## Generalized Cunningham chains

We call here  "generalized Cunningham chains" the sequence of primes  Pi such as Pi+1 = a*Pi+b, with a and b integers relatively prime. His length L is the number of successive primes in this sequence.

If  a+ b = 1 we can use an extension of the pseudo primality test "in cluster" in order to test in one pass the set of the pi of the chain (See Generalization of the Euler - Lagrange theorem and new primality tests).

The implantation of the corresponding algorithm allows us to obtain quickly the first following results,

with L >= 5,  P1 is the smallest prime of this chain of length L :

 L Pi+1 = 4*Pi - 3,  P1 13 60389563279 12 95472623 11 95472623 10 95472623 9 754451 8 408539 7 5869 6 523 5 331

 L Pi+1 = 6*Pi - 5,  P1 9 95807339 8 1273663 7 184409 6 1601 5 1237

 L Pi+1 = 9*Pi - 8,  P1 10 61637129 9 16908181 8 1627603 7 125399 6 13249 5 233

Moreover, for a = 2m  we can write Pi+1 = ((P1-1)/2)* 2m*i+1 + 1, so a Proth number with k= (P1-1)/2, and the exponent of the power of two is in arithmetic progression of difference m !

Example : for m=60 we find a chain of length L= 5 with :

P1 = 452227 = 226113*2 + 1,
P2 = 226113*261 + 1
P3 = 226113*2121 + 1
P4 = 226113*2181 + 1
P5 = 226113*2241 + 1

Some results for other m :

 L Pi+1 = (24)*Pi -(24-1) ,  P1 9 5676191 8 2218547 7 42701 6 607 5 467

 L Pi+1 = (26)*Pi -(26-1) ,  P1 9 6095693 8 3962213 7 9803 6 9803 5 2531

 L Pi+1 = (28)*Pi -(28-1) ,  P1 8 43105529 7 427591 6 156679 5 331

 L Pi+1 = (210)*Pi -(210-1) ,  P1 8 55469933 7 224563 6 37571 5 4679

 L Pi+1 = (212)*Pi -(212-1) ,  P1 8 25782037 7 11711033 6 15971 5 6029

 L Pi+1 = (2100)*Pi -(2100-1) ,  P1 4 67567

 L Pi+1 = (2240)*Pi -(2240-1) ,  P1 3 4127

 L Pi+1 = (2720)*Pi -(2720-1) ,  P1 3 8237

You can also consult the next links :

Created by  Henri Lifchitz : December, 15  1998, last modification: December, 17  1998.