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Conjectures about Cunningham Primes

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  • bonham bonham
    Can someone direct me to the history of the following conjectures and observations? 1.) All Cunningham primes of type 1 are contained in the sequences:
    Message 1 of 1 , Mar 31, 2004
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      Can someone direct me to the history of the following conjectures and observations?

      1.) All Cunningham primes of type 1 are contained in the sequences:
      2^q*(6*n-3)-1 with n>=1 , q>=0 except for those in the sequence 2^q-1. Both q and n are positive integers.

      2.) All Cunningham primes of type 2 are contained in the sequences:
      2^q*(6*n-3)+1 with n>=1 , q>=0 except for those in the sequence 2^q+1.

      3.)No Cunningham primes of type 1 are to be found in the sequences:
      2^q*(6*n-7)-1 with n>=2 , q>=0 and a necessary condition for the existence of a prime in this sequence is that q be an even integer. For type 2 the same can be said for the sequences 2^q*(6*n-11)+1 with n>=3 , q>=0.

      4.)No Cunningham primes of type 1 are to be found in the sequences:
      2^q*(6*n-11)-1 with n>=3 , q>=0 and a necessary condition for the existence of a prime in this sequence is that q be an odd integer. For type 2 the same can be said for the sequences 2^q*(6*n-5)+1 with n>=2 , q>=0.

      Observations:

      1.) In the Cunningham containing sequences the density of prime sequences appears to decrease exponentially in q for constant n. For n up to 4000 the largest Sophie Germain prime was less than 800 and about 700 for Cunningham triplets (Type 1).

      2.) For q<1000 the total number of Cunningham sequences shows a slight decrease for every 1000 increase in n.
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