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
$$D = \{\text{all possible nonnull finite digit strings}\},$$
$$D' = \{\text{all things in D that do not start with 0}\}.$$
Define a function $m: D' \times D -> N \cup {\infty}$$m: D' \times D \to N \cup {\infty}$ by: $m(A,B)= |${all prime members of the list AB, AAB, AAAB,
.$$m(A,B)= \min \{ k\geq 1 \colon A^kB \text{ is composite} \} - 1,$$
i.e.up until but not including, the first composite member}| (the sizenumber of consecutive primes at the set).Then:
Does m ever takebeginning of the value $\infty$list ? If not$AB, AAB, AAAB, \dots$. For example, is it an unbounded function?
(this$m(3,1)=7$.
Which values does $m$ take? Is it unbounded? Is it ever $\infty$?
This question has been posted to math. stackexchangestackexchange too, butand 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. So please, excuse me if it's posted wrongly, or if one shouldn't post to both channels).
