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-   -   Gaps between non-consecutive primes (https://www.mersenneforum.org/showthread.php?t=27301)

 mart_r 2021-12-31 11:18

New Year's Eve consolidation

1 Attachment(s)
Some data for maximal gaps in the file in close proximity. [SIZE=1][COLOR=LemonChiffon]In the next update, I'll include the data for p=2, I promise.[/COLOR][/SIZE]

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 by-product, a puzzle:
Given x, find the next three consecutive primes >= x. Denote the two gaps between them g[SUB]1[/SUB] and g[SUB]2[/SUB], and let g[SUB]1[/SUB] >= g[SUB]2[/SUB]. Let r = g[SUB]1[/SUB]/g[SUB]2[/SUB].
As x becomes larger, the geometric mean r[SUB]gm[/SUB] of values of r also become larger. Find an asymptotic function f(x) ~ r[SUB]gm[/SUB].

 mart_r 2022-01-03 20:42

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 in-between, 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.

 Bobby Jacobs 2022-01-09 14:14

[QUOTE=mart_r;595952]
[QUOTE=robert44444uk;595925]Cocky![/QUOTE]
What? What I wrote did look a little conceited to me:smile:
[/QUOTE]

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?

 mart_r 2022-01-09 20:37

[QUOTE=Bobby Jacobs;597473]What is going on?[/QUOTE]
Building mountains out of molehills, I guess :smile:
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?

 robert44444uk 2022-01-10 10:05

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

 robert44444uk 2022-01-10 10:22

[QUOTE=mart_r;592465]

Here's a tidbit of data of especially large gaps for k=1..19 and p<6*10[SUP]12[/SUP]:

[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[/CODE].....

Does anybody know of any further work on this topic?[/QUOTE]

I confirm Marts values for 17,18,19 as the largest average merits between 18,19 and 20 primes respectively,

It is worth looking at the minimum value found to date for these, as no-one has found the relevant all prime k-tuple 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

[/CODE]

 robert44444uk 2022-01-10 10:45

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]

And large, tested up to the same values, so it looks like the 24 and 25 records may go - no doubt somewhere in mart_r's file:

[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

[/CODE]

 robert44444uk 2022-01-10 10:53

[QUOTE=mart_r;597502]Building mountains out of molehills, I guess :smile:
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?[/QUOTE]

I already did a bit of work at k=1000 but I might concentrate at k=200 and 500 and see where that goes

 robert44444uk 2022-01-10 17:43

[QUOTE=mart_r;595755]

And this related result, 100 primes in the range p+[1..8349] while there are no primes in q+[1..8349], with q < p, still appears to be unmatched:
[URL]https://www.mersenneforum.org/showpost.php?p=479832&postcount=86[/URL]

[/QUOTE]

This is much harder than I anticipated - it really is an outstanding result. I have started to look at the next obvious candidate starting from 3483347771*409#/30 - 7016 (merit >39). I have only achieved 67 primes so far (after about 30 minutes of checking), so I am wondering if this can ever get to 100 primes

 mart_r 2022-01-10 21:40

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 k-tuplets. But some clusters are missing, see also post # 12. However, more data is always welcome!

[QUOTE=robert44444uk;597586]This is much harder than I anticipated - it really is an outstanding result. I have started to look at the next obvious candidate starting from 3483347771*409#/30 - 7016 (merit >39). I have only achieved 67 primes so far (after about 30 minutes of checking), so I am wondering if this can ever get to 100 primes[/QUOTE]

Though the difference seems little (merit 39.62 vs. 41.94), it's several times as hard to fill the gap with 100 primes larger than those surrounding the gap. I'd have to check the stats, but an admissible 1886-tuplet pattern (minimum width 15899) with no factors < 400-ish would be a good start for the search.

 robert44444uk 2022-01-11 10:04

[QUOTE=mart_r;597600]

Though the difference seems little (merit 39.62 vs. 41.94), it's several times as hard to fill the gap with 100 primes larger than those surrounding the gap. I'd have to check the stats, but an admissible 1886-tuplet pattern (minimum width 15899) with no factors < 400-ish would be a good start for the search.[/QUOTE]

I'm trying to understand the approach.

I've found a 1886-tuplet pattern width 15898 from the internet,[URL="https://math.mit.edu/~primegaps/tuples/admissible_1886_15898.txt"]https://math.mit.edu/~primegaps/tuples/admissible_1886_15898.txt[/URL] 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 1000-tuplet 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

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