Mersenne non-primes
 

Is there a list anywhere of the Mersenne numbers known not to be prime?

At the bottom of the status page
[url]http://www.mersenne.org/status.htm[/url]
there are links to different files making up the Mersenne database. The first three (exponents of factored numbers, double-checked numbers and numbers that have had only one Lucas-Lehmer test) would be what you are asking for.

Historically there has been an error rate of a few percent in the Lucas-Lehmer tests. Therefore the computation is repeated. The third file above can contain more than one test for an exponent if the tests are not matching.

Re: Mersenne non-primes
 

[QUOTE][i]Originally posted by Unregistered [/i]
[B]Is there a list anywhere of the Mersenne numbers known not to be prime? [/B][/QUOTE]

You can get this by combining the list of exponents with known factors (factors.zip) and the list of exponents that have had two Lucas-Lehmer tests with matching 64-bit residues (lucas_v.zip).

Currently 2,667,859 exponents have known factors and 256,339 exponents have two matching LL tests.

Strictly speaking, the third file (hrf3.txt) doesn't meet the criterion "known not to be prime". Each number in this file has been LL tested once, and for each number the result of that LL test indicates that the number is not prime. But until the LL test is repeated (doublecheck) and it is confirmed that the doublecheck result matches the first result, it cannot be stated for certain that the number is not prime. Until the doublecheck is complete, it's possible that the first LL test was incorrect.

Thanks for the answers
 

The main reason for my interest was that I thought it would be a neat objective test for anyone claiming supernatural or extraterrestrial knowledge. The factors of an unknown Mersenne Prime is hardly disruptive knowlegde: but if they really came back with one, they'd be worth taking seriously. Much more likely, they'd not come back at all, or come back with an answer that was demonstrably wrong.

I'm not qualified to produce such a list, nor to protect my home computer from the sort of hassle that it would attract if it attracted interest. Is there someone out there who feels like taking it up?

Don't forget that knowing a number is not prime is not the same as actually knowing its factors.

For instance, we know that M971 is not prime because it fails the Lucas-Lehmer test, but we currently don't know any factor for it.

A good test of extraterrestrial intelligence would be if they could tell us what, say, M45 is...

[QUOTE][i]Originally posted by GP2 [/i]
[B]Don't forget that knowing a number is not prime is not the same as actually knowing its factors.

For instance, we know that M971 is not prime because it fails the Lucas-Lehmer test, but we currently don't know any factor for it.

A good test of extraterrestrial intelligence would be if they could tell us what, say, M45 is... [/B][/QUOTE]


I know this is off the point you were making about m971 but your statement about no known factors got to me and I tried to use the ecm part of prime95 for the first time. My results file said this...

[Sat Jan 03 20:52:46 2004]
UID: lmurray/hp2200, M971 is not prime. Res64: F366174A235BF2B2. WZ1: 11540C62,504,00000000
UID: lmurray/hp2200, M971 no factor to 2^40, WZ1: 00100004
[Sat Jan 03 20:58:58 2004]
ECM found a factor in curve #1, stage #1
Sigma=8727776697676685, B1=1000000, B2=100000000.
UID: lmurray/hp2200, P971 has a factor: 495211
[Sat Jan 03 21:04:04 2004]
ECM found a factor in curve #4, stage #2
Sigma=8252620261253466, B1=1000000, B2=100000000.
UID: lmurray/hp2200, P971 has a factor: 3019967621847401
[Sat Jan 03 21:10:21 2004]
ECM found a factor in curve #10, stage #2
Sigma=905319878861278, B1=1000000, B2=100000000.
UID: lmurray/hp2200, P971 has a factor: 745797773625002825025979


Does this mean we now know factors for it or am I not understanding this?

What you have is [B][I]P[/B][/I]971, meaning 2[sup]971[/sup]+1. We are looking for factors of M971, of which none are known.

Note: P971=3*495211*3019967621847401*745797773625002825025979*P205 (a 205 digit prime).

Thanks for the answer. I got the part that the 971 I was using is the 2[sup]971[/sup]-1. I guess my problem is I don't understand what m971 is, and how you would go about looking for a factor for this. Thanks for any and all education... Although I've always loved math, it doesn't feel the same about me when it comes to understanding it:smile:

For a good understandable explanation of the math behind the project, look [URL=http://www.mersenne.org/math.htm]here[/URL] at the math page of the GIMPS website. There is also a Mersenne page [URL=http://www.utm.edu/research/primes/mersenne/]here[/URL] at The Prime Pages (a site you will probably want to explore.)

Could Prime95 if it allowed composite exponents be used to find factors, or is there some mathematical issue that precludes it ?

What about the power mod function is that valid for mersenne composites ? Would the factors still be in the form 2kp+1 ?

[QUOTE][i]Originally posted by dsouza123 [/i]
[B]Could Prime95 if it allowed composite exponents be used to find factors[/B][/QUOTE]

Prime95 works fine with composite exponents. The [URL=http://elevensmooth.com/Tour01.html]ElevenSmooth[/URL] [URL=http://elevensmooth.com/ElevenFAQ.html#Special]Special Project[/URL] uses Prime95 to search for factors of composite exponents. We've found [URL=http://elevensmooth.com/ElevenFactors.html]over fifty[/URL] factors with Prime95.

One small issue with using Prime95 is that automatically search the [URL=http://elevensmooth.com/ElevenFAQ.html#Algebraic]Algebraic Factors[/URL], too, so for each factor you must figure out which [URL=http://elevensmooth.com/ElevenFAQ.html#Primitive]Mersenne Primitive[/URL] the factor belongs too. Will Edgington collects Mersenne factors, so you should check with him about previously known factors and report to him any newly discovered factors. Reach him through his [URL=http://www.garlic.com/~wedgingt/mersenne.html]Mersenne Page[/URL].

William

So to use composite exponents it requires using ECM in Prime95 to find factors, is this correct ?

Is it (ECM) restricted to 2^p + 1 or can it test 2^p - 1 ( the mersenne numbers (prime exponent) that are tested by trial factoring, p-1 factoring and LL ) ?
============================

How/where do you find the status (factor(s), composite, prime) of a mersenne number ( with a prime exponent) ?
I checked a status.txt file from Dec 2003 and it starts with the 7 millions.

For the M971 ( 2^971 - 1) mentioned how/where would it's status be found ?

Prime95 will also do P-1 factoring of composite exponents.

I haven't tried it, but I think Prime95 will do trial factoring of composite exponents. I suspect that it would only find factors of the primitive part, not the algebraic factors.

I know Prime95 works with 2[sup]m[/sup]-1. I think it also works with 2[sup]m[/sup]+1 in all modes.

Will Edgington's lowM.txt file shows the following entries for M(971)

M( 971 )E: 413817700 50000000 0.56
M( 971 )H: 144115188077210231
M( 971 )c: 293
M( 971 )o: 4294000000 4294000000

The file mersfmt.txt explains these. The E line shows that Will knows that somebody has done ECM work with B1=50M, although no more than 8 curves have been tried at this level. The H line shows the highest trial factor attempted. The "o" line shows that somebody has tried P-1 factoring with those bounds, meaning no stage 2 was used. The c line shows the remaining unfactored composite is 293 digits. There are not "C" lines, so no factors are known for this primitive. 971 is prime, so there are no algebraic factors.

For small exponents it is often faster to use GMP-ECM than Prime95 - run some timing comparisons to find what works best for your machine and your exponent.

If you use Prime95 with composite exponents, you should look at Philmoore's information on [URL=http://www.mersenneforum.org/showthread.php?s=&threadid=1008]factoring highly composite Mersenne Numbers[/URL].

William