Perhaps I shouldn't have said "large" or "notable". The point is that when a prime number is discovered meeting certain criteria it is included in public databases. This makes the case of an "illegal" such prime interesting.
No. Even then they're not notable. For example there are 10^97.6 primes with 100 digits, of which not all have been found (not enough storage space in the universe), but it's trivially easy to generate one at random.
