So "known" primes could be invalidated?
This is all well past the math I studied, but if this is the 12th largest prime number, isn't there the possibility that we could find two new "lower ranked" primes that are the prime factors to a "higher ranked" prime?
Say we "know" Prime2 is a prime, but discover Prime12 and Prime7, and they are the two prime factors of Prime2, so that Prime12 x Prime7 = Prime2. We just invalidated Prime2 as a prime number.
We know 11 is a Prime number because we know all of the prime numbers below it. If we assume 221 is a prime but don't know 13 and 17 are primes we are wrong because of incomplete information. If we subsequently discover that 13 is a prime number, we then invalidate 221 as a prime.
Or is there a better proof that someone has come up with?