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Old 2020-07-15, 22:11   #23
mart_r
 
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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)
Attached Files
File Type: zip PGAP_data_CSG.zip (611.8 KB, 21 views)
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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
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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:

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
File Type: txt CSG_max @ 2^42.txt (66.3 KB, 17 views)
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Old 2020-08-08, 12:52   #27
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Quote:
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:
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).
Amazing, huh? Whadd'ya think? Huh?
Attached Files
File Type: txt PGAP_records per q 2020-09-07.txt (56.8 KB, 10 views)
File Type: txt PGAP_records - improvements 13-digit-p.txt (22.2 KB, 7 views)
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Old 2020-10-08, 20:21   #29
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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   692216 1.00114450
    35966731  2311616 1.00118004
    11351881  1198592 1.00118579
     5190293   255590 1.00119910
   253866997   237520 1.00121705
  3017101433  1773412 1.00123591
   213432451  2252666 1.00126905
   773018111  2677958 1.00127375
   144803717    31854 1.00127635
  2122715809  2088984 1.00136931
 16548370793    78492 1.00137881
    73300061  2005060 1.00138172
  2463669847  2024608 1.00141336
  3448559369  1258546 1.00146500
  8678039813   943310 1.00152356
  1166569531  1753414 1.00155117
     7891867   743460 1.00158817
  1282018477  2461888 1.00159889
  1826636507  2215840 1.00160267
   357489677   149234 1.00162602
    71708587  2371570 1.00165557
  8356079209   528868 1.00171646
    42526739   654196 1.00172866
  3693322679  1739580 1.00174615
   754287427  1955812 1.00175249
   786977909  2003266 1.00182987
    45047257  1511264 1.00183334
  1547165593   709524 1.00188299
   521898073   216620 1.00189487
    17239987   695108 1.00192691
     2887607   232298 1.00195140
   574687483  1748348 1.00198845
  1672469789   984180 1.00203138
    45349027  2421654 1.00206110
     9478433   585112 1.00207865
  1794917603   374278 1.00208604
   381129739  1222532 1.00212811
   489350311    97066 1.00213105
    45980497   974848 1.00213477
   460019653  2503452 1.00214113
   324271427  2129552 1.00220544
  3177249487   551038 1.00220997
  2850660019  1480204 1.00222379
   805019981  2674164 1.00223081
  7378248509   304240 1.00223705
   111036319   579394 1.00224222
    10926917  2644632 1.00228909
    52430479  2131010 1.00231980
  2336703133  1101006 1.00236313
  2475213493  1422620 1.00239342
   240932869   887596 1.00240285
   136809161  2024644 1.00241860
  3053615743  1672344 1.00244750
  1847877809   109584 1.00244941
     6891541  1314510 1.00248091
  2008573981  1246092 1.00249633
   663185953    89706 1.00249746
  1540935449   116900 1.00249804
    99646691  1197074 1.00252669
  1463512079  2247600 1.00254463
 48821476019    65514 1.00257633
  2666960237  2537754 1.00261311
  1863705377  2034734 1.00262241
  3654563879   715660 1.00264745
    16919813    61612 1.00266107
   648513517  1826268 1.00266870
    14155153  2235956 1.00270742
   530404363  2128050 1.00273011
    11492161   504756 1.00276737
  1347479131   231318 1.00277358
  3618594269  1223188 1.00281268
    28558199   694438 1.00281642
    23115893   897818 1.00282386
  3552650263   596294 1.00283500
    15781097  1999648 1.00284799
 69209038343    26384 1.00287275
  2142386327  2038742 1.00287392
  5360677823   652144 1.00289967
    74406719  2029602 1.00299353
    99960821   966132 1.00301795
   164742967  1534256 1.00301829
   292742119   730164 1.00301886
   509682361  2462590 1.00303627
  2487363163  1263446 1.00304545
    67314889  1921848 1.00306252
   838264771    66230 1.00308160
   998680231  1303280 1.00310773
   769650307   106028 1.00312129
   404876611  2945692 1.00314659
    84346579  1528432 1.00317304
   145465687    13512 1.00317902
   300750841  1020972 1.00319531
  8520266219   866420 1.00320704
     9531349    54290 1.00320939
   171639827  1308108 1.00321532
   121617919  2016980 1.00322659
   216750497  2717186 1.00323107
   195101233   195846 1.00326845
  4080819871  1023090 1.00327716
 13182668077   469872 1.00328809
    65548921   237878 1.00329484
  1876530017  1536608 1.00329915
   290131663  1450530 1.00330096
   914072521   606512 1.00334131
   787546453   337550 1.00334743
    17226889   759512 1.00334750
 13195552637    83346 1.00335182
   227878163   140828 1.00336134
   331960631   967454 1.00337003
  7397683367    67676 1.00341201
  2319088253  2110326 1.00343366
  5034766729   102882 1.00349145
  2762778617  1188402 1.00350118
  1747708063   663164 1.00352333
   300807289  2491206 1.00352700
   727591231  2224654 1.00352776
   456043963    32772 1.00353977
   583703311   146846 1.00354530
   953311367  2379626 1.00355197
 14964391577   362656 1.00355694
 15325816171    73028 1.00356455
    82541821    13814 1.00364502
  6563474651   277582 1.00364612
    25923263  2156606 1.00372844
  4858323163   741050 1.00377979
     5839843  1092184 1.00379472
   393144877   158658 1.00383254
    53553593  1735912 1.00383361
  8816753969   653068 1.00383746
   200911231  1775798 1.00386490
    44844973  1175904 1.00387015
   342854651   766098 1.00388808
   386906033  2800448 1.00389854
  4006574051   873592 1.00392011
  1133182793  1725328 1.00393163
    74385869  1997118 1.00395208
  9929834153   791494 1.00396140
  2569430399  2260272 1.00400958
     1789499   823694 1.00402571
  1994443151   981678 1.00403056
 27406095727   137004 1.00403542
   907407713  2323722 1.00405119
   312731233  1877704 1.00417858
  1245541321   419438 1.00422277
     2839381  2054468 1.00423291
    48057377  2230218 1.00423591
  1472835641  2921056 1.00424451
   565558423    75898 1.00424635
   254464237  2968800 1.00425960
  2927493631   244070 1.00428273
  1662544883  1465968 1.00429693
  1303019737   693414 1.00431036
  1581078533  1027444 1.00431819
  2963029759   631598 1.00437204
   113460497   116186 1.00439664
   265654813  1841442 1.00442120
    11609501   807568 1.00442835
   940166641  1906000 1.00447607
  1370634319  1583524 1.00450257
 13358017873   295048 1.00450699
  6537849419   215360 1.00450860
  8932347929   898798 1.00454102
  1168184821    92052 1.00457755
   432769123   346932 1.00461734
  2443778741  1811308 1.00465323
  2446718089   374960 1.00469660
   251058803  2704330 1.00471158
   313905113   182050 1.00471583
     3144419   221832 1.00472410
  1775362427  2416610 1.00477567
   128417521   766796 1.00479490
  1261108061  2797052 1.00481429
   396928111   947104 1.00486263
   556121471   383596 1.00491648
  3643488623  1568386 1.00491920
  4633461487   655168 1.00492399
  1582244249  2012464 1.00492507
 11728152599   496780 1.00493113
  1688577061  2381276 1.00495085
  2477778071  1622348 1.00498654
    54046873   202562 1.00502647
    54468217  1904298 1.00502741
   360343513  1942812 1.00503076
  1484014453  1286568 1.00509846
    68882887  2171158 1.00516859
  1854986369   790908 1.00518877
    53697283   976030 1.00523124
   986864401   187076 1.00523283
  2314055291  2234398 1.00523646
   338041493   539490 1.00526039
   119562493  1880908 1.00526149
  3786780001   898330 1.00527011
  7180821773   502044 1.00529630
  1625195219   455652 1.00529703
  6350571287   225330 1.00530058
    67224649   972054 1.00532477
  3240050263  1982078 1.00532736
  3105128137   248544 1.00537441
   235933801  1384810 1.00537762
   131740493  1317532 1.00539459
   939870067  2077258 1.00539996
  2760026917  1880788 1.00540491
 13158789733   313318 1.00541452
   560228653   369894 1.00545868
   155462149  1873864 1.00548094
   295424933  1513894 1.00548135
  7513487593   570892 1.00553027
   428249837  1101000 1.00556436
   788061331  1185266 1.00560496
   549084871   107728 1.00561455
   725453039  2693948 1.00567922
  1735084283   859682 1.00570974
   119114231  1873524 1.00571287
    98820391   756206 1.00572185
  2040751439  2240610 1.00573364
   263869679   995776 1.00573884
  2558096153  1930852 1.00577013
   212220587  2004574 1.00578840
    49012351   682826 1.00581329
  6225375587   167940 1.00583315
   657182531  1862818 1.00591617
 18762353227   104944 1.00595667
    21187823   548810 1.00595726
   302145839    55038 1.00598025
     7906807  2098098 1.00603903
  6896537899   808470 1.00608313
  1334505643  1216374 1.00617032
    12745541   550550 1.00617657
   146268803  1467806 1.00618182
    17292259  1679406 1.00619318
   911319859   366368 1.00623443
  2507174687   842308 1.00623600
    51529477  1486364 1.00628442
   542396123  2093716 1.00630562
   534139967  2400494 1.00632863
   407603563  2160940 1.00638304
  1719883861   633440 1.00638788
 77980184267    46102 1.00640830
  2829106853  1863352 1.00643234
    66349711  1735460 1.00644766
   439349837   155668 1.00645065
   223328951   859408 1.00653648
   660448757  2179810 1.00654170
   908026849  1204488 1.00654412
    34163341  1404240 1.00660910
  5912515871   735726 1.00661320
   108701029   181702 1.00662331
   668259359  2073468 1.00662795
   506401801   760340 1.00665237
   167677187   898864 1.00665564
  1358123359  1712640 1.00668890
    77639167   583948 1.00671129
    80442199  1141392 1.00676258
    26076107   492654 1.00677935
  1058830543  2539000 1.00681190
  2011075453  2462436 1.00681211
    52795381   113146 1.00681843
  1277046751  1941360 1.00683213
   776575799  1704856 1.00684096
  2165010473  1227416 1.00684609
    10803217   466666 1.00686958
    24931831   677832 1.00688666
  1941204971  1909724 1.00691263
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   849612919  1747002 1.16741936
     3216881   114340 1.16745025
     2839657  1153266 1.16758470
   566129467  1565372 1.16915694
   441988777  2135674 1.17420323
   582224303   979202 1.17439351
   150211753   232380 1.17532908
    19826693  1438480 1.17680552
   965985091  2792688 1.17810794
   683443393  2020918 1.17855025
      164663    16934 1.18522392
   292392073  1404430 1.18665808
    19019009   242974 1.19163582
    69589171  2592136 1.19412110
   722359397  1985124 1.19505249
  4517295317   758236 1.19518536
   799417357  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)
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Old 2020-10-21, 21:03   #30
mart_r
 
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Dec 2008
you know...around...

2×13×23 Posts
Default 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
File Type: zip _results_all_10^12-15×10^12.zip (238.5 KB, 2 views)
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