TITLE: Is a function of $\mathbb N$ known producing only prime numbers?
QUESTION [11 upvotes]: 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 $n=1,\ldots,14$.

REPLY [3 votes]: If there were such a function, then we wouldn't be talking about "largest prime yet discovered". We'd have an infinite number of them at our disposal. So no... we have no such closed form function.