View Single Post
2017-05-10, 09:07   #230
sweety439

Nov 2016

7×11×31 Posts

Quote:
 Originally Posted by gd_barnes I need some examples to understand this. It appears that the divisor would be different for every k on a single base. That doesn't make sense to me. How could it easily be searched? I need possibly two sets of examples: 1. An example of the divisor for several consecutive k's on a single base. 2. If I'm not understanding it and the divisor is the same for all k's on a single base, I need an example of the divisor for several consecutive bases.
This divisor is always gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel), e.g. for R7, k=197, the divisor is gcd(197-1,7-1) = 2, and for S10, k=269, the divisor is gcd(269+1,10-1) = 9. Besides, for SR3, the divisor of all even k is 1 and the divisor of all odd k is 2.

Thus, for example, for R13, the divisor of k = 1, 2, 3, ... are {12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, ...}, and for S11, the divisor of k = 1, 2, 3, ... are {2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, ...}.

In fact, this divisor is the largest number that divides k*b^n+-1 (+ for Sierpinski, - for Riesel) for all n. Thus, this divisor is the largest "trivial factor" of k*b^n+-1 (+ for Sierpinski, - for Riesel).

Last fiddled with by sweety439 on 2017-05-24 at 18:49