Regarding the discussion about the distribution of M_p:
Sam Wagstaff's results imply that the expected number of
Mersenne primes between 2^h and 2^2h is exp(gamma).
Thus, they DO get progressively rarer.
Further, by the PNT, the probability that a random integer
near x is prime is 1/log(x). *ASSUMING* that 2^p-1 behaves
like a random integer, the probability that it is prime should
be 1/p log(2).
Now, sum from 2 to k of 1/p is asymptotically
loglog k [this is easy; p_n ~ n log n from PNT, so
by Stieltje's integration (or Euler-Maclauren) on gets
sum from 2 to k of 1/p = integral from 2 to k of 1/(n log n) d [n].
Now integrate by parts. ]
Thus, one should expect that the number of Mersenne primes up to
k is O(log log k).
Be wary of what Richard Guy calls the law of small numbers...
Most number-theoretic phenomena only show their true behavior for
VERY large numbers and we are not there yet. To put it another way,
as John Selfridge said: although we know loglog n goes to infinity,
it has never actually been observed to do so...
BTW, there is nothing unique about base 2 in this regard. We
should expect that the number of primes of the form (a^p - 1)/(a-1)
up to k is O(log log k) for all a. The only thing that changes
is the implied constant.
Bob Silverman
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
