Puzzle 1223 Prime (n)^(Prime (k) - 1) + (Prime (k)

Puzzle 1223 Prime (n)^(Prime (k) - 1) + (Prime (k) - 1)

Sebastián Martín Ruiz sent the following puzzle:

Let F (n, k) = Prime (n)^(Prime (k) - 1) + (Prime (k) - 1)
I have found only three cases that F (n, k) is prime
Prime (n) Prime (k) F (n, k)
2 2 3
3 3 11
31 31 550618520345910837374536871905139185678862431
Note that in all cases
Prime (n) = Prime (k)
and 3 and 31 are Mersenne primes

I have tested it for k=1 to 300 and n=1 to 1000. I have not found any case with k=/=n. I have only found the three cases mentioned.

Q. Find another prime F (n, k) or prove it does not exist.