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For integers $a,b,k$ with $a \geq 1$ and $k\geq 2$, consider the shifted geometric progression $n_i = ak^i + b$. I would like to understand the set of integers (prime or otherwise) that divide at least one of the $n_i$'s. More precisely, let $$ D = D_{a,b,k} = \{ d \in \mathbb{N} \ : \ \text{there exists } i \geq 1 \text{ with } d \mid n_i \}, $$ $$ P = P_{a,b,k} = \{ p \ : \ p \text{ is prime and there exists } i \geq 1 \text{ with } p \mid n_i \}. $$ Questions:

  1. Is it known the set $P$ has positive relative density inside the primes?

  2. Barring a positive answer to 1., what can be said about the sizes of $D$ and $P$?

(It is certainly possible that $D$ has asymptotic density $0$, e.g. by letting $a,b$ and $k$ be squares. If the answer to the first question is positive then $\# D \cap [1,x] \geq \# P \cap [1,x] \gg x/\log x$. In the second question I'm hoping that perhaps a weaker asymptotic is true, such as $\# D \cap [1,x] \gg x^{1-\varepsilon}$.)

Context: It would follow from Artin's conjecture on primitive roots that $k$ is a primitive root for a positive proportion of the primes, and for each such prime $p$ there exists $i$ with $k^i \equiv -b/a$, whence $p \mid n_i$ (at least, as long as $p$ does not divide $ab$). Hence, if we believe Artin's conjecture then $P$ has positive density inside the primes.

One can also show by purely elementary methods that for each $C$ there exists $i$ such that $n_i$ has $>C$ prime divisors. Hence, $P$ is infinite. Here is a sketch: Assume, as we may do without loss of generality, that $b$ is coprime to $k$ (else, replace $a$ with $ak^{i_0}/d$, $b$ with $b/d$, and $i$ with $i-i_0$ where $d = \gcd(k^{i_0},b)$ for some large integer $i_0$). This assumption ensures that $n_i$ ($i \geq 1$) are coprime to $k$. Construct a sequence $i_j$ where $i_1$ is arbitrary, and $i_{j+1} = i_j + \varphi(n_{i_j}^2)$, so that $n_{i_{j+1}} \equiv n_{i_j} \bmod n_{i_j}^2$. This is set up so that $n_{i_{j+1}}/n_{i_j}$ is an integer coprime to $n_{i_j}$, and consequently $n_{i_j}$ has at least $j$ distinct prime factors.

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    $\begingroup$ You seem to be asking about the "two-variable Artin conjecture". Unfortunately the known (unconditional) results seem much weaker than what you are hoping for; I think the state of the art is arxiv.org/abs/1711.06410 $\endgroup$ – so-called friend Don May 1 at 0:07
  • $\begingroup$ @so-calledfriendDon: Thank you! Yes, it seems like what I'm asking is very closely related: If I understand correctly, the case of my question where $b/a \in \mathbb{Z}$ and we only care about prime divisors is corresponds precisely to Theorem 1.1 in the preprint you attached, while Theorem 2.1 is more general than my question (since $ak^i + b$ satisfies a binary recurrence). $\endgroup$ – Jakub Konieczny May 1 at 10:43
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This does not fully answer the question, but gives several links to some related literature. Most papers below study if the order of an element mod $p$ is odd or even, (or more general). Therefore the links below study the cases $a=1, b=1$ or $b=-1$ in your notation. Shparlinski's paper is more general, but primarily studies related sequences with more than 2 summands.

a) The following two papers by Hasse give at least some special cases such as prime divisors of the sequence $k^n+1$, ($k$ is fixed, $n \in \mathbf{N}$). In particular Hasse proved that the Dirichlet density of primes dividing an integer of the type $2^n+1$ is $17/24$. Note: if $p|(2^n+1)$, then $2^n\equiv -1 \bmod p$, and the order of $2$ modulo $p$ is even.

H. HASSE, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl $a\neq 0$ von gerader bzw. ungerader Ordnung mod p ist. Math. Ann. 166 (1966), 19-23. http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002296616

H. HASSE, Über die Dichte der Primzahlen p, für die eine vorgegebene rationale Zahl $a\neq 0$ von durch eine vorgegebene Primzahl $l \neq 2$ teilbarer bzw. unteilbarer Ordnung mod $p$ ist. Math. Ann. 162 (1965), 74-76 http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002295253

b) Odoni proved corresponding results for natural density. Journal of Number Theory Volume 13, Issue 3, August 1981, Pages 303-319 A conjecture of Krishnamurthy on decimal periods and some allied problems, https://www.sciencedirect.com/science/article/pii/0022314X81900160

In some cases he got positive relative prime density, in other cases 0-density.

c) On RH (related to Artin's conjecture Stephens, P. J. Prime divisors of second-order linear recurrences. I. J. Number Theory 8 (1976), no. 3, 313–332.

d) Shparlinski, Igor E. Prime divisors of sparse integers. Period. Math. Hungar. 46 (2003), no. 2, 215–222. https://link.springer.com/article/10.1023%2FA%3A1025996312037

e) Hooley's book "Application of sieve methods to the theory of numbers" has some information on prime factors of $2^n+b$.

f) Prime divisors of certain recurrent sequences have been studied e.g. by Ballot, and Moree.

Finally, I believe that the case $|b| \neq 1$ is more difficult, as there is less algebraic structure (such as order of an element).

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  • $\begingroup$ Thank you! That's very relevant - and the problem at the level of generality I was hoping for appears to be much harder than I anticipated. $\endgroup$ – Jakub Konieczny May 1 at 10:45

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