new options - factor to low limits, etc

[Jud McCranie]

The new version of Prime95 has some new options: trial factor to low limits, P-1 factoring, etc. I can't find any documentation on which of these to choose, especially guidelines according to what would be best for a particular computer. Is that information anywhere?

[mdettweiler]

Originally Posted by Jud McCranie

The new version of Prime95 has some new options: trial factor to low limits, P-1 factoring, etc. I can't find any documentation on which of these to choose, especially guidelines according to what would be best for a particular computer. Is that information anywhere?

Also what would be helpful is some documentation regarding exactly how the options on the web site correspond to those in the programming. For example, the web page doesn't have any "Trial factoring to low limits" option; instead it just has "Trial factoring" and "Trial factoring LMH".

[James Heinrich]

TF to low limits (or TF-LMH on the webpages) -- will assign trialfactoring assignments for low bit depths (currently this is around 2^61, the definition of "low" will change as more of the available workspace gets factored). Use this for old, slow machines (e.g. 5 years old, <1GHz).

P-1 will assign P-1 work just ahead of the range that is being assigned for LL testing. For this worktype to be useful you should have a generous amount of RAM allocated to Prime95 (i.e. at least 1GB assigned).

[Jud McCranie]

OK. I have one machine dedicated to TF. Since I upgraded to version 25, all of the factorizations seem to be a range of a factor of 2, which varies from exponent to exponent.

[James Heinrich]

TF to low limits will typically be up to anywhere from 0 - 2^59, maybe as high as 2^62 (for now). Standard TF work will typically fall in the range of 2^64 to 2^68. Remember, each higher bit depth takes twice as long to process (2^68 takes twice as long as 2^67; 2^68 takes 32 times as long as 2^63; etc)

[petrw1]

Originally Posted by James Heinrich

TF to low limits will typically be up to anywhere from 0 - 2^59, maybe as high as 2^62 (for now). Standard TF work will typically fall in the range of 2^64 to 2^68. Remember, each higher bit depth takes twice as long to process (2^68 takes twice as long as 2^67; 2^68 takes 32 times as long as 2^63; etc)

My TF-LMH (Low limits) goes to 2^64

[Jud McCranie]

Originally Posted by James Heinrich

TF to low limits will typically be up to anywhere from 0 - 2^59, maybe as high as 2^62 (for now). Standard TF work will typically fall in the range of 2^64 to 2^68. Remember, each higher bit depth takes twice as long to process (2^68 takes twice as long as 2^67; 2^68 takes 32 times as long as 2^63; etc)

If you like finding factors (as I do), then which is the most likely to find a factor? I have relatively new computers, and I could dedicate a GB of RAM.

[cheesehead]

Originally Posted by Jud McCranie

If you like finding factors (as I do), then which is the most likely to find a factor?

Short answer:

TF to low limits will find more factors per minute than any other type of factoring.

I have relatively new computers, and I could dedicate a GB of RAM.

TF won't require your GB of RAM !!

- - -

Long answer:

In trial factoring, there is a remarkable property --

The probability of finding a factor of 2^p-1 that is between 2^m and 2^m+1 is approximately 1/(m+1) (or maybe it's 1/m, I forget which).

That is, each power-of-2 interval has only a very slightly less chance (1/(m+1) vs. 1/m, or 1/(m+2) vs. 1/(m+1)) than the preceding power-of-2 interval of containing a factor.

Of course, each succeeding power-of-2 interval has twice as many potential factors as the preceding one, and so takes twice as long to test as that one does, so factors-found-per-minute is halved at each successively higher power-of-2. Thus, to maintain a high factors-found-per-minute, do TF to only low power-of-2 limits.

Notice that the exponent of the Mersenne itself doesn't matter in the approximated probability (but see below) - the chance that M1234577 has a factor between 2⁶³ and 2⁶⁴ is about the same -- 1/64 -- as the chance that M123456791 has a factor in that same 2⁶³-2⁶⁴ range. How can that be? Well, the larger Mersenne number is also likely to have _more_ factors in the power-of-2 ranges up to its square root than the smaller is likely to have in the (smaller number of) power-of-2 ranges up to its square root.

Now, this 1/(m+1) approximation only holds for sufficiently large m, such as in the ranges (59 and up) we commonly test. Since any factor of 2^p-1 must be of the form 2kp+1, the smallest possible factor of 2^p-1 is 2*1*p+1. If p = 1234577, the smallest possible factor is 2*1*1234577+1 = 2469155. That's not a prime, so it can't actually be a factor of 2^1234577-1, but it does show that no factor of 2^1234577-1 can be lower than 2²² = 2097152. 2^1234577-1 has no chance of having a factor in the ranges 2¹⁰-2¹¹ or 2²⁰-2²¹, in this example.

- - -

After all that, a caveat:

TF to low limits is one assignment type where old, slow systems can still make useful contributions. How would you feel about "taking candy from a baby" with your new, fast system? OTOH, the counter-argument is that there's an infinitude of exponents out there that need TF, enough for everybody!

Anytime you feel like doing something more challenging, you can just do your TF to higher limits!

... and when you feel guilty about not using that GB of RAM, switch to P-1.

[Jud McCranie]

It would be nice if some of this information about what is best for various computers were in the documentation.

[henryzz]

Originally Posted by Jud McCranie

It would be nice if some of this information about what is best for various computers were in the documentation.

that would certainly be nice

i also think that just the simple plain text format is getting a bit old hat
it should be available for people using old pcs though
something like a word document, pdf, html document or something else would be easier to read as more info keeps getting crammed in
a document with the potential for a good index would be great

[Jud McCranie]

I've been with the program since a few months after it went public. The program still does not have a help system. If you go to "help" on the menu and "help topics", it says that it can't launch help.