20211101, 05:46  #67 
Mar 2021
73_{8} Posts 
This approach is only useful when it is needed to guarantee that a number is prime. For prime gap searches that probably means only maximal gap searches where we want to make sure we haven't missed any large gaps. The approach requires knowing all primes up to sqrt(N) so it couldn't be used for very large numbers. It might be useful up to 2^{80}.

20220301, 23:49  #68 
May 2018
2·3·7^{2} Posts 
How far have you gotten in your search now? Have you found any new large prime gaps just above 2^{64}? Have you proven that 1552 and 1572 are maximal prime gaps?

20220302, 03:17  #69 
Jun 2015
Vallejo, CA/.
2·7·83 Posts 

20220507, 14:01  #70 
May 2018
2·3·7^{2} Posts 
Well, Craig has not been on this site in a while. Does anybody else know how to prove the new maximal prime gaps?

20220719, 17:34  #71  
Jun 2015
Vallejo, CA/.
10010001010_{2} Posts 
Quote:
As far as I know ATH had checked up to 2^{64} +1.05 X10^{16}. The first gaps that are still not known to be a first occurrence are
When the search is completed up to 18470057946260698231 ie 2^{64} +2.33 X10^{16}. then it will be (hopefully a new maximal gap. 

20220813, 18:29  #72 
May 2018
2·3·7^{2} Posts 
We should try to confirm these gaps ASAP. Where is the program to do it?

20230220, 18:34  #73 
Dec 2008
you know...around...
887 Posts 
I don't know whether this idea has been considered before or whether it's practical, i.e. faster than previous searches:
Searching only for primes of the form 6k+1 (or 6k1), and check for gaps > 1430. There should be one such gap every ~ 10^7 primes on average  or about 81 instances in a cycle of length 36*10^9. Then checking only these gaps for intermediate primes of the complementary form with a separate routine. A Pari program I wrote was about 7% faster with this strategy compared to a classical search, but with Craig's approach there probably would be a lot more to gain. At the very best case it would be almost twice as fast. Checking only e.g. 30k+1 would leave one gap every ~ 50th prime for a thorough check, which sounds like a great idea, but running a separate routine for every instance found  compared to a large sieve that covers the whole gamut  shows it's pretty much impractical. Maybe the best tradeoff would be achieved somewhere between the forms 6k+1 and 30k+{1,17}? 
20230220, 23:51  #74 
Jun 2015
Vallejo, CA/.
1162_{10} Posts 
To my understanding , (except for 2 and 3) all primes are of the form 6K+1 or 6k 1 so there must be something I did not quite get in the previous post.
Nevertheless the “things” that remain unsolved are a) Is 1370 the 81st Maximal gap b) is 1572 the 82nd Maximal gap c) Have any newer Smaller occurrences of first time gaps been found? I.e of these gaps? *1432 *1444 *1458 *1472 *1474 *1480 *1484 *1492 *1496 *1498 *1500 d) What is the current limit of exhaustive search over 2^{64}+10^{^16}? 
20230221, 11:59  #75 
Dec 2022
2^{6}×7 Posts 
He means sieving only one of the types mod 6 to find all possibilities for large enough gaps, then checking on the other side only within those  or something with that effect. It does look possibly attractive to do so.
I agree that it seems silly that those two maximal gaps haven't been confirmed and I'm not sure exactly what the reason is, but they should be at the top of the list for 'things to do related to prime gaps'. 
20230221, 16:50  #76 
Dec 2008
you know...around...
887 Posts 
@ rudy235: It's like Andrew said, looking only at one residue mod 6, since 1432 is a large gap that requires a merit of > 16 when looking only at those numbers. This should be rare enough that there shuold be a benefit in searching only this type, and only check those intervals for the other type (the other residue mod 6) when a large enough gap is found.
Realistically, I would expect a speedup of maybe 10 to 30%, but I'm not familiar enough with how Craig is checking the gaps after all primes are computed. Craig's code for 65bit numbers, which involves customized GPU usage, is fast  about 1.5*10^{11}/sec, there's a lot of stuff going on in parallel, but does only one Fermat test after sufficient sieving, which is not enough to call the search "exhaustive" (although the probability that one first occurrence gap has been overlooked is near zero, well, the < 10^{9} type of zero :). Doing all necessary prime tests would make the code about 10x slower. There's a lot of background info in this thread, but the search itself is on hiatus since 2021. Regarding c), beside the two new maximal gaps, Craig has only found one 1510 and one 1446, all others are smaller than 1432. Regarding d), ATH has searched up to 2^{64}+1.05*10^{16}, but even this search wasn't necessarily exhaustive, see post # 15 in the above mentioned thread. So strictly speaking we're still standing at the 64bit border with the exhaustive search. Last fiddled with by mart_r on 20230221 at 16:54 Reason: typo 
20230223, 17:38  #77 
May 2018
2×3×7^{2} Posts 

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