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Old 2020-03-27, 19:05   #10
mart_r
 
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Dec 2008
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Quote:
Originally Posted by Bobby Jacobs View Post
Interesting. What is the exact method for organizing the gaps in the first table? Why, for example, is the gap of 2 between 3 and 5 not in the list?
Recap time. In all mathematical clarity, I think:

For each positive integer k, there is a set of values {q,p} (q: positive integer, p: prime number) such that the value for CSG is a maximum. You can think of k as the number of steps in the arithmetic progression p+q*i, where p and p'=p+q*k are prime, and for each positive integer i<k, p+q*i is composite. If either q or p is larger than a certain threshold depending on k and CSG_max, then CSG cannot be larger than CSG_max for that respective k. Furthermore, see the explanation at the end of this post. I have to elaborate a bit more, using my most recent data...

Code:
 k        gap        q         p  CSG_approx (*)
 1          6        6         5  0.3022785196
 2          4        2         7  0.4198790468
 3         18        6        43  0.4580864612
 4        144       36        13  0.5078937563
 5         80       16        17  0.4945922381
 6        216       36       181  0.5127749768
 7        420       60       491  0.5785564482
 8        320       40        89  0.6047328717
 9        558      660       509  0.5563553702
10        660       66       941  0.6217267610
11        572       52        29  0.6957819573
12       1680      140       701  0.6322530512
13       4836      372       263  0.6532943787
14       1820      130       461  0.7120451019
15        690       46       197  0.7591592880
16      18240     1140      1933  0.7397723670
17      10098      594      1213  0.7408740922
18      79380     4410      1223  0.7223904024
19       3306      174      5413  0.7408652838
20      43680     2184      7841  0.7221957974
21      80262     3822       557  0.7356639423
22     115500     5250       887  0.8308343056
23       1610       70      7151  0.8202819418
24     121680     5070      2053  0.8291269738
25       1150       46      3109  0.7597500489
26      11856      456      7283  0.9098465138
27       2214       82      1553  0.8737278189
28     329280    11760     14759  0.8621806027
29      58812     2028     12109  0.8428999967
30     110880     3696      8539  0.9673158356
31     748650    24150    225077  0.8075674871
32      12800      400      9371  0.8520823935
33     391050    11850     76607  0.8108498874
34     119952     3528     59221  0.8737815728
35      61740     1764    159737  0.8267138843
36      15984      444     35257  0.9691356428
37      20646      558     58207  0.9212072554
38    1899240    49980    146117  0.9347761396
39    1236690    31710    593689  0.8794865058
40     176800     4420      3019  0.9204205455
41     212790     5190    479023  0.8758770439
42     294336     7008     15241  0.9157336406
43     128742     2994    113209  0.8850050549
44     194568     4422     62929  1.0347442307
45     754110    16758    333857  0.9208075667
46   11408460   248010    197963  0.8559649162
47    1639830    34890    130241  0.9537642386
48    2903040    60480   1828019  0.9360724738
49      66542     1358     29669  0.9501377450
50    8389500   167790   5943139  0.9150876319
51   14372820   281820  13354567  0.8816531164
52    2717520    52260   1431047  0.9780119273
53    2413620    45540    355417  1.1428167595
54    1343952    24888    135349  0.9670359549
55     229350     4170   1409633  1.0239918543
56    1172080    20930    801337  0.9276488991
57    1393650    24450   2403677  0.9627968462
58    5614980    96810  14224709  0.9172670989
59   19866480   336720    330791  0.9322437089
60   62546400  1042440   2426279  0.9655595927
61     570228     9348   1917871  0.9276525018
62    6145440    99120  14717069  0.9901575968
63     532602     8454    355339  1.0661299147
64    3225600    50400  21226511  0.9750409914
65     208650     3210   3415781  1.0821910171
66    1216512    18432    345577  1.0553714212
67   15812670   236010    800977  1.0100437615
68     964512    14184    697979  1.0519847511
69    1820910    26390   2449313  1.0007068828
70    1016260    14518     71713  1.0299861032
71   12309270   173370   8843699  0.9783007566
72   89555760  1243830  28312943  0.9682330236
73   99430380  1362060  48296291  1.0123498941
74    4013316    54234   1929793  1.0197942618
75  126094500  1681260   3818929  1.0382605330
76   98090160  1290660   1729477  1.0909304152
77   31955154   415002   5752739  0.9847660715
78    2157480    27660  13074917  1.0809486020
79     316790     4010    726611  1.0553141458
80   17746560   221832   3144419  1.0047285893
81   20655000   255000   7827217  1.1632336984
82    2972664    36252   5323187  1.0695381429
83   43562550   524850   1901563  1.1083998142
84  117356400  1397100   1629601  1.0126819069
85   16106820   189492    270509  1.0396328686  (q>2e6 TBD)
86   47941560   557460  49222847  1.0463006323  (q>1e6 TBD)
87   13075056   150288   1108727  1.1084866852  (q>1e6 TBD)
88  130738608  1485666   7421363  1.0424690911  (q>3e6 TBD)
(*) For this table, the formula used is

CSG_{approx}\hspace{1}=\hspace{1}\frac{[R(p+gap+\frac{q}{2})-R(p+\frac{q}{2})]^2}{gap*\varphi(q)}

for comparative reasons. (You may notice the outcome is slightly different compared to my first table, as the formula is slightly different.) The sum of derivatives of R(x) as explained in earlier posts is to be preferred IMHO, but it slows down the searching process terribly; the approximation formula as given here is the best for this purpose. Perhaps I should make some error analysis though.

One issue I have to mention is that I only look at even values of q, since the values for the initial primes p are the same for odd q/2, except when the initial prime is 2. For 2 \equiv q (mod 4), this leaves the possibility to assign q=q/2 and concurrently k=2*k. Scanning the table, we see that CSG_max at k=60 is smaller than at k=30, and at k=72 it's smaller than at k=36, this would have had consequences if q (3696 and 444 respectively) would not be divisible by 4. For example, if q was 446 at k=36, I should list the same p with q=223 at k=72. But that's not the end of the story - with odd q, the values for CSG are slightly larger. If I should also include odd values of q in my search, a couple of values in the table might be different.
(I've been thinking about this problem of making the CSG values comparable for waaay too long, somebody please stop me...)



Using the formula above, the gap between 3 and 5 has CSG=0.2956906641, where q=2 and k=1, so it's smaller in terms of CSG than the one in the table. This gap can also be represented with q=1 and k=2, in this case CSG would be 0.3127125473, which is again smaller than the value for k=2. That's way it's not listed.
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