Am Thu, 31. March 2005 um 10:02:40 +0200 schrieb Dr. med. Roland Linder: > My hypothesis: Starting with one-digit primes, you can use a Mersenne prime as > Mersenne exponent to yield a further Mersenne prime.
This sequence C_i = 2^C_{i-1} - 1 with C_0 = 2 is known as Catalan-
Mersenne sequence. I find these numbers very interesting, too, but
little seems to be known about them. The best pointers I have found are
http://mathworld.wolfram.com/Catalan-MersenneNumber.html
(plus the references stated there) and
http://www.utm.edu/research/primes/mersenne/#c .
C0...C4 are prime (2, 3, 7, 127 and 170141183460469231731687303715884105727),
but the status of all the others is unknown.
One thing is for sure: Either all C_i are prime, or only the first
few of them (C_0...C_j for some j). That's because if some C_i is
composite, C_{i+1} = 2^C_i - 1 can't be a (Mersenne) prime and so on.
It is believed by "Guy's strong law of small numbers" that only the
first few (only C_0...C_4?) are prime, but who knows? I think this must
be solved by some wise number theorist rather than sheer computing
power. In any case I don't know how it could be done to run the Lucas-
Lehmer test in parallel, if possible at all. But maybe there is a way
to simplify the Lucas-Lehmer test just for this special case, or
something like that?
By the way, this sequence is a subset of the so called Double
Mersenne numbers MM_p = 2^(2^p-1)-1 with 2^p-1 prime. The first four of
them are prime, the next four are composite, and the status of the rest
is unknown.
And I think I've seen some prove that the decimal representation of
all C_i with i >= 2 ends with "7", that of all C_i with i >= 3 ends with
"27", and so on, but I don't remember where.
- Stephan Niemz.
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