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2019-01-12, 01:09   #265
GP2

Sep 2003

258210 Posts

Quote:
 Originally Posted by Dr Sardonicus Looking at the current status of the exponent 8191, one other prime factor is known, and the remaining cofactor is a PRP.
Uh... it's not.

 2019-01-12, 01:21 #266 philmoore     "Phil" Sep 2002 Tracktown, U.S.A. 100010111102 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.
 2019-01-12, 02:43 #267 LaurV Romulan Interpreter     Jun 2011 Thailand 22F916 Posts We know that. We watched you at the time , and we also did a lot of work on those DM's in 2012-2014 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 2019-01-12 at 02:44
2019-01-12, 03:05   #268
GP2

Sep 2003

2×1,291 Posts

Quote:
 Originally Posted by philmoore 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.
If we ever find a non-squarefree Mersenne number (with prime exponent), it would make headlines. The factor in question would be the third known Wieferich prime.

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.

2019-01-12, 13:24   #269
Dr Sardonicus

Feb 2017
Nowhere

23×5×97 Posts

Quote:
 Originally Posted by GP2 Uh... it's not.
Sorry about that. When I read "PRP Cofactor" in the "Status" column I thought it meant the cofactor was a PRP. Apparently it means "results of PRP test on cofactor" or some such.

Last fiddled with by Dr Sardonicus on 2019-01-12 at 13:28

2019-01-12, 13:53   #270
GP2

Sep 2003

50268 Posts

Quote:
 Originally Posted by Dr Sardonicus Sorry about that. When I read "PRP Cofactor" in the "Status" column I thought it meant the cofactor was a PRP. Apparently it means "results of PRP test on cofactor" or some such.
You're not the first one to be confused by that.

It really should be changed from "PRP Cofactor" to "Cofactor PRP test".

And then for consistency, "LL" to "LL test", "PRP" to "PRP test", "P-1" to "P−1 test".

2019-01-12, 17:43   #271
JeppeSN

"Jeppe"
Jan 2016
Denmark

2×34 Posts

Quote:
 Originally Posted by GP2 Never mind larger examples, there's no smaller example. The only other p=2^k+k which is a Mersenne prime exponent is k=1, p=3, but then W(k) = 1.
Not sure I know what you mean. With k=1 you are describing the smaller example. It gives $$2^k+k = 3$$ and $$2^k=2$$ and the prime (seven) is $M(3)=2^3-1=2\cdot 2^2 - 1=W(2)$
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 2019-01-12 at 18:42 Reason: adding W(512) for comparison

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