Hello, we all know that 31,331,3331,33331,333331,3333331,33333331 all are primes, and that 333333331 is not. Here we prepend the digit 3 to 31, to get a list of 7 primes.This gives me the following thought:

Let
$$D = \{\text{all possible nonnull finite digit strings}\},$$
$$D' = \{\text{all things in D that do not start with 0}\}.$$

Define a function $m: D' \times D \to N \cup {\infty}$ by:
$$m(A,B)= \min \{ k\geq 1 \colon A^kB \text{ is composite} \} - 1,$$
i.e., the number of consecutive primes at the beginning of the list $AB, AAB, AAAB, \dots$. For example, $m(3,1)=7$.

Which values does $m$ take? Is it unbounded? Is it ever $\infty$?

This question has been posted to math.stackexchange too, and I got one comment talking about that it might involve Tao-Ziegler extension to the Green-Tao theorem, and so I thought it might be more appropriate here. So please, excuse me if it's posted wrongly, or if one shouldn't post to both channels.