TITLE: Are most numbers of the form $a\cdot b^n+c$ composite?
QUESTION [5 upvotes]: It seems evident that for $a,b,c$ with $a>0$ and $b>1$ that there are only $o(x)$ primes of the form $a\cdot b^n+c$ with $n\le x.$ Has this been proven?
Hooley (Applications of Sieves to the Theory of Numbers) proves this for Cullen numbers, and apparently this proof generalizes to forms $n\cdot2^{n+a}+b$.
A reference would be great, if one can be found. Otherwise, I'll take what I can get!

REPLY [2 votes]: It seems that the problem is entirely open, as seen by this answer on MathOverflow.
The difficulty is that the Cullen form gets equidistribution in all odd moduli from the $n$, while the general case has more restricted residue classes that are harder to control.