M4219 completely factored?

[MatWur-S530113]

Originally Posted by Mini-Geek

...
By the way, in looking for that, I found that a large number of primes between 86137 and 200000, 4971 out of 9611 to be exact, are not listed in PrimeNet's ECM progress report. I'm sure a few of those are just because they're fully factored, and 3 are because they're prime, but I'm not sure about the rest. Why is this? Is it that only numbers with some ECM progress are reported there?

About 2 years ago I recognized the same fact for the primes>25000. Then I run about 50 ECM-curves with B1=11e3 for every prime<80000. Since that time all exponents<80000 are listet in the ECM progress report.
So yes, I think only the numbers with some ECM curves done are listet.
and only 1 Pfennig about trivial factors of Mersenne-numbers:
IMHO a composite Mersenne-number with prime exponent p is trivially factored iff p is a Sophie Germaine prime congruent 3 mod 4. Then q=2p+1 is a trivial p-factor of M(p). If the remaining cofactor is prime, then it is another trivial p-factor of M(p). If it is composite, then the primefactors of this trivial c-factor are not trivial.
Thus M(29) is the first Mersenne-number with non-trivial factors.
For the largest Mersenne cofactors, which are proven primes see: http://primes.utm.edu/top20/page.php?id=49

greetings
Matthias

[Prime95]

Originally Posted by WVU Mersenneer

If I may trouble you with another question, does the server receive notification of a possible prime cofactor when found, does it perform the PRP test itself on recently-discovered factors in order to determine if an exponent has been fully-factored, or do you rely on us to ask about our finds in order to verify an exponent being fully-factored?

The server is NOT notified. The server does not check the cofactor. It would be nice to add this feature someday, but it isn't high on the priority list.

Congrats on your second complete factorization

[TimSorbet]

Originally Posted by MatWur-S530113

About 2 years ago I recognized the same fact for the primes>25000. Then I run about 50 ECM-curves with B1=11e3 for every prime<80000. Since that time all exponents<80000 are listet in the ECM progress report.
So yes, I think only the numbers with some ECM curves done are listet.

Ah okay, thanks.

Originally Posted by Mini-Geek

the number must not be of a form in which all factors are trivially known, such as x!, and the number can not have been originally formed by multiplying known primes and/or composites together.

I'll amend this to "the number must not be of a form in which any factor is trivially known," to exclude things like 2^4031399+1, where the factor of 3 is trivially known, or 31^101111-1 with a trivial factor of 30; in these cases both cofactors are PRP, so they'd otherwise qualify. (2^p+1)/3 would qualify, since the trivial factor has been removed. Also, the number still qualifies if it was formed by multiplying known primes and/or composites if that knowledge could not have been used to factor it, so RSA challenge numbers would count.

Originally Posted by MatWur-S530113

and only 1 Pfennig about trivial factors of Mersenne-numbers:
IMHO a composite Mersenne-number with prime exponent p is trivially factored iff p is a Sophie Germaine prime congruent 3 mod 4. Then q=2p+1 is a trivial p-factor of M(p). If the remaining cofactor is prime, then it is another trivial p-factor of M(p). If it is composite, then the primefactors of this trivial c-factor are not trivial.
Thus M(29) is the first Mersenne-number with non-trivial factors.
For the largest Mersenne cofactors, which are proven primes see: http://primes.utm.edu/top20/page.php?id=49

greetings
Matthias

Oh, very interesting. My new definition excludes this, since if those conditions line up, we know there's at least one trivial factor.

I thought it might be likely that we could find this largest non-trivially-factored fully-factored Mersenne number by looking at the list of top PRPs, so that's what I did. The largest on the list that's a divisor of a Mersenne number is (2^684127-1)/23765203727. 684127 is prime, but not a Sophie Germaine prime, so it seems to match all criteria: it would appear that M684127 is the largest fully-factored Mersenne number without a trivial factorization, and it's also semiprime. The PRP is 205933 digits. It's #39 on the PRP list right now. If it could be proven, it'd be #2367 on the top 5000 primes, but that won't happen for quite a while.
The next two largest are:
(2^406583-1)/813167, but 406583 is a Sophie Germaine prime congruent 3 mod 4, so the 813167 and the PRP cofactor are trivial. It still appears to be the largest fully-factored composite Mersenne number.
(2^271549-1)/238749682487. 271549 is not a Sophie Germaine prime, so it also qualifies to my definition.

It's very likely that many larger prime cofactors, and therefore full factorizations, have been run across, but ignored due to the difficulty in testing it (and only PRP result) when we were just trying to find factors to eliminate Mersenne prime candidates. I wonder what progress has been done on testing large-ish Mersenne cofactors, especially ones with only small TF-found factors known. Are cofactors tested to see if they're PRP or composite before they're put in the list for PrimeNet to run ECM on them? I wonder how hard it'd be to find large PRPs that way.

[henryzz]

I have run 74 curves of B1=11000 B2=11000 on both composites N-1 and N+1 in addition to P-1 at B1=100000 B2=default
Thats 8.56% of t20.

[ixfd64]

How long would it take to run the M4219 cofactor through ECPP?

@WVU Mersenneer

Are you "James Hints" on PrimeNet?

[Batalov]

Originally Posted by ixfd64

How long would it take to run the M4219 cofactor through ECPP?

@WVU Mersenneer

Are you "James Hints" on PrimeNet?

Isn't the certificate attached to post #6?
(it contains running times, too; less than an hour).

[ixfd64]

Ah, I see.

When I saw the original post saying that the certificate would be available in half an hour, it wasn't there yet. I guess I forgot to check recheck the post!

[MatWur-S530113]

Originally Posted by Mini-Geek

Ah okay, thanks.

You are welcome^^

Originally Posted by Mini-Geek

...
The next two largest are:
(2^406583-1)/813167, but 406583 is a Sophie Germaine prime congruent 3 mod 4, so the 813167 and the PRP cofactor are trivial.
...

attention please: (2^406583-1)/813167 is a trivial PRP-factor of M406583, but it is not trivial, that this factor is a PRP. Thus for a 'largest known PRP cofactor of a Mersenne-number' it should still be qualified.

btw: I startet some PRP-Tests for all Mersenne-numbers with prime exponents 1.1e6<p<1.2e6 and known factors. Maybe I have luck and find another large PRP. But only this small range of numbers will last at least 2 weeks. I will tell you if I find a PRP (the first ~200 exponents say 'no prime'...)

greetings
Matthias

[WVU Mersenneer]

Originally Posted by MatWur-S530113

You are welcome^^


attention please: (2^406583-1)/813167 is a trivial PRP-factor of M406583, but it is not trivial, that this factor is a PRP. Thus for a 'largest known PRP cofactor of a Mersenne-number' it should still be qualified.

btw: I startet some PRP-Tests for all Mersenne-numbers with prime exponents 1.1e6<p<1.2e6 and known factors. Maybe I have luck and find another large PRP. But only this small range of numbers will last at least 2 weeks. I will tell you if I find a PRP (the first ~200 exponents say 'no prime'...)

greetings
Matthias

Matthias, what are you using for your PRP-Tests? I am very interested to know what you find, and thank you for running these tests as well as contributing to this thread.

Would anyone happen to know if there is a Windows-based PRP tester? I'd be happy to donate some of my dual-core's time to checking Mersenne exponents <1e5, though I've had no luck setting up GMP-ECM or the new "super" TFer which I've seen results from listed in the Primenet reports. My searches for Primo returns a PDF maker...

[MatWur-S530113]

Originally Posted by WVU Mersenneer

Matthias, what are you using for your PRP-Tests? I am very interested to know what you find, and thank you for running these tests as well as contributing to this thread.
...

For the PRP-test of the Mersenne-numbers with known factors I simply use prime95 (the server will be flooded with unneeded 'is not prime'-results in 2 weeks^^).
For other numbers I have a library to work with big Integers using Delphi7/Pascal programs. The library has a built-in function 'NIsProbablePrime' based on some pseudo-prime tests.
Or I'm using GMP-ECM on my old computer (I still was not able to get a working GMP/GMP-ECM on my new computer with Windows7, compiling with a C++-compiler seems to be much more complicate as with Delphi...).
I will post the results of the tests I make atm as soon as the work is done

greetings
Matthias

[WVU Mersenneer]

Originally Posted by MatWur-S530113

For the PRP-test of the Mersenne-numbers with known factors I simply use prime95 (the server will be flooded with unneeded 'is not prime'-results in 2 weeks^^).
For other numbers I have a library to work with big Integers using Delphi7/Pascal programs. The library has a built-in function 'NIsProbablePrime' based on some pseudo-prime tests.
Or I'm using GMP-ECM on my old computer (I still was not able to get a working GMP/GMP-ECM on my new computer with Windows7, compiling with a C++-compiler seems to be much more complicate as with Delphi...).
I will post the results of the tests I make atm as soon as the work is done

greetings
Matthias

Sounds like we're in the same boat trying to get GMP-ECM to work on Windows! The ease and speed with which Prime95 runs on Windows is further testament to Dr Woltman and his band of experts behind the scenes.

May I ask how you use Prime95 for PRP? I tried earlier this week by simply removing the largest found factor for an exponent and running curves on the exponent at a level at least 2 above the size of the factor, for example if the factor was 40 digits, I ran curves with B1=44e6, but most of the factors were much smaller. However, this barely worked at all as ECM is not a "sure thing" by any means and has often amazed me at how many curves need to be run in order to find a factor...as well as how few sometimes.

Again, thank you for your help and I am eager to know what you find.