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Old 2019-06-10, 11:39   #12
robert44444uk
 
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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.
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Old 2019-06-10, 15:52   #13
CRGreathouse
 
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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).
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Old 2019-06-10, 16:15   #14
robert44444uk
 
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Quote:
Originally Posted by CRGreathouse View Post
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
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Old 2019-06-11, 09:57   #15
robert44444uk
 
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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
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Old 2019-06-11, 14:38   #16
Dr Sardonicus
 
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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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