trial factoring and P-1

[jocelynl]

When doing P-1 factoring the b1 is actually b1*M
suppose we test M=67 to b1=1000
It actually look for factor of the form (2*....*M*b1*M+1)
If it is (2*...*M+1) then the factor is found right away.
Would it become faster to do P-1 testing rather than trial at higher mersenne numbers?
ex. 79299959 has been P-1 tested to 8000000
so 2*...*79299959*8000000*79299959+1 76bits+
trial shows 79299959,72

or Have I got the whole thing wrong

ps I never do stage 2 as it messes stage 1 save files. I keep my files as it continues to higher level from where it leftoff.

[cheesehead]

Originally Posted by jocelynl

When doing P-1 factoring the b1 is actually b1*M

What is your definition of M?

My understanding is that, in the Prime95 implementation of the P-1 algorithm, b1 is the upper limit on the prime factors of the "k" of potential factors 2kp+1 of 2^p-1 that are to be found by the P-1 method.

That is, stage 1 P-1 with b1 = 10000 performed on 2^p-1 will find any factor 2kp+1 of 2^p-1 in which the largest prime factor of k is less than (or equal to, if b1 were prime itself) 10000.

[cheesehead]

Originally Posted by cheesehead

My understanding is that, in the Prime95 implementation of the P-1 algorithm, b1 is the upper limit on the prime factors of the "k" of potential factors 2kp+1 of 2^p-1 that are to be found by the P-1 method.

That is, stage 1 P-1 with b1 = 10000 performed on 2^p-1 will find any factor 2kp+1 of 2^p-1 in which the largest prime factor of k is less than (or equal to, if b1 were prime itself) 10000.

Correction:

My understanding is that, in the Prime95 implementation of the P-1 algorithm, b1 is the upper limit on the power-of-a-prime factors of the "k" of potential factors 2kp+1 of 2^p-1 that are to be found by the P-1 method.

That is, stage 1 P-1 with b1 = 10000 performed on 2^p-1 will find any factor 2kp+1 of 2^p-1 in which the largest power-of-a-prime factor of k is less than (or equal to, if b1 were prime itself) 10000.

Example:

59704785388637019242567 is a factor of 2^6049993 - 1.

59704785388637019242567 = 2 * 4934285493275531 * 6049993 + 1.

Prime95's P-1 stage 1 with b1 = 4000 would find this factor because the largest prime-power factor of 4934285493275531 is less than 4000.

4934285493275531 = 61² * 593 * 983 * 1153 × 1973.

61² = 3721.

In this example the factor 59704785388637019242567 could have been found in stage 1 with b1 as low as 3721.

Also, Prime95's P-1 stage 2 with b1 = 2000 and b2 = 4000 would find this factor because the largest prime-power factor of 4934285493275531 is less than 4000 and all other prime-power factors are less than 2000. (In fact, b1/b2 as low as b1 = 1973, b2 = 3721 would have worked.)

[axn]

Originally Posted by cheesehead

Also, Prime95's P-1 stage 2 with b1 = 2000 and b2 = 4000 would find this factor because the largest prime-power factor of 4934285493275531 is less than 4000 and all other prime-power factors are less than 2000. (In fact, b1/b2 as low as b1 = 1973, b2 = 3721 would have worked.)

AFAIK, stage 2 only considers primes and not prime powers. So this would not be found.

[jocelynl]

Originally Posted by cheesehead

Correction:

My understanding is that, in the Prime95 implementation of the P-1 algorithm, b1 is the upper limit on the power-of-a-prime factors of the "k" of potential factors 2kp+1 of 2^p-1 that are to be found by the P-1 method.

That is, stage 1 P-1 with b1 = 10000 performed on 2^p-1 will find any factor 2kp+1 of 2^p-1 in which the largest power-of-a-prime factor of k is less than (or equal to, if b1 were prime itself) 10000.

Example:

59704785388637019242567 is a factor of 2^6049993 - 1.

59704785388637019242567 = 2 * 4934285493275531 * 6049993 + 1.

Prime95's P-1 stage 1 with b1 = 4000 would find this factor because the largest prime-power factor of 4934285493275531 is less than 4000.

4934285493275531 = 61² * 593 * 983 * 1153 × 1973.

61² = 3721.

In this example the factor 59704785388637019242567 could have been found in stage 1 with b1 as low as 3721.

Also, Prime95's P-1 stage 2 with b1 = 2000 and b2 = 4000 would find this factor because the largest prime-power factor of 4934285493275531 is less than 4000 and all other prime-power factors are less than 2000. (In fact, b1/b2 as low as b1 = 1973, b2 = 3721 would have worked.)

I stand corrected!

[cheesehead]

Originally Posted by axn1

AFAIK, stage 2 only considers primes and not prime powers. So this would not be found.

Oops, I forgot that I had a weakness in my stage 2 understanding, and got carried-away with my prime-power correction. Thank you.

jocelynl, I presume you've noted axn1's correction to my erroneous paragraph about stage 2.

[cheesehead]

BTW, let this be a lesson.

I failed to actually TRY running
Pminus1=6049993,2000,4000,0,0
to confirm that stage 2 would find the 61² = 3721 prime-power factor of 4934285493275531 before I made my erroneous posting:

Originally Posted by cheesehead, in error!

Also, Prime95's P-1 stage 2 with b1 = 2000 and b2 = 4000 would find this factor because the largest prime-power factor of 4934285493275531 is less than 4000 and all other prime-power factors are less than 2000. (In fact, b1/b2 as low as b1 = 1973, b2 = 3721 would have worked.)

It would've taken just 4 minutes, including set-up time, to run that on my Athlon 1200. But that didn't occur to me before I threw in that erroneous last paragraph about stage 2, which was an idea I got just as I was about to post all the preceding (and correct) part about stage 1, which was itself a correction of my first hasty posting ...

[alpertron]

Originally Posted by jocelynl

I never do stage 2 as it messes stage 1 save files. I keep my files as it continues to higher level from where it leftoff.

Please notice that in general the factor is found in step 2. You can read how the p-1 works at MersenneWiki.

[jocelynl]

Originally Posted by alpertron

Please notice that in general the factor is found in step 2. You can read how the p-1 works at MersenneWiki.

I find that step 2 is a wast of time since all your effort are thrown to garbage when you test step 1 a little further. It would be nice if step 2 could test past 2^32 but I'm not complaining. It's fun to have a factor pop up once in a while.