Pick Your Poisson: P-1 Bounds

[James Heinrich]

P-1 and ECM are very different.

P-1 with given bounds will always find the factor(s) that meet the criteria of being within those bounds.
ECM bounds constrain the size of the factor found, but do not determine if a particular factor is found or not on that run.

P-1 with larger bounds is a continuation of P-1 with smaller bounds. If you have the savefile from a previous P-1 run and wish to extend the bounds, Prime95 will just need to run the portion between the old bound and the new bound.

[Ensigm]

Originally Posted by James Heinrich

P-1 with larger bounds is a continuation of P-1 with smaller bounds. If you have the savefile from a previous P-1 run and wish to extend the bounds, Prime95 will just need to run the portion between the old bound and the new bound.

My naive question: when utilizing the save file, is it only possible to extend Stage 2, or is it possible to extend Stage 1 and Stage 2 at the same time?

[James Heinrich]

Originally Posted by Ensigm

My naive question: when utilizing the save file, is it only possible to extend Stage 2, or is it possible to extend Stage 1 and Stage 2 at the same time?

Stage 1 can be extended, stage 2 will need to be restarted if B1 has been extended.

[storm5510]

Originally Posted by James Heinrich

P-1 and ECM are very different.

P-1 with given bounds will always find the factor(s) that meet the criteria of being within those bounds.

ECM bounds constrain the size of the factor found, but do not determine if a particular factor is found or not on that run.

I am not sure I follow the part in bold above. Consider the following:

M83621 has a factor: 350415918423517661715474499722683369441 (ECM curve 1, B1=1000000, B2=100000000)

I found this a while back. Clearly, the factor is way outside the bounds. It is pretty obvious bounds in ECM do not work the same as they do in P-1. So, how do they work?

[S485122]

Originally Posted by storm5510

...
It is pretty obvious bounds in ECM do not work the same as they do in P-1. So, how do they work?

Wikipedia is your friend

[James Heinrich]

Originally Posted by storm5510

bounds in ECM do not work the same as they do in P-1. So, how do they work?

It is beyond my mathability to explain (or understand). Wiki if you want to try.
In a crude analogy, it's something akin to picking a random card from a specific deck of cards (bounds) -- if you're lucky it might be the Ace of Spades (you find a factor) but you can pull 50 other cards (ECM curves run) from the same deck and not find a factor.

[kriesel]

Originally Posted by storm5510

I am not sure I follow the part in bold above. Consider the following:

I found this a while back. Clearly, the factor is way outside the bounds. It is pretty obvious bounds in ECM do not work the same as they do in P-1. So, how do they work?

It is quite routine in P-1 to find factors that are much larger than B1, B2, or B1 x B2 x 2p. Take a look at the test candidates, factors, and bounds here. B1 needs to cover most of the factors of k in f=2kp+1, and B2 the largest factor of k. See https://www.mersenne.ca/exponent/502000027 for its k's factors and its minimum bounds as an example.

[kriesel]

Originally Posted by James Heinrich

If you have the savefile from a previous P-1 run and wish to extend the bounds, Prime95 will just need to run the portion between the old bound and the new bound.

All well and good, if you have the prime95 stage 1 save file and the prime95 ram and horsepower. But if someone else did the first P-1, generally there's no access to the save files. And they would be no good in gpuowl anyway, both because of file incompatibility, between applications, and because gpuowl does not support P-1 run extension. (Same applies to CUDAPm1, but this is a gpuowl subforum thread.)

[storm5510]

Originally Posted by James Heinrich

It is beyond my mathability to explain (or understand). Wiki if you want to try.
In a crude analogy, it's something akin to picking a random card from a specific deck of cards (bounds) -- if you're lucky it might be the Ace of Spades (you find a factor) but you can pull 50 other cards (ECM curves run) from the same deck and not find a factor.

I looked at the link. It only took about a half-second to sail way over my head. I do not have that skill.

The deck of cards analogy is based on random chance. I believe what I am unable to grasp is how a small bound produces such a large factor. There must be some other mathematical function going on.

[Aramis Wyler]

Lord knows that I am not a mathematician - I grew up with everyone telling me how good at math I was, and how impressed they were, and then I went to college and met people who are actually good at math. As a computer science major, I was introduced to the bare minimum of mathematical theory that I could take and still complete the degree, so you can take all of the following with a grain of salt. It is somewhat more accurate than the deck of cards analogy.

https://upload.wikimedia.org/wikiped..._animation.gif

Here we have a view of the plane of the ecliptic, from the perspective of the solar orbit around the galactic core. The little moving dot is Haley's comet. The planets and Haley's comet are all orbiting the sun on elliptic curves, and are at any given point closer to or farther away from the sun. The numbers for the ECM are not bounds the way bookends are bounds - they describe the shape of the curve from nearly round like the planetary orbits to very sharp like Haley's comet orbit here. They also describe what I'm going to call the comet's perihelion, or the location when it is at it's closest point to the sun. So what you have here are a couple of small numbers (and your target number) from which can be derived the GCD (the perihelion) and the shape of the curve (narrow or broad).

The numbers they are testing are points on the elliptic curve. They might be small if they are points on the curve near the perihelion there, or very large if they are at the aphelion point. The bigger the curve you have the more likely your orbit is to intersect a factor, but it is only so likely, which is why the deck of card analogy, for practical purposes, is sufficient most of the time. Even a tiny difference in the number can affect the shape of the curve and send your little comet to different parts of the mathematical space where it might intersect a factor.

[storm5510]

Originally Posted by Aramis Wyler

Lord knows that I am not a mathematician - I grew up with everyone telling me how good at math I was, and how impressed they were, and then I went to college and met people who are actually good at math. As a computer science major, I was introduced to the bare minimum of mathematical theory that I could take and still complete the degree, so you can take all of the following with a grain of salt. It is somewhat more accurate than the deck of cards analogy.

https://upload.wikimedia.org/wikiped..._animation.gif

Here we have a view of the plane of the ecliptic, from the perspective of the solar orbit around the galactic core. The little moving dot is Haley's comet. The planets and Haley's comet are all orbiting the sun on elliptic curves, and are at any given point closer to or farther away from the sun. The numbers for the ECM are not bounds the way bookends are bounds - they describe the shape of the curve from nearly round like the planetary orbits to very sharp like Haley's comet orbit here. They also describe what I'm going to call the comet's perihelion, or the location when it is at it's closest point to the sun. So what you have here are a couple of small numbers (and your target number) from which can be derived the GCD (the perihelion) and the shape of the curve (narrow or broad).

The numbers they are testing are points on the elliptic curve. They might be small if they are points on the curve near the perihelion there, or very large if they are at the aphelion point. The bigger the curve you have the more likely your orbit is to intersect a factor, but it is only so likely, which is why the deck of card analogy, for practical purposes, is sufficient most of the time. Even a tiny difference in the number can affect the shape of the curve and send your little comet to different parts of the mathematical space where it might intersect a factor.

This is very good. If I had not been a NASA nut when I was a kid, a lot of this I would not understand. If I follow this correctly, the bounds determine the shape of the curve, length and width. Matching B1 and B2 would produce a circle. An increase of B2 relative to B1 would produce an ellipse. There is an ECM standard, if I remember correctly. B2 = B1 * 100. This would make a very tight ellipse. A guess: The sigma value used, like a random number generator, picks points on the curve to test. Small bounds, and the ellipse may reach Venus. Larger ones would reach out past Pluto and still maintain the shape.

There is theory and the application of the theory. The latter, I think of as the mechanics. How something is actually done in practice as opposed to the idea of how it should be done. This, I can grasp. Theory, very little. The Wiki page, to me, makes it appear like a person going from Washington, D.C. to Baltimore, via Tokyo.