Let $f(n)$ be the smallest integer $k\ge 1$ such that $$n^k+k!+1$$ is prime or undefined if no such $k$ exists.
I determined the values $f(n)$ for the even numbers $2,4,6,\cdots $ and $f(56)$ turned out to be $2138$ giving the huge prime $$56^{2138}+2138!+1$$ with $6194$ digits found by the PFGW-program (a software checking large numbers for primality).
The first even $n$ for which I do not know whether $f(n)$ is defined is $n=68$
Is there a prime of the form $68^k+k!+1$ with integer $k\ge 1$ ? If such a $k$ exists , it must be greater than $24\ 000$.