TITLE: Is there a prime of the form 711...117?
QUESTION [10 upvotes]: The integers
$$77,\,717,\,7117,\,71117,\,711117,\,\ldots$$
appear to all be composite. However, the lack of a small covering set suggests that there might eventually be a prime.
Algebraically, these numbers are given by
$$a_k=\frac{10^k\cdot64+53}{9}$$
for $k\geq1$.
Here is some partial progress:

A computer search confirms that $a_k$ is composite for all $1\leq k\leq5000$.
Even though there isn't a full covering set, you can still narrow down the possibilities for $k$. For example, looking modulo the primes $\{3,11,13\}$ shows that if $a_k$ is prime then $k\equiv4\pmod{6}$.

REPLY [10 votes]: As mentioned in the comments I found the following website where the problem is discussed: https://stdkmd.net/nrr/7/71117.htm
The author mentions two (probable) primes, which I assume means they were checked using a probabilistic algorithm
$\frac{10^{10906}\cdot 64+53}{9}=7(1)_{10905}7$ found by Jens Kruse Andersen in 2002 and $\frac{10^{499210}\cdot 64+53}{9}=7(1)_{499209}7$ found by Serge Batalov in 2015.