It is well known that a polynomial $$f(n)=a_0+a_1n+a_2n^2+\cdots+a_kn^k$$ is composite for some number $n$.
What about the function $f(n)=a^n+b$ ?
Do positive integers $a$ and $b$ exists such that $a^n+b$ is prime for every natural number $n\ge 1$ ?
I searched for long chains in order to find out whether there is an obvious upper bound. For example $4^n+4503$ is prime for $n=1,\ldots,14$.