TITLE: Prepending strings to primes.
QUESTION [9 upvotes]: 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.

REPLY [2 votes]: To supplement David's answer, the 2nd option, I can add
the following heuristics.
If we fix a positive integer $n$ and take irreducible polynomials
$$
f_k(x)
=A\frac{x^{k+1}-1}{x-1}+B
=A(x^k+x^{k-1}+\dots+x)+B,
\qquad k=1,\dots,n,
$$
subject to the condition that $f_1(x)\dots f_n(x)/m$ is not
an integer-valued polynomial for all $m>1$ (take, for
example, $A=3$ and $B=1$), then
Schinzel's Hypothesis H
implies that the numbers $f_1(R),\dots,f_n(R)$ are simultaneously prime for infinitely many
positive $R\in\mathbb Z$. There is no guarantee however that $R$ assumes the form $10^k$.
I owe this idea to Frictionless Jellyfish
who removed his/her related comment to
this post
(which I would definitely accept as answer).