I looked at a table of primes and observed the following:

If we choose $7$ can we concatenate one digit to the left so as to form a new prime number? Yes, concatenate $1$ to obtain $17$. Can we do the same with $17$? Yes, concatenate $6$ to obtain $617$. And with $617$? Yes, concatenate $2$ to obtain $2617$. Then we can form $62617$. And I could not continue since table gives primes with the last entry $104729$.

Now some terminology. Call a prime number $a_1...a_k$ a *survivor of order $m$* if there exist $m$ digits $b_1,...,b_m$ (all different from zero) so that the numbers $b_1a_1...a_k$ and $b_2b_1a_1..a_k$ and... and $b_mb_{m-1}...b_1a_1...a_k$ are all prime numbers.

Call a prime number $a_1...a_k$ a *survivor of order $+ \infty$* if $a_1...a_k$ is a *survivor of order $m$* for every $m \in \mathbb N$.

I would like to know:

Does there exist a

survivor of order $+ \infty$?

(This question, with exactly the same title and content, was asked on MSE about an hour ago, and I think that I should apologize for asking here and there a same question in so a little time-interval, but, as I thought that somebody will come up very fast with an argument with which question would be decided, and that did not happen, I decided to ask it here also, so that this question receives an attention here also. Yes, it has a recreational flavor, but I hope that you like it.)

**Edit:** I do not know what exactly to do with this question. At this moment, this question is "on hold". And, it is written: "This question appears to be off-topic for this site. While what’s on- and off-topic is not always intuitive, you can learn more about it by reading the help center. The users who voted to close gave this specific reason:

"This question does not appear to be about research level mathematics within the scope defined in the help center." If this question can be reworded to fit the rules in the help center, please edit your question."

I have a question: If this question is not about research-level mathematics, why we do not have an answer that settles this question? I am of the opinion that with present-day level of mathematics we have the tools to set this question, it is just a question of how to put things together in order to obtain an answer. Also, I am of the opinion that any and every question that generates some discussion, and has an attempts of a solution, and is within reach (or even if it is not within reach) is good for this site, because if at least one person likes a question, that means that there is at least one person for which it is good that a question is here on the site, and not closed. Probably an edit should be about mathematical content of the question, but, I think that I do not need to add anything to mathematical content of this question. Now I am going to drink a coffee and leave a destiny of this question to you.