Conjectures with one k remaining

[gd_barnes]

Originally Posted by gd_barnes

Hum, I wonder if the program "nash" sieves slightly higher than P=511? I'll try different P-depths as an experiment.

Gary

Karsten,

This is not jiving at all with your rieselprime.de site. Can you do me a favor? Run the "Nash" program on 5*2^n-1. Then do a sieve for n=100001 to 110000 to P=511 for the same. Your site shows a weight of 1650. The sieve leaves 2132 candidates remaining. A substantial difference. Something is not right.

I don't want to post these weights if there is not a consistency with how they are calculated for base 2.

I also tried the sieve for 301*k^n-1 with different parameters. To get closer to the Nash weight shown on your site of 2158, I increased the sieve depth to P=600 for n=100001 to 110000. That came in at 2166, which was as close as I could get. (P=601 drops 90 candidates.) I then tried tweaking the n-range. In no case for any n=10K range for n<200K could I get that low by only sieving to P=511. But using a sieve depth of P=600, I was able to get exactly 2158 candidates remaining for n=80001 to 90000.

I wonder what is going on here? Clearly the Nash program is doing something different than just pure sieving. I wonder if it is doing a mathematical expectation based on the long-term density and overlap of various factors up to some depth. (Perhaps in a manner similar to the way sr1sieve/sr2sieve very accurately calculate the expected # of factors in a sieving range.)


Gary

[kar_bon]

Originally Posted by gd_barnes

Run the "Nash" program on 5*2^n-1. Then do a sieve for n=100001 to 110000 to P=511 for the same. Your site shows a weight of 1650. The sieve leaves 2132 candidates remaining. A substantial difference. Something is not right.

The command with srsieve (as mentioned) gives 2132 candidates left after sieving.

The nash-command (WIN-exe, 237189 bytes) gives 2180 candidates (take the first value given!).

I've noticed, that some of the small-k-weights are different from this one: k=5 is quite too low. Perhaps the program I used in 2007 was one of the first and was changed from Thomas then. As I can say for now, only for k=5 the difference is so high. I will check this further today.

The source-code for 'nash' is available at the given link, i try to figure the difference out.

[kar_bon]

The original program from Chris Nash to calculate the 'weight' named 'psieve' is available here.

Example for Riesel-Base2 k=5:

Calling "psieve3 -e -r -b2 5" reports 2180 candidates unsieved.

This is the same value with the command 'nash 5'!

Calling 'srsieve -q -n 100001 -N 110000 -P 511 -G "5*2^n-1"' left 2132 candidates unsieved.


Example for Proth-Base 2 k=5:

"psieve3 -e -p -b2 5" reports 891 pairs left.

srsieve -q -n 100001 -N 110000 -P 511 -G "5*2^n+1" gives 922 pairs left.

'nash' is only for the Riesel-side!

I will use 'nash' further to obtain the weight (it's easier in calling) and check the values given at www.rieselprime.de.

Other bases:

psieve3 -e -p -b9 2036 gives 448 pairs left for 2036*9^n+1 (srsieve 502).
psieve3 -e -r -b23 404 gives 590 pairs left for 404*23^n-1 (srsieve 580).

Note: psieve3 -e -r -b22 3656 gives no result! k=3657 gives 3802!
I don't know, why psieve3 won't give a result for 3656*22^n-1.

PS:
Testing with NewPGen:

The smallest p-value to search up to is p=1000 so this is not compareable with 'nash.exe'.
But: srsieve and NewPGen generates the same pairs left up to p=1000 (Tested with 5*2^n+/-1)!

[gd_barnes]

It appears to me that you made an error ONLY on k=5 on your pages. In looking over about 10 others, they all looked correct or close to the actual candidates remaining on a sieve to P=511 using srsieve. I had always thought that weight seemed a little low after running RPS's k=5 drive for quite a while.

Based on your analysis here, what do you think is most accurate to show here? I'm now thinking that the simple sieve using srsieve to P=511 is close enough and still gives people a good idea of the weight and hence chance of finding a prime. But if there is some form of the Nash or psieve programs that give a little more accurate result, then we can certainly use those.

[henryzz]

Originally Posted by gd_barnes

It appears to me that you made an error ONLY on k=5 on your pages. In looking over about 10 others, they all looked correct or close to the actual candidates remaining on a sieve to P=511 using srsieve. I had always thought that weight seemed a little low after running RPS's k=5 drive for quite a while.

Based on your analysis here, what do you think is most accurate to show here? I'm now thinking that the simple sieve using srsieve to P=511 is close enough and still gives people a good idea of the weight and hence chance of finding a prime. But if there is some form of the Nash or psieve programs that give a little more accurate result, then we can certainly use those.

I think psieve will do other bases.

[kar_bon]

psieve3 and srsieve will do for every base and k.
I don't know the restrictions in values to choose for psieve3, but I think srsieve got higher possible values for k and b! And another advantage: srsieve will/could developed further/changed if any issues/requirements occurs/needed!

[gd_barnes]

OK, I'll list all the weights later today using srsieve for a sieve to P=511 for n=100001 to 110000. In other words, just using Karsten's list. I'll then run a few myself for the new k's since he made the list.

If something better comes along that can be used for all bases, I'll be glad to change them in the future.

[gd_barnes]

Recent work by Serge (Batalov) and Paleseptember have now put all bases < 220 with one 1k remaining at n>=50K. Nice work everyone.

I'm only seeing 4 bases < 200 with one 1k remaining that are at n<100K (one was recently reserved to n=100K). Having all 1k b<200 at n>=100K would be a good next target to shoot for.

[mdettweiler]

Originally Posted by gd_barnes

I'm only seeing 4 bases < 200 with one 1k remaining that are at n<100K (one was recently reserved to n=100K). Having all 1k b<200 at n>=100K would be a good next target to shoot for.

Ooh, I can help with that! The only problem is that all my resources are 32-bit, which makes for rather less than optimal sieving, not to mention that only my dualcore is readily available for sieving. Gary, I remember you offered previously to help me out in situations like this by offering some 64-bit sieving CPU time...mind if I take you up on that offer now? Essentially, if you could just start sieving through some of the bases <200 with 1 k remaining that are at n<100K from the low end up, I could run them through my quad for PRPing as soon as you're done.

Right now my quad doesn't have any active large-scale efforts going, so tackling these bases seems as good a next project as any.

Edit: After looking at the lists in this thread, I see that the only two such bases with this criteria that remain unreserved are R181 and R182, both at 50K. Those, therefore, would be the ones I'd wish to tackle if sieve files could be provided.

[gd_barnes]

Originally Posted by mdettweiler

Ooh, I can help with that! The only problem is that all my resources are 32-bit, which makes for rather less than optimal sieving, not to mention that only my dualcore is readily available for sieving. Gary, I remember you offered previously to help me out in situations like this by offering some 64-bit sieving CPU time...mind if I take you up on that offer now? Essentially, if you could just start sieving through some of the bases <200 with 1 k remaining that are at n<100K from the low end up, I could run them through my quad for PRPing as soon as you're done.

Right now my quad doesn't have any active large-scale efforts going, so tackling these bases seems as good a next project as any.

Edit: After looking at the lists in this thread, I see that the only two such bases with this criteria that remain unreserved are R181 and R182, both at 50K. Those, therefore, would be the ones I'd wish to tackle if sieve files could be provided.

OK, I'll start sieving R181 and R182 for n=50K-100K later today. Go ahead and post a regular reservation in the bases 101-250 thread. I'll show them as reserved by you in this thread.

These bases won't keep your quad busy for very long. The natural progression of things on your effort would make S208, R214, and R221 for n=50K-100K the next ones to tackle. S208 is heavier weight so would take longer unless a prime is found quickly. Let me know if you might be interested in those.

Another thing that I'd like to see tackled is the bases <= 250 that have 2 or 3 k's remaining that are only at n=25K. There are quite a few of those. It would be nice to see if we could add them to this thread (or prove them) by knocking out a k or 2 on them. That's something I'll probably start tackling myself over the next few months if others haven't done most of them already.

Admin edit: Moved follow up response/discussion about 2k & 3k remaining bases to "recommended bases/efforts" thread.


Gary

[gd_barnes]

I moved the discussion about the 2k & 3k remaining bases to the recommended bases thread.