20200715, 22:11  #23 
Dec 2008
you know...around...
1001000000_{2} Posts 
No major news in the last few weeks, but in the attachment there's the data for CSG_{max} up to p=2^{n} for n=[16..41] and q<=2690. Data for p=2^{42} might follow next week. Minor differences in conduct compared to the data for the older analysis on 20200501: here the endpoints p' of the gaps are key for clustering in powers of two, and CSG is determined using the Risumofderivatives formula as first considered in post # 2  has anyone got a catchy name for it, perchance?
(where k=(p'p)/q  in other words, there are k1 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) . (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) 
20200719, 09:48  #24 
Dec 2008
you know...around...
240_{16} Posts 
Umm... are my posts even visible to anybody other than Bobby?

20200719, 19:55  #25 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{2}·7·307 Posts 
Yes, they are.

20200805, 21:11  #26 
Dec 2008
you know...around...
240_{16} Posts 
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:
Code:
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)) Anywho, I've attached the data for p'<2^{42}. 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. 
20200808, 12:52  #27  
Dec 2008
you know...around...
2^{6}·3^{2} Posts 
Quote:
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/(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. 

20200908, 15:57  #28 
Dec 2008
you know...around...
2^{6}×3^{2} Posts 
number crunching and munching...
The computation has now reached 10^{13} for all q<=1000.
Summary for the 13digit range of p: No new record CSG was found between p=10^{12} and 10^{13}. The largest CSG in that range was 0.9687994043 for q=104, ranking # 5 in largest CSG over all p<=10^{13}. CSG_{max} has increased for 157 values of q. The arithmetic mean of CSG_{max} for q<=1000 has increased from 0.8258133580 to 0.8355509072. The next interesting thing is the observation of a quickanddirty cluster analysis of all the gaps with CSG > 1. For most of the gaps it looks like the number of gaps G_{d} 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 wouldratherbeconstant 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: Code:
#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 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 nonconventional 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 top100CSG 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? 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Prime gaps  Terence Schraut  Miscellaneous Math  10  20200901 23:49 
Gaps between maximal prime gaps  Bobby Jacobs  Prime Gap Searches  52  20200822 15:20 
The new record prime gaps  Bobby Jacobs  Prime Gap Searches  6  20181207 23:39 
Prime gaps above 2^64  Bobby Jacobs  Prime Gap Searches  11  20180702 00:28 
Residue classes  CRGreathouse  Math  4  20090312 16:00 