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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?

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  • $\begingroup$ Unconditionally this is known iff $b = \pm 1$. For the other $b$ it is a pleasant elementary exercise to prove that there are $\gg \log{X}$ prime divisors $< X$, and I don't think any improvement has been made over this basic bound. $\endgroup$
    – Vesselin Dimitrov
    May 27, 2016 at 1:34

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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.

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