If we concatenate the numbers $5^n$ and $31^n$ , where $n\ge 1$ is an integer, then we get the number $$a_n:=5^n\cdot 10^m+31^n$$ where $m$ is the number of digits in the decimal expansion of $31^n$
Is $a_n$ prime for some $n\ge 1$ ?
I checked upto $n=10^4$ without finding a prime.
There doesn't seem to be any obvious reason it should not be prime. Heuristically, the probability of a "random" integer $x$ being prime is approximately $1/\log(x)$, and $\log(a_n) \approx n \log (5\cdot 31)$.
Since the series $\sum_n 1/n$ diverges, we should expect infinitely many $a_n$ to be prime. But this is not a proof.