# Are there infinitely many Smarandache-Wellin primes and does anyone care?

Definition: The $n$th Smarandache-Wellin number in base $m$, denoted $S_n^m$ (or just $S_n$ for $m=10$), is the concatenation of the first $n$ base-$m$ primes.

So, for example, we would have $S_4 = 2357,$ or $S_7 = 235791113$.

Sometimes, the Smarandache-Wellin numbers are prime; for instance 2, 3, and 2357. There are currently 7-8 Smarandache-Wellin primes known - but are there infinitely many?

Of course this is almost certainly an open problem, but googling Smarandache-Wellin surprisingly says nothing even remotely about whether there are infinitely many. Are the numbers just not well-studied enough to even ask the question?

Heuristically (if I did not mess up my mental arithmetic) I think there should be infinitely many, about $\log \log x$ up to $x$. (But I am tired, so I do not vouch for that last bit.)