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?


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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