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 2019-06-10, 11:39 #12 robert44444uk     Jun 2003 Oxford, UK 111011001112 Posts Here is a graph of the average gap as a merit for ranges of 1e7 from 0 to 1e11 - i.e. 10,000 data points. I'm not sure how to express the best fit curve - on there is a Bill Gates "best fit" assuming a simple logarithmic relationship. Attached Thumbnails
 2019-06-10, 15:52 #13 CRGreathouse     Aug 2006 2·2,963 Posts Of course for things growing logarithmically you want to look at big things, not small things. Here are the gaps following the first 100 twin primes after 10^500: 532, 834, 2908, 96, 100, 1402, 2670, 948, 456, 612, 606, 2964, 186, 312, 268, 2580, 658, 2754, 2610, 466, 1468, 94, 108, 648, 1362, 1128, 2956, 2382, 3306, 210, 1600, 892, 996, 492, 336, 1318, 1536, 1570, 466, 664, 408, 348, 592, 3880, 712, 1590, 570, 376, 1920, 150, 1690, 888, 1276, 1194, 2890, 880, 450, 768, 1446, 150, 478, 2178, 1620, 2278, 294, 262, 12, 906, 600, 64, 1686, 1320, 226, 486, 2680, 1140, 556, 660, 270, 282, 2290, 2574, 1048, 1078, 1128, 112, 2176, 1708, 48, 970, 1038, 90, 1084, 1678, 1344, 570, 810, 804, 1356, 1864 The average length is 96.79% of the expected length, which is about a third of a standard deviation away from the expected length (using the exponential distribution). In other words, things seem just about as ordinary as they could be up here. Between this and your large numbers this seems like fairly strong evidence for a low-order term (big effect at small numbers, small effect at large numbers).
2019-06-10, 16:15   #14
robert44444uk

Jun 2003
Oxford, UK

5·379 Posts

Quote:
 Originally Posted by CRGreathouse Of course for things growing logarithmically you want to look at big things, not small things. Here are the gaps following the first 100 twin primes after 10^500: 532, 834, 2908, 96, 100, 1402, 2670, 948, 456, 612, 606, 2964, 186, 312, 268, 2580, 658, 2754, 2610, 466, 1468, 94, 108, 648, 1362, 1128, 2956, 2382, 3306, 210, 1600, 892, 996, 492, 336, 1318, 1536, 1570, 466, 664, 408, 348, 592, 3880, 712, 1590, 570, 376, 1920, 150, 1690, 888, 1276, 1194, 2890, 880, 450, 768, 1446, 150, 478, 2178, 1620, 2278, 294, 262, 12, 906, 600, 64, 1686, 1320, 226, 486, 2680, 1140, 556, 660, 270, 282, 2290, 2574, 1048, 1078, 1128, 112, 2176, 1708, 48, 970, 1038, 90, 1084, 1678, 1344, 570, 810, 804, 1356, 1864 The average length is 96.79% of the expected length, which is about a third of a standard deviation away from the expected length (using the exponential distribution). In other words, things seem just about as ordinary as they could be up here. Between this and your large numbers this seems like fairly strong evidence for a low-order term (big effect at small numbers, small effect at large numbers).
That sounds right to me

 2019-06-11, 09:57 #15 robert44444uk     Jun 2003 Oxford, UK 5×379 Posts Connected to this is the non-twin primes between twin primes sequence A048614. The sequence is quite interesting as it is quite lumpy. Look at the values for twin prime pi 21, 30, 35 and 50 for example and the 101 primes between 850351 and 851801 in the results shown below, which corresponds to the first merit 10 twin prime gap in our table of (242,141725,Fischer,10.284072) An extension of the series for n<1e8 and first instance of a given number of primes between twin primes is given below - this series is not on OEIS. The first value not appearing is 110 primes. The largest number of primes is 149, between 32822371 and 32825201. The values are given for (a) the larger of the smaller twins, (b) the smaller of the larger twin, and the number of intervening non-twin primes, rather than the twin prime pi shown in the OEIS page. I have excluded the special case of 3,5,7 as these are two adjoining twins. Code: 7 11 0 19 29 1 43 59 2 109 137 3 73 101 4 2269 2309 5 1093 1151 6 463 521 7 1321 1427 8 1153 1229 9 349 419 10 5743 5849 11 3001 3119 12 5281 5417 13 10141 10271 14 1489 1607 15 9463 9629 16 883 1019 17 661 809 18 13009 13217 19 9043 9239 20 15361 15581 21 8629 8819 22 28753 29021 23 83719 84059 24 13399 13679 25 18541 18911 26 14629 14867 27 44773 45119 28 54013 54401 29 60259 60647 30 59671 60089 31 142159 142589 32 77713 78137 33 61561 61979 34 178933 179381 35 26263 26681 36 122869 123377 37 293263 293861 38 89071 89519 39 24421 24917 40 167863 168449 41 137341 137867 42 384481 385079 43 289243 289841 44 367651 368231 45 751633 752201 46 120079 120689 47 682699 683477 48 1022509 1023227 49 1663549 1664459 50 813301 814061 51 62299 62927 52 938059 938879 53 1442071 1442921 54 2118301 2119259 55 698419 699287 56 2319433 2320361 57 439429 440177 58 926671 927629 59 2465539 2466491 60 1409791 1410707 61 224131 224909 62 251971 252827 63 4113931 4115051 64 1770493 1771421 65 6036271 6037391 66 687523 688451 67 2186839 2187959 68 1935343 1936397 69 187909 188831 70 7980061 7981319 71 7491853 7493249 72 688453 689459 73 5408989 5410289 74 3447643 3448799 75 3261829 3263081 76 2913331 2914487 77 6701803 6703001 78 10908661 10909991 79 27120409 27121889 80 2041201 2042399 81 13613713 13615109 82 11846143 11847497 83 2635909 2637197 84 17267911 17269409 85 2400163 2401547 86 18990781 18992261 87 2868961 2870471 88 5544559 5545997 89 13174321 13175879 90 4498369 4499837 91 43118083 43120079 92 21774451 21776141 93 19909951 19911509 94 21501373 21503357 95 9923989 9925709 96 4869913 4871441 97 18694903 18696521 98 23982559 23984069 99 30068149 30069929 100 850351 851801 101 45486073 45488117 102 28215619 28217507 103 29767303 29769011 104 87559039 87560969 105 39126961 39129071 106 31811629 31813547 107 42534601 42536909 108 80723563 80725691 109 Non consecutive first instances are: Code: 61759333 61761449 111 99745201 99747377 112 48898561 48900767 113 72766699 72768977 114 37813579 37815707 115 49613953 49616339 116 93782419 93784841 117 80680423 80682659 118 70722781 70725071 119 61177201 61179359 122 53262361 53264681 123 93096823 93099131 124 50630623 50633027 125 17382481 17384669 126 74240911 74243591 127 96947113 96949547 128 30752233 30754487 129 75672343 75675011 133 78794059 78796691 135 32822371 32825201 149 Last fiddled with by robert44444uk on 2019-06-11 at 10:05
 2019-06-11, 14:38 #16 Dr Sardonicus     Feb 2017 Nowhere 2·52·71 Posts Mindlessly playing with the formulas, we get an "expected" gap of length k*log2(x) between consecutive pairs of twin primes of size x (x large), for some k (which is probably around either 1.5 or .75). The "expected" number of primes in (x, x + k*log2(x)) would be k*log(x). I'm sure there's wide variation.

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