Lucky 13 (M51 related)

[GP2]

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.

[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. Currently, the ECM status shows that it probably has no other factors less than around 50 digits.

[LaurV]

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.

[GP2]

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.

[Dr Sardonicus]

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.

[GP2]

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".

[JeppeSN]

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:

   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)
      .   .
      .   .
      .   .