If you have a prime, and a claim that it's the nth prime, is there a fast way to check?

Question

I know there are fast ways to rule it out for special cases, but assuming it is actually true, how fast can you verify it?

I also wonder about sequences the primes are a subset of such as OEIS A050376 and A000961, whether verifying an index is any easier in those.

Answer 1

I presume your question is that you know that the given number is a prime and you only want to know if it is the $n$-th prime or not. Here is an approach.

As starting point you can use explicit bounds on the $n$-th prime such as Dusart's

$$ n\log n + n\log\log n - n < p_n < n\log n + n\log\log n $$

which holds for $n \ge 6$ or use new recent tighter bounds by Christian Axler. This will narrow down your search range to less than $n$ integers which can then be handled by a brute force computer search.