User blog comment:Rpakr/Value of 2↑↑5/@comment-32876686-20170913214027/@comment-3427444-20170914053920

We could sieve the first 100,000 numbers or so after 2^^5 and PRP test the remainders, and if a PRP is found, we could use methods such as ECPP to prove that it's prime. As of the current writing, the largest prime number proven prime using ECPP has 34,093 digits, and because these numbers have 19,729 digits, it's definitely doable, although time consuming.