User blog comment:PsiCubed2/Offtopic question about a mathematical problem/@comment-1605058-20171017210411

To the best of my knowledge, given N = pq, we can deduce basically nothing about p and q without actually factoring the number (and factoring a 300-digit number is pretty much out of the question). It's always hard to say with certainty that something has never been done before, but I think a way to figure out a digit of a prime factor out of the number's digit would be a breakthrough we would've heard of.

By the way, make sure you don't take primes which are too close to each other (for example, consecutive primes are a no-no), since there are ways to factor them quickly. Making one of them a digit longer should be enough.