20190112, 01:21  #266 
"Phil"
Sep 2002
Tracktown, U.S.A.
2137_{8} Posts 
I found the larger of the two known prime factors of M8191 in 2003 and did the computations at the time to show that the cofactor was not only composite, but was also not a power of a single prime factor, so we know that the cofactor has at least two distinct prime factors. Currently, the ECM status shows that it probably has no other factors less than around 50 digits.

20190112, 02:43  #267 
Romulan Interpreter
Jun 2011
Thailand
3×13^{2}×19 Posts 
We know that. We watched you at the time , and we also did a lot of work on those DM's in 20122014 or so, with the mmff fever, but stopped for a while. That mother is composite. But from the amount of work done on it, no new factor under (about) 45 digits should exits.
Last fiddled with by LaurV on 20190112 at 02:44 
20190112, 03:05  #268  
Sep 2003
5·11·47 Posts 
Quote:
I run a script to check for this every few days. Takes a fraction of a second. Very tiny effort, very huge payoff, astronomical odds. 

20190112, 13:24  #269  
Feb 2017
Nowhere
4,649 Posts 
Quote:
Last fiddled with by Dr Sardonicus on 20190112 at 13:28 

20190112, 13:53  #270  
Sep 2003
5×11×47 Posts 
Quote:
It really should be changed from "PRP Cofactor" to "Cofactor PRP test". And then for consistency, "LL" to "LL test", "PRP" to "PRP test", "P1" to "P−1 test". 

20190112, 17:43  #271  
"Jeppe"
Jan 2016
Denmark
2^{3}·3·7 Posts 
Quote:
The other example k=9 written the same way, since \(2^k+k = 521\) and \(2^k=512\), is \[M(521)=2^{521}1=512\cdot 2^{512}  1=W(512)\] For the fun of it, we can merge the lists of Mersennes and Woodalls like this: Code:
M(2) M(3) = W(2) W(3) M(5) M(7) W(6) M(13) M(17) M(19) M(31) W(30) M(61) W(75) W(81) M(89) M(107) W(115) M(127) W(123) W(249) W(362) W(384) W(462) M(521) = W(512) M(607) W(751) W(822) M(1279) M(2203) M(2281) M(3217) M(4253) M(4423) W(5312) . . . . . . Last fiddled with by JeppeSN on 20190112 at 18:42 Reason: adding W(512) for comparison 

20210618, 07:00  #272 
Jun 2021
41 Posts 
Good job!

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