At 01:42 AM 6/23/99 -0700, Alan Simpson wrote:
>
>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.
I was aware of Gilles conjecture about 20 or 21 years ago when I graphed them
and saw that 3/2 was a much better fit, but I'm not aware of Wagstaff's paper.
Anyhow, both of these are reasons for expecting an infinite number of Mersenne
primes (and there may be others).
>(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?
There's no heuristic argument that I know of for 3/2 (it just fits known data
well). Wagstaff's equation is a vast overestimate for small n, but maybe
better for large n, or an upper bound.
+----------------------------------------------+
| Jud "program first and think later" McCranie |
+----------------------------------------------+
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