I was searching for primes of the form $n^{n^2}-k$ on PARI/GP and noticed that primes of this form for same $k$ are quite rare. The probability of finding a prime of this form is $\frac {1}{n^2 \log (n)}$, which makes me believe that the number of primes for same $k$ of this form would be finite.
I also noticed that the maximum number of primes of the form $n^{n^2}-k$ for the same $k$ seems to be three.
I searched for a range of $k \le 3 \times 10^5$ and $n \le 50$ on PARI/GP and didn't find more than three primes for same $k$. Here are the some values of $k$, for $k \le 10^4$ I found on PARI/GP, that have three primes of the form $n^{n^2}-k$. The first column displays the values of $k$ and the second column shows the values of $n$ for which $n^{n^2}-k$ is prime.
1409 [4, 6, 8]
3083 [4, 6, 8]
5496 [5, 7, 13]
7427 [4, 6, 8]
8902 [3, 9, 15]
Questions:
$(1)$ Can there not be more than three primes of the form $n^{n^2}-k$ for the same $k$?
$(2)$ If there can be more than three, then is there a maximum number of primes of the form $n^{n^2}-k$ for the same $k$?
Edit 1: Following the same process, I believe that the maximum number of primes of the form $n^{n^2}+k$, for same $k$ is four.
Edit 2: Peter showed provided an example with four primes for the same $k$ which solves my question $1$. So now I only look for answers of question 2.
For $$k=1996199$$ the numbers $$4^{4^2}-k$$ $$6^{6^2}-k$$ $$8^{8^2}-k$$ $$10^{10^2}-k$$ are all prime