New mersenne factor found, what to do with it?

[James Heinrich]

Originally Posted by kriesel

That's more computer cycles (and James cycles) than I anticipated but would do the job.

The James cycles were not that many, and the computer cycles were considerably fewer than I expected.
I first ran it through GIMPS range (<1000M):

Examined 43,381,507 square factors in 5.4 minutes
Found 0 cases where any known factors is a square factor.

Since it ran quickly I then ran it in the rest of the 1G-10G range:

Examined 242,435,609 square factors in 28.1 minutes

Still no examples found.

[Uncwilly]

For a factor to be square of another factor it would still have to be a factor in its own right.
So for q to be a factor it would have to both be (2kp+1)(2kp+1) and 2Жp+1

(2kp+1)(2kp+1) = 2Жp+1
4k²p² + 4kp + 1 = 2Жp + 1
4k²p² + 4kp = 2Жp
2k²p + 2k = Ж
2k(kp +1) = Ж

As there any to prove that this can't be?

[slandrum]

Originally Posted by Uncwilly

For a factor to be square of another factor it would still have to be a factor in its own right.
So for q to be a factor it would have to both be (2kp+1)(2kp+1) and 2Жp+1
...
As there any to prove that this can't be?

No obvious way, the same argument applies to any composite factor of a Mersenne number, and the composite factors are all of the form 2kp+1 (as are the factors 1 and the Mersenne number M(p) itself).

That's why it's still open as to whether a Mersenne number can have a square (>1) as a factor.