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 2020-12-21, 18:30 #34 mart_r     Dec 2008 you know...around... 11768 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) P.S., @ Bobby: comparison between three ways of calculating CSG, in the case of q=3,613,418 / p1=487,021: g/[phi(q)*log²(p2)]: 0.9241119774 my underappreciated formula: 1.0251848498 g/[phi(q)*log²(p1)]: 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...
2020-12-27, 17:41   #35
Bobby Jacobs

May 2018

22·53 Posts

Quote:
 Originally Posted by mart_r g/[phi(q)*log²(p2)]: 0.9241119774 my underappreciated formula: 1.0251848498 g/[phi(q)*log²(p1)]: 2.2178622671
I am sorry that your formula is underappreciated. I will appreciate it more. I hope more people use it.

2020-12-27, 17:59   #36
mart_r

Dec 2008
you know...around...

2×11×29 Posts

Quote:
 Originally Posted by Bobby Jacobs I am sorry that your formula is underappreciated. I will appreciate it more. I hope more people use it.

2021-01-21, 21:53   #37
mart_r

Dec 2008
you know...around...

27E16 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)
The most recent inventory contained 2,849 exceptionally large gaps by the sum(Ri') measure,
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...)
Attached Files
 PGAP_records per q 2021-01-21.txt (56.9 KB, 24 views)

 2021-03-05, 22:31 #38 mart_r     Dec 2008 you know...around... 2·11·29 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) Attached Thumbnails
2021-04-08, 20:47   #39
mart_r

Dec 2008
you know...around...

2·11·29 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 < 106. I've searched even q < 4.4*106 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.)
Attached Files
 results_all_15e12_q=[1002...2000].zip (828.3 KB, 3 views) PGAP_records per q 2021-04-07.txt (132.7 KB, 3 views) Least primes in AP.txt (7.0 KB, 2 views)

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