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 p_i 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 :
 

Moreover, for a = 2^m  we can write P_i+1 = ((P₁-1)/2)* 2^m*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 :

You can also consult the next links :

•  Prime Pages glossary Cunningham chain  by Chris K. Caldwell.
•  Yves Gallot's Proth.exe and Cunningham Chains  by Warut Roonguthai.
•  Proth.exe  by Yves Gallot.