Hello, we all know that 31,331,3331,33331,333331,3333331,33333331 all are primes. 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 $m: D' \times D -> N \cup {\infty}$ by: $m(A,B)= |${all prime members of the list AB, AAB, AAAB, ...up until but not including the first composite member}| (the size of the set).Then: Does m ever take the value $\infty$ ? If not, is it an unbounded function? (this question has been posted to math. stackexchange too, but I got one comment talking about that it might involve Tao-Ziegler extension to the Green-Tao theorem, and 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).