> <<Suppose M(x) is the number of primes p <= x for which 2^p - 1 is prime....
> Lenstra, Pomerance, and Wagstaff all believe this [an early conjecture by
> Gillies] and in fact suggest that ?? M(x) ~ e^gamma log x ?? where the log
> is to base 2.>>
> Hence, my new conjecture:
> ?? M(x) ~ e^gamma log[2] (x) + C ??
>
> Of course, I used 1.4615 to make my 3 conjectures to the Mersenne mailing
> list. In reality, I'm guessing it might be 1.5, or even 2^(1/e^gamma)! (In
> fact, I'd rather go with 2^(1/e^gamma), as Erhardt chose 1.5 for the e^gamma
> in the conjecture and now several mathematicians call the Erhardt Conjecture
I'm afraid that if you are correct, so is Wagstaff. The symbol "~",
at least in mathematics means that if f(x)~g(x) then f(x)/g(x)=1 as
x->infinity.
Your conjecture seems like it would yeild a better aproximation than
Wagstaff's (you could certainly argue that 2 is a special case, since
it's corresponding Mersenne is the next bloody prime, and there is 1
of the 1.4615 right there).
-Lucas
_________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
