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.
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| Jud "program first and think later" McCranie |
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