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2019-06-30, 18:19   #45
paulunderwood

Sep 2002
Database er0rr

64108 Posts

Quote:
 Originally Posted by lavalamp 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

 2019-06-30, 18:31 #46 lavalamp     Oct 2007 London, UK 22×52×13 Posts 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.
 2019-07-01, 00:28 #47 paulunderwood     Sep 2002 Database er0rr D0816 Posts 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. ... Last fiddled with by paulunderwood on 2019-07-01 at 00:38
2019-07-02, 11:56   #48
lavalamp

Oct 2007
London, UK

101000101002 Posts

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 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

2019-07-02, 12:14   #49
paulunderwood

Sep 2002
Database er0rr

23·3·139 Posts

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
 lehmer.log (325 Bytes, 28 views) lehmer.cpp.txt (1.1 KB, 26 views) lehmer.dat.txt (30 Bytes, 26 views)

Last fiddled with by paulunderwood on 2019-07-02 at 15:13

 2019-07-02, 13:54 #50 lavalamp     Oct 2007 London, UK 101000101002 Posts 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
 2019-07-02, 15:37 #51 paulunderwood     Sep 2002 Database er0rr 1101000010002 Posts 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])) ? Last fiddled with by paulunderwood on 2019-07-02 at 15:56
2019-07-02, 19:20   #52
GP2

Sep 2003

2,579 Posts

Quote:
 Originally Posted by paulunderwood 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 ?

2019-07-02, 19:39   #53
paulunderwood

Sep 2002
Database er0rr

23·3·139 Posts

Quote:
 Originally Posted by GP2 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;

Last fiddled with by paulunderwood on 2019-07-02 at 20:09

2019-07-02, 21:26   #54
paulunderwood

Sep 2002
Database er0rr

64108 Posts

Quote:
 Originally Posted by paulunderwood 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

2019-07-02, 22:16   #55
GP2

Sep 2003

2,579 Posts

Quote:
 Originally Posted by paulunderwood 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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