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 2022-04-29, 23:22 #45 Bobby Jacobs     May 2018 25810 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.
2022-05-01, 14:56   #46
mart_r

Dec 2008
you know...around...

2·373 Posts

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?

 2022-05-11, 16:28 #47 Bobby Jacobs     May 2018 2×3×43 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 pm be the mth prime. Suppose the k gaps before the (m+n)th prime are one of these forbidden k-tuples. If pm+n-pm+n-k<=pm-pm-k, then pm+n-k-pm-k>=pm+n-pm. 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 pm.
 2022-05-17, 17:02 #48 mart_r     Dec 2008 you know...around... 2×373 Posts CSGmax 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).
 2022-05-19, 17:28 #49 mart_r     Dec 2008 you know...around... 2·373 Posts 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.

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