Generalized Mersenne Primes

[Cyclamen Persicum]

I need an information about nongenuine (generalized?) Mersenne numbers
in terms of 2^a +/- 2^b +/- 2^c ... +/- 2^0,
for instance, 2^384-2^128-2^96+2^32-1 is prime.
I believe, they have all the advantages of regular mersenne numbers:
1) there is a fast deterministic Lucas-Lahmer-like test;
2) there is a way of simple modular reduction based on magic figure "nine"
properties (I don't mean IBWDFFT here...)

How can I perform the items 1 and 2 really?
For example, how can I get a residue from division by 9909 in the decimal numeration?

Sorry, I have forgotten completely, why are nongenuine mersenne numbers much
better than regular mersenne numbers? Because there are a lot of them,
contrary to 40-ty known regular ones.

[Cyclamen Persicum]

Besides holed generalized mersenne numbers such as 1111111111000001111111110000000000000000000000011111111111111...
there are so called CRANDALL numbers 2^n-c, where c is small.
Modular reduction is easy and worth nothing for these sweet honey numbers!!!
But one thing has not been seen: is there deterministic LL-test for crandall primes or one must use common rabin-miller test?