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
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