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Old 2019-06-30, 18:19   #45
paulunderwood
 
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Quote:
Originally Posted by lavalamp View Post
With GIMPS and Prime95 it is, due to how the prize is shared. I do not know if there is such an arrangement with gpuowl.
It is a legal minefield. Mlucas abides by GIMPS rules. If gpuowl doesn't, then I should claim I probabilistically tested with pari/GP on an N2 in less than a millisecond. The EFF has its own rules too...

Last fiddled with by paulunderwood on 2019-06-30 at 18:56
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Old 2019-06-30, 18:31   #46
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As far as I can tell, gpuowl is GPL v3, which means you can take all the code and do what you like with it. There is no mention of prize distribution in the readme, and I don't think such an agreement would be enforceable with the GPL license. Since it can interface with primenet, perhaps there is some kind of tie-in there, but that seems tenuous, especially as you would be directly testing your exponent(s) of choice instead of requesting assignments.
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Old 2019-07-01, 00:28   #47
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There is more "evidence" for this sub-sequence I have described: If one takes any start value from https://oeis.org/A018844 that is S := S0 == 0 mod 4 and apply:

Mod(Mod(1,p)*x,x^2-S*x+1)^(lift(Mod(2,p^2-1)^p-kronecker(S^2-4,p))%p) == 1 where p is a non-2^q-1 (and not 2) p|Si

one ends up with the same numbers; 607, 665972737, 60651732991. ...

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Old 2019-07-02, 11:56   #48
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There's something funny about these numbers. I looked through them and recognised a bunch of them as being near powers of 2. Many of them in fact ARE Mersenne primes.
Quote:
Originally Posted by paulunderwood View Post
Code:
[2, 5]
[3, 5]
[7, 5]
[31, 5]
[97, 5]
[127, 7]
[607, 5]
[8191, 13]
[12289, 10]
[22783, 8]
[131071, 17]
[265471, 8]
[524287, 19]
[592897, 8]
[1310719, 18]
[21757951, 18]
[29687809, 14]
[39845887, 21]
[665972737, 10]
[708158977, 6]
[2147483647, 31]
[2210398207, 22]
[2543310079, 8]
[58133053441, 23]
[60651732991, 21]
If you look at this list +/- 1, you will find an abundance of powers of 2.
Code:
          3 + 1 = 2^2
          7 + 1 = 2^3
         31 + 1 = 2^5
         97 - 1 = 2^5 * 3
        127 + 1 = 2^7
        607 + 1 = 2^5 * 19
       8191 + 1 = 2^13
      12289 - 1 = 2^12 * 3
      22783 + 1 = 2^8 * 89
     131071 + 1 = 2^17
     265471 + 1 = 2^8 * 17 * 61
     524287 + 1 = 2^19
     592897 - 1 = 2^10 * 3 * 191
    1310719 + 1 = 2^18 * 5
   21757951 + 1 = 2^18 * 83
   29687809 - 1 = 2^16 * 3 * 151
   39845887 + 1 = 2^21 * 19
  665972737 - 1 = 2^12 * 3 * 11 * 13 * 379
  708158977 - 1 = 2^9 * 3 * 7^2 * 97^2
 2147483647 + 1 = 2^31
 2210398207 + 1 = 2^22 * 17 * 31
 2543310079 + 1 = 2^8 * 5 * 109 * 18229
58133053441 - 1 = 2^24 * 3^2 * 5 * 7 * 11
60651732991 + 1 = 2^21 * 28921
708158977 is actually an interesting one, because:
708158977 - 1 = 2^9 * 3 * 7^2 * 97^2
708158977 + 1 = 2 * 31^2 * 607^2

7 and 97 both appear earlier in the list. 31 and 607 also both occur earlier in the list. It seems somehow incestuous.

Not entirely sure what to make of this, it seems to be a list of Mersenne primes and select other primes that are somewhat near powers of 2.

Edit: 708158977 - 1 = 2^3 * (97-1) * 97^2 * (97+1)

Last fiddled with by lavalamp on 2019-07-02 at 12:04
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Old 2019-07-02, 12:14   #49
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Robert Gerbicz has already pointed out factors of S[i] are of the form k*2^[i+2]+-1. However, I find the list fascinating!

What I said about 2^q-1 in my test and in particular MM19 and MM127 is untested with the likes of:

p=607;r=Mod(Mod(1,p)*x,x^2-4*x+1)^(lift(Mod(2,p^2-1)^p));if(r==1,print([p,r]))

but MM19 will not appear in the list because it has already been LL tested elsewhere.

I re-ran , with a test for 0, through the attached primesieve+GMP program, resulting in:

Code:
[127, 5]
[8191, 11]
[12289, 8]
[22783, 6]
[131071, 15]
[265471, 6]
[524287, 17]
[592897, 6]
[1310719, 16]
[21757951, 16]
[29687809, 12]
[39845887, 19]
[665972737, 8]
[2147483647, 29]
[2210398207, 20]
[2543310079, 6]
[21474836479, 30]
[58133053441, 21]
[60651732991, 19]
[87075848191, 19]
[220600496383, 6]
[1049179854847, 9]
607 and 708158977 should be in the list but the program starts with S[5].
Attached Files
File Type: log lehmer.log (325 Bytes, 28 views)
File Type: txt lehmer.cpp.txt (1.1 KB, 26 views)
File Type: txt lehmer.dat.txt (30 Bytes, 26 views)

Last fiddled with by paulunderwood on 2019-07-02 at 15:13
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Old 2019-07-02, 13:54   #50
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Some of these also factor in another interesting way.
Code:
          3 - 1 = 2 * (1-1)(1+1)
          5 + 1 = 2 * (2-1)(2+1)
          7 - 1 = 2 * (2-1)(2+1)
         31 - 1 = 2 * (4-1)(4+1)
         97 - 1 = 2 * (7-1)(7+1)
        127 - 1 = 2 * (8-1)(8+1)
       8191 - 1 = 2 * (64-1)(64+1) = 90 * 91
      12289 - 1 = 2^8 * (7-1)(7+1)
     131071 - 1 = 2 * (256-1)(256+1)
     265471 + 1 = 2^2 * (33-1)(33+1) * 61
     524287 - 1 = 2 * (512-1)(512+1)
     592897 - 1 = 2^2 * (385-1)(385+1)
    1310719 + 1 = 2^14 * (9-1)(9+1)
   21757951 - 1 = 2 * (226-1)(226+1) * 213
  708158977 - 1 = 2 * (18817-1)(18817+1) = 2^3 * 96 * 97^2 * 98
 2147483647 - 1 = 2 * (32768-1)(32768+1)
58133053441 - 1 = 2^24 * (34-1)(34+1) * 3
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Old 2019-07-02, 15:37   #51
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Cool

Code:
p=607;eu=eulerphi(p^2-1);r=Mod(Mod(1,p)*x,x^2-4*x+1)^(lift(Mod(2,p^2-1)^lift(Mod(2,eu)^p)));if(r==1,print([p,r]))
[607, Mod(Mod(1, 607), x^2 - 4*x + 1)]
What I am trying to do is show that 2^607-1 appears in the list (which it does as it is a know Mp.)

If the above holds and:

Code:
p=607;r=Mod(Mod(1,p)*x,x^2-4*x+1)^(lift(Mod(2,p^2-1)^p));if(r==1,print([p,r]))
can one imply that:

Code:
p=607;r=Mod(Mod(1,p)*x,x^2-4*x+1)^(lift(Mod(2,p^2-1)^p-kronecker(p,n))%p);if(r==1,print([p,r]))
?


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Old 2019-07-02, 19:20   #52
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Quote:
Originally Posted by paulunderwood View Post
one ends up with the same numbers; 607, 665972737, 60651732991. ...
In another thread, there's a possible improvement to a PARI script to allow some factoring of very large exponents beyond mfaktc's range.

Maybe it could be used to try to find some small factors for M60,651,732,991 ?
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Old 2019-07-02, 19:39   #53
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Quote:
Originally Posted by GP2 View Post
In another thread, there's a possible improvement to a PARI script to allow some factoring of very large exponents beyond mfaktc's range.

Maybe it could be used to try to find some small factors for M60,651,732,991 ?
factor5 tells me "M60651732991 has 0 factors in [2^1, 2^68-1]."

I have resumed the above attached GMP+primesieve program to try and find the next in the sequence,
and since we can factor up to S[6], I added the following line as the first line of "callback":

Code:
  if ( ( prime + 1 ) %256 != 0 && ( prime - 1 ) %256 != 0 ) return;

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Old 2019-07-02, 21:26   #54
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Quote:
Originally Posted by paulunderwood View Post
one ends up with the same numbers; 607, 665972737, 60651732991. ...
The next is M4,123,168,604,161. Over a trillion digits and the exponent is a factor of S[35].

Code:
./factor5 4123168604161 1 75 2

Factor5 v. 5.01 - December 27th, 2007 - AMD64/Win32 compile - GMP 6.1.2

Current date Tue Jul  2 22:30:10 2019

Factoring M4123168604161 from 2^1 to 2^75   6.597% completed.
Factoring M4123168604161 from 2^1 to 2^75  49.000% completed.
Factoring M4123168604161 from 2^1 to 2^75  91.155% completed.
No factor found

Performed    178706599 powmod operations since last restart.
Used 135468 primes, max. prime = 1805789
Current date Tue Jul  2 22:31:21 2019

Last fiddled with by paulunderwood on 2019-07-02 at 22:07
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Old 2019-07-02, 22:16   #55
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Quote:
Originally Posted by paulunderwood View Post
Current date Tue Jul 2 22:30:10 2019
Current date Tue Jul 2 22:31:21 2019
[/code]
71 seconds clock time... maybe you're not trying hard enough to find factors.
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