M7508981 has at least 12 prime factors(new factor # record) - Page 13

pdazzl · 2014-08-25, 23:10

Quote:

Originally Posted by Batalov

M3273488573 has 9 (and >=1 invisible one). It is rather easy to find them.

How do you know there's more than 1 invisible prime factor?

retina · 2014-08-25, 23:13

Quote:

Originally Posted by pdazzl

How do you know there's more than 1 invisible prime factor?

Hehe.

Quote:

Originally Posted by Batalov

M3273488573 has 9 (and >=1 invisible one). It is rather easy to find them.

pdazzl · 2014-08-25, 23:40

Quote:

Originally Posted by retina

Hehe.

Maybe an invisible = sign :)

Batalov · 2014-08-26, 00:21

M3930621659 is another tenner.

pdazzl · 2014-08-26, 01:16

Quote:

Originally Posted by Batalov

M3930621659 is another tenner.

Sweet, keep em coming...you obviously have some fire power. We'll see how rare these are.

LaurV · 2014-08-26, 02:24

Quote:

Originally Posted by Batalov

Drawing statistics from biased sets is a very unrewarding business.
(Goes without saying that one has to know statistics, first.)

I'll give you an example of the visible structure in this dataset. (Here, "structure" is the antonym of randomness; you cannot use most of the approximations when there's no randomness and i.i.d. premise is violated)

Observe http://www.mersenne.ca/exponent/630486799 . What can you learn by observation? Here is what. It is an example of how mfaktc was run en masse with default parameters. The default parameters include StopAfterFactor=1. Therefore, this is just one of a huge amount of candidates that was reported to have been run to a certain initial level... but wasn't. Factoring stopped after one or a few small factors and the rest of the tiny factors was found only at a rather ridiculous level of 80-81 bits. There is a huge amount of candidates that have unreported factors at the reported bit level. You can easily check this premise.

=> The statistics drawn from this set will be severely flawed unless a sophisticated model is built and fitted. I wholeheartedly recommend the home-grown statisticians (armed with Excel and four basic arithmetic operations) to stop plotting number towers and drawing barcharts and then kowtowing to them.

This is by far the best post in this thread, and I deliberately quoted it whole.

OTOH, I think that the number of factors of 2^p-1 for a prime p is roughly ln(p), and I might have a proof for it...

pdazzl · 2014-08-26, 02:36

Quote:

Originally Posted by LaurV

This is by far the best post in this thread, and I deliberately quoted it whole.

OTOH, I think that the number of factors of 2^p-1 for a prime p is roughly ln(p), and I might have a proof for it...

I agree, getting a true model is the right way to do it. But I don't think we really have one yet, unless I missed it in the thread. It seemed like we couldn't get basic agreement on some things amongst some rather knowledgable folks.

For example how many factors on average should one find with 1Ghz days of work on a particular exponent? Does it matter what size range the exponent is in?

bloodIce · 2014-08-26, 08:33

It might be easy to find an exponent with 12+ factors in the higher ranges indeed. However, M7508981 is the first known exponent with 12+, isn't it? That is at least a funny observation. There are no conclusions from it, but it is a fact that 11 factors are known and the remaining part is not a PRP.

alpertron · 2014-08-26, 11:40

Quote:

Originally Posted by bloodIce

It might be easy to find an exponent with 12+ factors in the higher ranges indeed. However, M7508981 is the first known exponent with 12+, isn't it? That is at least a funny observation. There are no conclusions from it, but it is a fact that 11 factors are known and the remaining part is not a PRP.

Since the cofactor is composite, it must have at least 2 prime factors. Adding that to the 11 known prime factors, the total is at least 13 prime factors for M7508981.

wblipp · 2014-08-26, 11:56

Quote:

Originally Posted by pdazzl

I'd be interested to hear from wblipp what the expected median is of the next factor.

Until you get large, the median of the next factor from any starting point is always the square of the starting point (aka twice as many digits). It doesn't matter if factors have been found or not.

"Very remote" was always hyperbole. The probability of a factor between 10¹⁸ and 10²⁵ was (25-18)/25 = 28%. Not great, but not "very remote."

CRGreathouse · 2014-08-26, 14:28

Where "very large" means "close to the square root", in the sense that if you stay below $N^{1/2-\varepsilon}$ for some fixed positive $\varepsilon$ then you're fine (asymptotically).

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2014-08-26, 00:21	#136
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2·47·101 Posts	M3930621659 is another tenner.

2014-08-26, 08:33	#140
bloodIce Feb 2010 Sweden 10101101₂ Posts	It might be easy to find an exponent with 12+ factors in the higher ranges indeed. However, M7508981 is the first known exponent with 12+, isn't it? That is at least a funny observation. There are no conclusions from it, but it is a fact that 11 factors are known and the remaining part is not a PRP.

2014-08-26, 14:28	#143
CRGreathouse Aug 2006 3·1,993 Posts	Where "very large" means "close to the square root", in the sense that if you stay below $N^{1/2-\varepsilon}$ for some fixed positive $\varepsilon$ then you're fine (asymptotically).