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2020-07-15, 22:11   #23
mart_r

Dec 2008
you know...around...

27F16 Posts

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)
Attached Files
 PGAP_data_CSG.zip (611.8 KB, 166 views)

 2020-07-19, 09:48 #24 mart_r     Dec 2008 you know...around... 63910 Posts Umm... are my posts even visible to anybody other than Bobby?
 2020-07-19, 19:55 #25 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 2·32·17·31 Posts Yes, they are.
2020-08-05, 21:11   #26
mart_r

Dec 2008
you know...around...

32×71 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))
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.
Attached Files
 CSG_max @ 2^42.txt (66.3 KB, 150 views)

2020-08-08, 12:52   #27
mart_r

Dec 2008
you know...around...

32×71 Posts

Quote:
 Originally Posted by mart_r 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.

2020-09-08, 15:57   #28
mart_r

Dec 2008
you know...around...

32·71 Posts
number crunching and munching...

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:
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
(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).
Attached Files
 PGAP_records per q 2020-09-07.txt (56.8 KB, 148 views) PGAP_records - improvements 13-digit-p.txt (22.2 KB, 132 views)

 2020-10-08, 20:21 #29 mart_r     Dec 2008 you know...around... 32·71 Posts yummy... Status of my search (max. p searched): Code:  q <= 1000: 1.365e+13 1000 < q <= 2690: 6.635e+12 2690 < q <= 1e+5: 8.135e+10 1e+5 < q <= 2e+5: 4.730e+10 2e+5 < q <= 5e+5: 1.6e+10 5e+5 < q <= 1e+6: 1.3e+10 1e+6 < q <= 2e+6: 4.0e+9 2e+6 < q <= 3e+6: 3.0e+9 (only even q are examined) Counting 1892 extraordinarily large gaps so far (CSG via Sum Ri'). Just so you know that I'm not making things up: Code:  p q CSG 4350095419 156150 1.00001500 46270519 1890892 1.00002125 37817111 1825382 1.00005501 1142152841 1618578 1.00007488 7875272441 954208 1.00009851 11658250151 406390 1.00010247 63922637 1570936 1.00011465 22047139 169294 1.00017979 1326737999 1172874 1.00018913 789694553 40708 1.00019144 19623837047 8708 1.00019466 823763011 290950 1.00023417 820748273 2932362 1.00025688 7749010229 693636 1.00026869 66130513 1515012 1.00028709 675998923 188848 1.00029195 1509250627 278660 1.00031850 199076393 61338 1.00036214 90615509 666848 1.00038040 14042321 437044 1.00039334 934158587 2066708 1.00039691 439253173 577662 1.00040556 1019089139 302760 1.00042186 139129589 649226 1.00042619 127255123 1423856 1.00043604 80736983 2269128 1.00044273 86291323 2729228 1.00048292 10288449727 777328 1.00049765 3884996657 1497626 1.00050038 29192413 589942 1.00051431 704422049 2518398 1.00051717 110891981 1717496 1.00054689 245721767 930730 1.00054896 2344766101 1050160 1.00055093 9222708863 283248 1.00055674 14642527 1022758 1.00056055 262203131 335750 1.00056438 277043069 1609944 1.00057567 663716701 284094 1.00058958 163504573 10532 1.00059172 38669039 843652 1.00059970 1076137 1786246 1.00064585 31635763 1028874 1.00065783 327111269 492272 1.00073608 109741259 1600558 1.00076234 513996313 2204670 1.00077407 334260763 122162 1.00081598 619494587 2851938 1.00084366 359016773 82068 1.00085218 626379683 1985996 1.00085557 31237681 1702306 1.00085951 22863764699 58478 1.00086811 1265347 932590 1.00089452 120420481 1845808 1.00093994 3196462091 890638 1.00095762 219245869 574780 1.00098164 356884973 171176 1.00099972 825353008489 152 1.00101325 958051 251598 1.00101687 404307503 2974994 1.00102293 270262721 2471116 1.00102390 8702067713 965476 1.00106940 812829763 2427612 1.00107048 989438663 405422 1.00107327 81810607 2187116 1.00107967 340116151 2608132 1.00112082 30240523 93386 1.00112787 46777513 540244 1.00113968 5595526991 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1237676 1.19779995 812219587 1265710 1.20349091 397778383 615768 1.22502035 33868663 2800258 1.23601051 209348411 2830474 1.28829147 Last fiddled with by mart_r on 2020-10-08 at 20:26 Reason: (only even q are examined)
2020-10-21, 21:03   #30
mart_r

Dec 2008
you know...around...

32·71 Posts
Hm, I should wait 18 more days for the next update here... but I won't

Attached are the first occurence gap results for q<=1000 between p=1×1012 and 15×1012.
See post # 16 dated 2020-05-09 for results up to 1×1012.

As for the quest for exceptional gaps, I found one with CSG>1.2, ranked 3rd in largest CSG conventionally and 4th in largest CSG unconventionally, but either way the largest with at least that ratio of p/q:
q=1,251,242 / p=5,120,249,753 / k=303 / CSG=1.20852 (1.20475 conv.)
Attached Files
 _results_all_10^12-15×10^12.zip (238.5 KB, 112 views)

 2020-11-21, 16:42 #31 mart_r     Dec 2008 you know...around... 10011111112 Posts TWIMC Code:  max. p searched: q <= 1000: 1.748e+13 1000 < q <= 2690: 8.734e+12 2690 < q <= 1e+5: 1.435e+11 1e+5 < q <= 2e+5: 4.730e+10 2e+5 < q <= 5e+5: 3.205e+10 5e+5 < q <= 1e+6: 2.050e+10 1e+6 < q <= 2e+6: 9.0e+9 2e+6 < q <= 3.3e+6: 3.0e+9 (only even q are examined) which included a new record conventional CSG: p = 1,961,096,147 q = 3,128,278 k = 293 g/$$\varphi$$(q)/log²(p+g) = 1.23529914455... (unconventionally 1.2558799381..., or 1.2559180140... if q = 1,564,139 (a prime number BTW))
 2020-11-22, 19:40 #32 Bobby Jacobs     May 2018 22·53 Posts Is there a difference in the ranks of the best CSG values if you use g/(phi(q)*ln2(p1)) and g/(phi(q)*ln2(p2))?
 2020-11-22, 20:06 #33 mart_r     Dec 2008 you know...around... 32×71 Posts Quite a difference, yes. The difference between my unconventional CSG and the formula with the initial prime p1 is even bigger than the one between my formula and the conventional which uses p2, and using p1 gives dramatically different results for small p1. On a side note, if p is big enough and q small enough, my CSG can also be smaller than the one with the conventional formula (using p2).

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