I know (or think) that most record primes these days are Mersenne primes due to the relative ease of testing their primality. What I wonder is if probablistic t

I know (or think) that most record primes these days are Mersenne primes due to the relative ease of testing their primality. What I wonder is if probablistic tests could be used to find candidates for very large non-Mersenne primes before testing them using AKS. Or is o̅(log⁷·⁵(n)) just infeasible for those sizes of numbers?