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 2021-11-01, 05:46 #67 CraigLo   Mar 2021 738 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 280.
 2022-03-01, 23:49 #68 Bobby Jacobs     May 2018 2·3·72 Posts How far have you gotten in your search now? Have you found any new large prime gaps just above 264? Have you proven that 1552 and 1572 are maximal prime gaps?
2022-03-02, 03:17   #69
rudy235

Jun 2015
Vallejo, CA/.

2·7·83 Posts

Quote:
 Originally Posted by Bobby Jacobs How far have you gotten in your search now? Have you found any new large prime gaps just above 264? Have you proven that 1552 and 1572 are maximal prime gaps?
I think that is a valid question. But I don’t think CraigLo would have forgotten to tell us.

 2022-05-07, 14:01 #70 Bobby Jacobs     May 2018 2·3·72 Posts Well, Craig has not been on this site in a while. Does anybody else know how to prove the new maximal prime gaps?
2022-07-19, 17:34   #71
rudy235

Jun 2015
Vallejo, CA/.

100100010102 Posts

Quote:
 Originally Posted by Bobby Jacobs Well, Craig has not been on this site in a while. Does anybody else know how to prove the new maximal prime gaps?
it is “simple” each and every prime gap over a certain number (over 1400 or 1300 is a good bet) needs to be verified from 264 to 264 +2.33 X1016.
As far as I know ATH had checked up to 264 +1.05 X1016.
The first gaps that are still not known to be a first occurrence are
• *1432
*1444
*1458
*1472
*1474
*1480
*1484
*1492
*1498
*1500
*1496
. If anyone know if any of this gaps has been proven to be a first occurrence, please let us know.

When the search is completed up to 18470057946260698231 ie 264 +2.33 X1016. then it will be (hopefully a new maximal gap.

 2022-08-13, 18:29 #72 Bobby Jacobs     May 2018 2·3·72 Posts We should try to confirm these gaps ASAP. Where is the program to do it?
 2023-02-20, 18:34 #73 mart_r     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 6k-1), 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 trade-off would be achieved somewhere between the forms 6k+1 and 30k+{1,17}?
 2023-02-20, 23:51 #74 rudy235     Jun 2015 Vallejo, CA/. 116210 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 264+10^16?
 2023-02-21, 11:59 #75 Andrew Usher   Dec 2022 26×7 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'.
 2023-02-21, 16:50 #76 mart_r     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 65-bit numbers, which involves customized GPU usage, is fast - about 1.5*1011/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 264+1.05*1016, 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 64-bit border with the exhaustive search. Last fiddled with by mart_r on 2023-02-21 at 16:54 Reason: typo
2023-02-23, 17:38   #77
Bobby Jacobs

May 2018

2×3×72 Posts

Quote:
 Originally Posted by rudy235 Nevertheless the “things” that remain unsolved are a) Is 1370 the 81st Maximal gap b) is 1572 the 82nd Maximal gap
1370 is the 73rd maximal prime gap. We really want to know if 1552 is the 81st maximal prime gap.

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