Number of distinct prime factors of a Double Mersenne number
I am trying to figure out a way to estimate the number of distinct prime factors of a Double Mersenne number. If I understand it rightly, for a specific number n the number of distinct prime factors x are:
ω (n) which is asymptotically equal to ln (ln n)
for a Double Mersenne number n=MMp:
ln(ln (2^(2^p1)1))
ignoring both "1" since those parts will be infinitesimally small with growing p.
ln(2^p * ln(2)) =
ln(ln(2)) + p*ln(2) =
0.367 + p*ln(2) =
0.367 + 0.693*p
i.e. for MM127 (i.e. p=127) x=87.64
Maybe it may be argued that since both p and Mp are prime x may be a little smaller?
Have I understood this rightly or have I done something wrong?
If this is right the nice thing is that the estimated number of distinct prime factors of a MMp are directely proportional to p.
Last fiddled with by aketilander on 20121109 at 20:09
