Primes in shifted geometric progressions

Question

Are there integers $a,b,r\in\mathbb Z$ such that there are infinitely many primes of the form $ar^n+b$ with $n\in\mathbb N$?

(I added the reference-request tag because I suspect that the answer to the above question is contained in the literature.)

Answer 1

In short, we don't know.

In some sense, we are very very far from knowing this. Many of our results of the shape "There are infinitely many primes in the sequence..." are based ultimately on sieving. To use these sieves, we care very much about how sparse/dense the sequence is.

For results like Dirichlet's Theorem on primes in arithmetic progressions, the sequences are of the form $an + b$, for fixed $a$ and $b$, as $n$ varies. These sequences have positive density in the integers.

That's about the limit of what we can prove. We don't know of a any polynomial $f(n) \in \mathbb{Z}[x]$ of degree larger than $1$ such that $f(n)$ takes prime values infinitely many times. These sequences are sparse among the integers. There are several conjectures in this area, none of them proven (Schinzel's Hypothesis H, Bateman-Horn, and others).

What you're asking about is an exponential function $f(n)$, which is even more sparse in comparison to polynomials. We should expect this to be much harder than polynomial results.

I will note that if you remove the requirement that everything be an integer, you can specially construct exponentially distributed functions that almost take infinitely many primes. See for example Mill's number, which is the smallest number $A$ such that $$ \lfloor A^{3^{n}} \rfloor $$ is always prime for $n \in \mathbb{N}$. (This number does in fact exist). But this is more a statement about how exponential functions grow really rapidly and primes are much more dense than exponentially distributed points, so with lots of fiddling (and relying on the floor function to allow wiggle-room, some sort of locally constant behavior) you can engineer this constant.