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

asked before?

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)