## A subset of the Mersenne primes (and superset of double Mersenne primes)

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• I factor the number M_11 = 23·89 and notice that 89 is the exponent of a Mersenne prime. I therefore consider the following: What are the primes p that are
Message 1 of 1 , Aug 25, 2002
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I factor the number M_11 = 23·89 and notice that 89 is the exponent
of a Mersenne prime. I therefore consider the following:

What are the primes p that are simultaneously the index of a Mersenne
prime M_p *and* a divisor of some composite Mersenne number M_q (with
q prime)?

Using conditions on Mersenne divisors (see e.g. Caldwell's pages), I
think I have established that the only known numbers of this kind are
89 and 23209:

M_89 is prime, and 89 divides M_11
M_23209 is prime, and 23209 divides M_967

Here is my question: Can you say anything (heurestically) about how
many numbers of the above kind exist? Maybe this question has been

If we do not require M_q to be composite, we get additional numbers,
namely the double Mersenne primes 3, 7, 31 and 127:

M_3 is prime, and 3 equals M_2
M_7 is prime, and 7 equals M_3
M_31 is prime, and 31 equals M_5
M_127 is prime, and 127 equals M_7

--
Jeppe Stig Nielsen <URL:http://jeppesn.dk/>. «

"Je n'ai pas eu besoin de cette hypothèse (I had no need of that
hypothesis)" --- Laplace (1749-1827)
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