I thought of a problem - are there infinitely many prime numbers in a sequence $17, 137, 1337, 13337, \ldots$. With no luck.
This sequence is given by a recurrence $a_0 = 17$, $a_{n+1} = 10 a_n - 33$. So a natural question comes to mind: for such sequences ($a_n = A \cdot a_{n - 1} + B$), what are the conditions for finding infinitely many primes?
If $A = 1$ we have an arithmetic sequence and Dirichlet solved this problem. But his approach does not generalize for 'my' problem, because as far as I know, his proof uses the divergence of a series of reciprocals of primes in some sequences. But for $A > 1$ sum of reciprocals of primes in our sequence can't be infinite, because $\sum a_i < \infty$.
Since 'my' problem seems quite natural, I believe there was some research done by someone, somewhere. Is anybody familiar with any results or tools that could allow to study such problems?
I do not think that it can actually be determined whether the given sequence has infinite many primes.
If a number of this form has $n$ digits, the general form is $$\frac{4\cdot 10^{n-1}+11}{3}$$
It is prime for the following $n$
We do not have the situation that we can prove that every number of the given form is composite, so we can do not much.
Considering the growth rate , I think there are infinite many primes, but it is probably impossible to disprove or prove it.
See here for more details :
https://factordb.com/index.php?query=%284%2A10%5E%28n-1%29%2B11%29%2F3&use=n&perpage=20&format=1&sent=1&PR=1&VP=1&EV=1&OD=1&VC=1&n=2501