Density of primes in sequences of the form $a^n+b$

Question

Suppose that someone proves that there exist a sequence (if that is not already known) of the form $a^n+b$ where $a \in \mathbb N \setminus \{1\}$ and $b \in \mathbb Z$ which has an infinite number of primes as its values.

Now let us define the set $S_k=\{1 \leq i \leq k: a^i+b\in \mathbb P\}$. Let us denote with $|S_k|$ the cardinality of $S_k$.

Do we have $\lim_{k \to \infty} \dfrac {|S_k|}{k} =0$?

In other words, even if there is no concrete example of the sequence of the form $a^n+b$ which has an infinite number of primes as its values can it be proven that the density of primes in such sequences always equals zero?

Answer 1

In the case of Mersenne primes $p=2^n-1$ it is known (by elementary factorisation) that the exponent $n$ is necessarily a prime. Therefore, in this special case $\lim_{k \rightarrow \infty} \frac{|S_k|}{k}=0$.

To the best of my knowledge the general case is not known.

The closest results in the literature that I am aware of are to be found in Hooley's book (Applications of sieve methods to the theory of numbers. Cambridge Tracts in Mathematics, No. 70. 1976) [http://www.ams.org/mathscinet-getitem?mr=404173]

As far as I remember, (without the book at hand), Hooley needs quite strong assumptions (one of these GRH) to say more. However, he studies the related cases of Cullen primes, $p=n 2^n+1$, and here he shows, unconditionally, that the corresponding limit is $0$. (This case is easier as one has more control over the distribution in residue classes).

There is some further work by Rieger (Rieger, G. J. Über Primzahlen und dünne Folgen. Arch. Math. (Basel) 28 (1977), no. 6, 600–602. http://www.ams.org/mathscinet-getitem?mr=447169 ).

The fact that there are very few results on this topic reflects the fact that such thin sequences lead to problems where current methods do not quite work.