A curious of generalized repunit primes

[sweety439]

For given base b and given prime p:

(b^p-1)/(b-1) is divisible by p if and only if p divides b-1
(b^p+1)/(b+1) is divisible by p if and only if p divides b+1
(b+1)^p-b^p is never divisible by p (or p will divide 1, which is impossible)
((b+1)^p+b^p)/(2*b+1) is divisible by p if and only if p divides 2*b+1

(b^p-1)/(b-1)^2 is prime for b = 3, 4, 6, 18, 4358, ...
(b^p+1)/(b+1)^2 is prime for b = 4, 18, ...
((b+1)^p+b^p)/(2*b+1)^2 is prime for b = 2, 6, 18, ...

f(p) = (b^p-1)/(b-1), then f(2*p)/f(p) = (b^p+1)/(b+1)
f(p) = (b+1)^p-b^p, then f(2*p)/f(p) = ((b+1)^p+b^p)/(2*b+1)

Mersenne primes: (b^p-1)/(b-1) for b = 2, (b+1)^p-b^p for b = 1
Wagstaff primes: (b^p+1)/(b+1) for b = 2, ((b+1)^p+b^p)/(2*b+1) for b = 1

Smallest b such that (b^p-1)/(b-1) is prime for p = 109 is 12
Smallest b such that (b^p+1)/(b+1) is prime for p = 109 is 12
Smallest b such that (b+1)^p-b^p is prime for p = 109 is 12
Smallest b such that ((b+1)^p+b^p)/(2*b+1) is prime for p = 109 is 12

Smallest b such that (b^p-1)/(b-1) is prime for p = 317 is 10
Smallest b such that (b+1)^p-b^p is prime for p = 317 is 10

Smallest b such that (b^p-1)/(b-1) is prime for p = 11 is 5
Smallest b such that (b+1)^p-b^p is prime for p = 11 is 5

(the b = 11 case is divisible by 23 for (b is QR mod 23 for (b^p-1)/(b-1), (b+1)/b is QR mod 23 for (b+1)^p-b^p)

b = 18, (b^p-1)/(b-1) is prime for p = 2, next such p is very large (25667)
b = 18, (b+1)^p-b^p is prime for p = 2, next such p is very large (1607)

b = 96, (b^p-1)/(b-1) is prime for p = 2, next such p is very large (3343)
b = 96, (b+1)^p-b^p is prime for p = 2, next such p is very large (1307)