find all primes of the form (2^2^n) + 5 where n is a nonnegative integer
It is always divisbile by 3 for n>0 (and $\displaystyle 2^{2^n}+5\neq{3}$ for all nonnegative integers n)
Just note that: $\displaystyle 2^{2^n}+5\equiv{(-1)^{2^n}+2}(\bmod.3)$
If $\displaystyle n>0$ we have that $\displaystyle 2^n$ is even and therefore $\displaystyle (-1)^{2^n}=1$ thus: $\displaystyle 2^{2^n}+5\equiv{3}\equiv{0}(\bmod.3)$
If n=0 we have: $\displaystyle 2^{1}+5=7$ that is prime
Therefore the only prime in that sequence is 7