Each row below gives, for given n, the smallest value of A such that there is some B (also taken to be minimal below) with 0<B<A, so that A^(2^n) + B^(2^n) is prime.

[n, A, B]
[0, 2, 1]
[1, 2, 1]
[2, 2, 1]
[3, 2, 1]
[4, 2, 1]
[5, 9, 8]
[6, 11, 8]
[7, 27, 20]
[8, 14, 5]
[9, 13, 2]
[10, 47, 26]
[11, 22, 3]
[12, 53, 2]
[13, 72, 43]
...


Generated with PARI program:

for(m=0, 30, for(a=2, 10^100, forstep(b=(a % 2)+1, a-1, 2, if(ispseudoprime(a^(2^m)+b^(2^m)), print([m, a, b]); next(3)))))