According to Paulo Ribenboim's book quoted below by Jud Euler's Constant
gamma=0.577215665... and working out the number of mersenne primes below p=7000000
in Mathematica 4.0 gives 39.5572 primes, so we must be missing a prime if
Wagstaffs' right.
Allan Menezes
Jud McCranie wrote:
> For those of us who don't have access to Wagstaff's 1983 paper "Divisors of
> Mersenne Numbers", it is nicely summarized in "The New Book of Prime Number
> Records", by Paulo Ribenboim, chapter 6, section V.A. (page 411-413 in this
> edition). He gives 3 statements:
>
> (a) The number of Mersenne primes < x is about log(log(x))*e^gamma/log(2)
>
> (b) the expected number of Mersenne primes between x and 2x is about e^gamma.
> (equivalent to part a)
>
> (c) the probability that Mq is prime is about c*log(aq)/q where
> c=e^gamma/log(2) and a=2 if q = 1 mod 4; a=6 if q=1 mod 4.
>
> It gives fours considerations upon which Wagstaff's conjecture is based. Of
> course, these imply that the nth Mersenne number is about [2^(-gamma)]^n, or
> 1.4759^n.
>
> He goes on to mention Eberhart's earlier conjecture of (3/2)^n, but states that
> there is no serious reason supporting this version of the conjecture.
>
> +----------------------------------------------+
> | Jud "program first and think later" McCranie |
> +----------------------------------------------+
>
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