Pick Your Poisson: P-1 Bounds

[storm5510]

My old friend, P-1 bounds, is tasking me again. I have an exponent in the area of 99,8xx,xxx which I am currently running. Below are the apparent default bounds based on the worktodo line:

Prime95: 710,000 and 11,182,500.
James' probability calculator at 2.5%: 720,000 and 13,680,000
CUDAPm1: 910,000 and 18,427,500
gpuOwl: 1,000,000 and 30,000,000

I suppose what I am looking for here is a rule-of-thumb. Which should I use? Being a long-used and trusted application, I am using what Prime95 suggests in gpuOwl's config file. I am wondering is this enough; should they be larger. I really do not know.

[henryzz]

Originally Posted by storm5510

My old friend, P-1 bounds, is tasking me again. I have an exponent in the area of 99,8xx,xxx which I am currently running. Below are the apparent default bounds based on the worktodo line:

I suppose what I am looking for here is a rule-of-thumb. Which should I use? Being a long-used and trusted application, I am using what Prime95 suggests in gpuOwl's config file. I am wondering is this enough; should they be larger. I really do not know.

The faster P-1 runs compared to LL the larger the bounds should be.

[preda]

Originally Posted by storm5510

My old friend, P-1 bounds, is tasking me again. I have an exponent in the area of 99,8xx,xxx which I am currently running. Below are the apparent default bounds based on the worktodo line:

I suppose what I am looking for here is a rule-of-thumb. Which should I use? Being a long-used and trusted application, I am using what Prime95 suggests in gpuOwl's config file. I am wondering is this enough; should they be larger. I really do not know.

I am just revisiting the P-1 probabilities.

Based on my intuition (which is based on a few articles, among others https://www.researchgate.net/publica..._probabilities which has a handy table on page 12);

The idea is that there is a lot of benefit from a very large B2. Either use the default GpuOwl bounds of 1M/30M, or lower the B1 to a value of choice between 500K and 1M, but keep B2 to 30M.

Recently PRP got "cheaper" (so P-1 got relativelly "more expensive"), so the P-1 bounds need not be increased if efficiency is the goal. That's why I suggested "lower B1" instead of "increase B2".

(the P-1 probability calculators we have now may be a bit off)

[storm5510]

Originally Posted by preda

I am just revisiting the P-1 probabilities.

Based on my intuition (which is based on a few articles, among others https://www.researchgate.net/publica..._probabilities which has a handy table on page 12);

The idea is that there is a lot of benefit from a very large B2. Either use the default GpuOwl bounds of 1M/30M, or lower the B1 to a value of choice between 500K and 1M, but keep B2 to 30M.

Recently PRP got "cheaper" (so P-1 got relativelly "more expensive"), so the P-1 bounds need not be increased if efficiency is the goal. That's why I suggested "lower B1" instead of "increase B2".

(the P-1 probability calculators we have now may be a bit off)

Cost. This has to me a measurement of time. I am not sure what else it could be.

I looked a George's "Math" page on mersenne.org. Specifically, P-1. It is not written in a way which I can understand. Still, I tried.

Somebody here told me a couple of years ago that multiple factors can be found between 0 (zero) and B1, but only one above B1. Would lowering B1 not decrease the possibility?

I suppose what I have been looking for is a relationship between bound sizes and powers of two. It seems like there would be one somewhere. About a yer ago, I found a 39-digit factor in the process of running a really small exponent using a B1 of 1,000,000.

By the way. I decided to give my antique Core2Duo a shot with a PRP-CF by running it with Prime95. Only 11 hours each. I was expecting much more.

[kriesel]

Use big enough bounds that PrimeNet will retire the need for P-1 from the exponent.
Otherwise, unless you find a factor, it's wasted cycles that someone else will have to duplicate in their run to big enough bounds to retire the P-1 need for the same exponent. The PrimeNet bounds on mersenne.ca are sufficient.

[storm5510]

Originally Posted by kriesel

Use big enough bounds that PrimeNet will retire the need for P-1 from the exponent...

I have decided to pull away from running any more P-1's for a while. From what I gather and in the light of Prime95 v30.x being available, there will be some reevaluations done regarding bound sizes for P-1. It was James Heinrich who mentioned this in another topic.

I am sure you remember M1277. In 2017, someone ran a P-1 on it with B1 at 5-trillion and B2 at 400-trillion. I suppose whoever did not mind the wait.

[preda]

Of course "optimal bounds" depends on the "factored up to" bits. A first approximation for 100M exponents may be:
factored-to, B1, B2, chance
77: 1.2M 40M 4.2%
78: 1M 35M 3.6%
79: 1M 30M 3.2%
80: 0.9M 25M 2.7%

So, if you use 1M/30M (the default) you should be "just fine" for any bit-level :)


PS: and don't use my previous recommended values of 500K/30M, B1 too low relative to optimal.

[storm5510]

It appears first-time P-1 tests from Primenet have passed 100,000,000 in magnitude. So, I have decided to run a test to see what the time cost would be with gpuOwl:

B1 = 1000000
B2 = 35000000

I put the same exponent into the newest Prime95 just long enough to see what bounds it would use:

B1 = 770000
B2 = 13475000

These do not seem to be much different than what I saw 5 years ago. It seems to me, that as the exponents grow in size, P-1 bounds should follow along. Nothing would please me more than finding a nice big fat factor. My record is 39 digits.

[preda]

There is a new P-1 calculator in the pm1/ folder in GpuOwl. It comes as a python script ("pm1.py") and a C++ executable ("pm1"). The C++ version is very simple and only outputs the computed probability of a factor given the exponent, factored-to bits, B1, B2. The python script has an effort model (that can be tweaked in the source if desired), and based on that can propose some sort of "good" bounds. Example:

~/gpuowl/pm1$ ./pm1 100000000 77 1000000 30000000
3.91% (first-stage 1.53%, second-stage 2.38%)

/gpuowl/pm1$ ./pm1.py 100000000 77                 
Min: B1=  260K, B2=  7000K: p=2.38% (0.80% + 1.58%), work=0.96% (0.45% + 0.52%)
Mid: B1=  600K, B2= 20000K: p=3.38% (1.22% + 2.16%), work=2.41% (1.04% + 1.40%)
Big: B1= 1000K, B2= 34000K: p=4.01% (1.53% + 2.47%), work=3.99% (1.73% + 2.31%)

What you see in the above output is the chance of finding a factor, and also the cost of running the P-1 with those bounds. Personally I would go with "big" bounds above -- even if they provide not much time benefit vs running just the PRP, they do provide a significant chance of a factor which is cool.

[storm5510]

Originally Posted by preda

There is a new P-1 calculator in the pm1/ folder in GpuOwl...

That's very interesting. Not in my downloaded archive though. v6.11-364-g36f4e2a. Windows x64. Unless you have something newer...

[kriesel]

Originally Posted by storm5510

That's very interesting. Not in my downloaded archive though. v6.11-364-g36f4e2a. Windows x64. Unless you have something newer...

I have not incorporated any of the tools folder in the Windows builds I've posted. Until recently it was all python or shell script, no c. I haven't identified a python compilation method yet that both (a) succeeds and (b) produces code small enough to fit in a forum attachment. (But I have eliminated several candidate approaches.)