Sorry if this has been mentioned before, but...
I was thinking (always dangerous!) about some of the smaller
Mersenne numbers, like 2^727-1, for which no factors are known.
2^727-1 has 219 digits, and we are pretty darn sure that it has no
prime factors less than 40 digits long. Therefore it would seem
sensible to "assume" that it is the product of only two prime factors.
This may be also a reasonable assumption for some other small
exponents for which no factors are known.
In this case, there exist (large prime) integers a, b such that
(2 * a * n + 1) * (2 * b * n + 1) = 2^n - 1
A bit of rearrangement leads to the equation
2 * a * b * n^2 + (a + b) * n - 2^(n - 1) = 0
which it looks "feasible" to solve - yielding directly the two prime
factors of 2^n-1, assuming the assumption is correct.
On the other hand, if the equation has no integer solutions for a &
b, then we know that the assumption is incorrect, and that there
must therefore be more than two factors of 2^(n-1).
Given that a 219 digit number can't have all that many factors each
bigger than 10^40, it seems that it might be worthwhile trying to
solve the equation, at least for n=727.
Regards
Brian Beesley
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