20190630, 18:19  #45  
Sep 2002
Database er0rr
6410_{8} Posts 
Quote:
Last fiddled with by paulunderwood on 20190630 at 18:56 

20190630, 18:31  #46 
Oct 2007
London, UK
2^{2}×5^{2}×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 tiein there, but that seems tenuous, especially as you would be directly testing your exponent(s) of choice instead of requesting assignments.

20190701, 00:28  #47 
Sep 2002
Database er0rr
D08_{16} Posts 
There is more "evidence" for this subsequence I have described: If one takes any start value from https://oeis.org/A018844 that is S := S_{0} == 0 mod 4 and apply:
Mod(Mod(1,p)*x,x^2S*x+1)^(lift(Mod(2,p^21)^pkronecker(S^24,p))%p) == 1 where p is a non2^q1 (and not 2) pS_{i} one ends up with the same numbers; 607, 665972737, 60651732991. ... Last fiddled with by paulunderwood on 20190701 at 00:38 
20190702, 11:56  #48  
Oct 2007
London, UK
10100010100_{2} 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:
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  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 * (971) * 97^2 * (97+1) Last fiddled with by lavalamp on 20190702 at 12:04 

20190702, 12:14  #49 
Sep 2002
Database er0rr
2^{3}·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^q1 in my test and in particular MM19 and MM127 is untested with the likes of: p=607;r=Mod(Mod(1,p)*x,x^24*x+1)^(lift(Mod(2,p^21)^p));if(r==1,print([p,r])) but MM19 will not appear in the list because it has already been LL tested elsewhere. I reran , 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] Last fiddled with by paulunderwood on 20190702 at 15:13 
20190702, 13:54  #50 
Oct 2007
London, UK
10100010100_{2} Posts 
Some of these also factor in another interesting way.
Code:
3  1 = 2 * (11)(1+1) 5 + 1 = 2 * (21)(2+1) 7  1 = 2 * (21)(2+1) 31  1 = 2 * (41)(4+1) 97  1 = 2 * (71)(7+1) 127  1 = 2 * (81)(8+1) 8191  1 = 2 * (641)(64+1) = 90 * 91 12289  1 = 2^8 * (71)(7+1) 131071  1 = 2 * (2561)(256+1) 265471 + 1 = 2^2 * (331)(33+1) * 61 524287  1 = 2 * (5121)(512+1) 592897  1 = 2^2 * (3851)(385+1) 1310719 + 1 = 2^14 * (91)(9+1) 21757951  1 = 2 * (2261)(226+1) * 213 708158977  1 = 2 * (188171)(18817+1) = 2^3 * 96 * 97^2 * 98 2147483647  1 = 2 * (327681)(32768+1) 58133053441  1 = 2^24 * (341)(34+1) * 3 
20190702, 15:37  #51 
Sep 2002
Database er0rr
110100001000_{2} Posts 
Code:
p=607;eu=eulerphi(p^21);r=Mod(Mod(1,p)*x,x^24*x+1)^(lift(Mod(2,p^21)^lift(Mod(2,eu)^p)));if(r==1,print([p,r])) [607, Mod(Mod(1, 607), x^2  4*x + 1)] If the above holds and: Code:
p=607;r=Mod(Mod(1,p)*x,x^24*x+1)^(lift(Mod(2,p^21)^p));if(r==1,print([p,r])) Code:
p=607;r=Mod(Mod(1,p)*x,x^24*x+1)^(lift(Mod(2,p^21)^pkronecker(p,n))%p);if(r==1,print([p,r])) Last fiddled with by paulunderwood on 20190702 at 15:56 
20190702, 19:20  #52  
Sep 2003
2,579 Posts 
Quote:
Maybe it could be used to try to find some small factors for M60,651,732,991 ? 

20190702, 19:39  #53  
Sep 2002
Database er0rr
2^{3}·3·139 Posts 
Quote:
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 20190702 at 20:09 

20190702, 21:26  #54  
Sep 2002
Database er0rr
6410_{8} Posts 
Quote:
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 20190702 at 22:07 

20190702, 22:16  #55 
Sep 2003
2,579 Posts 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
The "one billion minus 999,994,000" digits prime number  a1call  Miscellaneous Math  179  20151112 14:59 
question range 1 billion to 2 billion?  Unregistered  Information & Answers  7  20100812 06:25 
Billion digit prime?  lfm  Operation Billion Digits  6  20090107 01:17 
Factoring a 617digit number?  Shakaru  Factoring  2  20050223 19:22 
10,000,000 digit number  Unregistered  Software  3  20040303 19:20 