Hi,
Brian Beesley wrote:
>
> If p, 2p+1 are both prime and p = 4k+3 for some integer k then 2p+1
> is a factor of 2^p-1. Therefore, if there is an infinite supply of
> Sophie Germain primes congruent to 3 (modulo 4), there is an infinite
> supply of compound Mersenne numbers with prime exponents.
>
> However, there seems to be no proof that there is an infinite supply
> of Sophie Germain primes, let alone of the subclass of S-G primes
> congruent to 3 (mod 4). There is a strong heuristic argument that
> predicts the number of S-G primes less than any given limit -
> interestingly, the formula contains the "twin primes" constant.
>
> There are other heuristic arguments that there is an infinite number
> of composite Mersenne numbers with prime exponents , e.g. since there
> are an infinite number of primes, and the probability that a
> particular prime generates a prime Mersenne number decreases with the
> size of the prime, the expected total number of composite Mersenne
> numbers with prime exponents is the sum of an infinite series of
> terms, with the terms asymptotically approaching 1. The sum is, of
> course, infinite, "Q.E.D." (But this isn't a formal proof!)
>
> The other way of putting this is that, if there are only a finite
> number of primes which generate a compound Mersenne number, there
> must be a _largest_ prime p such that 2^p-1 is compound, with 2^q-1
> prime for all prime q > p. This state of affairs seems absurd, in
> view of the fact that one would expect the probability that 2^q-1 is
> prime for any particular prime q to be very small as q tends to
> infinity.
>
You are correct when you state that this seems absurd and I agree
that there are surely going to be an infinite number of composite
Mersenne numbers with prime exponents !?
I just wondered whether there is a 'formal' proof of this fact,
and if so where can I find it ?
Benny;
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Benny Van Houdt,
University of Antwerp
Dept. Math. and Computer Science
PATS - Performance Analysis of Telecommunication
Systems Research Group
Universiteitsplein, 1
B-2610 Antwerp
Belgium
email: [EMAIL PROTECTED]
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