TITLE: Is 75986⋯69 ever prime or is there a 6 padded prime for every prime $p > 5$?
QUESTION [8 upvotes]: I came across this question:
How can we turn any number into a prime number by simply adding more digits?
While trying different approaches to find an algorithm that increases the chances of finding a prime this way I discovered the six padded primes:
Start with an prime $p>5$ then add a digit 6 before the last digit, repeat the process until the result is prime.
This table shows the results for p up to 100:
\begin{align*}
7 && 67 \\
11 && 16661 \\
13 && 163 \\
17 && 167 \\
19 && 1669 \\
23 && 263 \\
29 && 269 \\
31 && 3666661 \\
37 && 367 \\
41 && 461 \\
43 && 463 \\
47 && 467 \\
53 && 563 \\
59 && 569 \\
61 && 661 \\
67 && 666667 \\
71 && 761 \\
73 && 76666663 \\
79 && 769 \\
83 && 863 \\
89 && 8669 \\
97 && 967 \\
\end{align*}
There were only 2 numbers below $200000$ for which I could not find a result:
$15731$ and $75989$. My computer finally found a solution for the first padded with 7460 sixes resulting in a 7465 digit prime number. $75989$ was checked up to 15000 digits.
Question: 

Is $75986\cdots69$ ever prime or can it be proved there is a six padded prime for every prime > 5

REPLY [4 votes]: If you insert an odd number of $6$'s, it is simple to check that the resulting number is a multiple of $11$ (just use the divisibility criterion).
For an even number $n$ of $6$'s, notice that
$$
759866\equiv 3\pmod{13},\quad 
1000000\equiv 1\pmod{13},\quad 
666666\equiv 0\pmod{13},
$$
and
$$
75986666\equiv 25\pmod{37},\quad 
1000000\equiv 1\pmod{37},\quad 
666666\equiv 0\pmod{37}.
$$
Hence when $n=2+6k$ the number is a multiple of $13$, and when $n=4+6k$ the number is a multiple of $37$.
At the moment, I can say nothing for the case $n=6k$.
EDIT.
Playing around with Mathematica I noticed some other regularities (which I didn't bother to prove):

for $n=6(1+7k)$ the number is a multiple of $43$;
for $n=6(4+8k)$ the number is a multiple of $17$;
for $n=6(4+5k)$ the number is a multiple of $31$.

On the other hand, there are several cases where the number has only two prime factors, both large. At the moment, however, I could find no trace of a prime number.