What is the smallest integer $n$>1 such that $n^{5000}+n^{2013}+1$ is prime?

Which is the smallest integer $n>1$, such that $$n^{5000}+n^{2013}+1$$ is prime ? Since $x^{5000}+x^{2013}+1$ is irreducible over $\mathbb{Q}$ and has value $1$ for $x=0$, there should be infinitely many such $n$, if Bunyakovsky's conjecture is true.

$n=23205$ produces the smallest prime value of the polynomial (aside from the trivial $n=1$). Interestingly, $23205=3\times 5\times 7\times 13\times 17$.

$n\in\{44579, 55754, 78120, 78515, 94154, 99045\}$ produce all the remaining primes with $n<10^5$. In all cases, OpenPFGW has been used to find the primes and prove primality using Brillhart-Lehmer-Selfridge method.