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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?

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It is very rare that the infiniteness or not of primes in such sparse sequences is known. Note that even for the very well-known Mersenne and Fermat numbers this is not known (for the former there are likely infinitely many prime, for the later likely not.)

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.)

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