P-1 vs. TF

[GP2]

Originally Posted by petrw1

And if I read the rest of if correctly then:

* in about 1% of the exponents tested, it DOES NOT matter whether we do TF 72–76 first or P−1 first, because BOTH methods can find a factor

Mind you we are now at 101% but then the 2, 8, 90 are rounded?

No, I should have phrased it better. In 2% of the cases, both methods find the same factor. There is no additional 1%.

[GP2]

I redid the calculation and the numbers are ever so slightly different, probably because I'm now relying on exact bit length rather than using log base 2 to approximate the bit size, which earlier led to rounding errors and misclassification of a few factors that are very near to powers of two. It really doesn't affect the conclusions at all.

No new factors of > 72 bit length have been found in the meantime, only a few additional 66-bit factors being found by TJAOI. The number of pending P−1 in the range is now only 180, as of this writing.

The corrected numbers are:

1596 known factors of > 72 bit length in the range 86.0M to 86.9M

266 factors found by TF 72–73 (of which 97 would be found with P−1 B1=700k, B2=13M)
254 factors found by TF 73–74 (of which 85 would be found with P−1 B1=700k, B2=13M)
234 factors found by TF 74–75 (of which 73 would be found with P−1 B1=700k, B2=13M)
217 factors found by TF 75–76 (of which 65 would be found with P−1 B1=700k, B2=13M)
625 factors found by P−1 (of which 604 would be found with strict B1=700k, B2=13M)

If we change the P−1 bounds to B1=875k, B2=21M instead, then the numbers found with P−1 change from 97, 85, 73, 65, 604 to 106, 90, 80, 69, 615. Again, this really doesn't affect the conclusions, the numbers change by no more than 10%.

As before, we estimate that there are about 17,200 exponents in this range where neither TF 72–76 testing nor P−1 testing found a factor.



The main conclusion remains that TF 72–76 testing and P−1 testing are much more independent of each other than some of us thought earlier. They mostly find different factors.

Many factors found by TF 72–76 have unsmooth k, for example M86001449 would require B2=1,018,592,637,113 so it would be hopeless to try to find this with P−1.

Conversely, many factors found by P−1 have large bit lengths that are hopeless to find with TF:

69 factors of bit length 77
50 factors of bit length 78
36 factors of bit length 79
52 factors of bit length 80
35 factors of bit length 81
and so forth, with a long tail all the way up to a factor of bit length 123.

I am attaching a file containing all 1596 currently-known factors of bit length greater than 72 in the range 86.0M–86.9M. Each line contains five comma-separated values:
p, the Mersenne exponent
f, a known factor of the Mersenne number with that exponent
bit length of that factor, i.e., the number of digits when written in base 2
B1, the minimum B1 bound needed to find this factor with certainty using P−1 testing
B2, the minimum B2 bound needed to find this factor with certainty using P−1 testing (i.e., without a lucky hit via Brent-Suyama)