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

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  • Jeppe Stig Nielsen
    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
      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)
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