On Wed, 23 Jun 1999 01:42:08 PDT, you wrote:
>
>hi everyone,
>
>there have been several messages lately about this conjecture that the n-th
>Mersenne prime is "around" (3/2)^{n}.
>
>However, no one seems to have mentioned Wagstaff's paper in Math. Comp.
>(1982 or 1983).
>
>He shows two things in this paper.
>
>(1)he shows that an earlier conjecture of Gilles (??) (something like the
>n-th Mersenne prime is "around" 2^{n} -- I can't remember what his base in
>this exponential expression was) runs contrary to known results concerning
>the distribution of primes.
>
>(2)he also gives a heuristic argument that the n-th Mersenne prime is
>"around" e^{gamma*n), where gamma is Euler's constant= 0.57721...
>
>I realise that some of you have tried to find a "best-fit" value for this
>base and that so far it appears near 3/2. But do people have any
>mathematical arguments (a la heuristic of Wagstaff) for supporting this 3/2
>value? And a pile of other related questions that I can't articulate this
>early in the morning.
>
>Alan Simpson
>
Of course, if you want a better match, you could try combining the
two:
the n-th Mersenne prime is "around" e^(2/3*gamma*n). That gives a base
of approximately 1.469.
Note - I have no mathematical arguments for this whatsoever. :-)
Cheers
Steve Whitaker
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