>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.
>...
>In this case, there exist (large prime) integers a, b such that
>(2 * a * n + 1) * (2 * b * n + 1) = 2^n - 1
Actually, all factors of any 2^n-1 can be expressed in the form 2*a*n+1,
including 2^n-1 itself (for prime n, obviously.)
-Lucas Wiman
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