Gaps between non-consecutive primes

[mart_r]

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 by-product, a puzzle:
Given x, find the next three consecutive primes >= x. Denote the two gaps between them g₁ and g₂, and let g₁ >= g₂. Let r = g₁/g₂.
As x becomes larger, the geometric mean r_gm of values of r also become larger. Find an asymptotic function f(x) ~ r_gm.

[mart_r]

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]

Originally Posted by mart_r

What? What I wrote did look a little conceited to me

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]

Originally Posted by Bobby Jacobs

What is going on?

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?

[robert44444uk]

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]

Originally Posted by mart_r

Here's a tidbit of data of especially large gaps for k=1..19 and p<6*10¹²:

 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

.....


Does anybody know of any further work on this topic?

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.

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

[robert44444uk]

Here are some results for 20..25

Small average merits and gaps:

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

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:

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

[robert44444uk]

Originally Posted by mart_r

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?

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]

Originally Posted by mart_r

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:
https://www.mersenneforum.org/showpo...2&postcount=86

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]

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

Originally Posted by robert44444uk

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

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]

Originally Posted by mart_r

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.

I'm trying to understand the approach.

I've found a 1886-tuplet 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 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