search for MMM127 small factors?

Orgasmic Troll · 2006-03-25, 08:13

Would the allure of MM127's possible primality be diminished if a Catalan-Mersenne number was found to be composite? Has anyone searched for small factors of the CM numbers greater than MM127?

mild tangent:
I'm not aware if attempting to find small factors of these monstrous numbers is even feasible, so, if I'm talking out of my arse, leave your inflammatory comments at the door. Frankly, I'm too busy with my other math classes to go learning computational mathematics from scratch and buying $70 books to do so, nor is the question compelling enough for me to go through that much toil. If you want to make toast but you're out of bread, you don't go out and plant wheat.

alpertron · 2006-03-25, 12:44

Travis,

It's not a bad idea. Since MM127 is almost surely composite, MMM127 factors should not have the form 2*k*MM127+1.

My program to compute factors of googolplexplex could be adapted to perform trial division by small factors (say less than 10^12). If MMM127 has such a small factor, we are sure that MM127 is not prime.

alpertron · 2006-03-25, 13:09

What I wrote in my previous post is not correct. If MM127 is equal to the product of primes a*b*c*..*z, then after dividing MMM127 by 2^a-1, 2^b-1, etc. (which are very large numbers whose factors have the form 2*k*a+1, 2*k*b+1, which are out of reach) we get the primitive factor.

The factors of this primitive factor have the form 2*k*MM127+1.

So if we cannot find a factor of MM127 we also cannot expect to find a factor of MMM127.

R.D. Silverman · 2006-03-25, 15:50

Quote:

Originally Posted by alpertron

What I wrote in my previous post is not correct. If MM127 is equal to the product of primes a*b*c*..*z, then after dividing MMM127 by 2^a-1, 2^b-1, etc. (which are very large numbers whose factors have the form 2*k*a+1, 2*k*b+1, which are out of reach) we get the primitive factor.

The factors of this primitive factor have the form 2*k*MM127+1.

So if we cannot find a factor of MM127 we also cannot expect to find a factor of MMM127.

This is correct.

Orgasmic Troll · 2006-03-25, 17:05

Ahh, okay.

Thanks to both :)

David John Hill Jr · 2006-05-02, 02:23

If one could show x|(x-1)! evenly and with multiples of 5,
you will be quaranteed of finding factors .
So also for m^x 127.

ewmayer · 2006-05-02, 16:28

Quote:

Originally Posted by alpertron

The factors of this primitive factor have the form 2*k*MM127+1.

Good luck doing arithmetic modulo numbers of this size.

ixfd64 · 2006-06-11, 15:38

I think that testing one trial factor of MMM127 (2k * MM127 +1) takes roughly the same amount of computation as LL-testing MM127.

2006-03-25, 08:13	#1
Orgasmic Troll Cranksta Rap Ayatollah Jul 2003 641 Posts	search for MMM127 small factors? Would the allure of MM127's possible primality be diminished if a Catalan-Mersenne number was found to be composite? Has anyone searched for small factors of the CM numbers greater than MM127? mild tangent: I'm not aware if attempting to find small factors of these monstrous numbers is even feasible, so, if I'm talking out of my arse, leave your inflammatory comments at the door. Frankly, I'm too busy with my other math classes to go learning computational mathematics from scratch and buying $70 books to do so, nor is the question compelling enough for me to go through that much toil. If you want to make toast but you're out of bread, you don't go out and plant wheat.

2006-03-25, 13:09	#3
alpertron Aug 2002 Buenos Aires, Argentina 1341₁₀ Posts	What I wrote in my previous post is not correct. If MM127 is equal to the product of primes abc..z, then after dividing MMM127 by 2^a-1, 2^b-1, etc. (which are very large numbers whose factors have the form 2ka+1, 2kb+1, which are out of reach) we get the primitive factor. The factors of this primitive factor have the form 2kMM127+1. So if we cannot find a factor of MM127 we also cannot expect to find a factor of MMM127. Last fiddled with by alpertron on 2006-03-25 at 13:10

2006-05-02, 02:23	#6
David John Hill Jr Jun 2003 Pa.,U.S.A. 2²·7² Posts	one assurence numerically speaking If one could show x\|(x-1)! evenly and with multiples of 5, you will be quaranteed of finding factors . So also for m^x 127.

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2006-03-25, 12:44	#2
alpertron Aug 2002 Buenos Aires, Argentina 53D₁₆ Posts	Travis, It's not a bad idea. Since MM127 is almost surely composite, MMM127 factors should not have the form 2kMM127+1. My program to compute factors of googolplexplex could be adapted to perform trial division by small factors (say less than 10^12). If MMM127 has such a small factor, we are sure that MM127 is not prime.

2006-03-25, 17:05	#5
Orgasmic Troll Cranksta Rap Ayatollah Jul 2003 281₁₆ Posts	Ahh, okay. Thanks to both :)

2006-06-11, 15:38	#8
ixfd64 Bemusing Prompter "Danny" Dec 2002 California 100100101101₂ Posts	I think that testing one trial factor of MMM127 (2k * MM127 +1) takes roughly the same amount of computation as LL-testing MM127.