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Old 2020-07-15, 22:11   #23
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No major news in the last few weeks, but in the attachment there's the data for CSGmax up to p=2n for n=[16..41] and q<=2690. Data for p=242 might follow next week. Minor differences in conduct compared to the data for the older analysis on 2020-05-01: here the endpoints p' of the gaps are key for clustering in powers of two, and CSG is determined using the Ri-sum-of-derivatives formula as first considered in post # 2 - has anyone got a catchy name for it, perchance?

Ri(x)=1+\sum_{n=1}^\infty \frac{\log^nx}{n\hspace{1}n!\hspace{1}\zeta(n+1)}+\frac{\arctan\frac{\pi}{\log x}}{\pi}-\frac{1}{\log x}

CSG=\frac{q}{k\hspace{1}\varphi(q)}[\sum_{i=1}^k Ri'(p+iq)]^2
(where k=(p'-p)/q - in other words, there are k-1 consecutive composites in the arithmetic progression p+i*q)

I get some more or less meaningful numbers out of the data, but nothing that would justify a new conjecture or other breakthrough. Or is it just me being cautious? Maybe someone else has a better idea how to commercialize the data.

Right now I'm unable to decide whether or not to believe that CSG has a global maximum, I mean, as (p,q) \rightarrow \infty. (Of course maths is not about beliefs, but a good deal of conjectures were or are based merely on beliefs, also known as SWAGs...)

With already more than 500 extraordinarily large gaps, there's still no second example where q and r are both prime, but I'm close, and confident that I'll find at least one more of those before the end of July. There should be on average one in about 150 in the range I'm searching.

There are two values of q for which two extraordinarily large gaps are known:
q=28388, r=5859 and r=11949 (p=5088100651 and p=366870073)
q=389104, r=88289 and r=258931 (p=461954737 and p=2176128499)
Also, and quite unusually, there's one value of r with two values of q:
r=197077, q=223808 and q=502458 (p=2317057493 and p=267504733)
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Old 2020-07-19, 09:48   #24
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Umm... are my posts even visible to anybody other than Bobby?
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Old 2020-07-19, 19:55   #25
6809 > 6502
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Yes, they are.
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Old 2020-08-05, 21:11   #26
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From day to day I anticipated another result where q and r would both be prime, but to no avail. These are some of the most frustratingly near misses:

   q      phi     k       p           CSG     k*q/phi/log²p'
 792886  396442  204   445124227  1.012127792  0.997549475   (q/2 & p mod q/2 prime, simple CSG criteria not met)
 994102  497050  255  2001318337  1.10553142   1.09958173    (q/2 prime, p mod q/2 = 191011 = 251 × 761)
1933454  966726  219   686778133  1.031408014  1.009688338   (q/2 prime, p mod q/2 = 401963 = 541 × 743)
1889254  942480  228   954854339  1.048783531  1.031510191   (p mod q/2 prime, q/2 = 944627 = 617 × 1531)
 780814  390406  204   201321937  1.079546597  1.050952883   (p mod q prime, p mod q/2 not (1 of 4 examples))
Over one thousand candidates with CSG>1 (sum Ri'-method) - it's not a question of if, it's a question of when. The search continues.

Anywho, I've attached the data for p'<242. It seems I was even more lucky than I originally thought with that gap at q=152. It gets increasingly hard to find gaps even with CSG>0.9.
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File Type: txt CSG_max @ 2^42.txt (66.3 KB, 9 views)
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Old 2020-08-08, 12:52   #27
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Originally Posted by mart_r View Post
It seems I was even more lucky than I originally thought with that gap at q=152. It gets increasingly hard to find gaps even with CSG>0.9.
Specifically, between 5*1012 and 7.3*1012 for q<=1000 the largest CSG is about 0.8849, which puts a large question mark behind ever finding another gap that is larger than both q² and \varphi(q) * log²p. I'm kind of heuristically challenged there.

From what I gather, the conjectures of Granville et al. for gaps between consecutive primes only, and at best, apply to prime gaps in AP with common difference q if the bounding prime p' = p+q*k is no larger than (p+n)² for very small n (n<2, say).
On the other hand, primes in AP may behave more random, as it were, than the primes themselves. Even if the gaps with CSG>1 is finite for any fixed q, is it plausible to expect that CSG can be arbitrarily large, maybe even as a function of q, how do you write it, f(q) > O(1)?

But I won't leave unanswered questions here today without providing a new record gap with both (q, r) prime:
p = 7,302,961,447
q = 214,451
r = 47,093
k = 576
g/\varphi(q)/log²p' = 1.1150385797428...
CSG (via sum Ri') = 1.1158450619299...
Which is also quite remarkable as it's in the upper decile of all gaps with CSG>1.
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Old 2020-09-08, 15:57   #28
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The computation has now reached 1013 for all q<=1000.

Summary for the 13-digit range of p:
No new record CSG was found between p=1012 and 1013.
The largest CSG in that range was 0.9687994043 for q=104, ranking # 5 in largest CSG over all p<=1013.
CSGmax has increased for 157 values of q.
The arithmetic mean of CSGmax for q<=1000 has increased from 0.8258133580 to 0.8355509072.

The next interesting thing is the observation of a quick-and-dirty cluster analysis of all the gaps with CSG > 1. For most of the gaps it looks like the number of gaps Gd with CSG > 1+d is approximately \(G\cdot{e^{\frac{-d}{\omega}}}\) (BTW, thanks kruoli!) with \(\omega\) being a constant around 0.036, but for gaps with CSG > 1.17 or 1.18, the actual number of gaps drops significantly compared to the expected number (and with it the would-rather-be-constant 0.036).

This is where I came from: at the moment I have 1684 gaps with CSG > 1. Sorted by CSG, the increase in the minimum CSG of the upper half of gaps, taken repetitively, remains more or less constant up to CSG ~ 1.17, but drops significantly above that level:
#gaps  CSG >
1684   1        incr.
 842   1.02524  0.02524
 421   1.05108  0.02584
 210   1.07634  0.02526
 105   1.1000   0.02366
  53   1.1251   0.0251
  26   1.1485   0.0234
  13   1.1743   0.0258
   7   1.1912   0.0169
   3   1.204    0.013
   2   1.22     0.016
(The above mentioned 0.036 equals ~ the increase per "cluster" divided by log(2).)
So, 50% of the gaps with CSG > 1 have CSG > 1.025, 25% have CSG > 1.051 and so on.
This, again, is made with CSG via the non-conventional Sum(Ri')-formula. Taking the conventional formula gap/phi(q)/log²p, the tabulated values would fluctuate much more and are generally more "out of tune".

All a bit strange, this. Until I find a satisfying explanation, the next thing I do: analysing a list of top-100-CSG gaps between consecutive primes, checking how the 100th (99th etc.) largest CSG behaves with respect to p.

Also, after long last, a new record CSG @ p=209,348,411 / q=1,415,237: CSG = 1.28848055169164 (conventional: 1.2174453913778).
Amazing, huh? Whadd'ya think? Huh?
Attached Files
File Type: txt PGAP_records per q 2020-09-07.txt (56.8 KB, 3 views)
File Type: txt PGAP_records - improvements 13-digit-p.txt (22.2 KB, 1 views)
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