[TheRealPomax]

That list would be as long as the list of "all even numbers" because any list of primes immediately gives you a next prime that should be on the list, but isn't, in the same way that having a list of even numbers immediately gives you the next even number that should be on that list but isn't:

Given an ordered list of primes {2,3,5,...,n} one (of several) immediately known next prime number is simply (2 x 3 x 5 x ... x n) + 1. We know that number's prime because it cannot be cleanly divided by 2, or 3, or 5, or ..., or n.

As such, even if it might be useful to have a list of all known primes, annotated with what kind of prime each number is, such a list cannot be constructed: even just the subset of all prime numbers generated based on the simple above rule would be infinitely long, just like the list of all even numbers is infinitely long.