What is the smallest prime of the form $n^n+5$?
Solution 1:
$444^{444}+5$ is prime, and is the smallest of that form. The next is $3948^{3948}+5$.
perl -Mntheory=:all -Mbigint -E 'for (1..1e5) { say if is_prime((0+$_)**$_+5); }'
It's a little faster using -Mbigint=lib,GMP or -MMath::GMP=:constant. A bit under 0.3 seconds to find the first, albeit this is uses a robust PRP test rather than doing a proof.
See: factordb entry for a primality certificate.