Problems & Puzzles: Puzzles Puzzle 60. Generalized Cunningham chain (By Felice Russo) A Cunningham chain of length k (of the first kind) is a sequence of k primes, each which is twice the proceeding one plus one. A Cunningham chain of length k (of second kind) is a sequence of k primes, each which is twice the proceeding one minus one. A nice extension of the previous definitions can be: Find a chain of k>=2 primes such that: 1) P_{k} = k*P_{k1}  (k1) = k*(P_{k1}  1) + 1 = k!*(P_{1}  1) + 1 where p1 is the first term of the chain. 2) P_{k} = k*P_{k1} + (k1) = k*(P_{k1} + 1) 1 = k!*(P_{1} + 1)  1 where p1 is the first term of the chain. Below are the results of my search: Pk = k!*(P_{1}  1) + 1 (First kind) Larger chain known up to now is for k=8: 2506981 Pk = k!*(P_{1} + 1)  1 (Second kind) Larger chain known up to now is for k=9 1656251
*** Mike Oakes wrote (Nov 2010):
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