20201221, 18:30  #34 
Dec 2008
you know...around...
1176_{8} Posts 
Updates are tedious but necessary
My monthly tribute to the world of number theory.
Largest CSG found during the past 4.3 weeks: 1.264846947 (p=113,109,089 / q=3,745,830) Code:
max. p searched: q <= 1000: 1.908e+13 1000 < q <= 2690: 1.023e+13 2690 < q <= 1e+5: 1.770e+11 1e+5 < q <= 2e+5: 5.720e+10 2e+5 < q <= 5e+5: 4.600e+10 5e+5 < q <= 1e+6: 2.400e+10 1e+6 < q <= 2e+6: 1.20e+10 2e+6 < q <= 4e+6: 3.0e+9 (only even q are examined) g/[phi(q)*log²(p_{2})]: 0.9241119774 my underappreciated formula: 1.0251848498 g/[phi(q)*log²(p_{1})]: 2.2178622671 Finding a CSG above 2 by any other measure is, IMHO, impossible. But I have to be careful here since it's an open problem how large that value can actually be. Data might suggest that a global maximum depends on the ratio log(p)/log(q), in the sense that the largest CSG are attained when log(p)/log(q) is just a little above 1. It may be that CSG cannot be larger than, say, 1.2, if log(p)/log(q) is larger than 2 or thereabouts. All very sketchy at the moment, maybe I'll write a paper about it when the pandemic is over... 
20201227, 17:41  #35 
May 2018
324_{8} Posts 

20201227, 17:59  #36 
Dec 2008
you know...around...
638_{10} Posts 

20210121, 21:53  #37 
Dec 2008
you know...around...
2×11×29 Posts 
I won't give up, not yet
Search stats:
Code:
max. p searched: q <= 1000: 1.985e+13 1000 < q <= 2690: 1.211e+13 2690 < q <= 1e+5: 2.347e+11 1e+5 < q <= 2e+5: 6.245e+10 2e+5 < q <= 5e+5: 6.050e+10 5e+5 < q <= 1e+6: 2.400e+10 1e+6 < q <= 2e+6: 1.600e+10 2e+6 < q <= 5e+6: 3.000e+9 (only even q are examined) of which 1,689 meet the conventional criterion g/[\(\varphi\)(q) log²(p+g)] > 1. There is one new record by the conventional criterion: p = 938,688,203 q = 4,200,826 = 2×7×61×4,919 k = 239 g/[φ(q) log²(p+g)] = 1.239732926499... (unconventionally 1.2777045741...) 
20210305, 22:31  #38 
Dec 2008
you know...around...
1001111110_{2} Posts 
February was too short, no time for an update
Nothing to write home about anyway...
Code:
max. p searched: q <= 1000: 2.000e+13 1000 < q <= 2690: 1.489e+13 2690 < q <= 4566: 1.002e+12 4566 < q <= 1e+5: 2.755e+11 1e+5 < q <= 2e+5: 1.553e+11 2e+5 < q <= 5e+5: 8.300e+10 5e+5 < q <= 1e+6: 2.400e+10 1e+6 < q <= 2e+6: 2.100e+10 2e+6 < q <= 5e+6: 3.000e+9 5e+6 < q <= 1e+7: reserved (only even q are examined) 
20210408, 20:47  #39 
Dec 2008
you know...around...
1001111110_{2} Posts 
Ich hab noch mehr Daten...
More data, dedicated to science.
While I'm at it, I also search for the largest instance per q of the least prime in an arithmetic progression p=k*q+r. The most recent work I could find on this is from Li, Pratt, and Shakan: https://arxiv.org/abs/1607.02543. They searched all q < 10^{6}. I've searched even q < 4.4*10^{6} so far, so maybe this could be of use to somebody, anybody. (There's > 40 MB of raw data, so if a special analysis is needed, feel free to ask.) 
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