The AKS algorithm decides whether or not $n$ is prime in time $\tilde{O}((\log{n})^6)$. I am wondering if there is any faster algorithm to determine membership in some infinite set of primes.
What I have tried:
The Lucas-Lehmer test for Mersenne primes runs in $\tilde{O}((\log{n})^2)$, but we cannot prove there are an infinite number of Mersenne primes.
We could find and verify a Pratt certificate, for example if $n-1$ is easy to factor (perhaps it is $\log{n}$-smooth), and $a=2$ works, then this takes time $\tilde{O}((\log{n})^3)$. But I don't know how to prove there are an infinite number of cases where $n$ is prime and these conditions are satisfied.
The AKS algorithm can also return quickly in some cases (if $r$ is small) but again I am unable to show this happens infinitely often for prime $n$.
The Miller-Rabin test runs in $\tilde{O}((\log{n})^4)$ but this requires GRH.
We cannot possibly do better than $O(\log{n})$ since all bits of the input must be inspected.
How can we define an infinite set of primes and a decision algorithm for it that runs in time $O((\log{n})^k)$ for some $k \lt 6$?
