Suppose $a > 1, b \neq 0$ be two rational numbers. Is it known in general that the set of prime divisors of (the numerator of) $a^n + b$ has a positive relative density? 



Density of prime divisors of $a^n + b$
This is only known conditional on the generalized Riemann hypothesis. A prime $p$ dividing the numerator of $a^{n} + b$ is more or less equivalent to the statement that the subgroup of $\mathbb{F}\_{p}^{\times}$ generated by $a$ contains $-b$.


This problem is addressed in the 2000 Journal of Number Theory paper by Moree and Stevenhagen (titled *A two-variable Artin conjecture*) where it is proved (in a similar manner to Hooley's conditional proof of the Artin conjecture) that the set of primes has a positive density. The paper contains explicit formulas for that density.


