Gaps between non-consecutive primes - Page 7

mart_r · 2022-10-03, 12:45

Quote:

Originally Posted by Bobby Jacobs

You should get paid for this.

For that comment? Yeah, that was a tremendous outburst of creativity possibly worthy of a Pulitzer

But you lifted my mood, so I give a small update.
Prime scarcities with CSG > 1 are hard to find these days, but recently I got p_n = 205,465,264,987,331 which has
CSG = 1.0024950 for k = 927,
CSG = 1.0028949 for k = 941, and
CSG = 1.0063712 for k = 939.

Bobby Jacobs · 2022-10-16, 22:22

You should get paid for all of the work you do in finding prime numbers.

mart_r · 2022-10-17, 11:50

The replies that I get mean more to me than any amount of money (which is too tight to mention anyway

).

mart_r · 2022-11-25, 20:57

Not a record in terms of CSG, but a close contender - and the largest p for which a CSG (or "scarcity/paucity ratio") > 1 is known:
A region with exceptionally few k+1 consecutive primes for k=274..380 (listed only for CSG > 1)

Code:

k    k-gap  p start          CSG
274  12914  309292876045019  1.00720962
275  12950  309292876045019  1.00580100
276  12996  309292876045253  1.00890285
277  13050  309292876045019  1.01561044
278  13110  309292876045019  1.02503096
279  13156  309292876045093  1.02813121
280  13230  309292876045019  1.04390467
281  13244  309292876045019  1.03252630
282  13300  309292876044949  1.04014280
283  13318  309292876044931  1.03061882
284  13342  309292876044907  1.02384129
285  13398  309292876044731  1.03143202
286  13438  309292876044811  1.03184383
287  13518  309292876044731  1.05023223
288  13542  309292876044707  1.04343738
289  13566  309292876044683  1.03667817
290  13580  309292876044683  1.02549729
291  13590  309292876044673  1.01261680
292  13620  309292876044629  1.00864965
293  13662  309292876044731  1.00999419
294  13700  309292876044731  1.00957620
295  13728  309292876044731  1.00476373
313  14448  309292876045019  1.01377017
314  14462  309292876045019  1.00309794
315  14518  309292876044949  1.01051652
316  14552  309292876045019  1.00847219
317  14604  309292876045019  1.01416841
318  14634  309292876045019  1.01041245
319  14674  309292876044949  1.01095647
320  14736  309292876044731  1.02093989
321  14760  309292876044707  1.01462007
322  14784  309292876044683  1.00833143
323  14840  309292876044731  1.01572021
324  14892  309292876044731  1.02140336
325  14922  309292876044731  1.01768328
326  14960  309292876044731  1.01738801
327  14984  309292876044707  1.01114162
328  15008  309292876044683  1.00492570
331  15134  309292876045019  1.00919910
332  15180  309292876045253  1.01231622
333  15220  309292876045213  1.01289750
334  15256  309292876045177  1.01179308
335  15340  309292876045093  1.03098513
336  15414  309292876045019  1.04600439
337  15432  309292876045019  1.03721965
338  15484  309292876044949  1.04288334
339  15502  309292876044931  1.03413607
340  15526  309292876044907  1.02796453
341  15582  309292876044851  1.03529999
342  15622  309292876044811  1.03588507
343  15702  309292876044731  1.05337889
344  15726  309292876044707  1.04718529
345  15750  309292876044683  1.04102057
346  15768  309292876044683  1.03236619
347  15778  309292876044673  1.02041740
348  15804  309292876044629  1.01519709
349  15832  309292876044601  1.01083020
350  15850  309292876044601  1.00234451
351  15906  309292876044527  1.00959606
352  15924  309292876044527  1.00113860
379  16980  309292876043453  1.00929898
380  16998  309292876043453  1.00114922

mart_r · 2023-01-05, 22:04

New exceptional gaps, for p = 343,408,238,858,639 each:

Code:

k    gap    CSG
254  12174  1.0071454
255  12230  1.0147599
256  12248  1.0048780 (also for p-18)

Quote:

Originally Posted by mart_r

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:

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.

Since those numbers are taken from the underlying distribution process, it may also apply to primes in any admissible residue class r mod q.
As a heuristical reality check, here's a sample of 30 consecutive primes congruent to 7 mod 1983:

Code:

prime; mod 1983=7       gap    ratio    ratio    ratio    ratio
10^2023+24724407              2 gaps   3 gaps   4 gaps   5 gaps
10^2023+25533471     809064  max/min  max/min  max/min  max/min
10^2023+32002017    6468546  7.99510
10^2023+47417859   15415842  2.38320  19.0539
10^2023+53751561    6333702  2.43394  2.43394  19.0539
10^2023+59791779    6040218  1.04859  2.55220  2.55220  19.0539
10^2023+95803059   36011280  5.96192  5.96192  5.96192  5.96192
10^2023+95870481      67422  534.118  534.118  534.118  534.118
10^2023+106816641  10946160  162.353  534.118  534.118  534.118
10^2023+110156013   3339372  3.27791  162.353  534.118  534.118
10^2023+116216061   6060048  1.81473  3.27791  162.353  534.118
10^2023+117687447   1471386  4.11860  4.11860  7.43935  162.353
10^2023+126531627   8844180  6.01078  6.01078  6.01078  7.43935
10^2023+141193929  14662302  1.65785  9.96496  9.96496  9.96496
10^2023+145504971   4311042  3.40110  3.40110  9.96496  9.96496
10^2023+146421117    916146  4.70563  16.0043  16.0043  16.0043
10^2023+147230181    809064  1.13235  5.32843  18.1225  18.1225
10^2023+151517427   4287246  5.29902  5.29902  5.32843  18.1225
10^2023+154829037   3311610  1.29461  5.29902  5.29902  5.32843
10^2023+161190501   6361464  1.92096  1.92096  7.86275  7.86275
10^2023+167615421   6424920  1.00998  1.94012  1.94012  7.94118
10^2023+171010317   3394896  1.89252  1.89252  1.94012  1.94012
10^2023+172049409   1039092  3.26718  6.18321  6.18321  6.18321
10^2023+172398417    349008  2.97727  9.72727  18.4091  18.4091
10^2023+174893031   2494614  7.14773  7.14773  9.72727  18.4091
10^2023+177062433   2169402  1.14991  7.14773  7.14773  9.72727
10^2023+177895293    832860  2.60476  2.99524  7.14773  7.14773
10^2023+182031831   4136538  4.96667  4.96667  4.96667  11.8523
10^2023+190796691   8764860  2.11889  10.5238  10.5238  10.5238
10^2023+197411979   6615288  1.32494  2.11889  10.5238  10.5238

            geometric mean:  3.70023  7.86887  13.4903  20.6196

This can be generalized further to gaps between non-consecutive primes as well - either dependent on one another when running through consecutive primes (e.g. in the case k=2: 3-7, 5-11, 7-13, 11-17 etc.) or independent (by taking the differences at every other prime like 3-7, 7-13, 13-19, 19-29 etc.). Eventually these numbers appear for k=2:

Code:

   r_gm  r_gm  (geometric mean of ratio of maximal vs. minimal gap of n consecutive gaps between a prime and the k'th next prime (here k=2), limits as prime --> oo)
n  dep.  indep.
1  1     1
2  1.85  2.43
3  2.88  3.77
4  3.85  5.05
5  4.82  6.25
6  5.76  7.40
7  6.69  8.50
8  7.59  9.56
9  8.47  10.58
10 9.33  11.58
11 10.17 12.54
12 11.00 13.47
13 11.81 14.39

By now I think I've lost my audience for good... (not to mention my mind ;)

Bobby Jacobs · 2023-01-08, 23:53

You have not lost me. It is interesting that the first 3 numbers are integers:1, 4, 8. However, the next numbers are not integers. I wonder why that is.

mart_r · 2023-01-09, 19:18

Quote:

Originally Posted by Bobby Jacobs

You have not lost me. It is interesting that the first 3 numbers are integers:1, 4, 8. However, the next numbers are not integers. I wonder why that is.

You're truly my most loyal follower in this thread!
To be honest, I'm not entirely sure it's exactly 8 at n=3, rather 8.000000 ± 0.000007, ballpark. I still don't know how to pin the numbers down exactly.
I'm not so much wondering why the first few numbers are integers, there are a lot of sequences that start out with integers before fractional or even irrational numbers appear. What concerns me more is that I wasn't able to find a satisfying asymptotic approximation formula for larger n.

As a follow-up to post # 55, I've attached the data for 2*10^14 < p < 3.33333333333333*10^14, k <= 109.
Meanwhile I've also looked for T(38,16) up to 2*10^16 - assuming the attached list in post # 58 is complete -, to no avail.

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2022-10-16, 22:22	#68
Bobby Jacobs May 2018 286₁₀ Posts	You should get paid for all of the work you do in finding prime numbers.

2022-10-17, 11:50	#69
mart_r Dec 2008 you know...around... 2³·109 Posts	The replies that I get mean more to me than any amount of money (which is too tight to mention anyway ).

2023-01-08, 23:53	#72
Bobby Jacobs May 2018 436₈ Posts	You have not lost me. It is interesting that the first 3 numbers are integers:1, 4, 8. However, the next numbers are not integers. I wonder why that is.