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_k-1 - (k-1) = k*(P_k-1 - 1) + 1 = k!*(P₁ - 1) + 1 where p1 is the first term of the chain.

2)  P_k = k*P_k-1 + (k-1) = k*(P_k-1 + 1) -1 = k!*(P₁ + 1) - 1 where p1 is the first term of the chain.

Below  are the results of my search:

Pk = k!*(P₁ - 1) + 1 (First kind)

Larger chain known up to now is for k=8:

2506981
5013961
15041881
60167521
300837601
1805025601
12635179201
101081433601 

Pk = k!*(P₁ + 1) - 1 (Second kind) 

Larger chain known up to now is for k=9

1656251
3312503
9937511
39750047
198750239
1192501439
8347510079
66780080639
601020725759

Solution

Carlos Rivera found (15/06/99) a 'Generalized Cunningham Chain' of first order and k = 10 members starting with the prime P1 = 228698251. Two days later was found another second example for k = 10 and P1 = 378903601.