mersenneforum.org Gaps between non-consecutive primes
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 2022-08-01, 18:44 #56 mart_r     Dec 2008 you know...around... 11010000002 Posts Error terms, prime number scarcities, and trains Just an intermediate result that made me go "hmmmm...". Suppose we assume $$CSG=1+O(1)$$ for the gaps between non-consecutive primes, then, if I did the math right, this would imply that we also assume $$\pi(x)=Li(x)+O(\sqrt{x})$$, i.e. the error term is smaller by a factor log x compared to the RH prediction. Correct [y/n]? Code: Outline from my train of thought: p_1 = 2 (or set p_0 = 0, say) p_k = x k = pi(x)-1 ~ pi(x) gap = x-2 ~ x m = Gram(x)-Gram(2)-k+1 ~ Gram(x)-pi(x) CSG = m*|m|/gap - but for simplicity suppose that m is positive (means we assume a scarcity instead of an abundance of primes; the error term works both ways anyway): CSG = m^2/gap ~ (Gram(x)-pi(x))^2/x CSG ~ 1 --> (Gram(x)-pi(x))^2 ~ x --> Gram(x)-pi(x) ~ sqrt(x) OTOH, if Gram(x)-pi(x) = O(sqrt(x)*log(x)), then CSG = m^2/gap ~ O((x*log²x)/x) ~ O(log²x)
 2022-08-07, 19:05 #57 Bobby Jacobs     May 2018 52×11 Posts That is correct. By the way, you should submit the sequences in this thread to OEIS.
2022-08-24, 17:03   #58
mart_r

Dec 2008
you know...around...

15008 Posts

Quote:
 Originally Posted by mart_r 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 rgm of values of r also become larger. Find an asymptotic function f(x) ~ rgm.
$$\lim\limits_{\substack{x\to\infty}} r_{gm} = 4$$, if I may so conjecture (based on a random model similar to Cramér's). Is there a proof available?

I'd like to take the search for T(38,16) in A086153 up to 10^16, which will likely not be enough to find an example, but I still would like to see that case solved. It would take a bit more than a week with my program. I've identified 746 distinct constellations as shown in the attachment. I believe that list to be complete, albeit I'd be more content if that number was divisible by 4, so there's a slight possibility I have overlooked some constellations. If anyone with enough time on their hands feels inclined to do a quick double-check...

For good measure, here's a batch of 79 instances where CSG > 1 for k > 1000:

Code:
p                k     gap    CSG
123146152018999  1152  44280  1.0322718
123146152018933  1154  44346  1.0310658
123146152018999  1127  43378  1.0263032
123146152018999  1126  43342  1.0260889
123146152018933  1129  43444  1.0250767
123146152018933  1156  44394  1.0249425
123146152018933  1128  43408  1.0248616
123146152018921  1155  44358  1.0247205
123146152018999  1157  44428  1.0246186
123146152018999  1133  43582  1.0242866
123146152018993  1153  44286  1.0242773
123146152018999  1138  43758  1.0242719
123146152018933  1159  44494  1.0234272
123146152018933  1135  43648  1.0230694
123146152018933  1140  43824  1.0230603
123146152018823  1158  44456  1.0226588
123146152019071  1151  44208  1.0221944
123146152018999  1125  43288  1.0209057
123146152018999  1139  43780  1.0206462
123146152018933  1141  43846  1.0194401
123146152018933  1160  44514  1.0192932
123146152018933  1130  43458  1.0192328
123146152019521  1113  42856  1.0183353
123146152019521  1112  42820  1.0181224
123146152018853  1132  43524  1.0180185
123146152018853  1131  43488  1.0178006
123146152018999  1094  42184  1.0176670
123146152018999  1091  42078  1.0176016
123146152018921  1136  43660  1.0166975
123146152019521  1143  43906  1.0165963
123146152018853  1162  44574  1.0164815
123146152018933  1096  42250  1.0164132
123146152018933  1093  42144  1.0163444
123146152019521  1119  43060  1.0163041
123146152019521  1124  43236  1.0162807
123146152018993  1134  43588  1.0162592
123146152018999  1095  42210  1.0150911
123146152018823  1163  44604  1.0150789
123146152018823  1144  43934  1.0146314
123146152019071  1137  43686  1.0141775
123146152018933  1097  42276  1.0138418
123146152019419  1142  43860  1.0136414
123146152018823  1161  44528  1.0135410
123146152019521  1111  42766  1.0129308
123146152019507  1114  42870  1.0124697
123146152018801  1164  44626  1.0115109
123146152019521  1116  42936  1.0112458
123146152018801  1145  43956  1.0110334
123146152019507  1120  43074  1.0104594
123146152019521  1080  41662  1.0097438
123146152019521  1077  41556  1.0096842
123146152018993  1092  42084  1.0094570
123146152019483  1115  42894  1.0093765
123146152018853  1099  42330  1.0092682
123146152018999  1088  41938  1.0080624
123146152018823  1100  42360  1.0078183
123146152019419  1117  42958  1.0076073
123146152018801  1146  43978  1.0074438
123146152018801  1165  44646  1.0074105
123146152018921  1098  42288  1.0073935
123146152019483  1121  43098  1.0073785
123146152019071  1090  42006  1.0073695
123146152019521  1081  41688  1.0071605
123146152019521  1079  41616  1.0067457
123146152019419  1147  44008  1.0060364
123146152019167  1150  44112  1.0056258
123146152019419  1123  43162  1.0056203
123146152018823  1101  42386  1.0052645
123146152019207  1149  44072  1.0043139
123146152018999  1089  41958  1.0038262
123146152019461  1122  43120  1.0037558
123146152019507  1078  41570  1.0037494
123146152019521  1118  42976  1.0028794
123146152018583  1167  44696  1.0019334
123146152019419  1148  44028  1.0019166
123146152018801  1102  42408  1.0016116
123146152019507  1082  41702  1.0012434
123146152019521  1074  41416  1.0001307
123146152018793  1166  44654  1.0000862
Attached Files
 17primes76apart.txt (36.1 KB, 25 views)

2022-09-02, 18:14   #59
mart_r

Dec 2008
you know...around...

83210 Posts
Dancing with tears in my eyes...

Quote:
 Originally Posted by mart_r $$\lim\limits_{\substack{x\to\infty}} r_{gm} = 4$$, if I may so conjecture (based on a random model similar to Cramér's). Is there a proof available?
Is this not a well-known result? If so, I might try to tackle the proof myself...

They say that the gas pipeline Nord Stream 1 is kept shut indefinitely. So before power outrages become daily routine, I'd like to give an update on some numbers.

- Gaps between non-consecutive primes, for k=104 to 1024 step 4:
Code:
p <= 179133400000000
k    gap    CSG_max       p
104  5656   1.0781126752  36683716323847
108  5824   1.0843154811  36683716323847
112  5940   1.0555010733  36683716323847
116  6052   1.0251605182  36683716323619
120  6220   1.0326995729  36683716323283
124  6388   1.0404373460  36683716323283
128  6510   1.0185881717  36683716323161
132  6642   1.0039006107  36683716323167
136  6742   0.9697674279  36683716323109
140  7292   0.9521920888  175478559288359
144  6840   0.9648817776  17674627574369
148  6992   0.9668891162  17674627574141
152  7126   0.9577979013  17674627574083
156  7460   0.9745283792  30512335335437
160  8144   0.9792679542  175478559288359
164  7732   0.9570891069  30512335335319
168  7946   0.9949905766  30512335334951
172  8100   0.9972610993  30512335334797
176  8254   0.9996701497  30512335334797
180  8364   0.9766164520  30512335335299
184  8510   0.9748349809  30512335335059
188  8736   1.0192157620  30512335334927
192  8892   1.0231919672  30512335334771
196  9004   1.0021590389  30512335334797
200  9148   1.0144816568  28330683392731
204  9324   1.0303925866  28330683392659
208  9492   1.0417729138  28330683392597
212  9630   1.0362667509  28330683392353
216  9778   1.0365415158  28330683392371
220  9856   0.9982886575  28330683392371
224  9974   0.9826932475  28330683392147
228  10058  0.9492927463  28330683392129
232  8294   0.9483514306  185067241757
236  9700   0.9564124562  5185992136441
240  9850   0.9639477964  5185992136453
244  10626  0.9420515461  28330683392597
248  10818  0.9664456553  28330683392371
252  10596  0.9477134052  12666866223047
256  11310  0.9390807333  52248744686339
260  11476  0.9481806463  52248744686197
264  11604  0.9386620116  52248744686069
268  11724  0.9254797708  52248744686197
272  11264  0.9300152206  12666866223047
276  12106  0.9116757369  68182243872601
280  11752  0.9125108382  21947823205027
284  11920  0.9253516441  21947823205027
288  12096  0.9421630582  21947823204943
292  12310  0.9417896525  25698372297889
296  12460  0.9453382500  25698372297691
300  12704  0.9950535482  25698372297029
304  12920  1.0312457318  25698372297029
308  13170  1.0847521505  25698372297029
312  13308  1.0819497629  25698372297029
316  13482  1.0972915901  25698372297029
320  13616  1.0925708301  25698372296963
324  13728  1.0770435304  25698372296873
328  13878  1.0804891085  25698372297029
332  13986  1.0633678090  25698372297007
336  14136  1.0669387810  25698372296963
340  14234  1.0453848066  25698372296873
344  15204  1.0287670437  127946496635897
348  15390  1.0445817969  127946496635761
352  15540  1.0446056952  127946496635611
356  15692  1.0455495832  127946496635459
360  15798  1.0266221073  127946496635459
364  15044  1.0256385680  25698372296963
368  15222  1.0426511752  25698372295019
372  15360  1.0409959251  25698372294839
376  15546  1.0617275885  25698372295033
380  15694  1.0647408322  25698372294457
384  15832  1.0631372016  25698372294409
388  15968  1.0606657649  25698372294611
392  16158  1.0831680032  25698372294421
396  16242  1.0568222056  25698372294337
400  16344  1.0390881483  25698372294563
404  16536  1.0622824463  25698372294457
408  16678  1.0627769150  25698372294421
412  16762  1.0372222537  25698372294337
416  16852  1.0147699971  25698372294457
420  16974  1.0067230990  25698372295033
424  17160  1.0268837758  25698372294421
428  17302  1.0276653950  25698372294421
432  17396  1.0075134540  25698372294611
436  17586  1.0293131103  25698372294421
440  17724  1.0284358125  25698372294409
444  17810  1.0051347652  25698372294323
448  17886  0.9779763169  25698372294253
452  17972  0.9554989984  25698372294281
456  18114  0.9566926179  25698372293557
460  18234  0.9487559428  25698372293809
464  18390  0.9558193895  25698372293597
468  18536  0.9587388800  25698372293597
472  19656  0.9486426201  112364701413971
476  19770  0.9862855261  93152147737279
480  19878  0.9719230760  93152147737199
484  19954  0.9455466698  93152147737237
488  19192  0.9433486203  25698372294421
492  20322  0.9755220092  93152147736727
496  20490  0.9844049639  93152147736559
500  20598  0.9704056982  93152147736451
504  20748  0.9724525235  93152147736301
508  20850  0.9564344564  93152147736199
512  21004  0.9600442813  93152147736073
516  21260  1.0021068423  93152147735789
520  21390  0.9965846355  93152147735659
524  21478  0.9753892159  93152147735599
528  21592  0.9641382100  93152147735371
532  21726  0.9603975747  93152147735351
536  21874  0.9618568153  93152147735203
540  21964  0.9421058879  93152147735113
544  22076  0.9305879709  93152147734973
548  22224  0.9321622171  93152147733739
552  22486  0.9751346521  93152147732647
556  22628  0.9744504782  93152147734421
560  22792  0.9818076325  93152147734171
564  22958  0.9898950736  93152147734091
568  23130  1.0001758329  93152147733919
572  23346  1.0266184470  93152147733703
576  23524  1.0391401441  93152147733553
580  23610  1.0178636352  93152147733467
584  23706  1.0005048976  93152147733553
588  23912  1.0230990399  93152147733137
592  24068  1.0275448938  93152147732981
596  24240  1.0378113630  93152147732723
600  24436  1.0568436976  93152147732641
604  24540  1.0423472371  93152147732509
608  24676  1.0395621760  93152147732401
612  24798  1.0317744784  93152147732251
616  24880  1.0098055650  93152147732197
620  25008  1.0043765193  93152147732069
624  25164  1.0088849117  93152147731913
628  25264  0.9936835657  93152147731813
632  25348  0.9731085473  93152147731729
636  25500  0.9762666351  93152147731549
640  25578  0.9539550727  93152147731499
644  25696  0.9455489291  93152147731381
648  25860  0.9528438164  93152147731217
652  26004  0.9533334554  93152147731073
656  26252  0.9893564159  93152147730797
660  26412  0.9952890412  93152147730637
664  26606  1.0129446324  93152147730443
668  26706  0.9982265121  93152147730371
672  26826  0.9904861287  93152147730223
676  26938  0.9801091903  93152147730139
680  27094  0.9846872883  93152147729983
684  27186  0.9677069785  93152147729891
688  27276  0.9502720219  93152147729983
692  27368  0.9337153701  93152147729891
696  27516  0.9356922025  93152147729561
700  27582  0.9109248623  93152147729467
704  27698  0.9026102400  93152147729561
708  27820  0.8962995612  93152147729143
712  27948  0.8919623539  93152147729143
716  28048  0.8787786737  93152147729143
720  27710  0.8927568473  54116590394771
724  27860  0.8963650091  54116590394621
728  27998  0.8960609520  54116590394483
732  28172  0.9075013936  54116590393157
736  28332  0.9143775739  54116590394149
740  28536  0.9357024342  54116590393991
744  28666  0.9327253951  54116590393861
748  28800  0.9310848576  54116590393777
752  28982  0.9451815024  54116590393499
756  29130  0.9481253683  54116590393447
760  29370  0.9814398420  54116590393157
764  29456  0.9638702316  54116590393121
768  29630  0.9753745155  54116590392947
772  29706  0.9546906758  54116590393157
776  29826  0.9485339103  54116590392947
780  29964  0.9482566835  54116590392929
784  30076  0.9395966369  54116590392451
788  30192  0.9322967009  54116590392947
792  30288  0.9186909882  54116590392289
796  30456  0.9280597127  54116590392121
800  30654  0.9470376796  54116590391873
804  30740  0.9302478398  54116590391837
808  31974  0.9117548600  159316577936029
812  30990  0.9217161811  54116590391873
816  31176  0.9367352659  54116590391351
820  32550  0.9554814455  159316577935453
824  31516  0.9567140006  54116590391011
828  31596  0.9382012070  54116590391011
832  31734  0.9380755356  54116590391113
836  31852  0.9316943869  54116590391011
840  31936  0.9148047257  54116590391077
844  32062  0.9110448935  54116590391077
848  32158  0.8981057702  54116590391011
852  32880  0.8916246643  93152147732647
856  33006  0.8873627713  93152147732641
860  32594  0.9048332804  54116590389887
864  32714  0.8993611673  54116590389863
868  34120  0.9116132357  159316577935453
872  34218  0.8986902090  159316577935453
876  34398  0.9093907279  159316577935453
880  34498  0.8971206895  159316577935453
884  34592  0.8832711645  159316577933411
888  34758  0.8899100180  159316577935453
892  34934  0.8993959518  159316577936297
896  35074  0.8986493801  159316577936233
900  35262  0.9115556093  159316577935969
904  35406  0.9119383256  159316577935453
908  35630  0.9351767250  159316577935601
912  35784  0.9384016321  159316577935453
916  35880  0.9250502187  159316577935453
920  36024  0.9254373323  159316577935433
924  36102  0.9071761352  159316577935601
928  36288  0.9194386246  159316577935453
932  36460  0.9277658577  159316577935453
936  36576  0.9202752667  159316577935601
940  36744  0.9274602593  159316577935453
944  36906  0.9329559987  159316577935453
948  37068  0.9384527900  159316577935453
952  37140  0.9186590345  159316577935453
956  37282  0.9185545605  159316577935969
960  37384  0.9073477992  159316577935969
964  37650  0.9418370777  159316577935601
968  37800  0.9439561470  159316577935451
972  37914  0.9360034504  159316577935423
976  38014  0.9242274696  159316577935969
980  37812  0.9276107833  120293264372867
984  38382  0.9474156903  159316577935601
988  38538  0.9512197726  159316577935453
992  38680  0.9511189880  159316577935453
996  38796  0.9438031196  159316577935453
1000 38866  0.9238826098  159316577935453
1004 39040  0.9326313213  159316577935453
1008 39126  0.9172648307  159316577935453
1012 39348  0.9391390323  159316577935601
1016 39538  0.9523097203  159316577935453
1020 39680  0.9522520689  159316577935601
1024 39856  0.9615749322  159316577935453
- Gaps between primes in arithmetic progression, for q=4568 to 5004 step 2:
Code:
p <= 23388300000000
q     gap      CSG (conv.)   p
4568  1548552  0.8572222356  1677084447851
4570  1183630  0.8390238160  1196563621633
4572  676656   0.9313083686  3312086153
4574  864486   0.8348603294  1749438037
4576  617760   0.8078001685  464384941
4578  691278   0.8429669902  84048460189
4580  1108360  0.8026462441  890252180611
4582  1383764  0.8499365180  720395477939
4584  1141416  0.7964331851  21652890442697
4586  1371214  0.8127600152  606453831427
4588  1601212  0.8876812908  3550398242161
4590  867510   0.8724168546  5747636061659
4592  1428112  0.8464431695  7482931558789
4594  1745720  0.8491640934  9894775021751
4596  896220   0.8192055813  417572047247
4598  1549526  0.8296428911  21799960507001
4600  1283400  0.7977381722  13502858057147
4602  1090674  0.8445938334  16897337246939
4604  1749520  0.8362905821  12528155746207
4606  428358   0.8045087092  15779549
4608  926208   0.8881770694  207016317479
4610  1212430  0.8526133843  1183984521847
4612  894728   0.9475156489  618769103
4614  461400   0.8323251172  177577073
4616  1583288  0.8432163277  2500655788181
4618  1454670  0.8260526488  991355520937
4620  328020   0.8965801195  300426827
4622  1687030  0.8202913337  9089241698347
4624  550256   0.8642679215  26284901
4626  1050102  0.8210523101  3403774701511
4628  1596660  0.8144418081  17046531702059
4630  1361220  0.7957535391  16338258362707
4632  291816   0.8541090988  2708653
4634  1181670  0.8315601474  431161340839
4636  1070916  0.7908290917  74835258193
4638  834840   0.8481347282  92370239231
4640  389760   0.8488446783  8560609
4642  1425094  0.8402156795  2200086910369
4644  1114560  0.8332081592  8272678130681
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4978  1762212  0.8347191886  11084980562807
4980  806760   0.7651954247  2047787890429
4982  1509546  0.8050403273  1443995177347
4984  1046640  0.8968469262  16174195679
4986  917424   0.7982685913  276068375017
4988  1281916  0.8583537673  87830579833
4990  1347300  0.7979239899  4407482390587
4992  524160   0.8022670052  905148287
4994  1812822  0.8971218655  9687146951987
4996  1533772  0.8372621149  582820048477
4998  324870   0.8595695977  18853409
5000  1365000  0.8722366178  1407287251891
5002  750300   0.8398345295  238656883
5004  415332   0.8897546225  19150451

2022-09-09, 16:03   #60
mart_r

Dec 2008
you know...around...

15008 Posts

Quote:
 Originally Posted by mart_r I'd like to take the search for T(38,16) in A086153 up to 10^16,
No solution for T(38,16) for p < 10^16.
Meh.
Oh well...
Anyone else holding their breath for the 3rd season of "The Owl House"?

 2022-09-17, 18:29 #61 Bobby Jacobs     May 2018 52×11 Posts No. I have never heard of that show.
2022-09-18, 12:59   #62
mart_r

Dec 2008
you know...around...

83210 Posts

Quote:
 Originally Posted by mart_r 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 rgm of values of r also become larger. Find an asymptotic function f(x) ~ rgm.
What I have so far (without explicitly claiming to be correct):

Using a Poisson distributed random model, i.e. with random variables 0 < xn < 1 turned into a function equivalent to the merit mn = -log(1-xn), we want two consecutive values m1 and m2 such that m1 >= m2 and then the geometric mean rgm of (m1, m2).

For given x2, we use a function f(x2) that gives the geometric mean of m1/m2 for all m1 >= m2. We have
$$\log (f(x_2)) = \frac{\int_{x_2}^1 \log(-\log(1-y)) \: \text{d}y}{1-x_2} - \log(-\log(1-x_2)).$$

Taking all x2 into account, we would get
$$\log(r_{gm})=2 \cdot \int_0^{1-\varepsilon} (1-z) \log(f(z)) \: \text{d}z$$
and lim rgm = 4 for $$\varepsilon \to$$ 0 - numerically. I don't yet know how to prove that mathematically, but as I said, I'm sure the tools are available and I leave that as an exercise for those who are more comfortable working with integrals as I am.

Quote:
 Originally Posted by Bobby Jacobs No. I have never heard of that show.
To me it's one of the best Disney shows I've seen since Chip'n'Dale's Rescue Rangers in the early 90s.
I don't use streaming services, but I would be curious whether it's available on any of the popular platforms, and unabridged at that. I know that it's at least partially censored in some countries... but I don't want to spoiler anything Just, if you do, be careful to watch it in chronological order since it follows a single story plot.

 2022-09-19, 16:29 #63 mart_r     Dec 2008 you know...around... 26×13 Posts Where's the revolution? C'mon, people, you're letting me down! The "puzzle" with the geometric mean of consecutive gaps can be generalized: for n consecutive gaps the average ratio of the largest gap divided by the smallest gap appears to be as follows (rounded to three decimal places for n>=4): Code:  n r_gm 1 1 2 4 3 8 4 12.641 5 17.757 6 23.249 7 29.052 8 35.121 9 41.423 10 47.928 I'm afraid these numbers will give me headaches. 1.8 n (0.38 + log n) is an asymptotic facsimile for n<=1000, but we want more than this. What is it that I should ask myself?
2022-09-21, 21:52   #64
mart_r

Dec 2008
you know...around...

26×13 Posts
r_gm @ n=3 = 12.64200 ± 0.000015

All work and no pay makes me wish life wouldn't be so dull.

Here, I give you the first occurrence gaps for k=1000 up to p=2*10^14.

You'll find me in the kitchen.

PS: To this day I've never expressed my continual deep appreciation for the brilliantly derived joke that came via that calculation from this old idea of mine.
Attached Files
 GNCP_CFC_k=1000.txt (259.8 KB, 19 views)

2022-09-24, 11:57   #65
mart_r

Dec 2008
you know...around...

11010000002 Posts
I stand corrected

Quote:
 Originally Posted by mart_r r_gm @ n=3 = 12.64200 ± 0.000015

While I'm at it, here are the values up to n=13, corrected to be within +/- about 2 sigma:
Code:
 1  1
2  4
3  8
4  12.642007 ± 0.000009
5  17.75797 ± 0.00002
6  23.24943 ± 0.00003
7  29.05309 ± 0.00003
8  35.12109 ± 0.00004
9  41.42190 ± 0.00004
10  47.92674 ± 0.00005
11  54.62285 ± 0.00005
12  61.47478 ± 0.00006
13  68.49414 ± 0.00006

2022-10-01, 22:51   #66
Bobby Jacobs

May 2018

52·11 Posts

Quote:
 Originally Posted by mart_r All work and no pay makes me wish life wouldn't be so dull.
You should get paid for this.

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