Suppose that we take some composite number $n$ and factor it as $n=\prod_{k=1}^{m(n)} p_k^{a_k}$ where $p_1 < ... < p_{m(n)}$ and then we observe what happens when we concatenate the $p_1,...,p_{m(n)}$.
For example $70=2 \cdot 5 \cdot 7=257$ and $257$ is prime and the procedure can be stopped.
But, $121=11 \cdot 11=1111$ and $1111$ is not prime so we continue to obtain $1111=11 \cdot 101=11101=17 \cdot 653=17653=127 \cdot139=127139$ and $127139$ is a prime so the procedure can be stopped.
Surely, the question of whether there exists at least one composite number $c$ for which this procedure never stops can be asked, and I am interested if there really is at least one such $c$.
Are there some results with which we can prove that there is (or that there is no) such $c$?
The procedure that you describe is known as Home prime. As far as I know, the status for the starting value 49 is currently unknown (as it involves factoring huge numbers).