20110214, 18:14  #1  
Apr 2010
Over the rainbow
3^{2}×281 Posts 
stupid mersenne game
hello...
a new game... a stupid one, of course. take any mersenne factor, then consideer it as M(factor) then try to find a factor by tf in a 'short' time (30 bits above the start seem to be quick) Quote:
Quote:
the goal of this game is to get as high as possible Last fiddled with by firejuggler on 20110214 at 18:16 

20110215, 02:44  #2 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
4267_{10} Posts 
M5=31 is prime
M31=2147483647 is prime MM31 has a factor: 295257526626031 (48.07 bits) M295257526626031 has no factor to 2^80 MM31 has a factor: 87054709261955177 (56.27 bits) M87054709261955177 has a factor: 8322952533698486652263 (72.82 bits) M8322952533698486652263 has no factor to 2^101 MM31 has a factor: 242557615644693265201 (67.72 bits) M242557615644693265201 has no factor to 2^96 MM31 has a factor: 178021379228511215367151 (77.24 bits) M178021379228511215367151 has a factor: 3204384826113201876608719 (81.41 bits) M3204384826113201876608719 has no factor to 2^112 MM31 has no further factors to at least k=5105100000000, or about 2^76. (the chain M2 > M3 > M7 > M127 > MM127 gets boring pretty quickly: it's all primes until MM127, which has no known factors) The chance of a large (i.e. on the scale of this 'game') Mersenne number having at least one factor in the first 30 bits of TF is about 50% (for ~50 bit exponents) to 25% (for ~110 bit exponents). I'd suspect that chains with a length of 8 or more would be quite difficult to find. Even starting at M5, the most I could find just now was 4. Of course, searching many more candidates could find much longer chains, but each one individually would be significantly harder because of the starting size and the existing factoring depth of MM31. But some will be bound to have chains that grow quite slowly, keeping the probability of an easilyfound factor at each line relatively high. 
20110215, 03:00  #3 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
17×251 Posts 
M11=23*89
M23=47*178481 M47=2351*4513*13264529 M2351 has a factor: 4703 M4703 has no known factor (probably after lots of ECM) M4513 has a factor: 135391 M135391 has a factor: 1661565197287 (40.60 bits) M1661565197287 has no factor to 2^69 M13264529 (23.66 bits) has no factors to 2^64 (but it's proven composite ) M89 is prime MM89 (89.00 bits) has no known factors to at least k=6942936000000, or about 2^133. Longest known chain starting with M11: M11 > M23 > M47 > M4513 > M135391 > M1661565197287 Length: 6 Last factor size: 2^40.60 Longest known chain starting with M5: M5 > M31 > MM31 > M178021379228511215367151 > M3204384826113201876608719 Length: 5 Last factor size: 2^81.41 firejuggler's chain: M77224867 > M39977700267067630681 > M472813382664189639194777969 > M3388766988766086549622380450535561 Length: 4 Last factor size: 2^111.38 Only known M2 chain: M2 > M3 > M7 > M127 > MM127 Length: 5 Last factor size: 2^1271 
20110215, 05:10  #4  
Sep 2010
Annapolis, MD, USA
3^{3}×7 Posts 
Quote:
What software are people using to find factors in those ridiculously large numbers? I didn't realize that Prime95 did exponents over 999,999,999. 

20110215, 06:56  #5 
Apr 2010
Over the rainbow
100111100001_{2} Posts 
soooo...
M1019 has a factor : 2039 M2039 has a factor : 572478534119 (39.06 bits) M572478534119 has a factor : 1144957068239 (40.06 bits ) M1144957068239 has a factor : 99510508714787969 (56.47 bits) M99510508714787969 has no factor between 56 and 86 bits... damn it.... it looked good 
20110215, 11:17  #6 
Banned
"Luigi"
Aug 2002
Team Italia
2^{6}·3·5^{2} Posts 
I think Will Edgington may be interested in big Mersenne numbers' factors.
Luigi 
20110215, 11:21  #7  
Banned
"Luigi"
Aug 2002
Team Italia
2^{6}·3·5^{2} Posts 
Quote:
Luigi 

20110220, 18:32  #9 
Apr 2010
Over the rainbow
3^{2}×281 Posts 
so, an interesting sequence
Code:
Trialfactoring M100109 in [2^16, 2^461] M100109 has a factor: 7808503  Program: L5.0x Trialfactoring M7808503 in [2^22, 2^531] M7808503 has a factor: 78085031  Program: L5.0x Trialfactoring M78085031 in [2^26, 2^561] M78085031 has a factor: 293755886623  Program: L5.0x Trialfactoring M293755886623 in [2^38, 2^681] M293755886623 has 0 factors in [2^38, 2^681]. M78085031 has a factor: 1028379858271  Program: L5.0x Trialfactoring M1028379858271 in [2^39, 2^701] M1028379858271 has 0 factors in [2^39, 2^701]. M78085031 has a factor: 11109341641043161  Program: L5.0x Trialfactoring M11109341641043161 in [2^53, 2^831] M11109341641043161 has a factor: 95095964447329458161  Program: L5.0x Trialfactoring M95095964447329458161 in [2^66, 2^961] M95095964447329458161 has a factor: 760767715578635665289  Program: L5.0x M760767715578635665289 has a factor: 91558394569888802317531151  Program: L5.0x Trialfactoring M91558394569888802317531151 in [2^86, 2^1161] M91558394569888802317531151 has 0 factors in [2^86, 2^1161]. M760767715578635665289 has 1 factors in [2^69, 2^991]. M95095964447329458161 has a factor: 31220056099495204882845874297  Program: L5.0x Trialfactoring M31220056099495204882845874297 in [2^94, 2^1241] M31220056099495204882845874297 has a factor: 1672150138416031709921040265927481423  Program: L5.0x Trialfactoring M1672150138416031709921040265927481423 in [2^120, 2^1501] M1672150138416031709921040265927481423 has 0 factors in [2^120, 2^1501]. M31220056099495204882845874297 has 1 factors in [2^94, 2^1241]. M95095964447329458161 has 2 factors in [2^66, 2^961]. M11109341641043161 has a factor: 20530063352647761529  Program: L5.0x Trialfactoring M20530063352647761529 in [2^64, 2^941] M20530063352647761529 has 0 factors in [2^64, 2^941]. M11109341641043161 has 2 factors in [2^53, 2^831]. M78085031 has 3 factors in [2^26, 2^561]. M100109 has 1 factors in [2^16, 2^461]. 16.61(bits)>22.90>26.22>53.30>66.37>94.66>120.33 A 7 long sequence. Last fiddled with by firejuggler on 20110220 at 18:37 
20110220, 22:07  #10 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
17×251 Posts 
...and some notsointeresting (non?)extensions to that sequence:
M100109 has two more known factors: 1541125376370613303 (60.42 bits) and 29470214989842127217 (64.68 bits). M1541125376370613303 has no factor to 2^89, M29470214989842127217 has no factor to 2^93, and I extended the largest end a little: M1672150138416031709921040265927481423 has no factor from 2^150 to 2^151. 
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