Quote:
Originally Posted by JeppeSN
I should have said I was interested in odd primes only (and the number 1 is not prime).

There are some large "generalized Fermat" primes listed, e.g.
here; these have the forms
a^(2^20) + 1, a^(2^19) + 1, and a^(2^18) + 1.
Of course, they might not be the smallest prime a^(2^k) + b^(2^k) for their exponents.
In looking at smaller exponents, it did occur to me to look at cases
(a^(2^k) + b^(2^k))/2 with a*b odd. I noticed that (3^2 + 1)/2 and (3^4 + 1)/2 were primes, and, since I knew 2^32 + 1 isn't prime, I tried n = (3^32 + 1)/2. Pari's ispseudoprime(n) returned 1...