I looked at a table of prime numbers and observed the following:

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

Now, a little bit of terminology. Call a prime number $ a_1 … a_k $ a *surviving from the order $ m $* if there is $ m $ statistics $ 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 *surviving from the order $ + infty $* if $ a_1 … a_k $ is a *surviving from the order $ m $* for each $ m in mathbb N $.

I would like to know:

Is there a

surviving from the order $ + infty $?

(This question, with exactly the same title and content, was asked at MSE about an hour ago, and I think I should excuse myself for asking the same question here and there in such a short time interval, but, as I thought somebody will talk very quickly about the argument that the question would be settled, and that did not happen, I have decided to put it here too, for this question to get attention here too .. Yes, it has a recreational flavor, but I hope you like it.)