 354120799993 
20220110, 10:22  #17  
Jun 2003
Oxford, UK
2,039 Posts 
Quote:
It is worth looking at the minimum value found to date for these, as noone has found the relevant all prime ktuple at these sizes. Where 2 are listed, it shows the smallest gap and the smallest average merit in the gap. Code:
n gap p(n) p(n+k) ave merit 17 98 341078531681 341078531779 0.21708 18 114 1054694671669 1054694671669 0.22877 18 110 43440699011 43440699121 0.24948 19 126 1085806111031 1085806111157 0.23929 19 120 31311431897 31311432017 0.26134 Last fiddled with by robert44444uk on 20220110 at 10:46 

20220110, 10:45  #18 
Jun 2003
Oxford, UK
7F7_{16} Posts 
Here are some results for 20..25
Small average merits and gaps: Code:
n gap p(n) p(n+k) ave merit checked to 20 138 2037404713403 2037404713541 0.243448948 2.80E+12 20 136 1085806111021 1085806111157 0.245369164 21 144 2037404713397 2037404713541 0.241936843 2.81E+12 22 160 2037404713381 2037404713541 0.256599682 2.81E+12 22 156 325117822691 325117822847 0.267506235 23 174 2766595321597 2766595321771 0.264069002 2.81E+12 24 180 220654442209 220654442389 0.287137792 1.24E+12 25 190 220654442209 220654442399 0.290966296 8.22E+11 Code:
n gap p(n) p(n+k) ave merit 20 1582 968269822189 968269823771 2.866069068 21 1630 968269822189 968269823819 2.812408994 22 1680 968269822189 968269823869 2.766921063 23 1756 2137515911737 2137515913493 2.689187618 24 1740 752315299717 752315301457 2.651169565 25 1780 628177622389 628177624169 2.62091465 Last fiddled with by robert44444uk on 20220110 at 11:11 
20220110, 10:53  #19  
Jun 2003
Oxford, UK
3767_{8} Posts 
Quote:


20220110, 17:43  #20  
Jun 2003
Oxford, UK
2,039 Posts 
Quote:


20220110, 21:40  #21  
Dec 2008
you know...around...
2^{2}·11·17 Posts 
Thanks for your support!
Your results for maximum average merits are in accordance with my results in post # 12. I didn't look for minimum average merits as they are theoretically covered by the minimum widths of ktuplets. But some clusters are missing, see also post # 12. However, more data is always welcome! Quote:


20220111, 10:04  #22  
Jun 2003
Oxford, UK
2,039 Posts 
Quote:
I've found a 1886tuplet pattern width 15898 from the internet,https://math.mit.edu/~primegaps/tupl...1886_15898.txt so is the idea to get a Chinese Remainder (C) based on mods of primes <400, referenced the start prime of the large gap (P), and then to prp from P+n*C to P+n*C+15900, n integer? Or is there further sieving to do? Are the Chinese mods gotten by a greedy algorithm? Is such a large Chinese potentially inferior to a much smaller Chinese (c) based around say 1000tuplet where, if the prime count was high after testing, then it could be tested over the whole range. I'm thinking this trades off the greater chance of primes with ranges close to P, i.e. at P+c*n against the low chance at P+C*n Last fiddled with by robert44444uk on 20220111 at 10:12 

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