20230223, 23:27  #78 
Jun 2015
Vallejo, CA/.
1161_{10} Posts 

20230224, 19:41  #79 
Dec 2008
you know...around...
1566_{8} Posts 
SOE all over again?
Perhaps it would be expedient to consider a complete trial division sieve on large intervals in order for the search to remain exhaustive.
Not sure what's possible in terms of efficient computation, but I'm envisioning a large bitmap, for example 32 GB in size, which could contain almost 3e12 numbers if a wheel size of 30030 is used and only numbers of the form 6k+1 are being looked at (a gap of >= 1432 should appear every about 5e8 integers, only those would have to be checked again for 6k1 inbetween). To compete, we expect a throughput of at least, say, 3e10 per second. Trial division needs about 210 million primes (i.e. on top about 200 MB to store the halved differences between primes, one per byte), where I guess memory writing is the bottleneck for small primes, "mod"ing for large primes may be similarly slow (a simplification, I know, there's certainly more to it). Searching for gaps in the sieved interval is the next possible bottleneck, I don't know how much can be done in parallel. If one such interval could be dealt with in less than two minutes, we'd be happy. Is this worth any thought? What are the most critical hurdles we are facing? 
20230305, 17:14  #80 
May 2018
293 Posts 
Yes. This is a good idea. It does not seem to have any hurdles. You did primes up to 2^{64} quickly, so this should be quick, too.

20230306, 19:01  #81 
Dec 2008
you know...around...
1101110110_{2} Posts 
Me? Not so much, my performance as a participant back in 2018 was only mediocre, remember that I didn't find any first occurrences. Unlucky me
Also I don't know how to write code comparable to that of Robert Gerbicz's. It takes way more skill and dexterity than a few lines of Pari or Basic... 
20230320, 12:30  #82 
Dec 2022
2^{3}×5×11 Posts 
The obvious question is: to what extent can that code be reused in a search beyond 2^64? Also, because of the smaller interval required (less than 2^55), it doesn't need to be as efficient.

20230429, 19:44  #83 
May 2018
125_{16} Posts 
It should not be that much harder. When are we going to start the search?

20230518, 12:33  #84 
Dec 2022
2^{3}×5×11 Posts 
I still want to follow up this topic. What would count as an official verification, and who's deciding? I have written a segmented sieve but it's also limited to 2^64; as I said, though, efficiency hardly matters for this small range. It doesn't seem that it should be difficult to come up with something that should work, but again what would be necessary to have the results counted as official?

20230518, 14:35  #85 
Jan 2007
Germany
1010100101_{2} Posts 
I have a tiny idea...
Every prime p1 is divisible by 2 ...maybe so you can push up the limit to 2^65. Example: Range 11 to 41 > k= 5 to 20 ; we have limit = 20 Factor 3 kills: k=7,10,13,16,19 Factor 5 kills: k=7,12,17 So primes are: 2k+1, with k=11,14,15,18,20 > 23,29,31,37,41 > limit Largest GAP is (1815)*2=6 > 31,37 Last fiddled with by Cybertronic on 20230518 at 14:39 
20230519, 12:51  #86 
Dec 2022
2^{3}·5·11 Posts 
Yes, I probably thought of that, once ... but realistically, the bigger issue is again that I'm not going to spend time on a program that will be useful only for this unless I know it will be accepted as an official verification by whoever makes the decisions on this  after all, someone else already did write a program and found the new gaps, why duplicate that otherwise?
I would not go much past the limit required to verify the new gaps anyway; my resources are far short of the distributed effort that originally went to 2^64. 
20230519, 20:17  #87 
Dec 2008
you know...around...
2·443 Posts 
I am testing a SOE implementation in Excel/VBA, thus far I have the differences of the primes up to 75011^22 (which would suffice for a search up to about 3.166e19) packed into an array of 131469743 short integers (16bit), with two differences per integer.
Currently the division of the 65bit offset number by the upto33bit numbers is the bottleneck  it took 6 minutes in total to churn through an interval of 600 million! , so I'm wondering whether it would be faster to store the offsets for the sieve and the increment values for each interval to be sieved in another two array variables, so we get rid of the division step at the cost of a memory readout. Apart from the few 33bit numbers we would need, most of the numbers to do TDing with could be long integers (32bit), and we would need about 2 x 203 million of them (plus a couple more for the 33bit TD primes), so storing the prime differences, offset and increment values would take about 2 GB of RAM, which is already too heavy for my programming environment. My 6 min per 6e8 was a first test run where for the sieve I used an array of 300 million 16bit numbers with only one bit per number used. I wanted to rewrite it to use 32bit integers and all bits of them for the sieve, yet with the planned setup it can't possibly be much faster than 1e8 per second. I mean, if ever my code should work properly, I would run it, but be prepared that it would take a very long time for me to give updates on the progress... Again, my idea from before: if sieving is the main bottleneck it might make sense to only search one residue class mod 6 and look for gaps >= 716 (= 1432/2), and check the complementary residue class (possibly in a separate, parallel routine) if such a gap is found. If searching for the gaps in the sieved interval is the bottleneck, then it should be more efficient to sieve as usual, as larger intervals can then be skipped in the sieved array. 
20230519, 22:43  #88 
May 2018
293 Posts 
I cannot wait for the gaps above 2^{64} to be found. Let's get it started!

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