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Old 2011-02-14, 18:14   #1
firejuggler
 
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Default 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:
M77224867 has a factor
Quote:
: 39977700267067630681 (65.12 bits)
M39977700267067630681 has a factor: 2718403662760064751046639 (81.17 bits- TF from 81 to 110)
M39977700267067630681 has a factor: 472813382664189639194777969 (88.61 bits- 81 to 110)
M472813382664189639194777969 has a factor: 3388766988766086549622380450535561 (111.38 bits - 88 to 116)


the goal of this game is to get as high as possible

Last fiddled with by firejuggler on 2011-02-14 at 18:16
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Old 2011-02-15, 02:44   #2
Mini-Geek
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"Tim Sorbera"
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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 easily-found factor at each line relatively high.
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Old 2011-02-15, 03:00   #3
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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: 2^127-1
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Old 2011-02-15, 05:10   #4
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Quote:
Originally Posted by Mini-Geek View Post
M13264529 (23.66 bits) has no factors to 2^64 (but it's proven composite )
Claiming for some P-1, I've got another small P-1 I'm doing first, but this one should be done in 8 hours or so, according to mprime -s. Gonna do B1=400k, B2=10M. What's 0.64 GHz-days for a "stupid mersenne game", anyway?

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.
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Old 2011-02-15, 06:56   #5
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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
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Old 2011-02-15, 11:17   #6
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I think Will Edgington may be interested in big Mersenne numbers' factors.

Luigi
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Old 2011-02-15, 11:21   #7
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Quote:
Originally Posted by KingKurly View Post
Claiming for some P-1, I've got another small P-1 I'm doing first, but this one should be done in 8 hours or so, according to mprime -s. Gonna do B1=400k, B2=10M. What's 0.64 GHz-days for a "stupid mersenne game", anyway?

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.
Factor5 can easily handle such numbers.

Luigi
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Old 2011-02-15, 18:51   #8
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Quote:
Originally Posted by ET_ View Post
Factor5 can easily handle such numbers.

Luigi
Gotcha.

Anyway, my P-1 of M13264529 completed without finding any factors. If anyone is interested enough to do more, have at it. I think probably too much CPU time has gone into it already though...
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Old 2011-02-20, 18:32   #9
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so, an interesting sequence
Code:
Trial-factoring M100109 in [2^16, 2^46-1]
M100109 has a factor: 7808503 - Program: L5.0x
    Trial-factoring M7808503 in [2^22, 2^53-1]
    M7808503 has a factor: 78085031 - Program: L5.0x
        Trial-factoring M78085031 in [2^26, 2^56-1]
        M78085031 has a factor: 293755886623 - Program: L5.0x
            Trial-factoring M293755886623 in [2^38, 2^68-1]
            M293755886623 has 0 factors in [2^38, 2^68-1].    
        M78085031 has a factor: 1028379858271 - Program: L5.0x
            Trial-factoring M1028379858271 in [2^39, 2^70-1]
            M1028379858271 has 0 factors in [2^39, 2^70-1].
        M78085031 has a factor: 11109341641043161 - Program: L5.0x
            Trial-factoring M11109341641043161 in [2^53, 2^83-1]
                M11109341641043161 has a factor: 95095964447329458161 - Program: L5.0x
                    Trial-factoring M95095964447329458161 in [2^66, 2^96-1]
                        M95095964447329458161 has a factor: 760767715578635665289 - Program: L5.0x
                            M760767715578635665289 has a factor: 91558394569888802317531151 - Program: L5.0x
                                Trial-factoring M91558394569888802317531151 in [2^86, 2^116-1]
                                M91558394569888802317531151 has 0 factors in [2^86, 2^116-1].
                            M760767715578635665289 has 1 factors in [2^69, 2^99-1].
                        M95095964447329458161 has a factor: 31220056099495204882845874297 - Program: L5.0x
                            Trial-factoring M31220056099495204882845874297 in [2^94, 2^124-1]
                                M31220056099495204882845874297 has a factor: 1672150138416031709921040265927481423 - Program: L5.0x
                                    Trial-factoring M1672150138416031709921040265927481423 in [2^120, 2^150-1]
                                    M1672150138416031709921040265927481423 has 0 factors in [2^120, 2^150-1].
                            M31220056099495204882845874297 has 1 factors in [2^94, 2^124-1].
                    M95095964447329458161 has 2 factors in [2^66, 2^96-1].
                M11109341641043161 has a factor: 20530063352647761529 - Program: L5.0x
                    Trial-factoring M20530063352647761529 in [2^64, 2^94-1]
                    M20530063352647761529 has 0 factors in [2^64, 2^94-1].
            M11109341641043161 has 2 factors in [2^53, 2^83-1].
        M78085031 has 3 factors in [2^26, 2^56-1].
M100109 has 1 factors in [2^16, 2^46-1].
M100109->M7808503->M78085031->M11109341641043161->M95095964447329458161->M31220056099495204882845874297 ->M1672150138416031709921040265927481423
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 2011-02-20 at 18:37
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Old 2011-02-20, 22:07   #10
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...and some not-so-interesting (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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