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 2019-12-09, 22:13 #1 mart_r     Dec 2008 you know...around... 23·5·17 Posts Prime gaps in residue classes - where CSG > 1 is possible _Inspired by a paper of A. Kourbatov and M. Wolf (https://arxiv.org/pdf/1901.03785.pdf, brought to my attention via rudy235's post https://www.mersenneforum.org/showpo...34&postcount=3), I took a venture into the issue of gaps between primes of the same residue class mod q myself. _One of the first ideas was to make a list, similar to Dr. Nicely's list, for each q. We only have to look at even values of q, since the list looks practically the same for e.g. q=7 and q=14. _It would look something like this (g = gap size): Code: g/q q=2 q=4 q=6 q=8 ... 1 3 3 5 3 2 7 5 19 7 3 23 17 43 17 4 89 73 283 41 5 139 83 197 61 6 199 113 521 311 7 113 691 1109 137 8 1831 197 2389 457 9 523 383 1327 647 10 887 1321 4363 1913 11 1129 1553 8297 673 ... _But it's rather pointless to collect an arbitrary amount of data like that, so I thought it would be more interesting to take a sort of perpendicular approach and look for record values of merit and Cramér-Shanks-Granville (CSG) ratio in each row of the list above. _(Don't ask me why I wrote "perpendicular" here, it's just an image that conjured up in my head for the approach and I can't seem to think of any better word for it at the moment.) _This would give us a single list like the one for the ordinary prime gaps where record hunters can hunt for new heights in terms of merit and CSG ratio. _A maximum value for both merit and CSG ratio can be found for each g/q $\in N$ at certain q with prime p: Code:  record record g/q merit q p CSG ratio q p 1 1.044 30 7 0.35468 6 5 2 1.760 30 19 0.44989 4 5 3 2.267 30 61 0.46810 6 43 4 2.782 90 29 0.56356 36 13 5 3.118 210 503 0.52731 16 17 6 3.518 420 503 0.53135 240 47 7 4.103 420 379 0.58809 66 229 8 4.293 840 577 0.62318 40 89 9 4.676 840 1129 0.58533 62 19 10 5.030 1260 797 0.62602 66 941 11 5.326 1470 1559 0.72822 52 29 12 5.607 1890 2141 0.64058 140 701 13 5.962 2310 21211 0.67268 372 263 14 6.481 1050 5647 0.72290 130 461 15 6.542 9240 7621 0.77047 46 197 16 6.969 3150 2953 0.75173 1140 1933 17 7.267 30030 10037 0.75225 594 1213 18 7.630 4410 1223 0.73922 4410 1223 19 7.534 10920 62743 0.74289 174 5413 20 8.349 9240 24413 0.72892 2184 7841 21 8.395 11550 62597 0.75321 3822 557 22 9.039 5250 887 0.84885 5250 887 23 8.969 13860 88397 0.82106 70 7151 24 9.067 5070 2053 0.84321 5070 2053 25 9.126 117810 100003 0.76086 46 3109 26 9.708 9240 278459 0.91336 456 7283 27 10.044 16170 215077 0.87722 82 1553 28 10.329 11760 14759 0.87084 11760 14759 29 10.351 66990 341287 0.84785 2028 12109 30 11.239 43890 220307 0.97552 3696 8539 31 10.720 24150 225077 0.81048 24150 225077 32 10.647 330330 1929071 0.85442 400 9371 33 11.739 120120 655579 0.81450 11850 76607 34 11.541 53130 1877773 0.87622 3528 59221 35 11.542 35490 1155923 0.82734 1764 159737 36 12.640 131670 141587 0.97006 444 35257 37 12.140 92400 864107 0.92188 558 58207 38 12.884 49980 146117 0.93973 49980 146117 39 12.669 189420 906473 0.88131 31710 593689 40 13.387 60060 4654417 0.92989 4420 3019 ... 209 1.14919 18692 190071823 (largest CSG ratio found by Kourbatov and Wolf) ... _Here, for consistency and because otherwise the numbers for smaller p tend to be "skewed", I used Gram's variant of Riemann's prime counting formula $R(x)=1+\sum_{n=1}^\infty \frac{log^nx}{n\hspace{1}n!\hspace{1}\zeta(n+1)}$ for merit $M=\frac{R(p+g)-R(p)}{\phi(q)}$ and $CSG=\frac{M^2\hspace{1}\phi(q)}{g}$ _(That phi doesn't look quite right there... building TEX expressions is tedious.) _In English, this here is looking for large prime gaps of size g=k*q in terms of merit and CSG ratio for even q with smallest prime p such that p+g is also prime and p+i*q is composite for all 0