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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?

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    $\begingroup$ I doubt that there is a general way to determine if there are infinitely many primes in a recurence relation. For example, $a_{n}=2^{n}-1$ satisfies $a_{1}=1, a_{n+1}=2a_{n}+1$, but we don't know whether or not it has infinitely many primes. Primes in this sequence are referenced to as Mersenne primes, and the infinitude of Mersenne primes remain a conjecture. $\endgroup$ – S.C.B. Feb 8 '17 at 2:50
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    $\begingroup$ The sequence of numbers really has no particular significance other than a digital pattern. If there were infinitely many primes in the sequence you could probably publish a paper on that. As S.C.B. mentioned, there's no reason to expect that proving this is even remotely within our current capabilities. We don't know if there are infinitely many primes of the form $n^2+1$, to add an example. $\endgroup$ – Matt Samuel Feb 8 '17 at 3:10
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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 $n$

2 3 5 6 13 61 110 182 246 413 888 2478 2919

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

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