MM127

[Dougal]

Since there seems to be a large chance that this number is prime compared to other numbers,has anyone tried trial factoring it to a high bit depth?im new enough to this,so is there any way that you can stop in the middle of factoring and pick up where you started later on?

[ET_]

Originally Posted by Dougal

Since there seems to be a large chance that this number is prime compared to other numbers,has anyone tried trial factoring it to a high bit depth?im new enough to this,so is there any way that you can stop in the middle of factoring and pick up where you started later on?

The short answer is "yes" for both your questions

Check Will Edgington page: http://www.garlic.com/~wedgingt/MMPstats.txt
MathWorld page http://mathworld.wolfram.com/DoubleMersenneNumber.html

and if you still want to check them, you have the choice of Ernst Mayer's mfactor (if there is a version that can handle large exponents) ready to restart from any k in factors like 2kp+1, and my Factor5 multi-thread aware.

Luigi

[ewmayer]

FYI, I have tested MM127 up to 175 bits (k = 2^47 = 140737488355328, a few bits higher than the limits listed for Tony Forbes` search on the MMPstats.txt page) with no factors found. Once I finish work on my current updates to Mlucas I plan to revisit the double-Mersenne factoring code, perhaps release a multithreaded binary (the current version allows the factoring to be split into up to 16 independent passes, but that requires 16 executable images running separately) and see if we can't organize a search at least up to 180-odd bits.

And the fact that MM127 has no known factors only makes it roughly twice as likely to be prime as a randomly chosen Mersenne number in that size range, which - if we extrapolate the known plausible heuristics regarding the statistical distribution of M-prime exponents - still makes it exceedingly unlikely to be prime. But it sure would be nice to settle the question once and for all, just to shut up the cranks. [Who would then surely proceed to some other equally-unfounded speculation - but one would hope that at least the cranks would lose enough "credibility" thereby that they would have trouble wasting other folks' times in their future crankish endeavors.]

[rogue]

Originally Posted by ewmayer

FYI, I have tested MM127 up to 175 bits (k = 2^47 = 140737488355328, a few bits higher than the limits listed for Tony Forbes` search on the MMPstats.txt page) with no factors found. Once I finish work on my current updates to Mlucas I plan to revisit the double-Mersenne factoring code, perhaps release a multithreaded binary (the current version allows the factoring to be split into up to 16 independent passes, but that requires 16 executable images running separately) and see if we can't organize a search at least up to 180-odd bits.

And the fact that MM127 has no known factors only makes it roughly twice as likely to be prime as a randomly chosen Mersenne number in that size range, which - if we extrapolate the known plausible heuristics regarding the statistical distribution of M-prime exponents - still makes it exceedingly unlikely to be prime. But it sure would be nice to settle the question once and for all, just to shut up the cranks. [Who would then surely proceed to some other equally-unfounded speculation - but one would hope that at least the cranks would lose enough "credibility" thereby that they would have trouble wasting other folks' times in their future crankish endeavors.]

Geoff Reynold's and I wrote new code for double-Mersenne factoring. I've tried to contact Tony to see if he was interested as our code can be run on many platforms, but Tony never responded to my inquiries. If you are interested I can send the source to you.

[CRGreathouse]

Originally Posted by ewmayer

And the fact that MM127 has no known factors only makes it roughly twice as likely to be prime as a randomly chosen Mersenne number in that size range, which - if we extrapolate the known plausible heuristics regarding the statistical distribution of M-prime exponents - still makes it exceedingly unlikely to be prime.

I'm familiar with the standard heuristic about the probability of 2^p-1 being prime, given that p is prime. But how do you scale that probability given that there are no primes under k that divide it?

[Mini-Geek]

Originally Posted by CRGreathouse

I'm familiar with the standard heuristic about the probability of 2^p-1 being prime, given that p is prime. But how do you scale that probability given that there are no primes under k that divide it?

http://web.archive.org/web/200711121...e.org/math.htm

A simple approach is to repeatedly apply the observation that the chance of finding a factor between 2X and 2X+1 is about 1/x. For example, you are testing 210000139-1 for which trial factoring has proved there are no factors less than 264. The chance that it is prime is the chance of no 65-bit factor * chance of no 66 bit factor * ... * chance of no 5000070 bit factor. That is:

	64   65         5000069
	-- * -- * ... * -------
	65   66         5000070

This simplifies to 64 / 5000070 or 1 in 78126.

Of course, it goes on to say that it's really uncertain and there's some slightly more accurate estimates, (but I don't know if those other ones can be adjusted for varying factor depths as easily) but there's an approximation for you.

[CRGreathouse]

Great, I combined that source with Chris Caldwell's simplified Wagstaff proof to come up with a reasonable heuristic. There is a funny bit about constants (wanting to be e^gamma at one end at 2 at the other), but I'll probably just interpolate smoothly.

This gives the odds that MM127 is prime as 1.8 x 10^-36, given that M127 is prime and it has no factors up to 175 bits.

[Dougal]

luigi, i have tried your factoring programs before and they are very good,but there seems to be a problem with the cygwin files so it wont open,could you help me with this problem?

with prime95 are you supposed to create your own work to do file?because my download doesnt have one,it also doesnt have a help file.

MM127 is MMMMM2,isnt it?i know essentialy this doesnt make it more likely to be prime but it would be important if it was proved that it doesnt have a factor,wouldnt it?

what other trial factoring programs are there that can be stopprd and restarted where it left off?

i dont know a lot about this subject but im looking to learn,does anyone have links to web sites with dumbed down explanations of,lets say, the lucas lehmer test?

thanks for all your help.

Don

[ewmayer]

Originally Posted by rogue

Geoff Reynold's and I wrote new code for double-Mersenne factoring. I've tried to contact Tony to see if he was interested as our code can be run on many platforms, but Tony never responded to my inquiries. If you are interested I can send the source to you.

Thanks, but I have my own code for this. ;)

Be happy to do some comparative timings, though, to see where each of us in the speed ballpark.

[10metreh]

Originally Posted by Dougal

MM127 is MMMMM2,isnt it?i know essentialy this doesnt make it more likely to be prime but it would be important if it was proved that it doesnt have a factor,wouldnt it?

Then there's MMM127 etc. Just like for Fermat's Last Theorem, proving one case is nothing.

[bsquared]

Originally Posted by Dougal

MM127 is MMMMM2,isnt it?i know essentialy this doesnt make it more likely to be prime but it would be important if it was proved that it doesnt have a factor,wouldnt it?

Proving that it doesn't have a factor by trial division up to some bound is essentially meaningless, as CR Greathouse showed (you could raise the trial factoring bit level several levels and it would still have only an infinitesimal chance of being prime). The hope is to prove that it *does* have a factor. Then the MM127 zealots can go pick a different number to rally around.