A dual Williams prime of 3rd kind base 123.

For a base b>=2:

Williams primes of 1st kind base b: primes of the form (b-1)*b^n-1.
Williams primes of 2nd kind base b: primes of the form (b-1)*b^n+1.
Williams primes of 3rd kind base b: primes of the form (b+1)*b^n-1.
Williams primes of 4th kind base b: primes of the form (b+1)*b^n+1. (not exist when b = 1 mod 3)

Dual Williams primes of 1st kind base b: primes of the form b^n-(b-1).
Dual Williams primes of 2nd kind base b: primes of the form b^n+(b-1).
Dual Williams primes of 3rd kind base b: primes of the form b^n-(b+1).
Dual Williams primes of 4th kind base b: primes of the form b^n+(b+1). (not exist when b = 1 mod 3)

Where n>=1 is an integer.

To be completed...