On 24 Jan 00, at 11:48, Paul Landon wrote:
> This is not new news to most people here, but I have to remind
> myself, it still hasn't been proved whether there are an infinite
> number of Mersenne Primes or an infinite number of Mersenne
> composites.
The latter conjecture looks very, very probable!
Note that it would be sufficient to prove that there are an infinite
number of Sophie-Germain primes, since there is a "well known"
theorem which states that, if p is a Sophie-Germain prime, then 2^p-1
is divisible by 2p+1.
Of course, we do have heuristics which tend to indicate that the
number of Mersenne primes is infinite. If this is not so, then the
number of Mersenne composites _must_ be infinite. The same heuristics
(or even just application of the Prime Number Theorem) suggest that
the probability that 2^p-1 is prime decreases with increasing p,
which is a strong indication that there are, indeed, an infinite
number of Mersenne composites.
The contrary would be amusing - if there are a finite number of
Mersenne composites, there must be an integer P which is the exponent
of the _largest_ composite Mersenne number, i.e. 2^(P+k)-1 is prime
for every positive integer k. The challenge then would not be to find
all the Mersenne primes, but to determine the value of P.
Regards
Brian Beesley
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