Gaps between non-consecutive primes - Page 5

Bobby Jacobs · 2022-04-29, 23:22

Let an n-prime gap be the gap between a prime p and the prime n primes after p. Then, 2 and 3 are always the start of a maximal n-gap for all n. 5 is the start of a maximal n-gap if and only if the (n+2)nd prime and the (n+3)rd prime are not twin primes. 7 is the start of a maximal n-gap if and only if the (n+3)rd prime and the (n+4)th prime are not twin primes. 11 is the start of a maximal n-gap if and only if the (n+4)th and (n+5)th primes have a gap greater than 4. 13 is the start of a maximal n-gap if and only if the (n+5)th and (n+6)th primes are not twin primes, and the last 2 gaps before the (n+6)th prime are not (2, 4). Basically, every prime has a set of "forbidden prime gap combinations" such that the mth prime is the start of a maximal n-gap if and only if the last gaps before the (m+n)th prime are not one of the forbidden gap combinations. Here are the forbidden gap combinations of the first few primes.

Code:

2
[]
3
[]
5
[[2]]
7
[[2]]
11
[[2], [4]]
13
[[2], [2, 4]]
17
[[2], [4]]
19
[[2], [2, 4]]
23
[[2], [4]]
29
[[2], [4], [6]]
31
[[2], [2, 4], [2, 6], [2, 6, 4], [2, 4, 6]]
37
[[2], [4], [6], [2, 4, 8], [2, 4, 2, 10]]
41
[[2], [4], [2, 6], [4, 6], [2, 4, 6, 6], [4, 2, 4, 8], [2, 4, 2, 10]]
43
[[2], [2, 4], [2, 6, 4], [4, 2, 6], [2, 4, 6], [2, 4, 6, 2, 6], [2, 4, 2, 4, 8]]
47
[[2], [4], [4, 2, 4, 6], [4, 2, 4, 6, 2, 6], [4, 2, 4, 2, 4, 8]]

Notice that 29 just has the forbidden gaps 2, 4, 6, but 37 has the extra combinations (2, 4, 8) and (2, 4, 2, 10). That is why 29 is more common than 37.

mart_r · 2022-05-01, 14:56

Quote:

Originally Posted by Bobby Jacobs

Here are the forbidden gap combinations of the first few primes.

Very good! That's the sort of analysis I was looking for.
Do you have a program for these gap combinations?

Bobby Jacobs · 2022-05-11, 16:28

Yes. I have a program, but it is slow for primes above 47. We basically want admissible k-tuples where the total of the gaps is less than or equal to the total of the k gaps before p. Let p_m be the mth prime. Suppose the k gaps before the (m+n)th prime are one of these forbidden k-tuples. If p_m+n-p_m+n-k<=p_m-p_m-k, then p_m+n-k-p_m-k>=p_m+n-p_m. Then, the (m-k)th prime will have at least as big of an n-gap as the mth prime. Therefore, the forbidden gaps are minimal admissible k-tuples >= the k gaps before p_m.

mart_r · 2022-05-17, 17:02

CSG_max for p<=10¹⁴:

Code:

  k   gap  CSG_max     p
  1   766  0.81776202  19581334192423
  2   900  0.89182288  21185697626083
  3   986  0.92092951  21185697625997
  4  1134  0.93874248  66592576389587
  5  1170  0.91718026  66592576389551
  6  1154  0.89752827  30103357357379
  7  1148  0.88499578  14580922576079
  8   790  0.92651781  11878096933
  9  1316  0.95316163  14580922575911
 10   726  0.95096666  866956873
 11   754  0.94094924  866956873
 12   784  0.93630856  866956873
 13  1448  0.95644952  5995661470529
 14  1496  0.95744289  5995661470481
 15  1322  0.95352216  396016668869
 16  1358  0.94653445  396016668833
 17  1688  0.98369275  8281634108801
 18  1722  0.97105216  8281634108767
 19  1812  1.01651543  8281634108677
 20  1830  0.98808150  8281634108677
 21  2134  1.02168813  78736011999913
 22  2148  0.99072269  78736011999913
 23  2166  0.96394446  78736011999913
 24  2310  1.04764008  78736011999913
 25  2322  1.01591301  78736011999901
 26  2338  0.98829568  78736011999913
 27  2376  0.98009540  78736011999847
 28  2432  0.98752862  78736011999791
 29  2454  0.96623635  78736011999769
 30  2494  0.96053115  78736011999913
 31  2478  0.97621396  38986211476403
 32  2524  0.97682408  38986211476357
 33  2560  0.96892955  38986211476321
 34  2286  0.97036452  2481562496471
 35  2320  0.96392716  2481562496437
 36  2616  0.98341715  17931997861517
 37  2396  0.98957750  1933468592177
 38  2444  0.99810203  1933468592129
 39  2472  0.98638661  1933468592101
 40  2538  0.98219566  2481562496219
 41  2760  0.98030051  10631985435829
 42  2380  0.99919669  327076778191
 43  2392  0.97198960  327076778179
 44  2442  0.98739166  327076778129
 45  2470  0.97842905  327076778101
 46  2762  0.97061179  2481562496219
 47  2520  0.95456660  327076778051
 48  2776  0.94157086  1933468592101
 49  3038  0.94152718  10026387088493
 50  3092  0.95310074  10026387088439
 51  2946  0.94609699  2796148447381
 52  2976  0.93822027  2796148447381
 53  3450  0.93208471  60681682061173
 54  3224  0.92791606  10026387088493
 55  3278  0.93965214  10026387088439
 56  3096  0.92379571  2481562495661
 57  3390  0.94611179  11783179421371
 58  3560  0.93957475  29077945916363
 59  3808  0.96141677  90210824580841
 60  3764  0.95339422  55956455554739
 61  3798  0.94719704  55956455554739
 62  3852  0.95602954  55956455554651
 63  3942  0.99181087  55956455554561
 64  3976  0.98566033  55956455554561
 65  4004  1.00012038  45921691543349
 66  4020  0.98072956  45921691543333
 67  4086  0.99893031  45921691543267
 68  4140  1.00814094  45921691543213
 69  3854  1.02429119  6215409275249
 70  4292  1.05955757  45921691543061
 71  4310  1.04178765  45921691543043
 72  4332  1.02721666  45921691543061
 73  4386  1.03648387  45921691543061
 74  4062  1.03664125  6215409275041
 75  4078  1.01808582  6215409275041
 76  4128  1.02764140  6215409275041
 77  4150  1.01426227  6215409275407
 78  4200  1.02384705  6215409275357
 79  4308  1.08099942  6215409275249
 80  4328  1.06590295  6215409275249
 81  4340  1.04448708  6215409275237
 82  4380  1.04597955  6215409275177
 83  4414  1.04265662  6215409275143
 84  4516  1.09443534  6215409275041
 85  4536  1.07968013  6215409275041
 86  4548  1.05867025  6215409275029
 87  4556  1.03473951  6215409275021
 88  4578  1.02218676  6215409275041
 89  4596  1.00663763  6215409275041
 90  4620  0.99596010  6215409275041
 91  4642  0.98385445  6215409275041
 92  5020  0.96845804  36683716323913
 93  5058  0.97814135  33994032583531
 94  5146  1.00067267  36683716323913
 95  5194  1.00631376  36683716323913
 96  5278  1.03712167  36683716324039
 97  5404  1.09772451  36683716323913
 98  5418  1.07925696  36683716323899
 99  5470  1.08766762  36683716323847
100  5482  1.06802709  36683716323847
101  5526  1.07087308  36683716323791
102  5590  1.08768345  36683716323913
103  5638  1.09332314  36683716323913
104  5656  1.07811268  36683716323847
105  5704  1.08378894  36683716323847
106  5758  1.09362393  36683716323913
107  5772  1.07585272  36683716323899
108  5824  1.08431548  36683716323847
109  5830  1.06128699  36683716323841

And just above 10¹⁴, these 22 new records showed up:

Code:

 10  1528  0.96314466  102591551174059
 11  1560  0.94298881  102591551174027
 50  3450  0.97333053  102267713449991
 51  3480  0.96260938  102267713449991
 52  3562  0.99122668  102267713449879
 53  3592  0.98063297  102267713449879
 54  3634  0.97918812  102267713449807
 55  3684  0.98379105  102267713449757
 56  3714  0.97357591  102267713449757
 57  3768  0.98125523  102267713449673
 58  3798  0.97126377  102267713449673
 59  3834  0.96582204  102267713449607
 60  3874  0.96340363  102267713449567
 61  3904  0.95381038  102267713449567
 62  3958  0.96169379  102267713449483
 66  4186  1.00199403  102267713449117
 68  4324  1.03945196  102267713449117
 69  4354  1.03013486  102267713449117
 76  4658  1.03478754  101562452774609
 77  4694  1.03029216  101562452774609
 92  5304  1.01634058  102267713449117
 93  5328  1.00471893  102267713449093

@ Bobby: I'm working on a program to look for the forbidden gap combinations. If it works, it should be fast enough for primes up to at least 97 (well at least I hope so).

mart_r · 2022-05-19, 17:28

It appears my VBA code for "forbidden gap combinations" (for getting a heuristic grip on the generalized maximal gap candidates) works as it should:

Code:

 5: [ 2]
 7: [ 2]
11: [ 2], [ 4]
13: [ 2], [ 2, 4]
17: [ 2], [ 4]
19: [ 2], [ 2, 4]
23: [ 2], [ 4]
29: [ 2], [ 4], [ 6]
31: [ 2], [ 2, 4], [ 2, 6], [ 2, 6, 4], [ 2, 4, 6]
37: [ 2], [ 4], [ 6], [ 2, 4, 8], [ 2, 4, 2, 10]
41: [ 2], [ 4], [ 2, 6], [ 4, 6], [ 2, 4, 6, 6], [ 4, 2, 4, 8], [ 2, 4, 2, 10]
43: [ 2], [ 2, 4], [ 2, 6, 4], [ 4, 2, 6], [ 2, 4, 6], [ 2, 4, 6, 2, 6], [ 2, 4, 2, 4, 8]
47: [ 2], [ 4], [ 4, 2, 4, 6], [ 4, 2, 4, 6, 2, 6], [ 4, 2, 4, 2, 4, 8]
53: [ 2], [ 4], [ 6], [ 2, 4, 2, 4, 8], [ 4, 2, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 10]
59: [ 2], [ 4], [ 6], [ 4, 8], [ 2, 10], [ 2, 4, 2, 4, 6, 8], [ 2, 4, 2, 4, 6, 2, 6, 10]
61: [ 2], [ 2, 4], [ 2, 6], [ 2, 6, 4], [ 2, 4, 6], [ 2, 6, 6], [ 2, 4, 8], [ 2, 6, 6, 4], [ 2, 6, 4, 6], [ 2, 4, 6, 6], [ 2, 4, 2, 10], [ 2, 4, 2, 4, 8, 6, 4], [ 2, 4, 2, 4, 6, 8, 4], [ 4, 2, 4, 2, 4, 8, 6], [ 2, 4, 2, 4, 6, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 10]
67: [ 2], [ 4], [ 6], [ 2, 4, 8], [ 2, 6, 4, 8], [ 2, 4, 6, 8], [ 2, 4, 2, 10], [ 2, 4, 2, 12], [ 2, 6, 4, 2, 10], [ 2, 4, 6, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12]
71: [ 2], [ 4], [ 2, 6], [ 4, 6], [ 2, 4, 6, 6], [ 4, 2, 4, 8], [ 2, 4, 2, 10], [ 4, 6, 2, 6, 6], [ 2, 6, 4, 6, 6], [ 4, 2, 4, 8, 6], [ 4, 6, 2, 4, 8], [ 4, 2, 6, 4, 8], [ 4, 2, 4, 6, 8], [ 2, 6, 4, 2, 10], [ 2, 4, 6, 2, 10], [ 4, 2, 4, 2, 12], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 10, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12]
73: [ 2], [ 2, 4], [ 2, 6, 4], [ 4, 2, 6], [ 2, 4, 6], [ 2, 4, 6, 2, 6], [ 2, 4, 2, 4, 8], [ 2, 6, 4, 6, 2, 6], [ 2, 4, 6, 6, 2, 6], [ 2, 4, 2, 10, 2, 6], [ 2, 4, 6, 2, 6, 6], [ 2, 4, 2, 4, 8, 6], [ 2, 4, 2, 4, 6, 8], [ 2, 4, 6, 2, 6, 6, 4], [ 2, 4, 2, 4, 8, 6, 4], [ 2, 4, 2, 4, 6, 8, 4], [ 2, 6, 4, 2, 6, 4, 6], [ 2, 4, 6, 2, 6, 4, 6], [ 2, 6, 4, 2, 4, 6, 6], [ 2, 6, 4, 2, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 8]
79: [ 2], [ 4], [ 6], [ 4, 2, 4, 8], [ 2, 4, 2, 10], [ 2, 4, 2, 4, 6, 8], [ 2, 4, 6, 2, 6, 4, 8], [ 2, 6, 4, 2, 4, 6, 8], [ 2, 4, 2, 4, 6, 2, 10], [ 2, 6, 4, 2, 4, 2, 12], [ 2, 4, 2, 4, 6, 2, 12], [ 2, 4, 6, 2, 6, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 8]
83: [ 2], [ 4], [ 2, 6], [ 4, 6], [ 4, 2, 4, 6, 6], [ 2, 4, 2, 4, 8], [ 4, 2, 4, 2, 10], [ 4, 2, 4, 6, 2, 6, 6], [ 4, 2, 4, 2, 4, 8, 6], [ 2, 4, 2, 4, 6, 2, 10], [ 4, 6, 2, 4, 6, 2, 6, 6], [ 4, 2, 4, 6, 6, 2, 6, 6], [ 2, 4, 6, 2, 6, 4, 6, 6], [ 4, 2, 4, 6, 2, 6, 6, 6], [ 2, 4, 2, 4, 6, 2, 10, 6], [ 4, 2, 4, 6, 2, 6, 4, 8], [ 2, 4, 6, 2, 6, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 8]
89: [ 2], [ 4], [ 6], [ 2, 4, 8], [ 4, 2, 10], [ 2, 4, 2, 4, 6, 8], [ 4, 2, 4, 6, 2, 10], [ 4, 2, 4, 6, 2, 6, 4, 8], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 4, 2, 4, 6, 6, 2, 6, 4, 8], [ 4, 2, 4, 2, 4, 8, 6, 4, 8], [ 4, 6, 2, 6, 4, 2, 4, 6, 8], [ 2, 4, 6, 2, 6, 4, 6, 2, 10], [ 2, 6, 4, 2, 4, 6, 6, 2, 10], [ 2, 6, 4, 2, 4, 2, 10, 2, 10], [ 2, 4, 2, 4, 6, 2, 10, 2, 10], [ 2, 4, 6, 2, 6, 4, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 6, 10], [ 4, 6, 2, 6, 4, 2, 4, 2, 12], [ 2, 4, 6, 2, 6, 4, 2, 4, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 8]
97: [ 2], [ 4], [ 6], [ 8], [ 2, 10], [ 2, 12], [ 2, 6, 10], [ 2, 4, 12], [ 2, 6, 6, 10], [ 2, 4, 8, 10], [ 2, 6, 4, 12], [ 4, 2, 6, 12], [ 2, 4, 6, 12], [ 4, 2, 4, 14], [ 2, 4, 2, 4, 8, 6, 10], [ 2, 4, 2, 4, 6, 8, 10], [ 4, 2, 4, 6, 2, 6, 12], [ 2, 4, 2, 4, 6, 2, 16], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 12], [ 2, 6, 4, 2, 4, 6, 6, 2, 6, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 12], [ 2, 4, 2, 4, 6, 2, 10, 2, 4, 14], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 14], [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 14], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 14], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 16], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 18], [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 8, 10], [ 2, 4, 6, 2, 6, 4, 2, 4, 2, 12, 10], [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 12], [ 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 12]

Computation time: less than a minute, but I believe it is possible to do it in less than a second with some really optimised code.

Bobby Jacobs · 2022-05-22, 15:15

Very good! It seems like the popularity of a prime as the start of a maximal gap is based upon the gaps before p. However, there are weird exceptions like 29 and 37. The 2 gaps before 29 are (4, 6) and the gaps before 37 are (2, 6), but 29 is more popular than 37. If (2, 8) was an admissible gap combination, then that would be a forbidden gap combination for 29, but not 37. However, (2, 8) is not admissible.

By the way, I meant to use <= instead of >= in my previous post. The forbidden gap combinations are minimal admissible k-tuples <= the k gaps before p_m. How do I make the correct symbols for <= and >=?

2022-04-29, 23:22	#45
Bobby Jacobs May 2018 7·37 Posts	Forbidden prime gap combinations Let an n-prime gap be the gap between a prime p and the prime n primes after p. Then, 2 and 3 are always the start of a maximal n-gap for all n. 5 is the start of a maximal n-gap if and only if the (n+2)nd prime and the (n+3)rd prime are not twin primes. 7 is the start of a maximal n-gap if and only if the (n+3)rd prime and the (n+4)th prime are not twin primes. 11 is the start of a maximal n-gap if and only if the (n+4)th and (n+5)th primes have a gap greater than 4. 13 is the start of a maximal n-gap if and only if the (n+5)th and (n+6)th primes are not twin primes, and the last 2 gaps before the (n+6)th prime are not (2, 4). Basically, every prime has a set of "forbidden prime gap combinations" such that the mth prime is the start of a maximal n-gap if and only if the last gaps before the (m+n)th prime are not one of the forbidden gap combinations. Here are the forbidden gap combinations of the first few primes. Code: 2 [] 3 [] 5 [[2]] 7 [[2]] 11 [[2], [4]] 13 [[2], [2, 4]] 17 [[2], [4]] 19 [[2], [2, 4]] 23 [[2], [4]] 29 [[2], [4], [6]] 31 [[2], [2, 4], [2, 6], [2, 6, 4], [2, 4, 6]] 37 [[2], [4], [6], [2, 4, 8], [2, 4, 2, 10]] 41 [[2], [4], [2, 6], [4, 6], [2, 4, 6, 6], [4, 2, 4, 8], [2, 4, 2, 10]] 43 [[2], [2, 4], [2, 6, 4], [4, 2, 6], [2, 4, 6], [2, 4, 6, 2, 6], [2, 4, 2, 4, 8]] 47 [[2], [4], [4, 2, 4, 6], [4, 2, 4, 6, 2, 6], [4, 2, 4, 2, 4, 8]] Notice that 29 just has the forbidden gaps 2, 4, 6, but 37 has the extra combinations (2, 4, 8) and (2, 4, 2, 10). That is why 29 is more common than 37.

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2022-05-11, 16:28	#47
Bobby Jacobs May 2018 103₁₆ Posts	Yes. I have a program, but it is slow for primes above 47. We basically want admissible k-tuples where the total of the gaps is less than or equal to the total of the k gaps before p. Let p_m be the mth prime. Suppose the k gaps before the (m+n)th prime are one of these forbidden k-tuples. If p_m+n-p_m+n-k<=p_m-p_m-k, then p_m+n-k-p_m-k>=p_m+n-p_m. Then, the (m-k)th prime will have at least as big of an n-gap as the mth prime. Therefore, the forbidden gaps are minimal admissible k-tuples >= the k gaps before p_m.

2022-05-22, 15:15	#50
Bobby Jacobs May 2018 103₁₆ Posts	Very good! It seems like the popularity of a prime as the start of a maximal gap is based upon the gaps before p. However, there are weird exceptions like 29 and 37. The 2 gaps before 29 are (4, 6) and the gaps before 37 are (2, 6), but 29 is more popular than 37. If (2, 8) was an admissible gap combination, then that would be a forbidden gap combination for 29, but not 37. However, (2, 8) is not admissible. By the way, I meant to use <= instead of >= in my previous post. The forbidden gap combinations are minimal admissible k-tuples <= the k gaps before p_m. How do I make the correct symbols for <= and >=?